Multi-path mitigation in rangefinding and tracking objects using reduced attenuation RF technology

[0001] This application is a continuation-in-part of U.S. patent application Ser. No. 12/502,809, filed on Jul. 14, 2009, entitled METHODS AND SYSTEM FOR REDUCED ATTENUATION IN TRACKING OBJECTS USING RF TECHNOLOGY, which is a continuation of U.S. patent application Ser. No. 11/610,595, filed on Dec. 14, 2006, entitled METHODS AND SYSTEM FOR REDUCED ATTENUATION IN TRACKING OBJECTS USING RF TECHNOLOGY, which claims priority to U.S. provisional Patent Application No. 60/597,649 filed on Dec. 15, 2005, entitled METHOD AND SYSTEM FOR REDUCED ATTENUATION IN TRACKING OBJECTS USING MULTI-BAND RF TECHNOLOGY, which are incorporated by reference herein in their entirety.

[0002] U.S. patent application Ser. No. 12/502,809, filed on Jul. 14, 2009, entitled METHODS AND SYSTEM FOR REDUCED ATTENUATION IN TRACKING OBJECTS USING RF TECHNOLOGY, to which this application claims priority, is a non-provisional of U.S. provisional Patent Application No. 61/103,270, filed on 7 Oct. 2008, entitled METHODS AND SYSTEM FOR MULTI-PATH MITIGATION IN TRACKING OBJECTS USING REDUCED ATTENUATION RF TECHNOLOGY, which are incorporated by reference herein in their entirety.

[0003] 1. Field of the Invention

[0004] The present invention relates to wireless communications and wireless networks systems and systems for a Radio Frequency (RF)-based identification, tracking and locating of objects, including RTLS (Real Time Locating Service).

[0005] 2. Background of the Related Art

[0006] RF-based identification and location-finding systems for determination of relative or geographic position of objects are generally used for tracking single objects or groups of objects, as well as for tracking individuals. Conventional location-finding systems have been used for position determination in an open outdoor environment. RF-based, Global Positioning System (GPS), and assisted GPSs are typically used. However, conventional location-finding systems suffer from certain inaccuracies when locating the objects in closed (i.e., indoor) environments, as well as outdoors. Although cellular wireless communication systems provide excellent data coverage in urban and most indoor environments, the position accuracy of these systems is limited by self-interference, multipath and non-line-of-sight propagation.

[0007] The indoor and outdoor location inaccuracies are due mainly to the physics of RF propagation, in particular, due to losses/attenuation of the RF signals, signal scattering and reflections. The losses/attenuation and scattering issues can be solved (see co-pending application Ser. No. 11/670,595) by employing narrow-band ranging signal(s) and operating at low RF frequencies, for example at VHF range or lower.

[0008] Although, at VHF and lower frequencies the multi-path phenomena (e.g., RF energy reflections), is less severe than at UHF and higher frequencies, the impact of the multi-path phenomena on location-finding accuracy makes location determination less reliable and precise than required by the industry. Accordingly, there is a need for a method and a system for mitigating the effects of the RF energy reflections (i.e., multi-path phenomena) in RF-based identification and location-finding systems that are employing narrow-band ranging signal(s).

[0009] As a rule, conventional RF-based identification and location-finding systems mitigating multipath by employing wide bandwidth ranging signals, e.g. exploiting wide-band signal nature for multi-path mitigation (see S. Salous, “Indoor and Outdoor UHF Measurements with a 90 MHz Bandwidth”, IEEE Colloquium on Propagation Characteristics and Related System Techniques for Beyond Line-of-Sight Radio, 1997, pp. 8/1-8/6). In addition, spatial diversity and/or antenna diversity techniques are used in some cases.

[0010] However, the spatial diversity may not be an option in many tracking-location applications because it leads to an increase in required infrastructure. Similarly, the antenna diversity has a limited value, because at lower operating frequencies, for example VHF, the physical size of antenna subsystem becomes too large. The case in point is the U.S. Pat. No. 6,788,199, where a system and method for locating objects, people, pets and personal articles is described.

