Dual synthetic aperture radar system

[0002] This is based on provisional application serial No. 60/280,779 for Dual Synthetic Aperture Radar System of Apr. 2, 2001 and upon provisional application serial No. 60/282,915 for Dual Synthetic Aperture Radar System of Apr. 10, 2001

[0003] The field of the invention is an airborne radar employing electronic scanned array with a transmission array and a dual receive arrays employing diplaced center antenna (DPCA) methodology to detect moving targets in and out of clutter and measure there speed and azimuth very accurately

[0004] 1. Background of the Invention

[0005] In the field of this invention clutter has been a major impediment. Clutter being the unwanted signals from stationary ground that mixes with moving targets that makes it very difficult to obtain the objective of this invention.

[0006] The background in the state of the art, in airborne radars that detect moving targets in the presence of ground clutter and measuring their range, radial velocity and azimuth, consists of the following:

[0007] 1. Delay line cancellor Systems

[0008] 2. Monopulse Systems

[0009] 3. Two subarray DPCA radar

[0010] 4. Three Subarray DPCA Radar

[0011] 1. The Delay Line cancellor subtracts the inputs from the radar of two consecutive pulses. If the moving target is in the presence of Main Beam clutter it looks just like clutter and cannot be detected and therefore cannot measure its parameters. The E2C Hawkeye-Grumman is an example of this system where a double delay line cancellor plus special filtering is implemented.

[0012] 2. Monopulse System Utilizing delay line cancellor at the input because of the system cannot determine the radial velocity and azimuth in the presence of clutter. It utilizes two subarrays, a sum (ΔΣ) array and a difference (Δ) array to process the data. The accuracy is limited by the ratio of the noise of the difference channel and ten to one ratio is about the limit. It has better accuracy than delay line cancellor but more processing and its notable to detect and process moving targets. Like the delay line cancellor it can not detect targets in main lobe clutter.

[0013] 3. The DPCA Radar with two subarrays using system in classical theory a moving targets precise angle and radial velocity can not be determined. The detection of the moving target may be obtained only. This is the reason for three segment antenna to attain the precise velocity and azimuth.

[0014] 4. The DPCA Radar with three subarrays attains the objective of detecting a moving target in the presence of clutter and measuring its radial velocity and azimuth position but takes three subarrays (large amount of hardware and processing since it has three subarrays it has to process and must perfom many logic and mathematical operations)

[0015] Example: Northrop Grumman JSTARS System.

[0016] 2. Description of Prior Art

[0017] Patent Ser. No. 766,309 Filed Aug. 16, 1985—Abandoned

[0018] Entitled: Two Port Clutter Suppression Interfermometry System For

[0019] Radar Detection of Targets

[0020] Inventor: Thomas J. Cataldo

[0021] Grumman Reference: 388-GAC; P-1(3783)

[0022] This patent has the same objective and the same inventor as the aforementioned invention. This invention is not obvious from the previous patent stated above with one of ordinary skill in the art. This invention has many essentially independent techniques to attain the objective.

[0023] The patent above discloses a vague geometric technique which takes advantage of the first order change in target relative to clutter (which was considered changing very slowly).

[0024] U.S. Pat. No. 5,563,601 Filed Oct. 18, 1996

[0025] Entitled: Two Port Clutter Synthetic Aperture Radar System For

[0026] Radar Detection of Targets

[0027] Inventor: Thomas J. Cataldo

[0028] Northrup-Grumman

[0029] The patent above has the same objective and inventor as in contrast to existing dual array radar systems, the present invention employs a different principle. The principle is that it depends on a moving target creating a black hole (lack of signal) behind it, as well as when the target moves it both suppresses clutter at where its moved and exposes clutter(increased signal) from where it hap moved.

[0030] The range doppler bins, with this technique have to be relatively small to be an effective technique.

[0031] U.S. Pat. No. 4,885,590 Filed Apr. 14, 1989 Issued—Dec. 5, 1989

[0032] Entitled: Blind Speed Elimination For Dual Displaced Phase Center

[0033] Antenna Radar Processor Mounted on a Moving Platform

[0034] Inventor: Hassan; Moh'd A

[0035] General Electric Company (Moorestown, N.J.)

[0036] The patent above employs the dual displaced phase center antenna to eliminate blind speeds of targets but not to not to attain precisely there relative radial velocity and azimuth position in the presence of main lobe clutter.

[0037] This patent develops the mathematical basis and the physical radar principles and computer simulation results are the basis for the many techniques developed.

[0038] This invention considers the whole system from the application and mode of operation for all the essential elements of the system and the many possible implementations. Each part of the system is analyzed and optimized as much as possible to aid any techniques to attain the objective of the invention. The following are essential parts of the system (there are a number of supporting depending implementations for the optimization of the system) as follows:

[0039] (a) Antenna Look Angle

[0040] (b) Antenna Configuration

[0041] (c) RF Frequency

[0042] (d) PRF

[0043] (e) Apertures

[0044] (f) Beamwidth of antenna arrays

[0045] (g) Groups of radar data to process

[0046] (h) Weighting functions for clutter and target

[0047] (i) Error reduction techniques

[0048] (j) Antenna match considerations

[0049] (k) Adaptive techniques for the cancellation of clutter

[0050] (l) Management, control and optimization of implementation

[0051] (m) Others

[0052] The many techniques developed to attain the objective are unique and based on extensive mathematical development and computer simulation of radar principles and employing real radar data.

[0053] This invention is special synthetic aperture radar. It is a dual synthetic aperture radar. It is implemented in a special array (which may be formed by combining the two identical receiving arrays). The motion of the platform carrying the receiving arrays is such that on some number of succeeding radar pulses, the second antenna travels ½ distance between two receiving (displaced phase center antenna) the same space as the first antenna array (FIGS. 1 to 3—Basic Concept of DSARS). This is continued until the two synthetic arrays are formed. When the radar data is spectrally processed and subtracted from each other, the clutter in both arrays being identical are cancelled and the target being different is detected (FIGS. 1 and 2). The target parameters are such as the relative radial velocity and angular position are not yet determined. There is not enough processing at this point.

[0054] The object of this invention is to further process the radar data from two synthetic arrays and determine very accurately the range, relative radial velocity and angular position of the target (within the resolution of a doppler bin or better). The target may be in the presence of clutter or out of clutter. The essential contribution of this radar system is to be more cost effective than the more hardware system such as those using three or more receive arrays and much more capable and accurate than the delay line or monopulse systems

[0055] FIG. 1 shows an Ideal Dual Sub array displaced phase center(DPCA) Radar System

[0056] FIG. 2 is a Block Diagram of Ideal System

[0057] FIG. 3 is a Block Diagram of Non-Ideal System

[0058] FIG. 4 shows Slant Looking θ Antenna Arrays

[0059] FIG. 5 shows a Forward Looking Array

[0060] FIG. 6 shows an Upward or Downward Looking System

[0061] FIGS. 7A-7C show different Antenna Configurations

[0062] FIG. 8 illustrates Techniques in Processing

[0063] FIG. 9 illustrates Zero-Fill Spectrum Analysis

[0064] FIG. 10 illustrates Low Clutter, Intermediate Clutter and High Clutter Processing Areas

[0065] FIG. 11 illustrates Ideal displaced phase center antenna (DPCA) Depiction

[0066] FIG. 12 illustrates Clutter and Mover Definition

[0067] FIG. 13 illustrates Non-Ideal Spectrum

[0068] FIG. 14 shows Time Only and Pulse Plus Time DPCA Compensation

[0069] FIG. 15 shows Time only and Pulse Plus Time DPCA Compensation Aligned With the Filter Response of Clutter

[0070] FIG. 16 depicts the Relationship of Ideal Clutter Filter and Ideal W_cResponse

[0071] FIG. 17 depicts Target Ideal Filter and W_MResponse

[0072] FIG. 18 depicts Clutter and Target Detection Diagram

[0073] FIG. 19 depicts Doppler Ambiguity and Clutter Analysis

[0074] FIG. 20 depicts DΦ_RError-vs-PH1 for Different GCR=|M/C|(Δ{overscore (K)}²+β²)/(ΔK²+K²)

[0075] FIG. 21 illustrates DΦ_RError-vs-PHI for ΔK and α Tolerance for Different GCR=|M/C|(Δ{overscore (K)}²+β²)/(ΔK²+K²)

[0076] FIG. 22A is a 32 Point Doppler Filter Diagram—1000 Hz where

[0077] a) Odd and Even Pulse Sequences D=1 and

[0078] b) Antenna—Doppler Diagram—Odd and Even Sequences D=1 and FIG. 22B where

[0079] c) Odd and Even Pulse Sequences D=2 and

[0080] d) Antenna—Doppler Diagram—Odd and Even Sequences D=2

[0081] FIG. 23 is a 64 Point Filter Diagram at 1000 Hz where:

[0082] a)Antenna Doppler Diagram—odd and even pulse sequences

[0083] b) Odd and Even Pulse Sequence

[0084] c) Combined Odd and Even Pulse Sequences—64 Pulse Sequences

[0085] FIG. 24 is a Ratio of Target at Different Positions of Overlapping Half-Beams of Antenna

[0086] FIG. 25 identified as TABLE I is an Ideal Clutter Filter Response Correlated With Ideal weighting function of clutter (WC)Response and Relative Clutter Change

[0087] FIG. 26 identified as TABLE II is Target Filter Response Correlated With Relative Target Filter Response and Relative Target Change

[0088] FIG. 27 identified as TABLE III shows weighting function of Target (W_M) Target Determination

[0089] FIG. 28 identified as TABLE IV shows For pulse repetition frequency1 (PRF1) of 1000 Hz and pulse repetition frequency2 (PRF2) of 750 Hz (64 large range Doppler bins(LRDB) and 256 fine range Doppler bins) (FRDB)-vs-Frequency response and phase change before and after “D” and Radial Velocity.

[0090] FIG. 29 identified as TABLE V shows

[0091] fine range Doppler bin (FRDB)-Vs-θ Error for Time Only displaced phase compensation antenna(DPCA)compensation (1000 Hz & 750 Hz)

[0092] fine range Doppler bin (FRDB)-Vs-θ Error for Pulse plus Time DPCA compensation (1000 Hz & 750 Hz) fine range Doppler bin (FRDB)-Vs-Phase and Amplitude Response

[0093] fine range Doppler bin (FRDB)-Vs-Change in Filter Amplitude and Phase Response

[0094] FIG. 30 identified as TABLE VI shows Target Detection in High Clutter Area

[0095] FIG. 31 identified as TABLE VII shows Possible D_φR-vs-Processing fine range Doppler bin (FRDB)

[0096] FIG. 32 identified as TABLE VIII shows GCR (|M/C|(Δ{overscore (K)}²+β²)/(ΔK²+K²))-vs-DΦ_Rand (DΦ_R)PHI ERROR-vs-DΦ_R

[0097] FIG. 33 identified as TABLE IX shows 64 Point Data—1000 Hz D=1 Regions of Clutter

[0098] FIG. 34 identified as TABLE X shows 32 Point Data—1000 Hz D=1 Regions of Clutter for Odd and Even Sequence Data

[0099] FIG. 35 shows Amplitude as function of DΦ_Rfor Multiple D (D=1 To D=6)

[0100] FIG. 36 shows a pictorial and system representation of the whole radar and the DUAL SYNTHETIC RADAR SYTEM (DSARS) portion.

[0101] The present invention has broad applications to many technical fields for a variety of articles. For illustrative purposes only, a preferred mode for carrying out the invention is described herein, wherein a radar utilizes the platform to obtain vital detection of moving targets and there speed and azimuth.

[0102] In the foregoing description, certain terms and visual depictions are used to illustrate the preferred embodiment. However, no unnecessary limitations are to be construed by the terms used or illustrations depicted, beyond what is shown in the prior art, since the terms and illustrations are exemplary only, and are not meant to limit the scope of the present invention.

[0103] It is further known that other modifications may be made to the present invention, without departing the scope of the invention, as noted in the appended claims.

[0104] It is to be noted that the description of this disclosure are illustrative and cannot show all possible implementations that may be employed from the information in the disclosure by a person of ordinary skill in the state of the art. Therefore this disclosure is illustrative and not limited of the scope of the proposed invention and not limited in the scope of the role in obtaining the objective of this disclosure.

[0105] I. Basic Concept of a Dual Subarry DPCA Radar System

[0106] A. Ideal System

[0107] Classical DPCA technique takes advantage of the fact that the doppler shift in the frequency of the returns received from the ground is due entirely to the aircraft's velocity. Specifically, this shift which is manifest as a progressive pulse-to-pulse shift in the phase of the returns from any one range is the result of the forward displacement of the radar antenna's phase center from one interpulse period to the next.

[0108] For any two successive pulses, therefore, the shift can be eliminated by displacing the antenna phase center by an equal amount in the opposite direction before the second pulse of the pair is transmitted. The second pulse will then be transmitted from the same point in space as the first.

[0109] And how does one displace an antenna's phase center? Generally, the radar is provided with a two-segment side-looking electronically steered antenna. The aircraft's velocity and the radar's PRF are adjusted so that during each inter-pulse period the aircraft will advance a distance precisely equal to that between the phase centers of the two antenna segments.

[0110] Successive pulses then are alternately transmitted by the two segments:

[0111] Pulse (n) by the forward segment, pulse (n+1) by the aft segment;

[0112] Pulse (n+2) by the forward segment, pulse (n+3) by the aft segment, and so on.

[0113] As a result, the pulses of every pair—e.g., (n) and (n+1)—are transmitted from exactly the same point in space.

[0114] The returns of each pulse are received by the antenna segment which transmitted the pulse. When the return from any one range, R, is received, of course, the phase center of that segment will have advanced a distance equal to the aircraft velocity, V, times the round-trip transit time, t_r, for the range R.

[0115] There is a wide clutter-free region in which to detect moving targets. Moreover, any target whose apparent doppler frequency falls within the mainlobe clutter can be periodically moved into this region by switching the PRF among several different widely separated values.

[0116] However, if the radial component of a target's velocity is so low that its true doppler frequency lies within the mainlobe clutter, no amount of PRF switching will move the target's returns out of the clutter. Consequently, in many applications a special “slow-moving-target' indication capability is needed. Conceptually, the simplest is Classical DPCA.

[0117] FIG. 36 shows a pictorial and system representation of the whole radar and the DUAL SYNTHETIC RADAR SYTEM (DSARS) portion. As indicated the front end of the radar is the electronic scanned arrays. This array consists of a full array on transmit and two half-length receive subarrays. The receive array data is amplified and converted to digital data and sent to the DUAL SYNTHETIC ARRAY SYSTEM which is the invention. It is processed there and moving targets are detected and there range, velocity and azimuth are measured very accurately. This target report is further processed for surveillance, tracking and weapons control mode. All these operation are computer controlled.

[0118] The ideal system as illustrated in FIG. 1 and FIG. 2 is a displaced phase center antenna (DPCA) radar system with a transmit antenna and two identical receiving subarrays. This is a modified DPCA system as will be described.

[0119] One of the basic concepts is a sidelooking radar antenna system where the antennas are mounted in line with the platform motion as shown in FIG. 1. The antenna arrays are moving with the platform motion. The antenna arrays move half the distance between the phase centers of the receiving arrays in one pulse repetition period (PRF).

[0120] In FIG. 1 the radar returns from patch 3aand patch 7ais a patch of ground returns observed by the radar receiver in a particular range doppler bin (RDB).

[0121] In radar patch 3awe have clutter 6aand a target moving from position to position 4a. In radar patch 7awe have no clutter and a target moving from position 8ato position 9a.

[0122] As shown in FIG. 1 the subarray 1 moves from position R1,1 to R1,N from time 1 to time N as subarray 2 moves from position R2,1 to position R2,N from time 2 to time (N+1). Both subarrays move half the distance between subarrays for each pulse. This is the distance required to cancel clutter in a modified classical DPCA system.

[0123] After “N” pulses have been filtered for each range doppler bin (RDB) for subarray 1 and 2 and the corresponding range doppler bins, RDBs, are subtracted where the targets are detected as in patch 3aand patch 7a.

[0124] After the subtraction the clutter cancels but the target is moving, but remains in the same RDB, the amplitude of the target is the same but the phase changes allowing for the detection of the target as shown in FIG. 1, 1C and 1D. This phase change is proportional to the relative radial velocity of the target (Φ) and the target does not cancel. The objective of this disclosure is to find this relative radial velocity and from this the angular position of the target is determined. The returns from the no clutter patch 7aof FIG. 1 is represented by vectors 15aand 16a. This is the result of subarray 2 subtracted from subarray 1, vectors 15aand 16arepresents the target vectors separated by a phase Φ (24a) which is proportional to the relative radial velocity of the target (target does not cancel; it has phase shift per pulse). This phase is measured and consequently the relative radial velocity and azimuth of the target is determined.

[0125] The returns from patch 3acontains clutter 6aand a target moving from position 5ato position 4a. The vector 14ashows the clutter return and vectors 7aand 8aindicating the target returns. The next diagram illustrates the addition of clutter vector 14aand the target vectors 17aand 18asince the clutter and target appear in the same range doppler bin. The angle (Φ)—24acannot be measured directly because the clutter vector appears with the target vectors 17aand 18a. What is detected, vector 19a, is the resultant vector of vector 22aand 23a. Therefore (Φ)—24ais indeterminate from this simple analysis.

[0126] The object of this patent is to determine the angle (Φ) and thereby determine the azimuth of the target in the presence of significant clutter.

[0127] FIG. 2, Block Diagram of Ideal System, illustrates the ideal system processing required.

[0128] B Non-Ideal System

[0129] The non-ideal system will have many factors which will make it more difficult to determine the angle (Φ).

[0130] The two receiving antenna subarrays will in many cases not be in the ideal positions to cancel clutter. The conditions will not be ideal; in fact, there may be a significant difference from the ideal position. When this occurs, a phase correction is required for each doppler bin where the target is detected to make the clutter cancel.

[0131] The real system will have a number of sources or errors that have to be considered to at least reduce the errors. In the system, to have the required accuracy, the following is what has to be considered:

[0132] 1. Antenna subarray positional error

[0133] 2. Antenna subarray 1 and 2 not matched exactly

[0134] 3. Sidelobes of Filters

[0135] 4. Sidelobes of Antennas

[0136] 5. Processing errors

[0137] 6. Non-Ideal Clutter Returns (slightly moving clutter)

[0138] 7. Non-Ideal Target returns

[0139] 8. Motion Compensation

[0140] 9. Isodop Correction

[0141] 10. Others

[0142] FIG. 3 illustrates the simplified processing for the non-ideal system showing the amplitude and phase correction role for the error sources.

[0143] Section III and IV will give a great deal more attention to the non-ideal system.

[0144] II. Systhetic Aperture Processing System Applicable Implementations

[0145] A. All Frequencies

[0146] 1. The system is able to utilize all radio frequencies limited on the low end by the size of the antennas and on the high end by the practical limitations of short waves.

[0147] B. Employing Different Antenna Looking Capabilities

[0148] 1. Sidelooking Dual Synthetic Aperture Processing System

[0149] This is explained in Section IA, FIG. 1, where T1 to TN is the positions of transmission at time 1 to N, while R11 to R1N and R21 to R2N are the array 1 and array 2 receiver points respectively. This is the modified classical system, the thought of best mode of operation.

[0150] 2. Slant Looking System

[0151] The angle looking antenna arrays are defined as the two receiving antenna arrays are mounted at an angle greater than zero to the motion of the platform and less than plus or minus ninety (90) degrees. Ninety (90) degrees being a sidelooking system and zero and 180 degrees is a front and rear looking system. The angle looking antenna arrays is illustrated for only one angle in FIG. 4. When the motion of platform 3dmoves the platform from position 4dto position 5d, there the antenna array 2 moves from position 7d, and antenna array 1 moves from position 7dto position 1d. There is a distance. The system operation is similar as far as the sidelooking classical DPCA system where the second antenna is at the first antenna position on transmission and receiving. At the first position the first array transmits and receives and at second position the second array transmits and receives. At each position the clutter looks identical while a moving target does not since it moved between transmissions.

[0152] 3. Forward Looking Antenna Array

[0153] The forward looking antenna arrays when the motion is forward as illustrated in FIG. 5. The antenna array 1 at position 1 is the same as antenna 2 at position 2. Therefore, this is the feature that keeps repeating itself at succeeding antenna positions. The clutter in these positions in the two receiving antenna cancel each other and target does not. If the antennas as can be seen in forward direction antenna 1 shadows antenna 2, they can be displaced in the height direction so this does not happen and a phase error correction performed as described in the previous paragraph.

[0154] Looking at radar patch A, and having the same antenna pattern C in all succeeding positions from 2 to (n+1), with the forward motion B, the clutter is seen as the same in these positions, the target is different as explained previously.