[0011] The proposed system employs an antenna array to mitigate the multi-path. The optionally system operates at UHF in the 902-926 MHz band. It is well known that the linear dimension of the antenna is proportional to the wave length of an operating frequency. Also, the area of an antenna array is proportional to the square and volume to the cube of the linear dimensions ratio because in an antenna array the antennas are usually separated by ¼ or ½ wave length. Thus, at VHF and lower frequencies the size of the antenna array will significantly impact device portability.

[0012] On the other hand, because of a very limited frequency spectrum, the narrow bandwidth ranging signal does not lend itself into multi-path mitigation techniques that are currently used by conventional RF-based identification and location-finding systems. The reason is that the ranging signal distortion (i.e., change in the signal) that is induced by the multi-path is too small for reliable detection/processing in presence of noise. Also, because of limited bandwidth the narrow bandwidth receiver cannot differentiate between ranging signal Direct-Line-Of-Sight (DLOS) path and delayed ranging signal paths when these are separated by small delays, since the narrow bandwidth receiver lacks the required time resolution, which is proportional to the receiver's bandwidth (e.g., the narrow bandwidth has an integrating effect on the incoming signals).

[0013] Accordingly, there is a need in the art for a multi-path mitigation method and system for object identification and location-finding, which uses narrow bandwidth ranging signal(s) and operates in VHF or lower frequencies as well as UHF band frequencies and beyond.

[0014] The track and locate functionality need is primarily found in wireless networks. The multi-path mitigation methods and systems for object identification and location finding, described in co-pending application Ser. No. 12/502,809, can be utilized in most of the available wireless networks. However, certain wireless networks have communications standards/systems that require integration of the techniques into the wireless networks to fully benefit from various ranging and positioning signals that are described in co-pending application Ser. No. 12/502,809. Typically, these wireless systems can provide excellent data coverage over wide areas and most indoor environments. However, the position accuracy available with of these systems is limited by self-interference, multipath and non-line-of-sight propagation. As an example, the recent 3GPP Release 9 standardized positioning techniques for LTE (Long Term Evolution) standard has the following: 1) A-GNSS (Assisted Global Navigation Satellite System) or A-GPS (Assisted Global Positioning System) as the primary method; and 2) Enhanced Cell-ID and OTDOA (Observed Time Difference of Arrival), including DL-OTDOA (Downlink OTDOA), as fall-back methods. While these methods might satisfy the mandatory FCC E911 emergency location requirements, the accuracy, reliability and availability of these location methods fall short of the needs of LBS (Location Based Services) or RTLS system users, who require highly accurate locating within buildings, shopping malls, urban corridors, etc.

[0015] It is well known that the A-GNSS (A-GPS) accuracy is very good in open spaces but is very unreliable in urban/indoor environments. Also, Enhanced Cell-ID and OTDOA accuracy is impacted by the effects of multipath and other radio wave propagation phenomena.

[0016] There are other location methods, including ones that are based on received signal strength. However, RF propagation phenomenon make the signal strength vary 30 dB to 40 dB over the distance of a wavelength which, in wireless networks, can be significantly less than a meter. This severely impacts the accuracy and/or the reliability of methods based on received signal strength.

[0017] Accordingly, there is a need in the art for more accurate and reliable tracking and locating capability for wireless networks.

[0018] The present invention relates to a method and system for a Radio Frequency (RF)-based identification, tracking and locating of objects, including Real Time Locating Service (RTLS) that substantially obviates one or more of the disadvantages of the related art. The proposed (exemplary) method and system use a narrow bandwidth ranging locating signal(s). According to an exemplary embodiment, RF-based tracking and locating is implemented on VHF band, but could be also implemented on lower bands (HF, LF and VLF) as well as UHF band and higher frequencies. It employs multi-path mitigation method including techniques and algorithms. The proposed system can use software implemented digital signal processing and software defined radio technologies. Digital signal processing can be used as well.