[0155] 4. Forward Looking Antenna Arrays—Motion Upward

[0156] This is a forward looking antenna system with the motion as upward as illustrated in FIG. 5. The antenna array 2, position 2e, is the same as antenna array 1 when the platform motion A is upward as shown in Figure.

[0157] Therefore, in all succeeding positions from position 2eon, we have the conditions for the receiving antenna arrays having occupied the same position in space in two succeeding positions of the platform. Therefore, the clutter will cancel in the two receiving arrays and the target will not.

[0158] The platform motion A is upward and the radar patch B will be illuminated by antenna pattern C which will be the same for the antenna arrays 1 and 2 for every antenna position from 2fto (n+2)f as illustrated in FIG. 6. This is the synthetic aperture affect for the effective operation of the system.

[0159] C. Different Antenna Configurations

[0160] All Antenna Configurations, of course, have the transmitting antenna as part of the system. The transmitting antenna may be the two receiving antenna arrays added together. But whatever transmitting antenna system such as is utilized, it must be calculated into the whole system.

[0161] 1. Two Equal Receiving Antenna Arrays

[0162] Arrays (FIG. 7A) A and B are built to be as exact as possible. Therefore, the elements 1gand the spacing Cg and the length Lg of each array should all be the same to give the best results for this application. The motion of the antenna arrays should be so that in some number of radar pulses the phase centers 2gof antenna array 3gand 4gobey the DPCA laws for canceling clutter as stated in Section IA. Spacing is determined by the application.

[0163] 2. Two Receiving Antenna Arrays with Trim Antenna Elements (FIG. 7B)

[0164] The equal receiving antennas with trim antenna elements 3gare implements where the various possible motions of the platform is such that the antenna trim elements 3gare employed for a better match. Both receiving antennas always have to be kept the same length L with all trim elements. The trim elements 3gplus the antenna arrangement would make the phase centers 2gof the two equal receiving antenna arrays more closely occupy the same space on a number of succeeding radar pulses. The technique may be utilized to make the two equal receiving antennas of different length L.

[0165] 3. Two Equal Receiving Arrays Formed Within an Adaptive Array (FIG. 7C). A few examples are illustrated in FIG. 7C (Antenna Configurations). The two equal receiving antenna arrays are formed from the large adaptive array 1ggto 3gg. This is determined in any way that your application is best implemented. But it is generally determined by the speed of the platform (Carrying the array) so that the phase center of antenna arrays will nearly coincide in some number of succeeding pulses. Overlapped arrays share the same antenna elements by forming digital adaptive arrays. With two receiving adaptive arrays much more flexibility and capability are able to be implemented on the two antenna arrays simultaneously. Anti-jamming techniques may be performed.

[0166] Shape antenna arrays in any way desired such as:

[0167] a) low sidelobes

[0168] b) preferential sidelobes

[0169] c) increase target change, decrease clutter change

[0170] d) space time processing may be performed

[0171] e) other

[0172] The angle looking system was illustrated to the rights (FIG. 4); of course it just as well could have been to the left. The forward looking system was shown as forward looking; it just as well could have been rear looking.

[0173] The basic comment the two receiving antenna arrays may be mounted at any angle and a system may be formulated as indicated.

[0174] The implementation may also be with two dimensional arrays as well as a one dimensional array.

[0175] 4. A Rotatable Scanned Electronically Array

[0176] This would make all azimuths accessible in one or a few antenna array systems.

[0177] 5. Comments on Array Configurations

[0178] Many types of antenna configurations are possible. The basic idea is to formulate two equal receiving antenna arrays and operate the system as described. This is also possible with two dimensional arrays where the two dimensional arrays follow similar rules as described previously for the one dimensional array.

[0179] D. Single or Multiple Pulse Repetition Frequency (PRF)

[0180] The utilization of multiple pulse repetition frequency has the ability to resolve Doppler and/or range ambiguity which in many cases helps the process of obtaining the value of Φ the relative radial of the target. This will later be described in detail. In this disclosure only doppler ambiguity was used as an example but range and/or doppler ambiguity may be handled.

[0181] E. Single or Multiple Pulses of Different Transmission Frequencies or Add “D” Plus “E”

[0182] This is basically for the same reason as Section III(D). It is harder to implement than the method in Section III(D) but has an additional advantage of helping to overcome jamming and for resolving Doppler ambiquity as in Section III(D).

[0183] Combining “D” and “E” meaning both PRF and transmisson frequency may be changed between apertures for its associated advantages.

[0184] F. Range and Velocity Ambiguity Considerations

[0185] All pulse repetition frequency (PRF) radar systems have a limitation on determining range and velocity unambiguously. Single PRF radar systems are always ambiguous in range and velocity. If they are acceptable, one PRF may be employed, if not two or more PRFs have to be implemented to have an acceptable system.

[0186] The following [are] the equations in determining range and velocity ambiguity:

[0187] Ru=Ru1[(Ru2/Ru1−Ru2)][Ru3/Ru3−Ru2)] . . . ]

[0188] Ru—Unambiguous Range of the System

[0189] Ru1—Ambiguous Range of PRF1

[0190] Ru2—Ambiguous Range of PRF2

[0191] . . . —“ “ “ “

[0192] Run—Ambiguous Range of PRFn

[0193] Ru1−Ru2—Difference in Ambiguous Range

[0194] Ru2−Ru3—Difference in Ambiguous Range

[0195] . . . “ “ “ “

[0196] Ru(n−1)−Run—Difference in Ambiguous Range

[0197] Vu=Vu1[(Vu2/Vu1−Vu2)][Vu3/Vu3−Vu2)] . . . ]

[0198] Vu—Unambiguous Velocity of the System

[0199] Vu1—Ambiguous Velocity of PRF1

[0200] Vu2—Ambiguous Velocity of PRF2

[0201] . . . —“ “ “ “

[0202] Vun—Ambiguous Velocity of PRFn

[0203] Vu1-Vu2—Difference in Ambiguous Velocity

[0204] Vu2-Vu3—Difference in Ambiguous Velocity

[0205] . . . —“ “ “ “

[0206] Vu(n−1)−Vun—Difference in Ambiguous Velocity

[0207] Examples:

[0208] Vu1—50 feet/sec. Vu2=37.5 feet/sec.

[0209] Vu=Vu1[Vu2/(Vu1−Vu2)]

[0210] Vu=50 feet/sec. [37.5 feet/sec/(50 feet/sec.−37.5 feet/sec.)]

[0211] Vu=150 feet/sec.

[0212] The ambiguous velocity of one PRF system went from 50 feet/sec. to 150 feet/sec. with a second PRF. As more PRFs are employed the unambiguous get larger (in general) until it is acceptable.

[0213] For the same PRFs chosen for the velocity example:

[0214] Ru1=80 miles Ru2=106 miles

[0215] Ru=Ru1×Ru2/(Ru2−Ru1)

[0216] Ru=80×106/(106−80)

[0217] Ru=328 miles

[0218] The ambiguous range increased significantly from a one PRF system to a two (2) PRF system.

[0219] G. Processing of Radar Data

[0220] When radar data is received, it is converted to digital data, stored and processed.

[0221] It is grouped together in various ways and processed. Here are a few ways, but by no means limited by these stated techniques. It is just meant to indicate some methods of grouping data and processing them. These methods of grouping and processing data are determined by the following:

[0222] Ease of processing

[0223] Aid in attaining objective of disclosure

[0224] Application

[0225] Others

[0226] 1. Group Processing

[0227] This takes the radar data in groups, some subdivision of the total number of data points received and processes it in groups as indicated in FIG. 8.

[0228] 2. Sliding Window Processing

[0229] Sliding window processing is also illustrated in FIG. 8. The data in one aperture is taken in one group and processed. The next group to be processed is a number of data consecutive points to be added to the end of the first group and the same number of consecutive data points are dropped at the beginning. This is performed until all the data points are processed.

[0230] 3. Circularly Correlated Data

[0231] Circularly correlated data is shown in FIG. 8. This is where the first set of data is processed and the data is shifted and first data point is replaced by the last data point and processed again. This continued until all “n” data groups are processed.

[0232] This data may be processed one data shift at a time. The second set shows data shifted two data points at a time and processed (FIG. 8).

[0233] Also as inferred, a number of data points may be shifted at a time and processed until all data is processed.

[0234] 4. Comments on Processing Radar Data

[0235] As shown, all groups of data were taken from left to right (start to end) but just as well taken from right to left (end to start) in all processing techniques.

[0236] Combinations of batch, sliding window and Circularly correlated data may be performed to get the desired results of obtaining Φ relative radial velocity. Data may be taken in any kinds of groups and processed. Data does not have to be in any logical sequence. According to your application, you may process in any way desired to attain the objective since all data may be in storage before processing.

[0237] H. Data Weighting Function

[0238] In FIGS. 1 and 2, as illustrated, in processing data for the system, a spectral processing of the stored radar data is performed. Before spectral processing is performed a weighting function is applied to the data. This is a common technique in processing data to reduce spectral leakage and enhances other characteristics. Weighting functions may be used to reduce sidelobes of spectral filters (which is analogous to the sidelobes of the Antennas). Weighting functions may be used to enhance target energy and/or to reduce clutter energy.

[0239] One or a number of successively different weighting functions are applied to the basic radar data for aiding in the technique in finding a solution for Φ. The weighting function may be to obtain the following characteristics or trade offs in the following characteristics:

[0240] Reduce amplitude of clutter

[0241] Reduce mth order change in clutter where m=1 to 4

[0242] Increase amplitude of target

[0243] Increase mth order change in target where m−1 to 4

[0244] Reduce sidelobes of filters especially in the region of high clutter

[0245] Reduce the affects effects of clutter in sidelobes of antennas

[0246] Reduce the affects effects of mainlobe clutter in filter sidelobes

[0247] The above characteristics are most desirable for the figure of merit and least mean square technique and the other techniques.

[0248] The characteristics that would be most desirable for clutter and/or target correlation technique between apertures would be to have between apertures the target and clutter the ability to correlate as much as possible. This would involve making the filter in both apertures the same size (performed by weighting function) and reduce the time between processing of both apertures. This may be performed by shifting the data one quarter of the data points to the left (assuming Circularly correlated data) in the first aperture and processing and shifting the data in the second aperture one quarter of the data points to the right and processing. This would cut the time in half between the processing of data in both apertures since weighting functions reduce the affect about one quarter of data points at the beginning and end of the total number of data points as least significant in processing.

[0249] The weighting function required for the aperture to aperture clutter and/or target correlation technique should have the following characteristics:

[0250] Make filters in both apertures same size

[0251] Reduce filter sidelobes for clutter reduction

[0252] Reduce time between adjacent data in both apertures to as small as possible for best correlation

[0253] Reduce change in target and clutter change from aperture to aperture for the best correlation possible.

[0254] I. Target Bandwidth Characteristics

[0255] Ground targets have bandwidth and not line frequencies and are approximately fifteen (15) hertz wide at the half power point which represents reality. Targets do not appear as single line frequencies for a number of reasons such as the following:

[0256] targets do not travel at uniform velocity for the total dwell time of the aperture

[0257] acceleration or deceleration of targets

[0258] motor vibrations

[0259] tangential velocity

[0260] aspect angle change

[0261] radar beam bounce

[0262] rough roads

[0263] others

[0264] Therefore when the target is detected according to how closely spaced the spectrum filters are the target is detected in one or a number of doppler filters.

[0265] The number of outer doppler filters the target detected therefore depends on the filter width, how closely spaced the spectrum filters are and the spectrum width of the target.

[0266] An example is a ground target turning (accelerating or decelerating) changing doppler bins within an aperture time. The number of doppler filters the target is detected are many.

[0267] J. Calculate Max-Max Amplitude of Target for Target Correlation Technique

[0268] Calculate maximum target amplitude in the range-axis by detecting the amplitude of target in the two or more range bins the target is detected in, and the maximum amplitude of target in the range-doppler bin where detected.

[0269] Based on the assumed parabolic shape of the pulse in the range axis interpolate the maximum amplitude of the target in the range axis. Of course any assumed shape of the target that best fits the application that may be implemented or other interpolation techniques.

[0270] Also calculate the maximum interpolated target amplitude in the doppler axis by detecting the amplitude of the target in the two or more doppler bins the target is detected from the maximum amplitude of the target in the range doppler bins where detected.

[0271] Based on the assumed parabolic shape of the frequency spectrum of the return from the target in the doppler axis, interpolate the maximum amplitude of the target in the doppler axis. Of course, any assumed shape of the spectrum of the target that best fits the application may be implemented.

[0272] To calculate the max-max of the target in both the Doppler and range axis, multiply the interpolated maximum value in the Doppler axis by the interpolated maximum in the range axis.

[0273] This will tend to reduce the error of calculating the ratio of the amplitude of the target in two different apertures for moving targets in range or doppler changes.

[0274] K. The Effect of Detecting the Same Target in Different Range Bins and Different Doppler Bins

[0275] 1. Detecting a Target in the Same Doppler Bins But Different Range Bins

[0276] If there is no significant movement in the range for the target throughout the aperture time, target characteristics should be very similar such as doppler characteristics, amplitude and phase change for the different range bins. The clutter in each range bin will be different and the amplitude of the target is different in each range bin.

[0277] 2. Detecting a Target in the Same Range Bin but Different Doppler Bins

[0278] If the range of the target is changing only slightly, then the frequency components of the target will be detected in two or more doppler bins. The target characteristics will vary according to the spectrum of the frequency components of the target, the center frequency of the filter and the width of the filter. If the filters are a filter away from the central frequency of the target, there will be a phase shift proportional to how far the filter is away and an amplitude attenuation.

[0279] If the filter is one filter away in the high frequency direction, the target phase shift will be phase shift equivalent to a positive phase shift of one doppler bin giving the target the appearance of having a larger relative radial velocity and the same angular position. This is to be dealt with in the processing.

[0280] Also, if the filter is one filter away on the low side, analogously, the target will appear to have a lower relative radial velocity and the same angular position.

[0281] L. Alignment of Antenna Patterns in a Multi-Aperture System

[0282] In a two or more aperture system where the antenna pattern is moved as well as the pulse repetition frequency is changed, it may be beneficial to have precise control of where the antenna pattern is moved so that the spectrum filters in both apertures are overlapping as much as possible. This would be most desirable in the clutter and/or correlation technique.

[0283] M. Computer Generated Radar Mapping

[0284] An aid to determining where and how the targets may be located and the nature and measure of clutter detected in the real radar will be correlated to the computer generated clutter. An example may be ground clutter cannot be low in mountainous areas. Clutter of large amplitude cannot be in a desert, etc.

[0285] N. Histographic Mapping

[0286] Computer generated map of where the radar shadows would occur is called a histographic map. Of course, no ground targets can ever originate from a radar shadow and of course if a target appeared in a radar shadow there would be no clutter to compete with the target and relative radial velocity can easily be determined

[0287] O. Cartographic Mapping

[0288] Computer generated map of the area illuminated by the radar to indicate where all the roads are located. For ground targets, it will aid in locating where targets are positioned. For ground targets cannot appear in water, mountains or areas in dense trees, etc. Where there is targets that are traveling at high speed they must be on roads and not in terrain not support such speeds.

[0289] P. Comments on Section II

[0290] Sections II A to P are dependent characteristics that apply to implementation of the system and aid greatly in attaining the objective when employed efficiently. Sections II M, N, and O are aids in the implementation of the system and are not necessary in obtaining the desired results.

[0291] III. Basic Operations, Equations and Methodology Fundamental to All of the Systems Employed in This Invention

[0292] The mathematical fundamental equations, the radar analysis and the computer simulation for all many systems to obtain the objective of the invention are presented. The many techniques are as follows:

[0293] 1) Basic System is with one aperture, one PRF and one transmission frequency. If the ambiguous range and or velocity is to be increased another PRF and/or transmission frequency is implemented in the second aperture. The basic techniques that are employed are the following:

[0294] a) Change in time (ΔT)

[0295] b) Change in frequency (ΔF)

[0296] c) Change in delay (ΔD)

[0297] d) Interleaved pulses

[0298] e) And/or any combinations of the techniques above and correlated

[0299] 2) Figure of merit system (FM) is employed with the basic system especially with ΔT and ΔF techniques.

[0300] 3) Least Mean Square System (LMS) is employed with the basic system especially with ΔT and ΔF techniques.

[0301] 4) Same as basic system but with dividing up the input set of of pulses of data into a number of interleaved data sets.

[0302] 5) Same as 4) except that with each set of interleaved pulse the aperture is changed, the same PRF and transmission frequency. The second aperture overlaps the first aperture.

[0303] 6) Same as 5)except the transmission frequency is changed with each aperture change. The second aperture overlaps the first aperture.

[0304] 7) Same as system 1 except with same PRF and same transmission with a different aperture and antenna position. The second aperture overlaps the first aperture.

[0305] 8) Same as 7) except in the second aperture the PRF is changed.

[0306] 9) Same as 7) except in the second aperture the transmission frequency is changed.

[0307] 10) Same as 7) except in the second aperture the PRF and the transmission frequency is changed.

[0308] A. Basic System Parameters

[0309] The system is best understood and explained by taking a specific example to illustrate the principles, equations and processing involved but is not limited to this example. The following is a sidelooking radar with the following radar parameters are taken as the specific example:

[0310] 1) Operational Radar Frequency is 10,000 Megahertz

[0311] 2) Wavelength is 0.1 foot

[0312] 3) First Pulse Repetition Frequency (PRF) is 1000 Hertz

[0313] 4) Second Pulse Repetition Frequency (PRF) is 750 Hertz

[0314] 5) Beamwidth of Transmit Antenna is ⅔ degree

[0315] 6) Beamwidth of First and Second Subarray {fraction (4/3)} degree

[0316] 7) Length of Transmit Antenna is eleven (11) feet

[0317] 8) Length of First and Second Subarray is 5.5 feet

[0318] 9) Velocity of the platform is 500 feet/second

[0319] 10) Nominal antenna pointing angle is 90 degrees from direction of motion

[0320] 11)Electronic Scanned Array (One Dimensional)

[0321] 12)Ambiguous Velocity of First PRF—48 miles/hr

[0322] 13)Ambiguous Velocity of Second PRF—32 miles/hr

[0323] 14)Unambiguous Velocity—160 ft/sec.—96 miles/hr

[0324] 15)Unambiguous range of first PRF—83.3 miles

[0325] 16)Unambiguous range of second PRF—100 miles

[0326] 17)Unambiguous range of system—320 miles

[0327] 18)Basic 64 Pulse Operation

[0328] 19)Detecting ground targets and measuring their relative radial velocity and angular position and range very accurately.

[0329] B. Clutter Real Bandwidth Considerations With an Electronic Scanned Array (Mounted In-Line With the Platform Motion)

[0330] The transmitting antenna beamwidth together with the receiving antenna beamwidth determines the overall resulting beamwidth of the radar antenna system. The system performs well with many beam-widths as long as there is no significant clutter frequency fold over in the main lobe of the antenna system.

[0331] The antenna beam-width of the antenna system, the velocity of the platform (antenna mounted on) and the pointing angle of the antenna system are the main factors that affect the bandwidth of the main beam clutter pattern. The equation that governs it, is as follows:

[0332] C_BW=[2V_P(C0S( )+Ć))−C0S( )−Ć))]/(9 SIN Ψ)

[0333] C_BW≈(2V_P/λ)(2ΔΨ)

[0334] Where

[0335] 2ΔΨ=(Ψ+ΔΨ)+(Ψ−ΔΨ)

[0336] Ψ—pointing angle of antenna from the direction of the platform in radians

[0337] V_P—Velocity of the platform

[0338] λ—Wavelength of transmission frequency

[0339] Substituting in the equation for CBW we have the clutter bandwidth equal to 116 HZ.

[0340] For sixty-four data points at 1000 HZ the number of large range Doppler bins (LRDBs) per beamwidth is 7.4. The number of fine range doppler bins FRDBs is (7.4×4≈30) thirty in the main beam of the antenna.

[0341] For thirty-two data points at 500 HZ the number of LRDBs per beamwidth is 7.4, since the filters are the same width as the sixty-four data point case. The phase shift per filter is twice that of 64 data point case.

[0342] C. Zero Fill Spectrum Analysis

[0343] The zero fill spectrum processing allows for the basic sixty-four (64) data points (I and Q data) that would normally be processed and yield 64 spectrum lines to yield more spectrum lines. One way to perform this spectrum processing is by Fast Fourier Transfer (FFT). If a greater number of spectrum lines were required or desired then the method of obtaining this would be to take the 64 data points and filling or adding zeros (0), the number added would be equivalent to the number of spectrum lines required plus 64 (original data points).