[0019] The system of the exemplary embodiment can be constructed using standard FPGAs and standard signal processing hardware and software at a very small incremental cost to the device and overall system. At the same time the accuracy of the RF-based identification and location-finding systems that are employing narrow-band ranging signal/s can be significantly improved.

[0020] The transmitters and receivers for narrow bandwidth ranging/locating signal, for example VHF, are used to identify a location of a person or an object. Digital signal processing (DSP) and software defined radio (SDR) technologies can be used to generate, receive and process a narrow bandwidth ranging signal(s) as well as perform multi-path mitigation algorithms. The narrow bandwidth ranging signal is used to identify, locate and track a person or an object in a half-duplex, full duplex or simplex mode of operation. The Digital signal processing (DSP) and software defined radio (SDR) technologies are used in the multi-path mitigation processor to implement multi-path mitigation algorithms.

[0021] The approach described herein employs a multi-path mitigation processor and multi-path mitigation techniques/algorithms described in co-pending application Ser. No. 12/502,809 that increase the accuracy of tracking and locating system implemented by a wireless network. The present invention can be used in all wireless systems/networks and include simplex, half-duplex and full duplex modes of operation. The exemplary embodiment described below operates with wireless networks that employ OFDM modulation and/or its derivatives. Thus, the exemplary embodiment described below operates with LTE networks and it is also applicable to other wireless systems/networks.

[0022] The approach described herein is also applicable to other wireless networks, including WiMax, WiFi, and White Space. Other wireless networks that do not use reference and/or pilot signals may employ one or more of the following types of alternate modulation embodiments of present invention as described in co-pending application Ser. No. 12/502,809: 1) where a portion of frame is dedicated to the ranging signal/ranging signal elements as described in co-pending application Ser. No. 12/502,809; 2) where the ranging signal elements (see co-pending application Ser. No. 12/502,809) are embedded into transmit/receive signals frame(s); and 3) where the ranging signal elements (described in co-pending application Ser. No. 12/502,809) are embedded with the data.

[0023] These alternate embodiments employ multi-path mitigation processor and multi-path mitigation techniques/algorithms described in co-pending application Ser. No. 12/502,809 and can be used in all modes of operation: simplex, half-duplex and full duplex.

[0024] The integration of multi-path mitigation processor and multi-path mitigation techniques/algorithms described in co-pending application Ser. No. 12/502,809 with OFDM based wireless networks, and other wireless networks with reference/pilot signals, can be done with little or no incremental cost to the device and overall system. At the same time the location accuracy of the network and system will be significantly improved. As described in the exemplary embodiment, RF-based tracking and locating is implemented on 3GPP LTE cellular networks will significantly benefit from the localization of multi-path mitigation method/techniques and algorithms that are described in co-pending application Ser. No. 12/502,809 application. The proposed system can use software- or hardware-implemented digital signal processing.

[0025] Additional features and advantages of the invention will be set forth in the description that follows, and in part will be apparent from the description, or may be learned by practice of the invention. The advantages of the invention will be realized and attained by the structure particularly pointed out in the written description and claims hereof as well as the appended drawings.

[0026] It is to be understood that both the foregoing general description and the following detailed description are exemplary and explanatory and are intended to provide further explanation of the invention as claimed.

[0039] Reference will now be made in detail to the preferred embodiments of the present invention, examples of which are illustrated in the accompanying drawings.

[0040] The present invention relates to a method and system for RF-based identification, tracking and locating of objects, including RTLS. According to the exemplary embodiment, the method and system employs a narrow bandwidth ranging signal. The exemplary embodiment operates in VHF band, but can be also used in HF, LF and VLF bands as well as UHF band and higher frequencies. It employs multi-path mitigation processor. Employing multi-path mitigation processor increases the accuracy of tracking and locating implemented by a system.

[0041] The exemplary embodiment includes small, highly portable base units that allow users to track, locate and monitor multiple persons and objects. Each unit has its own ID. Each unit broadcasts an RF signal with its ID, and each unit is able to send back a return signal, which can include its ID as well as voice, data and additional information. Each unit processes the returned signals from the other units and, depending on the triangulation or trilateration and/or other methods used, continuously determines their relative and/or actual locations. The preferred embodiment can also be easily integrated with products such as GPS devices, smart phones, two-way radios and PDAs. The resulting product will have all of the functions of the stand-alone devices while leveraging the existing display, sensors (such as altimeters, GPS, accelerometers and compasses) and processing capacity of its host. For example, a GPS device with the device technology describe herein will be able to provide the user's location on a map as well as to map the locations of the other members of the group.

[0042] The size of the preferred embodiment based on an FPGA implementation is between approximately 2×4×1 inches and 2×2×0.5 inches, or smaller, as integrated circuit technology improves. Depending on the frequency used, the antenna will be either integrated into the device or protrude through the device enclosure. An ASIC (Application Specific Integrated Circuit) based version of the device will be able to incorporate the functions of the FPGA and most of the other electronic components in the unit or Tag. The ASIC-based stand-alone version of the product will result in the device size of 1×0.5×0.5 inches or smaller. The antenna size will be determined by the frequency used and part of the antenna can be integrated into the enclosure. The ASIC based embodiment is designed to be integrated into products can consist of nothing more than a chipset. There should not be any substantial physical size difference between the Master or Tag units.

[0043] The devices can use standard system components (off-the-shelf components) operating at multiple frequency ranges (bands) for processing of multi-path mitigation algorithms. The software for digital signal processing and software-defined radio can be used. The signal processing software combined with minimal hardware, allows assembling the radios that have transmitted and received waveforms defined by the software.

[0044] Co-pending application Ser. No. 11/670,595 discloses a narrow-bandwidth ranging signal system, whereby the narrow-bandwidth ranging signal is designed to fit into a low-bandwidth channel, for example using voice channels that are only several kilohertz wide (though some of low-bandwidth channels may extend into a few tens of kilohertz). This is in contrast to conventional location-finding systems that use channels from hundreds of kilohertz to tens of megahertz wide.

[0045] The advantage of this narrow-bandwidth ranging signal system is as follows: 1) at lower operating frequencies/bands, conventional location-finding systems ranging signal bandwidth exceeds the carrier (operating) frequency value. Thus, such systems cannot be deployed at LF/VLF and other lower frequencies bands, including HF. Unlike conventional location-finding systems, the narrow-bandwidth ranging signal system described in co-pending application Ser. No. 11/670,595 can be successfully deployed on LF, VLF and other bands because its ranging signal bandwidth is far below the carrier frequency value; 2) at lower end of RF spectrum (some VLF, LF, HF and VHF bands), e.g., up to UHF band, conventional location-finding systems cannot be used because the FCC severely limits the allowable channel bandwidth (12-25 kHz), which makes it impossible to use conventional ranging signals. Unlike conventional location-finding systems, the narrow-bandwidth ranging signal system's ranging signal bandwidth is fully compliant with FCC regulations and other international spectrum regulatory bodies; and 3) it is well known (see MRI: the basics, by Ray H. Hashemi, William G. Bradley . . . —2003) that independently of operating frequency/band, a narrow-bandwidth signal has inherently higher SNR (Signal-to-Noise-Ratio) as compared to a wide-bandwidth signal. This increases the operating range of the narrow-bandwidth ranging signal location-finding system independently of the frequency/band it operates, including UHF band.

[0046] Thus, unlike conventional location-finding systems, the narrow-bandwidth ranging signal location-finding system can be deployed on lower end of the RF spectrum—for example VHF and lower frequencies bands, down to LF/VLF bands, where the multipath phenomena is less pronounced. At the same time, the narrow-bandwidth ranging location-finding system can be also deployed on UHF band and beyond, improving the ranging signal SNR and, as a result, increasing the location-finding system operating range.