[0344] For example, if 256 spectrum lines were desired then 192 zeros would be tagged onto the 64 original data points to make a total of 256 data points. Spectrum processing would be performed on all 256 data points and the result would be 256 spectral lines.

[0345] As shown in FIG. 9, “1i,” shows the spectrum lines due to the original 64 data points and 2ishows all other lines are due to the add zero fill data. As observed in the diagram the number of spectrum lines corresponds to the number of zero fill data. The resolution of each doppler filter has not changed but spacing of the filters has decreased to one-quarter, inversely proportional to the total number of data points. The 64 data points of the data is shown and the spectrum of the original 64 data points plus that of that of the 192 additional spectrum lines.

[0346] The significance of this operation is the spacing of the doppler filters has decreased by a factor of four. The application and usefulness of this operation will be evident in the following write up in this disclosure.

[0347] D. DPCA Theory and Application and Example

[0348] 1. Detecting A Moving Target (MT) Signal Processing and Measuring Its Velocity in the Low Clutter Region.

[0349] FIG. 11 shows a conceptual diagram of the basic approach used in this technique. On transmit, the full phased array is used to illuminate the target area of interest, but the signal echoes are received through two adjacent sub-arrays into which the full array is subdivided. These sub-arrays are shown as nominally aligned with the platform's velocity vector, although actually significant crab and pitch angle generally exits between the array axis and the direction of motion. The clutter spectrum as seen through any one of the two sub-arrays is virtually identical, and is determined by the product of the full array pattern on transmit multiplied by the broader sub array pattern on receive. A typical spectrum is sketched in FIG. 10.

[0350] Looking at FIG. 10, notice that the primary (i.e., non-aliased) clutter spectrum is divided into two multiply connected regions. In the low frequency region “2j,” the clutter energy within a single range-doppler cell is large enough to mask or seriously degrade the detection of moving targets, whereas in the region labeled “3j” the clutter is sufficiently small so as to be virtually indistinguishable from system receiver noise. One of the basic moving target detection philosophy employed is to perform simple envelope and threshold detections on those range-doppler cells falling in the pulse doppler region, while more sophisticated interferometry techniques are employed in the clutter region. In the pulse doppler region phase (Φ) can be measured directly as the phase difference between the two target vectors.

[0351] 2. Detecting Targets in Clutter Region (Region 2jof FIGS. 10 and 4jis the region within the bandwidth of the antenna) Normal Pulse Doppler processing is not able to reliably detect moving targets in the clutter region. For those filters within the clutter subvector output of the spectrum processing (FFT), the unique phase relationship between clutter returns in adjacent phase centers is exploited to cancel these signals while permitting moving ground target echoes to persist. In order to understand this process, consider the nature of echoes from motionless ground clutter as received through any pair of subarrays. FIG. 11 shows a two-dimensional simplification of the ideas involved. FIG. 11 illustrates a two dimensional simplification of the Displaced Phase Center Antenna (DPCA) ideas involved, “1k” being the X-axis and “2k” the Y-axis. Since “4k” is transmit phase center for the full array at t=0, the forward receive phase center is (R1) “5k”.

[0352] 3. DPCA Principle Involved in the Dual Receiving Array System

[0353] The path length is the following:

[0354] L (R1, t=0) equals the path length of discrete clutter from point “4k” to “5k” through point “3k” which is located at point (X,Y) from origin “4k”.

[0355] The path length of the same discrete clutter from “4k”, the transmit phase center, at t>0 where “7k” is the velocity of the platform is such that the array travels the distance between points “4k” and “5k” (half the length between receiver sub arrays). We have the following path length to the second receive sub array (R2) “6k”.

[0356] L (R2, t>0)=L (R1, t=0) The same path length.

[0357] This may be seen easily in FIG. 11. Hence we have:

[0358] L (R1, t=0)=L (R2, t=/2V) The same path length

[0359] D=the distance between the receive sub-arrays (“5k” and “6k”)

[0360] V=the velocity of the platform

[0361] The time (t) is to travel the distance D/(2V).

[0362] FIG. 15 illustrates the filters with the phase slopes “7p” (T1=D/2v) and “4n” (T1−k/fr) from FIG. 14 superimposed (8pof FIG. 15). With or without the time domain correction, the correction is exact at the center of the filter “2p”. Also shown is the difference in error as “4p” for T1=D/2V and “5p” for T1−k/fr. A relatively large difference in error for a discrete clutter “3p” which making the combination of time domain and phase compensation more effective. 9pis the center of filter where the error is zero (0) regardless of what delay is implemented. 6p is the theta axis and the vertical axis is the amplitude of the filter response.

[0363] This relationship is the fundamental basis for all displace phase center processing techniques, of which this is one example. It says that the signal received from a scatterer at (X,Y) through the trailing phase center R2 is simply a time-lagged version of the signal received through phase center R1, with the lag equal to the time it takes the aircraft to fly one half the distance between phase centers. Furthermore, since the value of the lag does not depend on the location (X,Y) of the scatterer, this relationship holds for all scatterers, which includes all main lobe clutter for a low PRF radar. Conceptually then, we could cancel all stationary clutter echoes by simply sampling the outputs of R1 and R2 at times t and t+(D/2V) respectively, then subtracting the one signal from the other. A true moving vehicle target, on the other hand will change its (X,Y) position during the elapsed time D/2V, thereby changing its round-trip path length relative to that of a fixed scatterer. Most such motions will cause the relative phase between the R1 and R2 receptions to be non-zero, hence complete cancellation will not occur. In fact, if the path length changes by an odd integer multiple of one-half wavelength, constructive interference will actually double the amplitude of the signal and four times the power of the “canceled” target signal.

[0364] The previous discussion suggests that the way to implement a displace phase center concept is to choose a radar PRF whose interpulse period is equal to the ideal lag D/2V, that is, PRF=fr=2V/D. For this case, we simply combine consecutive pulse returns from phase centers R1 and R2 in order to cancel clutter, and then follow up with sufficient coherent integration to achieve the desired signal-to-noise ratio. Unfortunately, such an approach imposes severe limitations on the overall system design. In the first place, it may result in a highly inefficient utilization of a peak power limited transmitter whose optimal PRF may be several times higher than the “matched” PRF. Although this may be circumvented to some extent by choosing a higher PRF whose interpulse period divides the value D/2V, the basic problem still remains that the allowable PRF values are determined by relatively uncontrollable parameters of the platform and antenna. This fact clashes directly with the need to choose multiple PRFs according to some pattern that leads to the resolution of doppler (and in some systems, range) ambiguities or both.

[0365] 4. P-DPCA Compensation (P-DPCA) Operation

[0366] The approach taken in this technique is to ignore the “matched” PRF requirement, choosing PRFs only on the basis of efficient transmitter utilization and ambiguity constraints. The displaced phase center clutter cancellation is achieved under these conditions using signal processing techniques, which will now be described. The basic idea is to “synthesize” the desired lag compensation between phase centers by a combination of time and frequency domain processing. The PRFs for this technique are generally several times as large as the ideal matched PRF, 2V/D. Consequently, several radar pulses will be transmitted and received during the time period D/2V. Dividing this time by the radar inter pulse period fr gives frD/2V, the number of actual radar pulse repetition intervals (PRIs) during one desired lag time. If this number turns out to be an integer such as 2, then a simple way to achieve the DPCA condition is simply to “skip over” the first 2 pulses in the output from the trailing phase center R2. This is trivially accomplished in the programmable signal processor by simply offsetting the addresses of data memory references by two locations. In order to preserve the batch size for subsequent spectrum processing (FFT), 2 extra pulses would be added to each data processing interval. In this technique, this number of inter pulse periods is almost never a pure integer. Whatever the value might be, it is computed in real time by the system and rounded to the nearest integer value. This integer is then used to offset the memory addresses on the leading phase center, as described earlier. This process leads to a residual uncompensated lag whose process leads to a magnitude which is less than or equal to half the radar inter pulse period. The technique used to compensate for the remaining fractional inter pulse lag is based on the fact that a time domain lag “t” is equivalent to a linear phase versus frequency exp (j2 Πft) superimposed on the Fourier Transform of the lagged signal. The actual implementation of the compensation involves first performing an N point FFT on each of the received phase centers after the integer time slip on the trailing channel. This gives us access to the Fourier transform of the signal at a discrete set of frequencies spaced by fr/N. Let fk=Kfr/N be the center frequency of the K^thdoppler filter. Then the relative phase between receiver channels from main lobe clutter lying within this filter will be exp (jkfrT/N) at the center of the filter. The basic idea then is to multiply the filter output in one of the channels by the conjugate of the predicted phase value for that filter. The two channels are then subtracted, leading to excellent cancellation of the clutter.

[0367] 5. Q-DPCA Compensation (Q-DPCA)

[0368] FIGS. 12 through 15 illustrate some of the ideas discussed above. FIG. 12 shows what the phase relationship between receiver channels would look like if no attempt were made to correct for the lag, either by pulse reindexing or by doppler filter corrections. The slope of the (“4l”) linear phase-versus-frequency, where 11 is the frequency and 2lis the phase, locus is proportional to the lag value D/2V between channels. Note also in FIG. 12, “3l” that individual clutter cells are represented as scattered about the line rather than lying right on it. This is due to the uncorrelated noise that also lies within each range/doppler cell and perturbs the phase of the clutter signals. It is interesting to note that the larger the clutter cross-section within the cell, the more closely the relative phase will conform to the line. Since distance from the line determines the residual phase error that will remain after pre-filter compensation, and this error determines cancellation ratio, this illustrates the fact that the larger a scatterer is, the greater will be its cancellation. In fact, if hardware imperfections in the radar are neglected, the cancellation ratio would be in direct proportion to the size of the scatterer, and all clutter would cancel to the same residue level. As will be shown later, the previous statement also neglects the effect of non-zero filter width, since exact phase compensation holds only at the center of the filter.

[0369] FIG. 12, “5l” also illustrates the interchannel phase for a typical moving target signal. Since it lies well off the clutter phase line, it will not cancel like clutter will when phase-compensated subtractions are performed in each doppler filter. FIG. 13 shows actual data for uncompensated mainlobe clutter data. Averaging the phase in each filter over a number range cells so that the linear nature of the lag induced phase is evident so that the scatter about the straight line has been drastically reduced. FIG. 12 and FIG. 13 indicate another typical characteristic of real world data in the fact that the phase-versus-frequency line does not pass through the origin. This non-zero intercept is not predicted by the simplified theoretical development here, but is due to the somewhat imperfect nature of platform motion compensation. Such imperfections are estimated on the signal processor from range averaged ensembles of clutter data such as that shown in FIG. 13, 2mis the amplitude and 1mis the frequency and 3mis the response, and are accounted for in the compensation weights applied to each filter prior to cancellation.

[0370] 6. Results Of Q-DPCA Compensation versus (P-DPCA). In looking at FIGS. 12 and 13, it might seem as though we could cancel clutter using phase corrections alone without having to time slip the trailing phase center by several pulses. It will now be shown that such an approach results in severe limitations on the cancellation ratio that can be achieved. To see this, first examine FIG. 14, which illustrates the effect of slipping the trailing phase center by k^thinterpulse period. The initial phase slope is proportional to T1=D/2V, whereas that of the time slipped signal is proportional to T1−k/fr. Now zero-in on a single doppler filter for which the phase has been corrected to the value at the line of the filter center frequency. FIG. 15, where 1mis the phase response and 2mis the amplitude response, illustrates the filter with the two phase slopes from FIG. 14 superimposed. With or without the time domain correction, the phase correction is “perfect” at the filter center (“9p”). Notice, however, away from this center we begin to build up a small but measurable phase error. Consider a discrete piece of clutter (“3p”), whose doppler frequency places it a significant fraction of the filter's width away from the center. Without pulse delay (“7p”), compensation prior to the FFT, the phase error between channels after phase correction will be proportional to the original lag T1=D/2V (FIG. 15 “4p”) between phase centers. For a displacement of one half the filter spacing and lag T1=k/fr (FIG. 15 “5p” the resulting phase error will have a value Φ₀, given by Φ₀=ΠDf_r/2NV.

[0371] For typical parameter values, TDPCA-FIG. 15 “7p”, this error would limit cancellation of the scatterer to somewhere between 15 and 20 db. The signal that has been pulse plus time-domain corrected, FIG. 15 “5p”−P-DPCA however, suffers a much smaller uncompensated phase error (T1−D/2V). In fact, the worst possible case in which the fractional portion of the lag is exactly one-half the interpulse period lead to a phase error Φ₁given by: Φ₁=2Π/N.

[0372] Typical values of the improvement factor 4v/(frD) in the above equation imply a reduction of as much as five to one in phase error, which equates to an increase of 14 db in cancellation ratio. The increase will be even more dramatic for lag values that are closer to being integer values, so that the fractional lag is less than 0.5 interpulse periods. By this analysis we can conclude that the pulse plus time domain lag correction is an essential precursor to post FFT frequency domain compensation. This will be especially true for large discrete clutter, where large cancellation ratios are required.

[0373] The analysis presented so far has been based on the idea of a simple lag correction using a combination of time domain and pulse “slipping” and post-FFT implementation of this technique involves estimation of the slope and intercept of the interchannel phase line. This is accomplished by a combination of micro-navigation or motion compensation calculations and regression techniques based on range averaged cross-correlations between main lobe clutter in adjacent phase centers. Having estimated the mean clutter phase line, cancellation phase values are picked off this line for frequencies corresponding to the center of each Doppler filter. A separate amplitude imbalance estimation is performed to derive a relative gain correction to be applied uniformly across all filters in one of the channels. This latter correction accounts for uncertainties in the precise gain of each receiver channel.

[0374] 7. Weighting Function for Clutter (W_C) and Weighting Function for Target Analysis (W_M)

[0375] The actual approach used in this technique differs in one significant aspect from that described above, and that is in the method used to determine the phase and amplitude weights to be applied to each doppler filter. The motivation for departing from the simple theory presented is the fact that small imperfections may exist in the hardware used which leads to small but significant departures from the linear phase, constant gain model assumed. For example, suppose that the nominally identical antenna amplitude and phase patterns of adjacent sub arrays show small variations from one another due to manufacturing tolerances, differential radome effects, failed phase shifters, etc. This lack of a perfect “template match” between channels will manifest itself as an amplitude and phase “ripple” in azimuth, which by virtue of the one-to-one correspondence of azimuth with doppler for main lobe clutter will result in a superposition of this ripple on the nominally linear inter channel phase and constant gain ratio per filter. To account for such real world effects, a more robust approach to weight estimation has been employed. Basically, we remove the “a priori” assumption of a known model for filter-to-filter variation in the phase and amplitude of mainlobe clutter between channels. Instead, we seek an independent solution in each filter for the “best” complex weight unique to that filter which maximizes clutter cancellation. A separate solution is then obtained for each Doppler filter, which intersects a significant portion of the main lobe clutter spectrum. The optimality criterion employed is the minimization of the mean squares residues after cancellation when averaged over all range bins within a range swath.

[0376] In general, small range modes will utilize the entire swath as the range ensemble, while large range mode swaths are subdivided into contiguous sub-swaths each of which leads to an independent solution for cancellation weights.

[0377] Consider such an ensemble consisting of M consecutive range bins and N Doppler filters. Let Unk and Vnk be the complex signals at the output of the k^thDoppler filter in the N^thrange bin for the leading and trailing phase centers respectively. The goal of weight estimation then is to find a complex number Wc unique to the kth Doppler filter that satisfies the following complex least mean squares condition: $W_C={\left(\sum _M^{M=1}{\left|\mathrm{Vnk}\right|}^2\right)}^{-1}\sum _M^{M=1}{u}_{\mathrm{nk}}{V}_{\mathrm{nk}}^{*}$ $\sum _M^{M=1}{\left|u_{\mathrm{nk}}-W_C{V}_{\mathrm{nk}}\right|}^2=\min \left(W_C\right)\sum _M^{M=1}{\left|u_{\mathrm{nk}}-W_K\mathrm{Vnk}\right|}^2$ $\sum _M^{M=1}{\left|u_{\mathrm{nk}}-W_C{V}_{\mathrm{nk}}\right|}^2=\min \left(W_C\right)\sum _M^{M=1}{\left|u_{\mathrm{nk}}-W_K\mathrm{Vnk}\right|}^2$

[0378] The well-known solution to this problem is given by: $W_C={\left(\sum _M^{M=1}{\left|\mathrm{Vnk}\right|}^2\right)}^{-1}\sum _M^{M=1}{u}_{\mathrm{nk}}{V}_{\mathrm{nk}}^{*}$

[0379] Equation for Wc represents the solution to be implemented in a programmable signal processor for note that it is optimal from a phase and an amplitude point of view, and so replaces both the linear phase regression and the gain imbalance estimation of the previously described approach.

[0380] The nature of the least mean squares solution described above necessitates two passes through the data. In the first pass, the two statistics are generated cumulatively for each doppler filter lying in mainlobe clutter. The ratio of the statistics is calculated between passes to determine the optimum weight Wc for each filter, after which the second data pass proceeds to use these weights to perform clutter cancellation. Since weight estimation is performed on the very same data set on which the weights will be used, there is no question of the optimality of these weights in a range averaged sense.

[0381] The above equations may be employed for the optimization of weights for targets (Wm) as well as for clutter. The solution of system equations will give a solution for the azimuth angle for the target, and the corresponding clutter filter (Wc) that is at this same azimuth will be the weight for the target Wm. This will be utilized in the system objective.

[0382] As indicated by the previous discussion on DPCA techniques and evaluation, the importance of getting close to a matched PRF and to detect the target and/or clutter as close to the center of the filter as possible to reduce the effect of mismatch. This will be manifest in the subsequent development of system solutions. A tradeoff of PRF for match as close as possible with other factors as discussed previously should be attempted.

[0383] 8. DPCA Calculations Based on Basic Parameters for the Radar Assumed for the Purpose of Illustration

[0384] The basic radar assumed for the purpose of illustration is tabulated in section III A. Basic System. The length of transmit antenna is eleven (11) feet and the length of the two receive antenna sub arrays are five and one-half feet (5.5′) each. Therefore the travel length for ideal cancellation of clutter is two and three quarter feet (2¾ feet).

[0385] Assuming the platform has a velocity of 500′/sec. and the first PRF is 1000 HZ, the number of pulses for best cancellation of clutter will be calculated as follows:

[0386] V_pN(1/PRF)=Dist

[0387] V_p×5×{fraction (1/1000)} HZ=500×10^−3×5=2.5 feet

[0388] Vp—Velocity of platform

[0389] N=“5”—number of pulses of 1000 HZ=PRF

[0390] Dist—Distance Travelled

[0391] 2.75′ feet is the ideal distance to travel to cancel clutter and is therefore 0.25′ short which is half the distance between the fifth (5) and sixth (6) pulse.

[0392] The calculation for the T-DPCA of half the distance error (maximum error on each side of center frequency as described in the previous paragraphs is the following):

[0393] fr—Pulse Repetition Frequency—PRF—1000 HZ

[0394] D—Distance between sub arrays—5.5 feet

[0395] V_p—500 feet/second

[0396] N—5 pulses

[0397] Substituting in this basic equation derived in the previous paragraphs we have the following:

[0398] MAX Phase Error=ΠDf_r/2NV=3.14*5.5*1000 /2*64*500

[0399] M_PT=0.270 Radians=15.4°

[0400] Performing the calculation of the maximum phase error for the P-DPCA we have the following: $\begin{array}{c}\mathrm{Mpp}-\text{Maximum Phase Error}=\Pi /2N\\ =3.14159/\left(2*64\right)=1.4\mathrm{degrees}\\ =\mathrm{.0245}\mathrm{radians}\end{array}$

[0401] The ratio of (M_PT/M_PP)=11.0. The significance of this ratio is that the P-DPCA technique has less than one-tenth the error of T-DPCA technique.

[0402] The calculations for Q-DPCA at D=6 is the same as for P-DPCA but is opposite in sign since it is on other side of the pulse midway between the pulses.

[0403] M_PQ−Maximum Phase Error=−1.4 degrees=−0.0245 radians

[0404] The same calculations will now be performed for the PRF of 750 HZ. The number of pulses for the P-DPCA is four (4) instead of (5) as in the 1000 HZ case. The results of this calculation are as follows:

[0405] T—DPCA−M_PT=(Πf_rD)/(2NV_P)=0.202 radians=11.6 degrees

[0406] The factor of “≈¼” in the P-DPCA equation is determined from the following:

[0407] Distance traveled per pulse−D_p=500′/sec×1.333 MS per=0.665′/pulse

[0408] “4” pulses=0.665×4=2.66′

[0409] 2.75′−2.66′=0.09′

[0410] 0.09′/0.665 pulse=0.135≈⅛ or ⅛ distance between two pulses or ¼ of ½ distance between pulses or 0.270.

[0411] Therefore P-DPCA is as follows:

[0412] M_PP=(3.14/(2*64))*0.270)=0.006 radians=0.38 degrees

[0413] The ratio of (M_PT/M_PP)=11.6o/0.38=30.5

[0414] The significance is that the P-DPCA technique has less than {fraction (1/30)} of the phase error of the T-DPCA technique at 750 HZ.

[0415] The Q-DPCA at 750 HZ at D=5 error is the following:

[0416] M_PQ−Maximum Phase Error=1.03 degrees=0.017 radians

[0417] M_PQ/M_PP=1.03/0.38=2.71

[0418] 9. Comments on the DPCA Example

[0419] As can be readily seen in previous section “8” that the DPCA error due the PRF mismatch is considerably less in the PRF=750 HZ. At 750 HZ, PRF the Mpp=0.38° and at 1000 HZ—PRF, it is 1.4°, this is ¼ of the error. The processing may be performed at 750 HZ with the W_Cand W_Mterms (since the other term is very small) employed with only their offset values of amplitude and phase and simplifying the equation and the processing required to obtain the desired results.

[0420] E. Basic Radar Doppler Equations

[0421] The systems consist of two equal receiving sub arrays and a transmit array. The two receiving sub arrays will receive radar data that will have clutter only, target only and target plus clutter. The system is employed to detect targets and measure its relative radial velocity and angular position and range. Since range is easily measured from the range bin the target is detected, the other two parameters are the objective of this invention in and out of the presence of clutter.

[0422] The basic system endeavors to measure the moving targets relative radial velocity and angular position. A moving target which changes its position in space relative to the radar each PRF and relative to the transmission frequency and the PRF given by the equation as follows:

[0423] Φ_R=2ΠD(λ/2*PRF)

[0424] Φ_R—Phase change per PRF of target

[0425] D—Distance traveled per PRF relative radial to the radar

[0426] PRF—Pulse Repetition Frequency of the Radar

[0427] λ—Wavelength of transmission frequency

[0428] Φ_Ris proportional to the relative radial velocity of the target. The radar has motion compensation (since it is moving itself) to make the bore sight of its antenna look like it has zero (“0”) velocity (no motion), targets or clutter appearing at some angle will have an apparent motion relative to the radar proportional to the cosine of the angle off bore sight.

[0429] The equation that represents this phase change due to the angular position of the clutter or target off bore sight is the following:

[0430] Φ_R=2ΠD/(λ/2*PRF)

[0431] Where

[0432] D=(V_P×sin(BW))×(1/PRF)

[0433] Vp—Velocity of the platform

[0434] BW—Beamwidth of the transmit antenna

[0435] Other parameters defined in previous equation.

[0436] Taking an example for illustration. Let V_P=500 feet/sec and BW=⅔° and λ=0.1 and the PRF=1000 HZ, solving the above equation we have the phase change Φ_R=41.760 (0.728 radians).

[0437] Solving for the bandwidth of clutter in a beamwidth as follows:

[0438] CBW=(Φ_R/360)PRF

[0439] Substituting the values we obtained above:

[0440] CBW=(41.76/360)×1000=116 HZ

[0441] CBW—clutter bandwidth

[0442] F. Differentiating Target Doppler from Clutter Doppler

[0443] Since the objective of this invention is to measure target doppler in the presence of clutter doppler an analysis of this situation will help in attaining the objective of this invention.

[0444] The concept is of differentiating the doppler of the target and the doppler of the clutter into their respective component parts and as distinct and independent doppler. The system is analyzed and employed in that manner.

[0445] When clutter and target occur in the same LRDBs (Large Range Doppler Bins), careful attention must be made to find their individual characteristics and employ this to advantage. This will be further explored.

[0446] At the beginning of Section E both clutter and target have a doppler due to their position in the main beam of the antenna proportional to their angular position off bore sight of the antenna.

[0447] With clutter this is the only component of doppler. This doppler may be detected in the same LRDB as a target and hence difficulty obtaining doppler of the target due to its relative radial velocity.

[0448] Targets on the other hand have two doppler components. One component of target doppler like clutter has a component due to its angular position in the main beam of the antenna. The other component of the target doppler is due to its radial velocity relative to the radar.

[0449] This will be now expressed in equation form as follows:

[0450] Φ_D=Φ_A+Φ_R

[0451] Φ_D—Phase change of the target proportional due to its angular position plus its relative radial velocity in each receive antenna array.

[0452] Φ_A—Phase change of the target proportional to its angular position off bore sight of the antenna in each receive antenna array.

[0453] Φ_R—Phase change of target due to its relative radial velocity in in each receive antenna array.

[0454] Therefore, the target when detected in any receive array, its doppler bin or change in phase is due to two components; it's own azimuth position and that due to its relative radial velocity. This is the object of the invention, to determine the relative radial velocity. Once the relative radial velocity is determined, the target's azimuth position is determined and its doppler and change in phase is calculated.

[0455] The “D”—delay between the two receive antenna arrays where the second receive array, the “D” delay of pulse returns is implemented. The phase change of the target at the second receive array from the first receive array is DΦ_R. When “DΦ_R” is solved for it must be divided by “D” to determine Φ_R.

[0456] The determination of Φ_Ais therefore and as explained previously, this is the phase change due to the targets azimuth position and is fixed and does not change within the limited time of the target being illuminated.

[0457] Therefore when W_Mis measured, it is measured at the doppler bin or filter at this azimuth position as explained previously as if clutter resided at this position. The W_Mmeasure at this position is the following:

[0458] W_M=A_CMe^j(Ψ+K_CM^(s−y))

[0459] W_MA_CMe^j(Ψ+K_CMm^(s−y))

[0460] Acm—Offset Amplitude

[0461] Ψ_CM—Offset Angle

[0462] y—actual angular position of target

[0463] s—estimated angular position of target

[0464] s−y—difference between actual and estimated angular position of target

[0465] W_Mis dependent on s−y since the targets angular position does not change in the limited amount of time on target. The offset angle and offset amplitude is known dependent on angular position.

[0466] DΦ_R—Phase change of target due to its relative radial velocity between sub arrays (antenna 1 and antenna 2) and delay D.

[0467] If the target is not detected at its peak (FRDB) but DΦ_Ris determined by processing, in order to solve the previous equation for Φ_Aand Φ_D, the target has to be measured at the position where the target is detected at its peak. The DΦ_Rparameter is measured from the peak detection of the target. This term is divided by D to determine Φ_Rand consequently determine Φ_A, the azimuth position of the target.

[0468] Φ_C=Φ_Cand Φ_C=Φ_Dwith no target present

[0469] Φ_C—Phase change of clutter due to its angular position in the beam of the antenna

[0470] As illustrated the spectrum analysis for the 64 data points received from the radar would be 64 frequency components spaced equally apart 15.625 HZ with the same frequency resolution. These frequency components will be called (0₀, 1₀, . . . , 63₀) the large frequency components—the large range doppler bins (LRDB).

[0471] The frequency components performed on the zero filled data up to 256 data points and a spectrum analysis performed would be 0₀, 1₀. . . . 63₀to 0₁, 1₁, . . . , 63₁to 0₃, 1₃, . . . , 63₃. These frequency components would be called (FRDB) the fine frequency doppler bins components due to the zero filled operation.

[0472] As a result of the zero fill operation all the filters will have a resolution of 15.625 HZ and a spacing of 3.90625 HZ.

[0473] The number of filters have been increased by a factor of four (4).

[0474] The delayed data (256 points) spectrum analysis of sub-array 2 subtracted from the corresponding FRDBs in the sub-array 1 there is a resultant vector when target is present as shown in FIG. 1-19a. The FRDB that has the maximum resultant amplitude is where the target is at that frequency response of the filter.

[0475] The FRDB of clutter maximum amplitude is most probably in another position within plus or minus two (2) FRDB from where the target is detected at its peak.

[0476] Target and clutter are completely independent and consequently where the target amplitude is peak (frequency response) will be most probably different than the peak of clutter even though they are detected in the same LRDB but do not peak in the same position.

[0477] FIG. 3, BLOCK DIAGRAM OF NON IDEAL SYSTEM, may be implemented by a hardwired or general purpose or general purpose plus signal processor computer.

[0478] The target detection has one peak detection in a particular or near a particular FRDB and its affect is detected in adjacent FRDBs and LRDBs in range and doppler. There is assumed only one target. The target's angular position most of the time will be in some other position.

[0479] Clutter on the other hand is essentially independent in every LRDB. Its angular position is determined by where its peak is detected as mentioned in the previous paragraphs. Ifs independent from LRDB to LRDB but detected at its peak for its angular position.

[0480] G. Basic Mathematical Equations Developed for Processing a Target Detected in the Same Filter at Two Sequential Times

[0481] As illustrated in FIGS. 1 to 3 and FIG. 9, the vector diagram of the target and target plus mover, block diagram of the system and the large range doppler bins (LRDBs) in each sub-array, there is a fundamental set of equations that are to be developed and to be employed in all techniques in the following sections to attain the objective.

[0482] For each sub-array the radar data is stored and a weighting function applied to the data and a spectrum analysis obtained for “N” data points and produces “N” doppler bins for each of the “M” range bins. Sub-array #2 the data points are delayed “D” pulses for effective DPCA processing. (FIG. 10 to FIG. 17), For ×4 radar data there is “N×4” doppler bins for each of “M” range bins.

[0483] FIG. 9 illustrates this, it is performed by adding (256−64) zeros before doing the spectrum processing.

[0484] The sub-array 2 processed FRDBs are subtracted from the corresponding FRDBs of sub-array 1 and the vector diagram results as in FIG. 1. Illustrated in FIG. 1 are the conditions of target only and also with target plus clutter where a target is detected.

[0485] Where there is target only the angle DΦ_Ris measured between the two target vectors and determines the relative radial velocity and the angular positioned is also determined from Φ_Dand DΦ_R. There is no problem in attaining the objective when clutter is not present with the target.

[0486] When the target is detected in the presence of clutter, at this point, DΦ_Rcannot be measured directly and a technique has to be developed to measure DΦ_Rand hence the objective of the invention. The following section will develop the mathematical equations, which in Section IV will be utilized to solve this problem.

[0487] 1. Ideal System

[0488] Sub-array 1 equation when the target is detected in a particular FRDB is as follows:

[0489] V′₁₁=C+M  (1)

[0490] V′₁₁—Measured vector in sub-array 1 at time 1

[0491] C—Clutter vector in sub-array 1 at time 1

[0492] M—Mover vector in sub-array 1 at time 1

[0493] Sub-array 2 equation for ideal system when the target is detected in the same corresponding FRDB as in sub-array 1 at a delayed “D” number of pulses is as follows:

[0494] V′₁₂=C+MEXP(JDΦ_R)  (2)

[0495] V′₁₂—Measured vector in sub-array 2 at time “D”

[0496] C—Clutter vector in sub-array 2 at time “D”

[0497] M—Mover vector in sub-array 2 at time “D”

[0498] DΦ_R—The phase shift due to the motion of the target from pulse “1” in sub-array 1 to pulse “D” in sub-array 2

[0499] This equation is ideally almost the same as in sub-array 1 but due to the delay “D” and the phase shift of the target as stated in the aforementioned paragraphs.

[0500] For data point “2” in sub-array 1 delayed one pulse and phase shifted in value corresponding to the conjugate of the phase shift of the filter processed we have the following equation:

[0501] V″₁₁XC+YM  (3)

[0502] V′11—Measured vector in sub-array 1 for data point “2”

[0503] X—Change in clutter from data point “1” to data point “2”

[0504] C—Clutter vector of data point “1”

[0505] Y—Change in target from data point “1” to data point “2”

[0506] M—Mover vector of data point “1”

[0507] V″₁₂=XC+YMEXP(JDΦ_R)  (4)

[0508] V″₁₂—Measured vector in sub-array 2 for data point “D+1”

[0509] C—Clutter vector of data point “1”

[0510] Y—Change in target from data point “1” to data point “2”

[0511] M—Mover vector of data point “1”

[0512] Subtracting equation (2) from equation (1) we have the following:

[0513] M=(V′₁₁−V′₁₂)/(1−EXP(JDΦ_R)  (5)

[0514] Similarly subtracting equation (4) from equation (3) we have the following:

[0515] YM=(V″₁₁−V″₁₂)/(1−EXP(JDΦ_R)  (6)

[0516] Taking equation (6) and dividing by equation (5) it follows:

[0517] Y=^jB=(V″₁₁−V″₁₂)/(V′₁₁−V′₁₂)⁻  (7)

[0518] {overscore (K)}—Amplitude ratio change in target

[0519] B—Phase change in target

[0520] Multiplying equation (1) by EXP(JDΦ_R) and subtracting equation (2) and solving for clutter (C) we have the following:

[0521] C=(V′₁₁EXP(JDΦ_R)−V′₁₂)/(1−EXP(JDΦ_R))  (8)

[0522] Similarly multiplying equation (3) by EXP(JDΦ_R) and subtracting equation (4) and solving for (XC) changed clutter we have the following:

[0523] X C=(V″₁₁EXP(JDΦ_R)−V″₁₂)/(1−EXP(JDΦ_R)  (9)

[0524] Taking equation (9) and dividing by equation (8) we have:

[0525] X=(V″₁₁EXP(JDΦ_R)−V″₁₂)/(V′₁₁EXP(JDΦ_R)−V′₁₂)  (10)

[0526] We see by equation (7) we have solved for the change in target and it is independent of DΦ_R. In equation (10) solving for the change in clutter, it is dependent on DΦ_R.

[0527] If we take equation (10) and solve for DΦ_Rwe have the following:

[0528] EXP(JDΦ_R)=(XV′₁₂−V″₁₂)/(XV′₁₁−V″₁₁)  (11)

[0529] As previously developed was a set of equations where the clutter cancels ideally but as discussed previously ideal conditions of canceling clutter rarely is present and so consequently a set of equations will be developed for the non-ideal case. In section III-D-DPCA Theory and application of the practical implementation of the ideal case is rarely attained. There is almost never clutter that cancels completely.

[0530] This set of equations is for the P-DPCA compensation of section IIID.

[0531] 2. Non-Ideal System

[0532] The next set of equations will be for the non-ideal case where the clutter and target have to be corrected for. The sub-array 1 equation for the non-ideal case is the following:

[0533] V′₁₁=C+M  (12)

[0534] The same as equation (1).

[0535] Sub-array 2 for data point D equations are the following:

[0536] V′₁₂=C /W_C1+(M/W_M1)EXP(JDΦ_R)  (13)

[0537] W_C1—The factor that relates to clutter in both arrays which makes clutter cancel.

[0538] W_M1—The factor that relates to target in both arrays, which make a target correction in both arrays to make them the same amplitude and phase.

[0539] For data point “2” in sub-array 1 we have the following equation:

[0540] V″₁₁=XC+YM  (14)

[0541] The same as equation (3).

[0542] The equation for sub-array 2 at data point (D+1) is the following:

[0543] V″₁₂=XC/W_C2+(YM/W_M2)EXP(JDΦ_R)  (15)

[0544] Similar to equation (4) but with the factors W_C2and W_M2added W_C2factor that relates the clutter in both arrays which makes {overscore (K)} and B correct.

[0545] W_M2factor that relates to the target in both arrays which makes K and α correct.

[0546] Multiplying equation (13) by W_C1and subtracting it from equation (12) and solving we attain:

[0547] M=(V′₁₁−W_C1V′₁₂)/(1−(W_C1/W_M1)EXP(JDΦ_R))  (16)

[0548] Similar with equation (14) and (15) we get the following:

[0549] YM=(V″₁₁−W_C2V″₁₂)/(1−(W_C2/W_M2)EXP(JDΦ_R))  (17)

[0550] Solving for Y employing equation (16) and (17) we have:

[0551] Y=(V″₁₁−W_C2V″₁₂)/(V′₁₁−W_C1V′12)  (18)

[0552] Where

[0553] (1−(W_C1/W_M1)EXP(JDΦ_R))=(1−(W_C2/W_M2)EXP(JDΦ_R))

[0554] Assuming W_C1≈W_C2≈W_Cand W_M1≈W_M2W_Mwe have:

[0555] Y=^jB=(V″₁₁−W_C1V″₁₂)/(V′₁₁−W_C1V′₁₂)  (19)

[0556] Multiplying equation (12) by (EXP(JDΦ_R)/W_M1and subtracting equation (13) from (12) we have:

[0557] V′₁₁EXP(JDΦ_R)/W_M1−V′₁₂=C(EXP(JDΦ_R))/W_M1−1/W_C1  (20)

[0558] Similarly multiplying equation (14) by (EXP(JDJ_R)/W_M2) and subtracting equation (15) from (14) we have: (EXP(JDJ_R)/W_M2

[0559] (V″₁₁EXP(JDΦ_R)/W_M2−V₁₂″=XC(EXP(JDΦ_R))/W_M2)−1/W_C2  (21)

[0560] Solving for X employing equation (20) and (21) we have:

[0561] X=((V″₁₁EXP(JDΦ_R)/W_M2)−V″₁₂)/(V′₁₁EXP(JDΦ_R)/W_M1−V′₁₂)(EXP(JDΦ_R)/W_M1−1/W_C1)/((EXP(JDΦ_R)/W_M2)−1/W_C2)  (22)

[0562] Assuming W_C1≈W_C2≈W_Cand W_M1≈W_M2≈W_Mwe obtain:

[0563] X=Kexp(jα)=(V″₁₁EXP(JDΦ_R)−W_MV″₁₂)/(V′₁₁EXP(JDΦ_R)−W_MV′₁₂)  (22)

[0564] Solving for EXP(JD(DR) in equation (22) we have the following:

[0565] (EXP(JDΦ_R))/W_M=(XV′₁₂−V″₁₂)/(XV′₁₁−V″₁₁)  (23)

[0566] Looking at the assumption W_C1≈W_C2and W_M1≈W_M2

[0567] W_Cand W_Mis the LMS average (Section IIID) of the same FRDB between sub-arrays 1 and 2. W_Cand W_Mis in effect a ratio. When the next measurement taken is ≈one millisecond later that ratio W_Cand W_Mwill remain essentially equal, Summarizing the assumption why W_C1≈W_C2≈W_Cand W_M1≈W_M2≈W_Mis correct.

[0568] same FRDBs are measured

[0569] same measurement from sub-array 1 and 2

[0570] an effective ratio taken (diminishes changes)

[0571] only millisecond later in time taken (very little change)

[0572] data points are the same just shifted in time

[0573] The assumption is well founded in the facts and the conditions involved.

[0574] The next set of equations will be the condition different from the previous where the P-DPCA compensation will be utilizing time only (Section IIID). The development is very analogous to the P-DPCA compensation employing Q-DPCA. The basic equations will be stated and explained and then the important resultant equations given:

[0575] Sub array 1 for data point 1:

[0576] V′₁₁=C+M  (12)

[0577] The same as equation (12) and the same definition of terms

[0578] Sub array 2 for data point (D+1):

[0579] =C/+M/  (13)

[0580] W—the factor that relates to clutter in both arrays which cancels clutter in the mode where Q-DPCA compensation is used. This is different from W_C1for P-DPCA compensation (Section III-D).

[0581] W—the factor that relates to the target in both arrays, which makes the target, vector the same. This is different from W_M1for the same reason as stated in previous paragraphs for W (Section III-D).

[0582] V′₁₂—The measured vector in sub-array 2 for data point using Q-DPCA compensation.

[0583] For data point 2 in sub array 1 we have the following equation:

[0584] V″₁₁=XC+YM  (14)

[0585] The same as equation (3) and the same definition of terms.