[0047] To minimize multipath, e.g., RF energy reflections, it is desirable to operate on VLF/LF bands. However, at these frequencies the efficiency of a portable/mobile antenna is very small (about 0.1% or less because of small antenna length (size) relative to the RF wave length). In addition, at these low frequencies the noise level from natural and manmade sources is much higher than on higher frequencies/bands, for example VHF. Together, these two phenomena may limit the applicability of location-finding system, e.g. its operating range and/or mobility/portability. Therefore, for certain applications where operating range and/or mobility/portability are very important a higher RF frequencies/bands may be used, for example HF, VHF, UHF and UWB.

[0048] At VHF and UHF bands, the noise level from natural and manmade sources is significantly lower compared to VLF, LF and HF bands; and at VHF and HF frequencies the multi-path phenomena (e.g., RF energy reflections) is less severe than at UHF and higher frequencies. Also, at VHF, the antenna efficiency is significantly better, than on HF and lower frequencies, and at VHF the RF penetration capabilities are much better than at UHF. Thus, the VHF band provides a good compromise for mobile/portable applications. On the other hand in some special cases, for example GPS where VHF frequencies (or lower frequencies) cannot penetrate the ionosphere (or get deflected/refracted), the UHF can be a good choice. However, in any case (and all cases/applications) the narrow-bandwidth ranging signal system will have advantages over the conventional wide-bandwidth ranging signal location-finding systems.

[0049] The actual application(s) will determine the exact technical specifications (such as power, emissions, bandwidth and operating frequencies/band). Narrow bandwidth ranging allows the user to either receive licenses or receive exemption from licenses, or use unlicensed bands as set forth in the FCC because narrow band ranging allows for operation on many different bandwidths/frequencies, including the most stringent narrow bandwidths: 6.25 kHz, 11.25 kHz, 12.5 kHz, 25 kHz and 50 kHz set forth in the FCC and comply with the corresponding technical requirements for the appropriate sections. As a result, multiple FCC sections and exemptions within such sections will be applicable. The primary FCC Regulations that are applicable are: 47 CFR Part 90—Private Land Mobile Radio Services, 47 CFR Part 94 personal Radio Services, 47 CFR Part 15—Radio Frequency Devices. (By comparison, a wideband signal in this context is from several hundred KHz up to 10-20 MHz.)

[0050] Typically, for Part 90 and Part 94, VHF implementations allow the user to operate the device up to 100 mW under certain exemptions (Low Power Radio Service being an example). For certain applications the allowable transmitted power at VHF band is between 2 and 5 Watts. For 900 MHz (UHF band) it is 1 W. On 160 kHz-190 kHz frequencies (LF band) the allowable transmitted power is 1 Watt.

[0051] Narrow band ranging can comply with many if not all of the different spectrum allowances and allows for accurate ranging while still complying with the most stringent regulatory requirements. This holds true not just for the FCC, but for other international organizations that regulate the use of spectrum throughout the world, including Europe, Japan and Korea.

[0052] The following is a list of the common frequencies used, with typical power usage and the distance the tag can communicate with another reader in a real world environment (see Indoor Propagation and Wavelength Dan Dobkin, WJ Communications, V1.4 Jul. 10, 2002):

[0053]

915 MHz	100 mW	150 feet
2.4 GHz	100 mW	100 feet
5.6 Ghz	100 mW	75 feet

[0054] The proposed system works at VHF frequencies and employs a proprietary method for sending and processing the RF signals. More specifically, it uses DSP techniques and software-defined radio (SDR) to overcome the limitations of the narrow bandwidth requirements at VHF frequencies.