[0586] The equation or sub-array 2 at data point (D+1) is the following:

[0587] =XC/+(YM/)EXP(J(D+1)Φ_R)  (15)

[0588] V″ Similar to equation (15) but with the factors W substituted for W_C2and W substituted for W_M2and V″ substituted for V″₁₂

[0589] XC and YM same definition as stated previously

[0590] W—the same as W_C2except this term is for the Q-DPCA

[0591] W—the same as W_M2except this term is for the Q-DPCA

[0592] V″₁₂—similar to V″₁₂but for Q-DPCA instead of P-DPCA

[0593] Employing all the same algebraic manipulations as in the previous derivation, we have all analogous results as previously except the following terms are different as follows:

[0594] W substitutes for W_C1

[0595] W substitutes for W_C2

[0596] V′₁₂substitutes for V′₁₂

[0597] V″₁₂substitutes for V″₁₂

[0598] W substitutes for W_M1

[0599] W substitutes for W_M2

[0600] (D+1)Φ_Rsubstitutes for DΦR

[0601] All other terms are the same. Some key equations to illustrate this we have the following:

[0602] M=(V′₁₁−)/(1−(/)EXPJ((D+1)Φ_R)  (16)

[0603] Y=exp(jβ)=(V″₁₁−)/(V′₁₁−)  (19)

[0604] X=Kexp(jK)=(V″₁₁EXP(J(D+1)J_R)/−)/(V′₁₁EXP(J(D+1)J_R)/)  (22)

[0605] e^j(D+1)Φ)/W_M1=(X−)/(X−V″₁₁)  (23)

[0606] Observing the set of equations developed on the same set of data employing P-DPCA and Q-DPCA compensation and equating equation (19) with equation (19) we have the following

[0607] (V″₁₁−)/(V′₁₁−)=(V″₁₁−W_C2V″₁₂)/(V′₁₁−W_C1V′₁₂)  (24)

[0608] Taking the assumption W=W and W_C1=W_C2and the following definition of W_Cterms:

[0609] W_C1=W_C2=A_CMEXP(J(Ψ_CM+K_CM(s−x)))  (25)

[0610] ==e^{j(Ψ+(s−x))}  (26)

[0611] (V″₁₁−)/(V′₁₁−)=(V″₁₁−W_C1V″₁₂)/(V′₁₁−W_C1V′₁₂)  (26A)

[0612] ((V″₁₁−A_CMEXP(J(Ψ_CM+K(s−x))V″₁₂)/((V′₁₁−A_CMEXP(J(Ψ_CM+(s−x))V′)=

[0613] ((V″₁₁−A_CMEXP(J(Ψ_CM+K_CM(s−x))V″₁₂)/((V′₁₁−A_CMEXP(J(Ψ_CM+K_CM(s−x))V′₁₂)  (26B)

[0614] Acm=A where Acm or A is the offset amplitude of the term that makes the amplitude of the clutter in both arrays equal. This is assumed to be equal, which should be equal regardless of the type of DPCA compensation employed. This term is calculated on the same filter.

[0615] Ψ_CM=Ψ_CMwhere Ψ or Ψ_CMis offset angle that makes the phase of the clutter in both arrays equal. The reasoning is the same as in the previous paragraph for Acm.

[0616] “s” is the FRDB processed and is known.

[0617] x is the FRDB in which the clutter amplitude is at its peak (at the center of the filter response—FIGS. 16—3q)—which is unknown.

[0618] All the “V” terms are as defined in the previous paragraphs and all are measured known terms

[0619] K_CMis the slope of the phase compensation term for clutter for “D” delay P-DPCA compensation (FIG. 16). This is known and calculated a shown in Section III-D.

[0620] Making A_CM=A and is known because it is measured in the clutter FRDB at or near the FRDB of where the clutter resides. The amplitude of the offset should remain approximately equal over a number of FRDBs.

[0621] Making Ψ_CM=Ψ and is known for the reasons as stated in the previous paragraph.

[0622] K_CMand K_CMare known and are calculated as stated in the section III D on DPCA compensation.

[0623] Taking equation (26B) and substituting W_C1and W as in equations (25) and (26) the only unknown is x and therefore solvable in x. This determines the FRDB where clutter is at its maximum amplitude. Since x is closely the same for both sets of equations. The value x determines where the peak of clutter is located corresponding to “3q” in FIG. 16, Ideal Clutter Filter and Ideal W_CResponse. The value “s” is the FRDB processed and according to this value, the distance from the peak of clutter on “8q”, the phase response of clutter is the phase. The corresponding clutter change is X=KαEXP(Jα) in time. FIG. 16 indicates 2qas the amplitude response of clutter and the ideal phase response of clutter and 4q, 2q, 3qand 10qfor the respective delays. The X axis depicts the FRDB and the Y axis the amplitude.

[0624] X_F=Kα_FEXP(Jα_F) is the clutter response from filter to filter (FRDB to FRDB) in FIG. 26—Table I and FIG. 17 illustrates a similar set of components for target as in FIG. 16 for clutter. The amplitude response of target as a function of FRDB and the phase response as function of azimuth. 3ris the target peak and 2ris the target amplitude response, 4ris the phase response, 7rand 9rare phase responses for P-DPCA and Q-DPCA respectively. At 3ris the target peak amplitude response equal to one and phase response equal zero.

[0625] It is to be noted, in theory, by observing in FIG. 16, Ideal clutter filter response and clutter change in time and W_Cresponse are related. Once one factor is known (clutter change in time or clutter filter response or clutter response for P-DPCA or Q-DPCA) all other factors are estimated closely or known. This enables these techniques to be effective. Therefore a priori processing must be performed to establish the relationship between all these factors. A priori knowledge of the filter response and K_CMand K and other factors are calculated and known.

[0626] Also the factors A_CMand Ψ_CMmay be determined by processing all the filters as in Section III D-7.

[0627] Equation (24) with W_C1=W_C2and W=W with A_CM=A and Ψ_CM=Ψ we have the following:

[0628] (V″₁₁−)/(V′₁₁−)=(V″₁₁−W_C1V″₁₂)/(V′₁₁−W_C1V′₁₂)  (24)

[0629] ((V″₁₁−A_CMEXP(J(Ψ_CM+(s−x))))/((V′₁₁−A_CMEXP(J(Ψ_CM+(s−x))))=

[0630] ((V″₁₁−A_CMEXP(J(Ψ_CM+K_CM(s−x)))V″₁₂)/((V′₁₁−A_CMEXP(J(Ψ_CM+K_CM(s−x)))V′₁₂)  (24A)

[0631] Taking equation (24A) and separating out all the known parameters from the unknown parameter, x, and taking all the known parameters and making constants out of them we have the following

[0632] (V″₁₁−A_CMEXP(J(Ψ_CM+(s−x))EXP(−J))V″₁₂))/(V′₁₁−A_CMEXP(J(Ψ_CM+K_CMS))EXP(−Jx)))=

[0633] (V″₁₁−A_CMEXP(J(Ψ_CM+K_CM(s−x))EXP(−JK_CMX))V″₁₂))/(V′₁₁−A_CMEXP(Ψ_CM+K_CMS))EXP(−JK_CMX))V′₁₂)  (25)

[0634] Taking all the known parameters and making them constants out of them we have the following:

[0635] K′₁−K′₂EXP(−JK_CMx))/(K′₃−K′₄EXP(−JK_CMx))=K₁−K₂EXP(−JK_CMx))/(K₃−K₄EXP(−JK_CMx))  (26)  (26)

[0636] K_CM=−  (26A)

[0637] v=e^jKx  (26B)

[0638] Substituting equations (26A) and (26B) into equation (26) and making constants out of known terms we have the following:

[0639] K_Av²+K_Bv+K_C=0  (26C)

[0640] This is a second order equation that is solvable in “v” and therefore solvable in “x”. The “x” that is within plus or minus three FRDB (x) is the solution and those where the solution is other is not.

[0641] Another solution is the following:

[0642] Multiplying out equation (26) and taking all known parameters and making constants out of them we have the following:

[0643] K_A+K_BEXP(−JX)+K_CEXP(−JK_CMX)+K_DEXP(−JX(K_CM−)=0  (27)

[0644] K_A=K_AX+JK_AY:

[0645] K_B=K_BX+JK_BY:

[0646] K_C=K_CX+JK_CY:

[0647] K_D=K_DX+JK_DY

[0648] K_Bexp(−Jx)=

[0649] K_BXcos(−x)−K_BYsin(−x)+j((K_BYcos(−x)−

[0650] K_BXsin(−x))K_DXcos(−K_CMx+x)−K_DYsin(−K_CMx+x)+

[0651] K_BXcos(−x)−K_BYsin(−x)+j((K_BYcos(−x)−K_BXsin(−x))

[0652] K_Cexp(−JK_CMx)=K_CXcos(−K_CMx)−K_CYsin(−K_CMx)+j((K_CYcos(−x)−K_CXsin(−K_CMx))

[0653] K_Dexp(−J(K_CM−K_CM)X)=K_DXcos(−K_CMx+x)−K{overscore ( sin(−K_CMx+x)+

[0654] K_DXcos(−K_CMx+x)−K_DYsin(−K_CMx+x)+

[0655] j((K_DYcos(−K_CMx+x)−K_DXsin(−K_CMx+x))

[0656] j((K_DYcos(−K_CMx+x)−K_DXsin(−K_CMx+x))j((K_DYcos(−K_CMx+x)−K_DXsin(−K_CMx+x))

[0657] The sum of the x and components of equation (27) is the following:

[0658] Σx=0  (27A)

[0659] K_AX+K_BXcos(−x)−K_BYsin(−x)+K_CXcos(−K_CMx)−K_CYsin(−K_CMx)+K_DXcos(−K_CMx+x)−K_DYsin(−K_CMx+x)=0

[0660] ΣY=0  (27B)

[0661] K_AY+K_BYcos(−x)−K_BXsin(−x)+

[0662] K_CYcos(−K_CMx)−K_CXsin(−K_CMx)+

[0663] K_DYcos(−K_CMx+x)−K_DXsin(−K_CMx+x)=0  (27B)

[0664] Since all parameters are known except x, the equations (27A) and (27B) give two solutions to x and the solution that is within plus or minus three (3) is the solution. As stated x is the location of the peak of clutter and hence W_Cis as indicated previously. Rather than solve (27A) and (27B) the all the possible locations of x can be substituted since there are only a limited number of locations are possible and consequently may solve for x.

[0665] Another method for attaining x is to detect the target at its peak where Y={overscore (K)}e^jb=|1| at phase angle=0. Therefore at P-DPCA and Q-DPCA we have the following:

[0666] Y=1e^j(0)=((V″₁₁−W_C1V″₁₂)/(V′₁₁−W_C1V′₁₂)

[0667] Y=1e^j(0)=(V″₁₁−)/(V′₁₁−)

[0668] Equating the above equations and solving for x we have the following: $\begin{array}{c}\underset{\_}{W_{\mathrm{C1}}}/W_{\mathrm{C1}}=A_{\mathrm{CM}}{e}^{j\left({\Psi }_{\mathrm{CM}}+\underset{\_}{{{\rm K}}_{\mathrm{CM}}}x\right)}/A_{\mathrm{CM}}{e}^{j\left({\Psi }_{\mathrm{CM}}+{{\rm K}}_{\mathrm{CM}}x\right)}\\ =e^{j\left(\underset{\_}{{{\rm K}}_{\mathrm{CM}}}-{{\rm K}}_{\mathrm{CM}}\right)x}\\ =\left(V_{12}^{″}-V_{12}^{\prime }\right)/\left(\underset{\_}{V_{12}^{″}}-\underset{\_}{V_{12}^{\prime }}\right)\end{array}$

[0669] From the above set of equations the only unknown parameter is x and easily solved. This will be the location of the peak of clutter.

[0670] From the solution for x or that from equation (24) x is solved for (the peak location of clutter and this leads to the solution of W_C1and W which give the X=KαEXP(Jα) from the FRDB distance from the FRDB processed. This is illustrated in FIG. 16 where W_C1and W are employed to obtain X=KαEXP(Jα).

[0671] H. Basic Mathematical Equations Developed for Processing a Target Detected at the Same Time for Two Different Filters—ΔF Technique

[0672] 1. Ideal system

[0673] Previous derivation of a set equations were performed between sub-array 1 and sub-array 1 with the same FRDB at two successive times. This derivation will be between sub-array 1 and 2 at the same time with two different filters. For examples we will take filter FRDB, 0₀, at time 1, data point 1 and also different filter FRDB, 0₁, at time 1 and derive a set of equations. Target is detected at FRDB 0₀. (FIG. 18) illustrating the processing performed.

[0674] Taking filter FRDB, 0₀, at time 1 in sub-array 1 we have the following

[0675] V′₁₁=C₁+M₁  (1′)

[0676] V′₁₁—measured vector in sub-array 1 at time 1 for filter 0₀.

[0677] C₁—clutter vector in sub-array 1 at time 1 for filter 0₀.

[0678] M₁—target vector in sub-array 1 at time 1 for filter 0₀.

[0679] The sub-array 1 equation for the ideal system where the target is detected in the same corresponding FRDB as in sub-array 1 at a delayed (D) number of pulses is as follows:

[0680] V′₁₂=C₁+M₁EXP(JDΦ_R)  (2′)

[0681] V′₁₂—measured vector in sub-array 1 at time “D” for filter 0₀.

[0682] C₁—clutter vector in sub-array 1 at time “D” for filter 0₀.

[0683] M₁—target vector in sub-array 1 at time “D” for filter 0₀.

[0684] DΦ_R—the phase shift due to this motion of the target from pulse “1” in sub-array 1 to pulse “D” in sub-array 2.

[0685] This equation is ideally almost the same as in sub-array 1 but due to delay “D” and the phase shift of the target as stated as in the aforementioned paragraphs.

[0686] For data point “1' and filter 0₁, and time 1 we have the following equation:

[0687] T′₁₁=C₂+M₂  (3′)

[0688] T′₁₁—measured vector in sub-array 1 at time “1” for filter 0₁.

[0689] C₂—clutter vector in sub-array 1 at time “1” for filter 0₁.

[0690] M₂—target vector in sub-array 1 at time “1” for filter 0₁.

[0691] For data point “D” in sub-array 1 where the target is detected in the same corresponding FRDB 0₁as in sub-array 1 we have the following equation:

[0692] T′₁₂=C₂+M₂EXP(JDΦ_R)  (4′)

[0693] T′₁₂—measured vector in sub-array “2' for pulse delay “D” at time “D” for filter 0₁

[0694] C₂—clutter vector in sub-array “2’ for pulse delay “D” at time “D” for filter 0₁

[0695] M₂—target vector in sub-array “22 for pulse delay “D” at time “D” for filter 0₁

[0696] Subtracting equation (2′) from equation (1′) we have the following:

[0697] M₁=(V′₁₂−V′₁₁)/(1−EXP(JDΦ_R))  (5′)

[0698] Similarly subtracting equation (4′) from equation (3′) we have the following:

[0699] M₂=(T′₁₂−T′₁₁)/(1−EXP(JDΦ_R))  (6′)

[0700] Taking equation (6′) and dividing by equation (5′) it follows:

[0701] M₂/M₁=Y_F=K_FEXP(J_F)=(T′₁₂−T′₁₁)/(V′₁₂−V′₁₁)  (7′)

[0702] Y_Fis the filter response of target between FRDB 0x(0₀) and FRDB 0_(X+Y)(0₁).

[0703] K is the filter response of the target amplitude between FRDB 0x(0₀) and FRDB 0_(X+Y)(0₁).

[0704] B_Fis the filter response of the phase between FRDB 0x(0₀) and FRDB 0_(X+Y)(0₁).

[0705] Multiplying equation (1′) by EXP(JDΦ_R) and subtracting equation (2′) and solving for C₁we have the following:

[0706] C₁=(V′₁₁EXP(JDΦ_R)−V′₁₂)/(EXP(JDΦ_R−1)  (8′)

[0707] Similarly multiplying equation (3′) by EXP(JDΦ_R) and subtracting equation (4′) and solving for C₂we have the following:

[0708] C₂=(T′₁₁EXP(JDΦ_R)−T′₁₂)/(EXP(JDΦ_R−1)  (9′)

[0709] Taking equation (9′) and dividing by equation (8′) we have:

[0710] C₂/C₁=X_F=K_FEXP(Jα_F)=(T′₁₁EXP(JDΦ_R)−T′₁₂)/(V′₁₁EXP(JDΦ_R)−V′₁₂)  (10′)

[0711] X_F—is the filter response of clutter between FRDB 0x(0₀) and FRDB 0_(x+Y)(0₁).

[0712] K_F—is the filter response of amplitude of clutter between FRDB 0x(0₀) and FRDB 0_(X+Y)(0₁).

[0713] α_F—is the filter response of phase of clutter between FRDB 0x(0₀) and FRDB 0_(X+Y)(0₁).

[0714] We see by equation (7′) we have solved for the target filter response between filters and it is independent of DΦ_R. On the other hand equation (10′) solving for the clutter filter response between filters is dependent on DΦ_R.

[0715] If we take equation (10′) and solve for DΦ_Rwe have:

[0716] EXP(JDΦ_R)=(X_FT′₁₁−T′₁₂)/(X_FV′₁₁−V′₁₂)  (11′)

[0717] 2. Non-Ideal System

[0718] As previously developed was a set of equations where the clutter cancels ideally but as discussed previously ideal conditions do not exist for canceling clutter and so consequently a set of equation will be developed for the non-ideal case. In section III D-DPCA theory and application that the practical implementation of the ideal case is rarely attained and there is almost never a case where clutter cancels completely.

[0719] The next set of equations will be for the non-ideal case where the clutter and target have to be corrected for. The sub-array 1 equation for the non-ideal case is the following:

[0720] V′₁₁C+M  (12′)

[0721] The same as equation (1′).

[0722] The sub-array 2 equation for the non-ideal case where the target is detected in the same corresponding FRDB as in sub-array 1 at a delay “D” number of pulses is as follows:

[0723] V′₁₂=C/W_C1+(M/W_M1)EXP(JDΦ_R)  (13′)

[0724] This equation is the same as equation (2′) except for W_C1and W_M1.

[0725] W_C1—the factor that relates clutter in sub-array 1 to sub-array 2 to cancel clutter as effectively as possible (making K_βFand B_Faccurate).

[0726] W_M1—the factor that relates target in sub-array 1 to sub-array 1 to make the target as accurate as possible (making K_Fand α_Faccurate).

[0727] For data point 1, sub-array 1 and a different filter where the target is detected we have the following equation:

[0728] T′₁₁=C₂+M₂  (14′)

[0729] This is the same as equation (3′).

[0730] The equation for sub-array 2, at data point “D” and the same filter as sub-array 1 is the following:

[0731] T′₁₂=C₂/W_C2+(M₂/W_M2)EXP(JDΦ_R)  (15′)

[0732] Similarly to equation (4) but with the factors W_C2and W_M2added.

[0733] W_C2—the factor that relates clutter in sub-array 1 to sub-array 2 to cancel clutter as effectively as possible (making K_Fand B_Faccurate.

[0734] W_M2—the factor that relates the target in sub-array 1 to sub-array 1 to make the target vectors as accurate as possible (making K_Fand α_Faccurate).

[0735] Multiplying equation (13′) by W_C1and subtracting it from equation (12′) and solving for M we obtain the following:

[0736] M=(V′₁₁−W_C1V′₁₂)/(1−(W_C1/W_M1)EXP(JDΦ_R))  (16′)

[0737] Similarly with equation (14′) and (15′) we obtain the following:

[0738] M₂=(T′₁₁−W_C2T′₁₂)/(1−(W_C2/W_M2)EXP(JDΦ_R))  (17′)

[0739] Solving for Y_Femploying equation (16′) and (17′) we have:

[0740] M₂/M=((T′₁₁−W_C2T′₁₂)/(1−(W_C2/W_M2)EXP(JDΦ_R)))/(V′₁₁−W_C1V′₁₂)/(1−(W_C1/W_M1)EXP(JDΦ_R)))  (18′)

[0741] Multiplying equation (12′) by EXP(JDΦ_R)/W_M1and subtracting equation (13′) from (12′) we have:

[0742] V′₁₁EXP(JDΦ_R)/W_M1−V′₁₂=C₁(EXP(JDΦ_R)/W_M1−1/W_C1)  (20′)

[0743] Similarly multiplying equation (14′) by EXP(JDΦ_R)/W_M2and subtracting equation (15′) from (14′) we have:

[0744] T′₁₁EXP(JDΦ_R)/W_M2−T′₁₂=C₂(EXP(JDΦ_R)/W_M2−1W_C2)  (21′)

[0745] Solving for C₂/C₁employing equation (20) and (21) we have:

[0746] X_FEXP(JpK_CM)=(T′₁₁EXP(JpK_CM)EXP(JDJ_R)/W_M1−T′)/(V′₁₁EXP(JDJ_R)/W_M1−V′₁₂)C₂/C₁=X_F=K_FEXP(Jα_F)=

[0747] ((T′₁₁EXP(JDΦ_R))/W_M2−T′₁₂)/(EXP(JDΦ_R)/W_M2−1/W_C2))/((V′₁₁EXP(JDJ_R))/W_M1−V′₁₂)/(EXP(JDJ_R)/W_M1−1/W_C1))  (21A)

[0748] Where

[0749] (e^jDΦ/W_M1−1/W_C1)/(e^jDΦ/W_M1e^jpK−1/W_C1)≈1 since e^jpK≈1

[0750] X_FEXP(JpK_CM)=(T′₁₁EXP(JpK_CM)EXP(JDJ_R)/W_M1−T′₁₂)/(V′₁₁EXP(JDJ_R)/W_M1−V′₁₂)

[0751] The previous set of equations was developed with pulse plus time P-DPCA compensation. This set of equations will be developed for Q-DPCA compensation.

[0752] The equation for sub-array 1 for data point 1 is the following:

[0753] V′₁₁=C+M  (12′)

[0754] (the same as equation (12′) and the same definition of terms)

[0755] The equation for sub-array 2 for data point D+1 is the following:

[0756] V′₁₂=C/+(M/)EXP(J(D+1)Φ_R)  (13′)

[0757] This equation is the same as equation (13′) except for W and W and V′{overscore (K)}V′{overscore (K)}V′ which are the result of the Q-DPCA compensation.

[0758] {overscore (K)}W—the factor that relates clutter in sub-array 1 to sub-array 1 to cancel clutter as effectively as possible (making {overscore (K)}_Fand β_Fas accurate as possible)

[0759] W—the factor that relates target in sub-array 1 to sub-array 1 to cancel clutter as effectively as possible (making K_Fand α_Fas accurate as possible)

[0760] V—this is the the vector that is measured in sub-array 2.

[0761] For data point 1, sub-array 1, and a nearby filter where the target is detected, we have the following equation:

[0762] T′₁₁=C₂+M₂  (14′)

[0763] (the same as equation (14′) and the same definition of terms)

[0764] The equation for sub-array 2, at data point D+1 and the same filter as in equation (14′) sub-array 1 we have the following:

[0765] T′=C₂/+(YM/W)EXP(J(D+1)Φ_R)  (15′)

[0766] This is similar to equation (15′) but with the factors T, W and W as the result of the Q-DPCA compensation.

[0767] W—the factor that relates clutter in sub-array 1 to sub-array 1 to cancel clutter as effectively as possible.

[0768] T—this is the sum of all the vectors, which is measured in sub-array 2

[0769] Employing all the same algebraic manipulations as in the previous derivation we have all analogous results as previously except the following analogous terms are different as follows:

[0770] W substitutes for W_C1

[0771] W substitutes for W_C2

[0772] W substitutes for W_M1

[0773] W substitutes for W_M2

[0774] V—substitutes for V₁₂

[0775] T′—substitutes for T′₁₂

[0776] All the other terms are the same. The following is a list of some of the analogous key equation to illustrate this:

[0777] M₂/M₁=((T′₁₁−)/(1−

[0778] (/)EXP(J(D+1)(Φ_R)))/(V′₁₁−)/(1−(/)EXP(J(D+1)(Φ_R))  (18′)

[0779] W_C1≈W_C2and W_M1≈W_M2we have the following:

[0780] W_C1≈W_C2and W_M1≈W_M2we have the following:

[0781] Y_F=_FEXP(Jβ_F)=(T′₁₁−W)/(V′₁₁−)  (19′)

[0782] C₂/C₁=X_F=K_FEXP(Jα_F)=(T′₁₁EXP(J(D+1)Φ_R))((V′₁₁EXP(J(D+1)(Φ_R))/−)/(EXP(J(D+1)(Φ_R)/e^jpK≈1  ()

[0783] Therefore we have the following:

[0784] X_F=K_FEXP(Jα_F)=EXP(−Jp)((T′₁₁EXP(J(D+1)Φ_R)/)−T′₁₂))/((V′₁₁EXP(J(D+1)Φ_R)/)−)  (22′)

[0785] EXP(JDΦ_R)=((X_FEXP(−Jp)−)/((X_FEXP(Jp))/)V′₁₁−T′₁₁/)  (23′)

[0786] EXP(JDΦ_R)=((X_FEXP(Jp))₁₂−)/((X_FEXP(Jp))V′₁₁−T′₁₁EXP(−Jp)  (24′)

[0787] M₁=(V′₁₁−)/(1−()EXP(JDΦ_R))  (16′)

[0788] M₂=(T′₁₁−)/(1−()EXP(JDΦ_R)  (17′)

[0789] Evaluation of the following assumptions:

[0790] The first assumption as follows:

[0791] (1−(W_C1/W_M1))=(1−(W_C2/W_M2)EXP(JDΦ_R))  (A)

[0792] W_M1=A_CMEXP(J(Ψ+K_CM(t−y)); W=A_CMEXP(J(Ψ+K(t−y))

[0793] W=W_M2=WW_M1=A_CMEXP(J(Ψ) at the peak of mover

[0794] W_M2=A_CMEXP(J(Ψ+K_CM(t+p−y));

[0795] W=A_CMEXP(J(Ψ+K_CM(t+p−y))

[0796] W_C1=A_CMEXP(J(Ψ+K_CM(s−x)); WA_CMEXP(J(Ψ+K(s−x))

[0797] W_C2=A_CMEXP(J(Ψ(+K_CM(s+p−x))

[0798] W=A_CMEXP(J(Ψ+K(s+p−x))

[0799] s—is the FRDB processed

[0800] t—is the FRDB relative to where the target is processed

[0801] y—is the FRDB the targets angle position is relative to where target is processed

[0802] x—is the FRDB the clutter peak position is at

[0803] It is assumed that the amplitude offset within the extent of FRDBs for target angle and clutter angle the change is very small. It is even more appropriate for the target angle alone or the clutter angle.

[0804] It is assumed that offset angle within the extent of the FRDBs for target angle and clutter angle the change is very small. It is ever more appropriate for target angle alone or clutter angle alone or clutter angle.

[0805] It is assume that slope of the angle is constant within the extent of the FRDBs for target and clutter. It is more appropriate for target alone or clutter alone. It is noted the factors above for P-DPCA or Q-DPCA compensation does not equal to the factors for time only DPCA compensation (reference section III D).

[0806] Looking at the terms W_C1/W_M1and W_C2/W_M2in equation (A) we have the following: $\begin{array}{c}{W}_{\mathrm{C1}}/W_{\mathrm{M1}}=A_{\mathrm{CM}}\mathrm{EXP}(J\left(\Psi +K_{\mathrm{CM}}\left(s-x\right)\right)/\\ A_{\mathrm{CM}}\mathrm{EXP}(J\left(\Psi +K_{\mathrm{CM}}\left(t-y\right)\right)\\ =\mathrm{EXP}(J\left(K_{\mathrm{CM}}\left\{\left(s-t\right)+\left(y-x\right)\right\}\right)\\ W_{\mathrm{C2}}/W_{\mathrm{M2}}=A_{\mathrm{CM}}\mathrm{EXP}(J\left(\Psi +K_{\mathrm{CM}}\left(s+1-x\right)\right)/\\ A_{\mathrm{CM}}\mathrm{EXP}(J\left(\Psi +K_{\mathrm{CM}}\left(t+1-y\right)\right)\\ =\mathrm{EXP}(J\left(K_{\mathrm{CM}}\left\{\left(s-t\right)+\left(y-x\right)\right\}\right)\end{array}$

[0807] Therefore the assumption in equation (A) that the left side of the equation is equal to the right side of the equation is correct.

[0808] Looking at the two terms in equation (21A) as follows:

[0809] ((EXP(JDΦ_R)/W_M1)−1/W_C1)/((EXP(JDΦ_R)W_M2)−1/W_C2)=W_M1/WM₂((EXP(JDΦ_R)−W_M1/W_C1)/(EXP(JDΦ_R)−W_M2/W_C2))  (B)

[0810] From the previous example (W_M1/W_C1)=W_M2/W_C2) therefore equation (B) becomes the following:

[0811] (B)=W_M1/W_M2and this is equal to the following: $\begin{array}{c}{W}_{\mathrm{M2}}/W_{\mathrm{M1}}=(A_{\mathrm{CM}}\mathrm{EXP}\left(J\left(\Psi +K_{\mathrm{CM}}\left(t-y\right)\right)\right)/\\ (A_{\mathrm{CM}}\mathrm{EXP}(J\left(\Psi +K_{\mathrm{CM}}\left(t+p-y\right)\right)\\ =\mathrm{EXP}\left(\mathrm{Jp}{K}_{\mathrm{CM}}\right)\end{array}$

[0812] p is the number of FRDBs that W_M1and W_M2is measured from each other or processed from each other. Therefore we have the following equation:

[0813] (B)=EXP(JP K_CM)

[0814] if p=1 Then

[0815] (B)EXP(JK_CM)

[0816] IV System Employed to Attain the Objective of the Disclosure

[0817] A. Basic System and Objectives

[0818] This disclosure presents the fundamental radar principles and from them the mathematically developed equations and methodology employed in attaining the object of the disclosure. It illustrates a number of possible implementations but not all possible implementations. It is therefore considered with people familiar in the state of the art, implementations in a different manner could be employed but with the same fundamental principles and basic equations.

[0819] This is an airborne system employing a one dimensional electronically scanned phased array implementing a transmit antenna and dual receive sub-arrays employing displaced phase center antenna (DPCA) techniques. The transmit beam, which may be the sum of the two (2) sub-arrays, and the receive sub-arrays are aligned along the same bore-sight of the antennas and are moved with the platform after each transmitted pulses. Pulse Doppler operation will be implemented in the low or near zero clutter area to detect targets and measure their relative radial velocity and angular position as well as range (Reference Section III, D, 1).

[0820] Targets detected in the significant clutter region where pulse doppler techniques are ineffective, DPCA techniques (as are described in Section III, D in the disclosure) will be implemented to detect targets and measure their relative radial velocity and angular position and range.

[0821] 1. Specific System to Illustrate Principles, Equations and Objective

[0822] In the following disclosure a specific radar system will be employed as an example. This will be a real system, with real parameters that illustrate the system (Reference Section III, A).

[0823] A transmit and two receiving subarrays are aligned to have the same coverage. The boresights of the antennas are aligned. A number of “N” pulses illuminate the targets in the coverage. The “N” data points are processed in the two receiving subarrays to attain the objective of this disclosure. Range and Doppler ambiguity are resolved to the desired level by the transmission of additional one or more pulse repetition frequencies. Example is Section III A, radar parameters. Included in the example is the zero-fill spectrum of multiply by four (4) analyses as illustrated in FIG. 9.

[0824] FIG. 3 illustrates the basic block Diagram of the system.

[0825] Block Diagram of non-ideal system in FIG. 3. Subarray 1 processor receives the “N” radar data points (In phase & Quadrature data), which are equal to 64 I,&Q data points in our illustrative example. They are N×S zero fill that is X4 in the illustrative example as shown in FIGS. 3 and 9. It processes the radar data by applying a weighting functions and then performing a spectrum analysis (such as fast fourier transform (FFT)). Obtaining (64×4) 256 doppler bins with a frequency resolution 1000 Hz164=15,625 Hz spaced 3,9063 Hz apart. This is illustrated in FIG. 9. FIG. 9 the (large range Doppler bins)LRDBs are O₀to 63₀and the (fine range Doppler bins)FRDBs are O₁, O₂, O₃to 63₁, 63₂, 63₃. FIG. 9 also illustrates the spectrum for subarray 1 and subarray 2. Subarray 2 is processed in the same manner and starts with the “D” pulse (5 pulse delay) or delayed to pulse “6”, or (6 pulse delay) or delayed to pulse “7” so that the distance subarray 2 travels is as close as possible to the distance (Section III, D) required to cancel clutter and to be able to detect a target, in any FRDB.

[0826] Target detected in the very low clutter region is discussed in Section III, D, 1 detecting the target and measuring its radial velocity and azimuth position in the low clutter region is simple and straightforward. This is also shown in FIG. 1.

[0827] 2. Clutter Weighting Function (W_c) and Target Weighting Function (W_m) Analysis.

[0828] a) W_cWeighting Function Analysis

[0829] The equation for W_crepresents the weighting function between subarray 1 and 2 that maximizes clutter cancellation. Based on the angular location of clutter, W_cis calculated. W_cis a least mean square vector that multiplied by the FRDB of subarray 2 and subtracted from the corresponding FRDB of subarray 1 makes the clutter equal in amplitude and phase so when subtracted the clutter is close to zero. This is the weighting function of clutter between corresponding FRDBs of subarray 1 and 2.

[0830] Similarly the target vectors detected in sub-array 1 and 2 for more precise results should be as accurate as possible.

[0831] If the precise angular position of the target were known in an analogous way the W_Mof the target would be calculated to make the target vectors of the correct amplitude and phase. The azimuth of the target is not known at this point and therefore this W_Mmay not be calculated as yet but is a function of φ_A=φ_D−φ_R.

[0832] Since the target has to be in the main beam of the antenna (assumed only targets within the 3 db points of the main beam are detected).

[0833] The W_Cmay be calculated for every FRDB in the main beam and where ever clutter is of significant amplitude may apply to clutter or target at that particular angular position.

[0834] Usually W_Cis calculated at a particularly FRDB to cancel clutter, but in this case it is employed also to ascertain the filter response of clutter. If W_cis ideally equal to one (1) at an angle “0”, that is the term applied to the clutter in that particular FRDB processed to cancel clutter. If the W_Cis employed to determine target filter response ({overscore (K)}_Fand B_F), it has a value set by equation (19) of Section III, E.

[0835] The W_Cterm to cancel clutter, when clutter is detected at its peak ideally is equal to amplitude response “1” and phase response “0”, Section II, D—depicts the error is proportional to where the clutter is detected in the clutter filter. The phase error is “0” at the center of the filter and larger as the clutter is detected toward the edge of the filter.

[0836] FIG. 16 and FIG. 25—Table I shows the phase response error as a function of where the peak response of clutter (filter response). FIG. 16 depicts the amplitude and phase response of clutter detected in a particular FRDB where 3qis the peak of clutter, 2qis the amplitude and 4qis the phase response. Also is shown the ideal phase response T-DPCA(7q) and 8qthe phase response for P-DPCA and 10qfor Q-DPCA. The X-axis is the FRDBs. FIG. 25—Table I shows the same information with numbers instead of graphical form as in FIG. 16. The amplitude and phase response of clutter within a filter is (K_Fand K_F). The amplitude and phase response of clutter due to a change in time of clutter is K and K. It also shows the ideal phase response for P-DPCA and Q-DPCA as a function of FRDB. It is essential to note the correlation between all these responses that if one is known all are known or estimated accurately. Therefore as shown W_Cis related to where in the clutter filter response curve is the peak of clutter located and the also determines the change in clutter X=KEXP(Jα). If W_Cis known the clutter response and the change in clutter is known (FIG. 16).

[0837] It is also determined that the targets azimuth position decides the response related to target (W_M-amplitude and phase response and the change in target). (FIGS. 17 and 18 and FIG. 26—Table II and FIG. 27—Table III). FIG. 17 is analogous to that of FIG. 16 is for clutter, the same information is obtained, the difference is the target azimuth position is not where the target is detected as with clutter. As shown the target azimuth position is at different position. FIG. 173ris the target peak, 2ris the amplitude response and 4r, 7r, and 9rare phase responses for their respective DPCA delays. The X-axis is FRDBs. FIG. 26—Table II shows the same information with numbers instead of graphical form as in FIG. 17. Analogous to clutter the amplitude and phase response of target within a filter is ({overscore (K)}_Fand B_F). The amplitude and phase response of target due to a change in time of target is ({overscore (K)} and B_F). It also shows the ideal phase response for P-DPCA and Q-DPCA as a function of FRDB. It is essential to note the correlation between all these responses that if one is known all are known or estimated accurately. FIG. 27—Table III is a table of W_Mas a function of FRDB. FIG. 18 illustrates the example of target detection in clutter. Since the zero-fill employed in the implementation each LRDB filter has four (4) FRDB filter within its resolution (FIG. 9—Zero Fill Spectrum Analysis), the W_Mdetermination is so calculated. If the position of the target angle is determined the W_Mmay be calculated and computed more precisely.

[0838] When the target is detected in a particular FRDB the angular position of the target (φ_A—phase shift proportional to angular position) is unknown at this point and the W_Min equation (2) in Section II, F. The angular position of the target may be determined relative to the detected target (φ_D) and the relative radial velocity of the target (Dφ_R) as follows: φ_A=φ_D−φ_R.

[0839] Since the angular position of the target (φ_A) from above is determined the filter (FRDB) is determined.

[0840] The weighting function of clutter measured at a particular FRDB is as follows:

[0841] W_CA_CMe^{j(Ψ+K(s−x))}W_C=A_CMe^{j(ΨK(s−x))}

[0842] W_c=weighting function of clutter

[0843] A_CM=Offset Amplitude of clutter (due to mismatch of receiving antennas and other affects)

[0844] Ψ_CM=Offset Angle of clutter (due to mismatch of receiving antennas and other affects)

[0845] K_CM=P-DPCA slope of phase error of clutter

[0846] x=position of peak of clutter

[0847] s=actual position of clutter processed relative to the peak of clutter

[0848] The measurement of W_cat any particular FRDB just measures the A_CMand Ψ_CMsince the measurements obtains the least mean square of a number of range bins. What is to be determined is whether that particular piece of clutter is at its peak and (s−x), and from this term and the known K_CMthen W_Cis determined.

[0849] Therefore to summarize measuring a particular FRDB obtains the offset angle and offset amplitude of the clutter weighting function. Measuring all FRDBs, with clutter, would obtain all the offset angle and offset amplitude and would determine if there are any significant differences as a function of FRDB.

[0850] The other term K_cm(s−x) is obtained by determining (s−x) since K_CMis known. The (s−x) term is determined by knowing where the peak of clutter is relative to the FRDB processed or by calculation in an equation where it can be determined.

[0851] b) W_M—Weighting Function Analysis

[0852] The target is detected at a particular FRDB but the value of W_Mis determined at the targets angular position. The targets angular position is the position at which the target resides. Because the radar detects the target based on its phase shift change per pulse or frequency response, it will detect the target based on the total phase shift per pulse that due to angular position from the bore sight of the antenna plus that due to the relative radial velocity of the target. Clutter on the other hand has only one component of velocity (phase shift per pulse) due to its angular position from the bore sight of the antenna). If we had a perfect DPCA compensation between array 1 and array 2 clutter would cancel completely since it has no relative radial velocity of its own. This cancellation would be based on its angular position (filter response) would be identical in both arrays and therefore cancel.

[0853] The target on the other hand would not cancel due to the two components of filter response, the component of the target due to its angular position and the component due to its relative radial velocity are not known, but the sum of the two are known in the terms of Φ_Dwhich has both components. If the target is stationary, it would be detected with its component of doppler proportional to off bore sight of the antenna and would act just like clutter and cancel when the outputs from both arrays are subtracted. When the target moves (moves radially), the component of target due to its angular position is the same and could be cancelled and its angular position was known and cancelled like if it were clutter at that angular position (therefore make the amplitude and phase of the two mover vectors equal), but the component of the target that moves radially would not cancel because this would produce a phase between the two mover vectors in both arrays proportional to the velocity of the target.

[0854] Therefore the target components of velocity due to its angular position would cancel if its angular position were known, and behaves like clutter at that position. The target components are not known at this point, therefore if the relative radial velocity is known the angular position could be calculated and the W_Mcould be determined by measure of the W_Cof clutter at that angular position and this would be its value. Like the W_Cof clutter makes the amplitude and phase of clutter in array 1 and array 2 equal therefore cancel each other when array 2 is subtracted from array 1. The W_Mwould make the target (mover vector) amplitude and phase equal, but due to the angular position, that is due to its relative radially velocity have a phase shift between the two mover vectors in array 1 and 2. Therefore this would make the two mover vectors equal in amplitude and phase due to its angular position and the difference in phase of the mover vectors would be only that due to the relative radial velocity.

[0855] c) Clutter and Target Weighting Function Analysis

[0856] The analysis of Section III the DPCA theory and the calculations for “(D+1)” time delay (Q-DPCA) error and the (“D”) time delay (P-DPCA) (FIGS. 16-17 and FIGS. 25-27, Table I--III).

[0857] In the succeeding analysis of this system in this section the various systems will be treated as follows:

[0858] 1) Ideal System

[0859] a) Two receiving sub-arrays are matched perfectly (W_Cand W_Mare negligible).

[0860] b) The target and clutter are detected at the peak of the filter response.

[0861] 2) Non-Ideal System

[0862] a) The two receiving subarrays are not exactly matched.

[0863] b) W_Cand W_Mcalculations must be performed and determined.

[0864] c) W_Cand W_Mresponse has an (A_CM) gain offset that is at least constant for a few FRDBs.

[0865] d) W_Cand W_Mresponse has (K_CMOR K_CM) phase response that is linear but has a phase offset (Ψ_cm) that is constant for at least a few FRDBs.

[0866] e) Peak of the target is detected at the peak of the filter response and corresponds to W_MA_CMe^{j(Ψ+K(t−y))}. This will be defined later in the section.

[0867] f) Peak of the clutter is detected at the peak of the filter response and corresponds to W_C=A_CMe^{j(Ψ+K(s−x))}. This will be defined later in the section.