[0055] Operating at lower (VHF) frequencies reduces scatter and provides much better wall penetration. The net result is a roughly ten-fold increase in range over commonly used frequencies. Compare, for example, the measured range of a prototype to that of the RFID technologies listed above:

[0056]

216 MHz	100 mw	700 feet

[0057] Utilizing narrow band ranging techniques, the range of commonly used frequencies, with typical power usage and the distance the tag communication range will be able to communicate with another reader in a real world environment would increase significantly:

[0058]

	From:			To:
	915 MHz	100 mW	150 feet	500 feet
	2.4 GHz	100 mW	100 feet	450 feet
	5.6 Ghz	100 mW	75 feet	400 feet

[0059] Battery consumption is a function of design, transmitted power and the duty cycle of the device, e.g., the time interval between two consecutive distance (location) measurements. In many applications the duty cycle is large, 10× to 1000×. In applications with large duty cycle, for example 100×, an FPGA version that transmits 100 mW of power will have an up time of approximately three weeks. An ASIC based version is expected to increase the up time by 10×. Also, ASICs have inherently lower noise level. Thus, the ASIC-based version may also increase the operating range by about 40%.

[0060] Those skilled in the art will appreciate that the exemplary embodiment does not compromise the system long operating range while significantly increases the location-finding accuracy in RF challenging environments (such as, for example, buildings, urban corridors, etc.)

[0061] Typically, tracking and location systems employ Track-Locate-Navigate methods. These methods include Time-Of-Arrival (TOA), Differential-Time-Of-Arrival (DTOA) and combination of TOA and DTOA. Time-Of-Arrival (TOA) as the distance measurement technique is generally described in U.S. Pat. No. 5,525,967. A TOA/DTOA-based system measures the RF ranging signal Direct-Line-Of-Site (DLOS) time-of-flight, e.g., time-delay, which is then converted to a distance range.

[0062] In case of RF reflections (e.g., multi-path), multiple copies of the RF ranging signal with various delay times are superimposed onto the DLOS RF ranging signal. A track-locate system that uses a narrow bandwidth ranging signal cannot differentiate between the DLOS signal and reflected signals without multi-path mitigation. As a result, these reflected signals induce an error in the estimated ranging signal DLOS time-of-flight, which, in turn, impacts the range estimating accuracy.

[0063] The exemplary embodiment advantageously uses the multi-path mitigation processor to separate the DLOS signal and reflected signals. Thus, the exemplary embodiment significantly lowers the error in the estimated ranging signal DLOS time-of-flight. The proposed multi-path mitigation method can be used on all RF bands. It can also be used with wide bandwidth ranging signal location-finding systems. And it can support various modulation/demodulation techniques, including Spread Spectrum techniques, such as DSS (Direct Spread Spectrum) and FH(Frequency Hopping).

[0064] Additionally, noise reduction methods can be applied in order to further improve the method's accuracy. These noise reduction methods can include, but are not limited to, coherent summing, non-coherent summing, Matched filtering, temporal diversity techniques, etc. The remnants of the multi-path interference error can be further reduced by applying the post-processing techniques, such as, maximum likelihood estimation (like.g., Viterbi Algorithm), minimal variance estimation (Kalman Filter), etc.

[0065] The exemplary embodiment can be used in systems with simplex, half-duplex and full duplex modes of operation. Full-duplex operation is very demanding in terms of complexity, cost and logistics on the RF transceiver, which limits the system operating range in portable/mobile device implementations. In half-duplex mode of operation the reader (often referred to as the “master”) and the tags (sometimes also referred to as “slaves” or “targets”) are controlled by a protocol that only allows the master or the slave to transmit at any given time.

[0066] The alternation of sending and receiving allows a single frequency to be used in distance measurement. Such an arrangement reduces the costs and complexity of the system in comparison with full duplex systems. The simplex mode of operation is conceptually simpler, but requires a more rigorous synchronization of events between master and target unit(s), including the start of the ranging signal sequence.

[0067] In present invention the narrow bandwidth ranging signal multi-path mitigation processor does not increase the ranging signal bandwidth. It uses different frequency components, advantageously, to allow propagation of a narrow bandwidth ranging signal. Further ranging signal processing can be carried out in the frequency domain by way of employing super resolution spectrum estimation algorithms (MUSIC, rootMUSIC, ESPRIT) and/or statistical algorithms like RELAX, or in time-domain by assembling a synthetic ranging signal with a relatively large bandwidth and applying a further processing to this signal. The different frequency component of narrow bandwidth ranging signal can be pseudo randomly selected, it can also be contiguous or spaced apart in frequency, and it can have uniform and/or non-uniform spacing in frequency.

[0068] The embodiment expands multipath mitigation technology. The signal model for the narrowband ranging is a complex exponential (as introduced elsewhere in this document) whose frequency is directly proportional to the delay defined by the range plus similar terms whose delay is defined by the time delay related to the multipath. The model is independent of the actual implementation of the signal structure, e.g., stepped frequency, Linear Frequency Modulation, etc.

[0069] The frequency separation between the direct path and multipath is nominally extremely small and normal frequency domain processing is not sufficient to estimate the direct path range. For example a stepped frequency ranging signal at a 100 KHz stepping rate over 5 MHz at a range of 30 meters (100.07 nanoseconds delay) results in a frequency of 0.062875 radians/sec. A multipath reflection with a path length of 35 meters would result in a frequency of 0.073355. The separation is 0.0104792. Frequency resolution of the 50 sample observable has a native frequency resolution of 0.12566 Hz. Consequently it is not possible to use conventional frequency estimation techniques for the separation of the direct path from the reflected path and accurately estimate the direct path range.

[0070] To overcome this limitation the invention uses a unique combination of implementations of subspace decomposition high resolution spectral estimation methodologies and multimodal cluster analysis. The subspace decomposition technology relies on breaking the estimated covariance matrix of the observed data into two orthogonal subspaces, the noise subspace and the signal subspace. The theory behind the subspace decomposition methodology is that the projection of the observable onto the noise subspace consists of only the noise and the projection of the observable onto the signal subspace consists of only the signal.

[0071] The super resolution spectrum estimation algorithms and RELAX algorithm are capable of distinguishing closely placed frequencies (sinusoids) in spectrum in presence of noise. The frequencies do not have to be harmonically related and, unlike the Digital Fourier Transform (DFT), the signal model does not introduce any artificial periodicity. For a given bandwidth, these algorithms provide significantly higher resolution than Fourier Transform. Thus, the Direct Line Of Sight (DLOS) can be reliably distinguished from other multi-paths (MP) with high accuracy. Similarly, applying the thresholded method, which will be explained later, to the artificially produced synthetic wider bandwidth ranging signal makes it possible to reliably distinguish DLOS from other paths with high accuracy.

[0072] In accordance with the exemplary embodiment, the Digital signal processing (DSP), can be employed by the multi-path mitigation processor to reliably distinguish the DLOS from other MP paths. A variety of super-resolution algorithms/techniques exist in the spectral analysis (spectrum estimation) technology. Examples include subspace based methods: MUltiple SIgnal Characterization (MUSIC) algorithm or root-MUSIC algorithm, Estimation of Signal Parameters via Rotational Invariance Techniques (ESPRIT) algorithm, Pisarenko Harmonic Decomposition (PHD) algorithm, RELAX algorithm, etc.

[0073] In all of the abovementioned super-resolution algorithms the incoming (i.e., received) signal is modeled as a linear combination of complex exponentials and their complex amplitudes of frequencies. In case of a multi-path, the received signal will be as follows:

[0074] $\begin{array}{cc}r\left(t\right)=\beta \times e^{\mathrm{i2\pi }f\times t}\sum _{k=0}^{k=L-1}{\alpha }_k\times e^{-\mathrm{i2\pi }f\times {\tau }_{\kappa }},& \left(1\right)\end{array}$