[0868] B. Processing a System Where the Target is Detected in the Same FRDB at Two Sequential Times

[0869] System Processing The Target In The Same Filter at Two Sequential Times (ΔT processing) Reference Section III, G.

[0870] As depicted in FIGS. 16 to 19 and FIG. 25—Table I to FIG. 29—Table VI, the various parameters that make up the system are illustrated. This assumes the radar parameters as stated in the previous paragraphs and the mathematical development of Section III.

[0871] The key role that makes the system amenable to a solution is the relationship between the clutter filter response, K_Fand α_f, and the clutter change in time, K and α, and the clutter weighting function, W_C. These relationships are fixed and illustrated in FIG. 16, “Ideal Clutter Filter Response and Time change and W_CResponse,” and FIG. 25—Table I, “Ideal Clutter Response Correlated with Ideal Clutter Time change and Ideal W_CResponse.” If any one of these parameters is known, the other two are known or estimated very closely. For instance if clutter weighting function, W_C, is measured ideally at amplitude “1” and phase shift “0” then the relative clutter response is the amplitude is a maximum and equals maximum and relative phase response is zero (“0”). The change in clutter from one data point to next in time is illustrated as K_x=1.0 and α_x=0.0 FIG. 16 illustrates all other conditions and there values. Therefore if one knows one parameter it knows all others (shown in FIG. 16 and FIG. 25—Table I.)

[0872] The filter response is the spectrum analysis performed on the data.

[0873] The change in clutter with time is also performed for the filter response of clutter.

[0874] The weighting function of clutter (W_C) is performed on real data and determined as described in Section III.

[0875] Another key role that makes the system amenable to a solution is the relationship between the target filter response, the relative target filter response ({overscore (K, B_F), and the target change in time ({overscore (K)}, B) and the target weighting function (W_M). This relationship is fixed and illustrated in FIG. 17, “Target Ideal Filter Response,” and FIG. 18, “Target Azimuth—W_mDetermination” as well as FIG. 26—Table II, “Table of Target Ideal Filter Response Correlated With Target Relative Filter Response and Relative Target Change with Time.” This is analogous to that of clutter in the previous paragraphs.

[0876] The detection of the target at its peak amplitude is at its maximum and the filter response is proportional to the total velocity of the target. This velocity has a component due to its angular position and that due to its relative radial velocity. Clutter has a filter response only due to its angular position off bore-sight of the antenna. At this point the angular position of the target is unknown. The targets relative radial velocity is also unknown, only the total velocity of the target is known the sum of the velocity due to angular position and that due to its relative radial velocity. Since phase shift is proportional to velocity, the total phase shift (filter response) of the target is the addition of the phase shift due to angular position added to that due to the relative radial velocity of the target.

[0877] The target filter response as shown in FIG. 17 and FIG. 26—Table II determines the relative target filter response and the target change with time but doesn't determine the weighting function for target (W_M). W_Mis determined by the angular position of the target that is unknown from the detection of target only. Unlike clutter, the detection of target that has two components which at this time are undetermined and one being the angular location of the target.

[0878] The (W_M) weighting function of the target has to await the determination of the angular position of the target. Once the angular position of the target has been determined FIG. 18 and FIG. 26—Table II will give the weighting function (W_M) of the target. W_Mcan be expressed in terms of [(φ_D−φ_R)=φ_A] which are the proportional to the angular position of the target where φ_Awhich is not known but is a function of φ_Dthat is known but that φ_Ris unknown at this point.

[0879] The “y” position off the peak of target is determined by the difference in position of the peak of its maximum amplitude and the position where it's processed.

[0880] W_M=|A_CM|EXP(J(Ψ_CMK_CM(s−y))W_M=|A_CM|EXP (J(Ψ_CM+K_CM(s−y))

[0881] The equation is as where K_CMand Ψ_CMand |A_CM| are determined by azimuth position (Φ_A) but assumed constant. If not an average value may be taken across the beamwidth of the antenna. When an estimate is close to the azimuth it will determine a closer estimate. A more accurate requirement will require an iterative process.

[0882] The other key element is the zero fill (×4) which four (4) times more Doppler bins where target and clutter are detected as illustrated and discussed in Section III, C. The four more doppler detections will be employed to determine the objective of the disclosure, to find the relative radial velocity of the target (φ_R).

[0883] B1. Employing P-DPCA

[0884] With the aforementioned information on the weighting function of clutter (W_C) and the associated information as well as that for the weighting function of target (W_M) and its associated information and the derived equations and information of Section III, the object of the disclosure will be attained. The procedure will be set forth with an example as follows:

[0885] The target and clutter is as in FIG. 18, “Clutter and Target Detection Diagram”.

[0886] Target Peak Amplitude Detection at FRDB-O₀

[0887] Clutter Peak Amplitude Detection at FRDB-O₁

[0888] Target Angular Position at FRDB 61₀(−3₀)

[0889] For Q-DPCA Compensation (P-DPCA) and employing equation (19)

[0890] Y=^jB=amplitude 1 and angle 0=(V_1,1¹¹−W_CV_1,2¹¹)/(V_1,1¹−W_CV_1,2¹)  (19)

[0891] Detecting a target at its peak is performed by subtracting corresponding filters from array 1 and 2 and detecting a target (which does not cancel to near zero like clutter). Performing this operation on all corresponding filters where the target is detected determines the FRDB where is the peak amplitude of target.

[0892] Estimating {overscore (K)}=1 and B=0 since the target was detected at its peak and from Table II target time change {overscore (K)}=1 and B=0.

[0893] With {overscore (K)} and B substituted in equation (19) of section III {overscore (K)}e^jB=(V″₁₁−W)/(V′₁₁−W) and solving for W_c. All other parameters are known.

[0894] Solving for x where W_C1=A_CMe^j(Ψ+KX)we have the following:

[0895] e^jB=(V″₁₁−W_C1V″₁₂)/(V′₁₁−W_C1V′₁₂) for P-DPCA and Ke^jB=1 and solve for x

[0896] W_C1=A_CMe^{j(Ψ+K(S−X))}=K₁=(V″₁₁−V′₁₁)/(V″₁₂−V′₁₂)

[0897] e^jKX=K₂=K₁/A_CMe^j(Ψ)e^jKS

[0898] |1|=|K₂|−K_CMX=K₂angle

[0899] x=K₂angle/(−K_CM)

[0900] The above solution is the location of the peak of clutter for P-DPCA. An analogous solution is for Q-DPCA as follows:

[0901] e^jB=(V″₁₁−)/(V′₁₁−) for Q-DPCA and Ke^jB=1 and solve for x

[0902] W_M=|A_CM|EXP(J(Ψ_CM+K_CMy)W_M=|A_CM|EXP(J(Ψ_CM+K_CMy))

[0903] =A_CMe^{j(Ψ+(S−X))}==(V″₁₁−V′₁₁)/(−)

[0904] e^jX==/A_CMe^j(Ψ)e^jS

[0905] |1|=||−X=angle

[0906] x=angle/(−K_CM)

[0907] =A_CMe^{j(Ψ+(S−X))}==(V″₁₁−V′₁₁)/(−)

[0908] e^jX==/A_CMe^j(Ψ)e^jS

[0909] |1|=||−X=angle

[0910] x=angle/(−)

[0911] The solution “x” for P-DPCA and Q-DPCA should be the same or very close to each other.

[0912] The W_csolution will be as shown on FIG. 16 and FIG. 26—Table I which on the same table the change in time K=0.99 and α=−0.02 (X=Ke^jα). Since the clutter is one FRDB (O₁) to the right of processed FRDB (O₀).

[0913] Employing equation (23) we have the following:

[0914] e^jDφR/W_m=(XV′_1,2−V″_1,2)/ (XV′_1,1−V″_1,1)  (23)

[0915] substituting X in right side of equation and its all known quantities will set that equal to Ae^jDΦ′. Rewriting equation (23) hence the following.

[0916] e^jDφR=W_M|A|e^jDφ′R  (24)

[0917] W_M=A_CMe^{j[ΨCM+K(t−y)]}  (25)

[0918] Setting the exponentials or angles equal we have the following:

[0919] Dφ_R=Ψ_cm+K_cm(t−y)+Dφ′_R  (26)

[0920] φ_A=φ_D−φ_R  (27)

[0921] t—is the FRDB processed relative to where the target is at its peak which is −3₀

[0922] y—is the FRDB of the target angular position, which is at FRDB 61₀(−3₀).

[0923] All other parameters have been defined previously. The Dφ′_Rterm give the approximate location of the target, close enough to give a good estimate of Ψ_CMand A_CM. From the first estimated Dφ′_Ris taken as the closest FRDB equal to Dφ′_R. In this case Dφ′_R=Dφ_Rand the ideal W_Mphase will equal Ψ_CM+0. The term K_CM(t−y)=0, the ideal W_Mphase shift, since the angular position of target is FRDB 61₀(−3₀). The FRDB is at the same zero position so that (t−y)=0 (φ_A−proportional to the angular position of the target). If this does not occur on first estimate and iterate until the solution is attained.

[0924] As indicated in FIG. 31—Table VII and FIG. 18, example 1, there is only one correct solution, which is at FRDB (61₀) where the W_Mand Dφ_Rare correct, in all other possible solutions where the W_Mand Dφ_Rare incorrect.

[0925] The second example indicated in FIG. 18, if the peak of the target is not at the peak of a FRDB but somewhere in between FRDBs. The detection of the peak of the target, the peak of clutter, the Dφ_Rand the φ_Aprobably will not be at the FRDB or somewhere in between and to get greater accuracy is the estimation of Dφ_R, this has to be taken into account. Therefore an estimate of these parameters with an interpolation techniques and/or zero fill greater than (4) four could be employed. Also a specialized filter in the places where targets are detected to get as close an estimate of the parameters as necessary to the required accuracy. This would be true of example 2.

[0926] Performing the above operations determine the position of the peak of clutter. Performing the processing where the clutter is at its peak makes Ke^jα where K=1 and α=0 for (FRDB=0₁where processed) and an estimate of Y={overscore (K)}e^jBwhere {overscore (K)}=0.99 and b=−0.02

[0927] B2. Employing Q-DPCA Operation at D=6

[0928] The previous procedure was performed with the P-DPCA and this analogous procedure may be implemented with Q-DPCA and results compared.

[0929] B3. Employing P-DPCA and Q-DPCA (No Estimate of Y or X Necessary)

[0930] 1. Operation to attain X Equating Y to attain X

[0931] The approach would to equate equations (19) and (19) and solve as follows:

[0932] (V″_1,1−W_c1V″_1,2)/(V′_1,1−W_c1V′_1,2)=(V″_1,1−)/(V′_1,1−)  A)

[0933] and since from the previous development we have the following:

[0934] W_C1=A_CMe^{j[ψ+K(s−x)]}  B)

[0935] =A_CMe^j[ψ+  C)

[0936] Performing all the algebra for equation (A) and reducing all known quantities to constants (K₁, K₂etc.) we have the following:

[0937] (K₁/W_c1)+(K₂/)=1  D)

[0938] where K₁=|K_1m|e^jKand K₂=|K_2m|e^jK

[0939] Substituting in equation (D) K₁, K₂, W_c1and W we have the following:

[0940] (|K_1m|/|A_cm|)e^{j[K1a−[ψ+K(s−x)]]}+(|K_2m|/|A_cm|)e^j[K2A−[ψ+=1  E)

[0941] Substituting new constants K_Aand K_Bfor known quantities we have the following:

[0942] K_Ae^−jKcmx+K_Be^−j=1  F)

[0943] e^−jKcmx=1−jK_CMxand e^−j=1−jx

[0944] K_AK_AX+JK_AYand K_B=K_BX+JK_BY

[0945] K_Ae^−jKcmx+K_Be^−j=1  F

[0946] where K_CMx and Kcm x are small angles therefore we have the following:

[0947] e^−jKcmx=1−jK_CMxand e^−j=1−jx

[0948] Taking equation (F) and substituting for K_A=K_AX+jK_ayand K_B=K_BX+jK_byand equating the real and imaginary parts of the resulting equation we have the following:

[0949] K_AXcos(−K_CMx)−K_AYsin(K_CMx)+K_BXcos(−x)−K_BYsin(−x)=1  G)

[0950] K_AXsin(−K_CMx)+K_AYcos(K_CMx)+K_BXsin(−x)+K_BY=0  H)

[0951] Where x=(K_AY+K_BY)/(K_AXK_CM+K_BXK)

[0952] All quantities are known in equations (G) and (H) except x. There are two equations for the solution of x. They should both give the same value for x or very close or within the tolerance required to get a good solution.

[0953] Another approach to the solution is the following taken advantage of (F′)K_CM=−K and therefore (F″) K_Ae^jKX+K_Be^−JKX=1 and y=e^jKXwe have K_Ay+K_B/y=1 and K_Ay²−y+K_B=0. The last equation is a quadratic solution which solves x as above The two solutions of x one may be eliminated by a reasonableness test (a possible solution) or a comparison with the P-DPCA or Q-DPCA solution.

[0954] This also should agree with the value of x in the previous solutions for P-DPCA and Q-DPCA. There are three (3) methods of attaining x. This method does not rely on an estimate of Y.

[0955] With the last method a more easily attained W_ccan be obtained with only a few FRDB candidates from the FRDB processed. With only a few candidates considered, if more accurate x is required an interpolation may be performed or more detailed values of FRDB candidates performed.

[0956] From this value of x and employing FIG. 16 and FIG. 26—Table I and the value of X may be obtained and employed in the equation for x in P-DPCA and Q-DPCA solution for DΦ_R.

[0957] The procedure for Q-DPCA would be analogous to that of P-DPCA and the results should be the same as for P-DPCA (or be very close within the accuracy required).

[0958] 2. Employing Q-DPCA and P-DPCA and Equating X and solving for Φ_R.

[0959] Another approach would be to equate equation (22) and (22) and substitute W_mand W as follows:

[0960] (V″_1,1e^jDΦ−WmV″_1,2)/(V′_1,1e^jDΦR−WmV′_1,2)=(V″_1,1e^j(D+1)Φ−)/(V′_1,1e^j(D+1)Φ−)

[0961] After performing all the algebra we obtain the following:

[0962] e^j6φR/W_MK₁+e^j5φR/K₂+K₃=0  (A)

[0963] Everything is known in the above equation except φ_Rand W_Mand W. Notice the formula for W_Mand W is the following:

[0964] W_M=A_CMe^{j[ψ+K(t−y)]}  (B)

[0965] =A_cme^j[ψ+  (B′)

[0966] t—is the processed FRDB relative to where the target peak position

[0967] Assuming the target processed at the peak angular position then W=W_m=A_CMe^j[ψ

[0968] Then equation (A) may incorporate W and W_Mand we have the following:

[0969] e^j(D+1)Φ+e^jDΦK′₁+K′₂=0  (AA)

[0970] when the new constants contain W and W_m.

[0971] φ_R=φ_D−φ_A

[0972] φ_R=Dφ_R/D

[0973] and t−y is at its angular position proportional to (φ_D−φ_R)=_Awhere t=y

[0974] φ_Ris not known therefore not determined. The value of K_cm(t−y=0) is determined by the azimuth position of the target.

[0975] When substituting W_mand W in equation (A) we have only φ_Rthat is unknown.

[0976] The variable φ_Ris complicated to solve for but is uniquely solvable. A solution would be to substitute all candidate φ_RS that are a possible solution and the closest φ_Rthat fits the equation is the solution. Another way to approach the solution is to assume W_M1=W=A_CMe^jΨand solve. This will give a close enough solution to get an estimate of φ_Rfrom which the ψ_cmand A_cmcan be estimated closely and the number of candidate solutions reduced quite significantly since W_Mand W do not add significant error at the correct Φ_R.

[0977] If delay D=5 and delay D+1=6. Another solution is taking equation (AA) and substituting z=e^jΦand z⁵=e^j5Φand z⁶=e^j6Φand therefore equation (AA) becomes

[0978] K′₁z⁶+K′₂z⁵+K′₃=0  (BB)

[0979] FIG. 17, FIG. 18 and FIG. 31—Table VII illustrate the principles stated.

[0980] Computer simulation has shown the multiple candidate approach employed as effective. Test all Φ_Rand determine the solution.

[0981] 3. Equating X and solving for Φ_R.

[0982] In the previous B1 system, the calculation of φ_Rwas be equations (19) and (19) and (22) and (22). This technique will estimate Y (change in target from data point 1, to data point 2) and also X (change in clutter from data point 1 to data point 2). By P-DPCA and/or Q-DPCA methodology as performed in the previous section an estimate of Y and X are obtained.

[0983] We will repeat the ΔT basic equations as follows:

[0984] All terms are defined in Section III or IV or in the Glossary of Terms

[0985] (EXP(J5Φ_R))/W_M=(XV′₁₂−V″₁₂)/(XV′₁₁−V″₁₁) for P-DPCA  A)

[0986] (EXP(J6Φ_R))/W_M=(X−)/(XV′₁₁−V″₁₁) for Q-DPCA  A′)

[0987] Since W and W_Mare known therefore equation (A′) divided by equation (A) and solving for e^jΦwe have:

[0988] e^jΦ=(X−)/(XV′₁₂−V″₁₂)W_M= at y=0 W_M=  A1)

[0989] All terms on the right side of equation A1) are measured quantities therefore e^jΦis known.

[0990] Substituting X into equation (A) and (A′) and substituting constants K₁, K₂, and K₃solving for EXP(JDΦ′_R) we have:

[0991] (EXP(J5Φ_R))/W_M=(EXP(J5Φ′_R))=|K_A|EXP(JK_A)  A2)

[0992] EXP(J6Φ′_R)=(EXP(J6Φ_R))/=|K′_A|EXP(JK′_A)  A3)

[0993] W_Mand W are known and incorporated in right side of equation equating the amplitude and phase of vector equation A2) and A3) we have:

[0994] |1/A_CM|*|K_A|=1  A4)

[0995] 5Φ_R=5Φ′_R−Ψ_CM=K_A  A5)

[0996] |1/A_CM|*|K″_A|=1  (A6)

[0997] 6Φ_R=6Φ″_R−Ψ_CM=K″_A  A7)

[0998] The assumption made the difference between 5Φ_Rand 5Φ′_R, 6Φ_Rand 6Φ″_Rthe values of Ψ_CMand A_CMdoes not change. (This assumption is not necessary but thought true and simplifies the computations and the explanations).

[0999] 4. Solving for X Above and Equating X in Equation (A)and (A′)

[1000] We should obtain the same solution. The known terms are replaced with constants we have the following:

[1001] K₁e^j6Φ+K₂e^j5ΦK₃=0

[1002] Y=e^jΦ

[1003] K₁y⁶+K₂y⁵+K₃=0

[1004] We have a sixth order equation where a close approximate solution may be had from the previous solution or from the P-DPCA alone or the Q-DPCA alone solution to obtain the solution of the above sixth order equation.

[1005] At 1000 HZ the P-DPCA D=5 solution at half the filter is 1.4 degrees at ±¼ of the filter it is 0.7 degrees and at ⅛ of the filter it is 0.35 degrees(Section III D). At 1000 HZ the Q-DPCA D=6 solution at half the filter is −1.4 degrees, at ¼ of the filter it is −0.7 degrees and at ⅛ of the filter it is −0.35 degrees

[1006] At 750 HZ the Q-DPCA solution at half the filter is −0.35 degrees, at ¼ of the filter it is −0.175 deg. and at ⅛ of the filter it is −0.08 degrees. At 750 HZ the P-DPCA (Section III D) solution at half the filter is 0.35 degrees, at ¼ of the filter it is 0.175 deg. and at ⅛ of the filter it is 0.08 degrees

[1007] This indicates that clutter detection must be at least approximately ¼ of the filter distant from the peak of clutter to make a significant difference.

[1008] One requirement is therefore is to process clutter off its peak to make this significant difference but at same time to detect target to have as large a change as possible.

[1009] Another approach would be detecting clutter at its peak, where the change in clutter would be close to zero. Then the approach would be when this condition is met the P-DPCA or Q-DPCA operation would be implemented. The added feature is to have a large a change in target as possible.

[1010] B5. Evaluation of equations for P-DPCA and Q-DPCA for ΔT and removing the approximation W_C1=W_C2and W_C1=W_C2e^jΔKand W_M1=W_M2at correct Φ_R. Δ is the estimated difference in x from data point 1 to data point 2 for a nominal determination of s=x. The difference of K would be 0.001 from data point 1 to data point 2 that was determined from real data. Processing at the peak of clutter (s=x) we have the following:

[1011] Y=[G]e^jΔK/2∥sin((DΦ_R+ΔK_CM)/2)|/|sin(DΦ_R/2)∥A_R=∥sin((DΦ_R+ΔK_CM)/2)|/|sin(DΦ_R/2)∥  (G)

[1012] For Q-DPCA we have the following:

[1013] Y=[H]e^jΔ|sin(((D+1)Φ_R+Δ)/2)|/|sin(((D+1)Φ_R)/2)|A′_R=|sin(((D+1)Φ_R+Δ)/2)|/|sin(((D+1)(Φ_R)/2)|  (H)

[1014] For X and P-DPCA we have the following:

[1015] X=[I]e^−JΔK/2|sin(DΦ_R+ΔK_CM/2)|/|sin(DΦ_R/2)|  (I)

[1016] For X and Q-DPCA we the following: $\left(J\right)X=\left[J\right]e^{-J\Delta \underset{\_}{K_{\mathrm{CM}}}}\left|\sin \left(\left(D+1\right){\Phi }_R+\Delta \underset{\_}{K_{\mathrm{CM}}}/2\right)\right|/\left|\sin \left(\left(D+1\right){\Phi }_R/2\right)\right|$

[1017] Equation (K)=(G)/(I)=Y/X=[G]/[I]e^jΔK

[1018] Equation (L)=(J)/(H)=Y/X=[J]/[H]e^−jΔ

[1019] Equation (K)/(L)=[G]/[I]e^{jΔ(K+)=[J]/[H]}

[1020] With an estimate of ΔK_CMand ΔK being very small and all the parameters are known except Φ_R. This is a solution without knowing the value of X and Y (clutter and target). Processing at s=x for this solution but not necessary but a more complicated solution when s is not equal to x.

[1021] The procedure would be as follows:

[1022] Each candidate doppler bin (Φ_R) is tested so that when the above equation is satisfied that doppler bin (Φ_R) is the correct one.

[1023] Another solution is the following:

[1024] Taking the equation (I) and (J) dividing (I) by (J) we have the following:

[1025] [I]/[J]e^−jΔ(K−)=A_R/A′_R  (IJ)

[1026] With Δ having a negligible affect on the solution and with s=x everything is known except Φ_Rwe substitute all possible candidate solutions. The correct Φ_Rwill satisfy the above equation.

[1027] After obtaining the above results test equation (G) and (H) substituting all possible candidate solutions Φ_Rand the Y obtained in each equation should be equal or close in value.

[1028] Another solution with the aid of determining the clutter peak in a different way then in the previous paragraphs would be as follows: $\mathrm{LET}\begin{array}{c}{K}_A{e}^{{\mathrm{jK}}_A}=\left(\underset{\_}{V_{11}^{\prime }}-\underset{\_}{W_{\mathrm{C1}}}\underset{\_}{V_{12}^{\prime }}\right)/\left(V_{11}^{\prime }-W_{\mathrm{C1}}{V}_{12}^{\prime }\right)\\ =\left(1-\underset{\_}{W_{\mathrm{C1}}}/\underset{\_}{W_{\mathrm{M1}}}{e}^{j\left(D+1\right){\Phi }_R}\right)/\left(1-W_{\mathrm{C1}}/W_{\mathrm{M1}}{e}^{jD{\Phi }_R}\right)\\ =e^{j{\Phi }_R/2}\left|\mathrm{SIN}\left(\left(D+1\right){\Phi }_R/2\right)\right|/\left|\mathrm{SIN}\left(D{\Phi }_R/2\right)\right|\end{array}$

[1029] Therefore when s=x W_C1=W and W_M1=W and as follows:

[1030] |K_A|=|SIN((D+1)Φ_R/2)|/|SIN(DΦ_R/2)|

[1031] e^jK=e^jΦ/2Therefore K_A=Φ_R/2

[1032] The procedure would be as follows:

[1033] Each candidate doppler bin (Φ_R) is tested so that when the above equation is satisfied that doppler bin (Φ_R) is the correct one.

[1034] Other doppler bins may be processed in like manner such as on other side of the peak of target to obtain the peak of clutter. Also there are other range bins where the target is detected. They may also be processed and correlated to obtain the best solution.

[1035] Employing the same equations as for the previous solution for equating M we will equate C as follows:

[1036] C=(V′₁₁e^jDΦ/W_M1−V′₁₂)/(e^jDΦ/W_M1−1/W_C1) for P-DPCA

[1037] C=(V′₁₁e^j(D+1)Φ/−)/(e^j(D+1)Φ/−1/) for Q-DPCA

[1038] At peak of clutter s=x and W_C1≈W and equating C we have an equation where the only unknown is Φ_Rbut of nth order and difficult to solve but solvable since the solution is real and restricted to a range of values. It can also be checked with previous solution equating M.

[1039] As in the previous solution other doppler bins may be processed in like manner such as on other side of the peak of target to obtain the peak of clutter. Also there are other range bins where the target is detected. They may all be processed and correlated to obtain the best solution.

[1040] Since there a limited number of solutions possible all candidate doppler frequencies are processed in both P-DPCA and Q-DPCA that at the correct frequency

[1041] C,M,X and Y are equal.

[1042] B4. Determine Φ_Rby estimating Y and X

[1043] Estimate of Y_F, Y and X_F, X with P-DPCA and/or Q-DPCA

[1044] Taking equation (1) and (12) and solving for C and M we have the following:

[1045] X_FC=(YV′₁₁−V″₁₁)/(Y−X)^{XC=(YV′−V″)/(Y−X})

[1046] Y_FM=(XV′₁₁−V″₁₁)/(X−Y)Y_FM=(XV′₁₁−V″₁₁)/(X−Y)

[1047] 1. For P-DPCA employ equation (2) and solve for e^J5Φ/W_M1we have the following: (2p) e^j5Φ/W_M1=(V₁₂−X_FC/W_C1)/Y_FM with all parameters on right side of the equation are measured, estimated or calculated therefore known. W_M1known since at the correct “5Φ_R” W_M1=A_CMe^jΨtherefore we have a solution for “5Φ_R” and for Φ_Rand Φ_Aand the object of the disclosure.

[1048] 2. For Q-DPCA employ equation (2) and solve e^J6Φ/W we have the following: (2q) e^J6Φ/W=(V′−X_FC/W)/Y_FM with all parameters on right side of the equation are measured, estimated or calculated therefore known. W_M1and W are known therefore we have a solution for “6Φ_R” and for Φ_Rand Φ_Aand the object of the disclosure. The solution for P and Q-DPCA should be equal.

[1049] 3. Estimating Y and X and combining P-DPCA and Q-DPCA

[1050] Dividing equation (2q) by equation (2p) from the two previous sections we have the following:

[1051] e^jJ=(−X_FC/)/(V′₁₂−X_FC/W_C1)*K_M

[1052] All parameters on right side of equation are known therefore the object of disclosure is determined. K_M=W_M/W_M

[1053] Another solution or a check on the previous solution would be to process the data in the FRDB that clutter has peaked at (that would be x in the previous solution).

[1054] Correlation of the results of the two sets of data taken, all the resulting solutions and parameters should correlate closely.

[1055] For example the DΦ_Rand Φ_Asolution should be very closely correlate, the M calculated in both sets of data should be nearly equal and all other parameters should correlate, taken into account they are two sets of data closely spaced in relation to each other.

[1056] Note on the B4 Technique: The data processed in the B4 technique was where the target was detected at its peak and the other processed data was where the clutter was determined at its peak. This is special data thought to have significance and ease of processing but in reality any set of data or sets of data may be processed and correlated.

[1057] It is thought that to process data where clutter is at its peak may be the more meaningful or best data to process. This may be true due to clutter to be most unaffected by adjacent clutter and processing of data more accurately when clutter is at its peak. Also the target estimation of Y and Y_Fshould more accurate due to its more determinate nature than clutter.

[1058] 4. Correlation of Results

[1059] All the techniques involved in Section IV, B may be applied to a number of FRDBs and different PRFs and between the various techniques described and correlated for better results.

[1060] C. System Processing the Target in Two Adjacent Filters but at the Same time.

[1061] C1-1. System Processing the Target in Two Adjacent Filters but at the same time employing P-DPCA (ΔF)

[1062] With reference to the same discussion at the start of Section IV B1 this technique will process two or more filters where the target is detected. The basic equations for this technique are derived in Section III H.

[1063] With the previous technique a specific filter is processed at two or more times. With this technique two or more filters (FRDBs) are processed at the same time. With this technique two filters employing equations (1′) to (23′) and (1′) to (23′) in Section III were developed and is similar to that for the Section III, B technique.

[1064] With the aforementioned information on the weighting function of clutter (W_C) and its associated information as well as that for the weighting function of target (W_M) and its associated information and also the derived equations and information of Section III will be employed to attain the object of the disclosure.

[1065] Evaluation of equation (A1) and (A2) for P-DPCA and part of equation(18′) for Y_Fas follows:

[1066] 1−(W_C1/W_M1)e^jDΦ  (A1)

[1067] 1−(W_C2/W_M2)e^jDΦ  (A2)

[1068] W_M2=W_M1Φ_R

[1069] W_C2=W_C1EXP^jpK

[1070] Equation (Y_F) becomes the following:

[1071] Y_F=e^jB=((T′₁₁−W_C1EXP^JpKT′₁₂)/(V′₁₁−_WC1V′₁₂))Wh

[1072] ere p is the distance between filter 1 and filter 2. This distance should be as close as possible without being identical but be significant that

[1073] EXP ^JpK≈1 at angle≈0≈1+jpK_CMat small angles

[1074] The procedure will be set forth with an example as follows:

[1075] The target and clutter is as in FIG. 18, “Clutter and Target Detection Diagram”.

[1076] Target Peak Amplitude Detection at FRDB-0₀

[1077] Clutter Peak Amplitude Detection at FRDB-0₁

[1078] Target Angular Position at FRDB at 61₀or (−3₀)

[1079] Employing equation (19′) or (19′) to find W_c1and W_c2we have the following:

[1080] Y_F=_Fe^jBF=(T′_1,1−W_c2T′_1,2)/(V′_1,1−W_c1V′_1,2)  (19′)

[1081] Where the estimate of {overscore (K)}_Fand B_Ffrom where the target is detected at its peak is {overscore (K)}_F0=1.0 and B_F0=0. The filter response of the one filter to the right of the above is {overscore (K)}_F1=0.85 and B_F1=45°. Therefore the response from filter 0₀to fil{overscore (ter)}0₁is K_F=0.85 and B_F=−45° (Reference FIGS. 17 and 18).

[1082] Taking the following equations for W_c1and W_c2we will determine the relationship between W_c1and W_c2as follows:

[1083] W_C1=A_CMe^{j[ψ+K(s−x)]}=A_CMe^j[ψ+ks]e^−jKx  (A)

[1084] W_C2=A_CMe^{j[ψ+K(s+1)−x)]}=A_CMe^j[ψ+Ks]e^jKcme^−jKcmx  (B)

[1085] From the equation (A) and (B) we observe the following:

[1086] W_C2=e^jKW_c1Where p=1  (C)

[1087] Substituting equation (C) and Y_Fin equation (19′) and solving for W_c1we have the following:

[1088] W_C1=(T′_1,1−Y_FV′_1,1)/(e^jKT′_1,2−Y_FV′_1,2)  (D)

[1089] On the right side of the equation all the parameters are measured or estimated and therefore are known parameters we shall call it |K_m|e^jKaequal to the right side of the equation and solve for W_c1and x as follows:

[1090] W_C1A_cme^{j[ψ+K(s−x)]}=|K_m|e^jKa  (E)

[1091] Equating the exponentials of equation (E) and solving for x and setting x=x₁we have the following:

[1092] x₁=(K_A−Ψ_cm−K_cms)/K_cm  (F)

[1093] Setting x=x₂

[1094] x₂=x₁+1FRDB:x₂=(−K_A+Ψ_cm+K_cm(s+1))/K_cm  (F′)

[1095] All the terms on the right side of equation (F) is known or estimated therefore x₁and x₂is solved for. x₁and x₂is the position in number of FRDB the peak of clutter is from the FRDB processed (s) or (s+1).

[1096] From x₁X_F1is determined and from x₂is determined X_F2and X_F=X_F2/X_F1

[1097] All terms have been defined in Section III, H.

[1098] From x₁, is determined X_F1of clutter, as in this example the peak of clutter is one FRDB to the right of the processed FRDB(s). FIG. 16 and FIG. 26—Table I shows the value of X_F1is K_F1=0.85 and α_F1=−45°. K_F2=0.70 and α_F2=−90°. Therefore K_F=0.85/0.70=1.214 and α_F=−45−(−90)=45

[1099] Employing equation (24) of section III for P-DPCA ΔF technique we have:

[1100] Where |A_R| is approximately equal to 1(most of the time) and the phase shift is approximately equal to zero (0) (most of the time) and the first filter is detected at the peak of clutter we have the following technique.

[1101] e^j5Φ/W_M1=(X_FV′₁₂−T′₁₂)/(X_FV′₁₁−T′₁₁)

[1102] where all the parameters on the right side of the above equation are measured or estimated and making it to a known |A|e^j5Φ′value. |. W_M1=A_CMEXP^J(Ψ+Ky)and y is determined by the distance from the peak of mover and Ψ_CMand A_CMare determined by Φ′_R. This is close enough to Φ_Rto give a good estimate. Rewriting above equation inserting this we have:

[1103] EXP^J5Φ=|A||A_CM|EXP^{J(Ψ+Ky+5Φ′)}

[1104] where 5Φ_R=Ψ_CM+K_CMy+5Φ′_Rand |A||A_CM=|1

[1105] This is a solution to the object of the invention. From Φ_RΦ_Ais determined and consequently the targets relative radial velocity and azimuth.

[1106] C1-2. System Processing the Target in Two Adjacent Filters but at the Same Time Employing Q-DPCA

[1107] Analogous to the above development and the determining of x and X_Fwe have a parallel development employing equation (19′) for Q-DPCA where p=1, as follows:

[1108] Y_F=_Fe^jB=(T′_1,1−)/(V′_1,1−) and

[1109] =(T′₁₁−Y_FV′)/(e^j−Y_F)  (19′)

[1110] Solving in the same manner for equation (19′) we have the solution for x as follows:

[1111] =(−Ψ_cm−K_cms)}→X_F1and →X_F2  (G)

[1112] =+1FRDB}x_F=x_F2/x_F1  (G¹)

[1113] From x is determined X. x=x and X_F=X. This should occur or be very close to each other within the accuracy required.

[1114] x₁—is from P-DPCA compensation

[1115] x₂—is from P-DPCA compensation

[1116] x—is from T-DPCA compensation

[1117] x—is from Q-DPCA compensation

[1118] X_F—is from P-DPCA compensation

[1119] X—is from Q-DPCA compensation

[1120] This is two different set of equations where the value of x and X_Fare the same and are determined and will be a check on each other.

[1121] C2. Employing P-DPCA and Q-DPCA and solving for Φ_R

[1122] 1. Taking EXP(J6Φ_R) equation and divide by EXP(J5Φ_R) equation and solve for Φ_R

[1123] Taking equation (24′) of section III where EXP(JpK_CM) and EXP(JpK) where the amplitude is approximately equal to one (1) and phase shift is approximately equal to zero (0) we have:

[1124] EXP(J5J_R)/W_M1=(X_FV′₁₂−T′₁₂)/(X_FV′₁₁−T′₁₁) This is for P-DPCA  (24′)

[1125] EXP(J6J_R)/=(X_F−)/(X_FV′₁₁−T′₁₁) This is for Q-DPCA.  (24′)

[1126] Noting that at correct Φ_Rwe have W_M1=W and are known for reasons stated previously and X_F's are equal in the above equations at the true candidate Φ_R. A preliminary solution may be obtained for the P-DPCA or Q-DPCA alone technique and a more accurate solution combining them.

[1127] Therefore the object of disclosure attained.

[1128] 2. Equating X_Fand Solving for EXP(JDΦ).

[1129] From equation (21′) equating X_Fwe have the following:

[1130] X_F=[(T′₁₁EXP(J6J_R)−)/(V′₁₁EXP(J6J_R)−)]  (21′)

[1131] X_F=[(T′₁₁EXP(J5J_R)−T′₁₂)/(V′₁₁EXP (J5J_R)−V′₁₂)]  (21′)(21′)

[1132] Where the following:

[1133] W_M1and W are known at the correct Φ_Rfor the same reasons stated previously.

[1134] Equating the two equations for X_Fabove and utilizing the equations of section and solving for EXP(J5Φ_R)and EXP(J6Φ_R) and all other parameters as constants we have a sixth order equation with all parameters known except Φ_R.

[1135] The most effective way of solving is to obtain a solution as obtained with techniques of Section C1 and/or Section C2 and using this as a solution in the eleventh order equation and solve by candidate solutions and by synthetic division to be exact.

[1136] Therefore the object of disclosure is attained.

[1137] Correlation of Results

[1138] Correlate solutions for all techniques performed may yield better accuracy if that is required. Other FRDBs may be processed and correlate results.

[1139] C3. ΔF Processing Employing Estimate of Y_Fand X_FFor P-DPCA and Q-DPCA

[1140] This technique will estimate Y_Fand X_F(change in target or clutter from filter 1 (FRDB-0₀) to filter 2 (FRDB-0₁) and at the same time (Data Point 1) from the procedure in Section IV, C1-1 and/or C1-2 an estimate of Y_Fand X_Ffor the filters chosen as in FIGS. 16, 17, 18 and FIG. 25—Table I, FIG. 26—Table II, FIG. 27—Table III.

[1141] We will repeat the ΔF basic conditions as follows:

[1142] where p=1 is the spacing between filter 1 and filter 2.

[1143] All other terms are defined in Section III and IV. In the previous C2 system, the C3 system employs the same basic equations and filter examples. The only difference when X_Fand Y_Fare attained as explained and derived in Section IV, C1-1 and/or C1-2 we employ the following equations to attain M and C which are derived from filter 1 and filter 2 equations (1′) and (12′).

[1144] M=(X_FV′_1,1−T′_1,1)/(X_F−Y_F)

[1145] C=(Y_FV′_1,1−T′_1,1)/(Y_F−X_F)

[1146] For filter #1—FRDB-0₀processed the filter response of the target is {overscore (K)}_F1e^jBF1(where the target is detected at its peak in {overscore (K)}_F1e^jBF1=1└0) and for filter #2 FRDB-0₁the target response is {overscore (K)}_F2=0.85 and B_F2=−45° and therefore {overscore (K)}_Fe^jB={overscore (K)}_F2e^jBF2/{overscore (K)}_F1e^jBF1=0.85/└−45°/1.0└0=0.85└−45°.

[1147] Performing the same operations for clutter we have the following:

[1148] For filter 1-FRDB-0₀processed the filter response of clutter is K_F1e^jαF1where K_F1=0.85 and α_F1=45° and the for filter 2-FRDB-0₁processed the filter response of clutter is K_F2e^jαF2where K_F2=1.0 and α_F2=0° and therefore K_Fe^αF=K_F2e^jαF2/K_F1e^jαF1=1.0└0/0.85└−45°=0.85└−45°.

[1149] Taking equation (2′) we have the following:

[1150] e^j5ΦR/W_M1=[(V′_1,2−C/W_c1)/M]=|K_A|e^jK  (L)

[1151] where W_M1is known at the correct Φ_Rand the parameters on right side of above equation including W_M1therefore we have the following:

[1152] |K_A|=1 and Φ_R=(K_A+Ψ_CM)/5

[1153] Therefore the object of disclosure is attained as explained previously.

[1154] Analogously we solve for (L′)e^j6Φ/W=(V′−C/W)/M=|K′_A|e^jK′in equation (2′) |K′_A|=1 and Φ_R=(K′_A+Ψ_CM)/6

[1155] Therefore the object of disclosure is attained as explained previously. The result should agree with the previous result.

[1156] Dividing equation (L′) by (L) we have the following:

[1157] e^jJ=(−C/)/(V′₁₂−C/W_C1)

[1158] that is another solution for Φ_R.

[1159] C4. Correlation of Results

[1160] All technique and methods in Section IV C may be correlated for better results. Other FRDBs may be processed and other PRFs may be processed and correlated.

[1161] D. Employing D and D+1 For a Solution.

[1162] D1—Ideal Data Approach

[1163] W_Cand W_mare equal to 1└0.

[1164] Ideal solution to D=5 for M P-DPCA is equation (5) of Section III which is the following:

[1165] M=(V′₁₁−V′₁₂)/(1−e^j5Φ)  (1)

[1166] Applying this same equation to D=6 we have the following:

[1167] M=(V′₁₁−)/(1−e^j6Φ)  (2)

[1168] Equating equation (1) to (2) and solving we have the following:

[1169] e^jΦR/2{|sin(3Φ_R)/sin(5Φ_R/2)|}=(−)/(V′₁₁−V′₁₂)  (3)