[0075] where β×e^i2πf×t is the transmitted signal, f is the operating frequency, L is the number of multi-path components, and α_K=|α_K|×e^jθK and τ_K are the complex attenuation and propagation delay of the K-th path, respectively. The multi-path components are indexed so that the propagation delays are considered in ascending order. As a result, in this model τ₀ denotes the propagation delay of the DLOS path. Obviously, the τ₀ value is of the most interest, as it is the smallest value of all τ_K. The phase θ_K is normally assumed random from one measurement cycle to another with a uniform probability density function U (0,2π). Thus, we assume that α_K=const (i.e., constant value)

[0076] Parameters α_K and π_K are random time-variant functions reflecting motions of people and equipment in and around buildings. However, since the rate of their variations is very slow as compared to the measurement time interval, these parameters can be treated as time-invariant random variables within a given measurement cycle.

[0077] All these parameters are frequency-dependent since they are related to radio signal characteristics, such as, transmission and reflection coefficients. However, in the exemplary embodiment, the operating frequency changes very little. Thus, the above-mentioned parameters can be assumed frequency-independent.

[0078] Equation (1) can be presented in frequency domain as:

[0079] $\begin{array}{cc}A\left(f\right)=\sum _{k=0}^{k=L-1}{\alpha }_k\times e^{-i\left(2\pi \times {\tau }_{\kappa }\right)f},& \left(2\right)\end{array}$

[0080] where: A(f) is complex amplitude of the received signal, (2π×τ_K) are the artificial “frequencies” to be estimated by a super-resolution algorithm and the operating frequency f is the independent variable; α_K is the K-th path amplitude.

[0081] In the equation (2) the super-resolution estimation of (2π×τ_K) and subsequently τ_K values are based on continuous frequency. In practice, there is a finite number of measurements. Thus, the variable f will not be a continuous variable, but rather a discrete one. Accordingly, the complex amplitude A(f) can be calculated as follows:

[0082] $\begin{array}{cc}\hat{A}\left(f_n\right)=\sum _{k=0}^{k=L-1}{\alpha }_k\times e^{-i\left(2\pi \times {\tau }_{\kappa }\right)\times f_n},& \left(3\right)\end{array}$

[0083] where Â(f_n) are discrete complex amplitude estimates (i.e., measurements) at discrete frequencies f_n.

[0084] In equation (3) Â(f_n) can be interpreted as an amplitude and a phase of a sinusoidal signal of frequency f_n after it propagates through the multi-path channel. Note that all spectrum estimation based super-resolution algorithms require complex input data (i.e. complex amplitude).

[0085] In some cases, it is possible to convert real signal data, e.g. Re(Â(f_n)), into a complex signal (e.g., analytical signal). For example, such a conversion can be accomplished by using Hilbert transformation or other methods. However, in case of short distances the value τ₀ is very small, which results in very low (2π×τ_K) “frequencies”.

[0086] These low “frequencies” create problems with Hilbert transform (or other methods) i\ implementations. In addition, if only amplitude values (e.g., Re(Â(f_n))) are to be used, then the number of frequencies to be estimated will include not only the (2π×τ_K) “frequencies”, but also theirs combinations. As a rule, increasing the number of unknown frequencies impacts the accuracy of the super-resolution algorithms. Thus, reliable and accurate separation of DLOS path from other multi-path (MP) paths requires complex amplitude estimation.

[0087] The following is a description of a method and the multi-path mitigation processor operation during the task of obtaining complex amplitude Â(f_n) in presence of multi-path. Note that, while the description is focused on the half-duplex mode of operation, it can be easily extended for the full-duplex mode. The simplex mode of operation is a subset of the half-duplex mode, but would require additional events synchronization.

[0088] In half-duplex mode of operation the reader (often referred to as the “master”) and the tags (also referred to as “slaves” or “targets”) are controlled by a protocol that only allows the master or the slave to transmit at any given time. In this mode of operation the tags (target devices) serve as Transponders. The tags receive the ranging signal from a reader (master device), store it in the memory and then, after certain time (delay), re-transmit the signal back to the master.

[0089] An example of ranging signal is shown in FIG. 1 and FIG. 1A. The exemplary ranging signal employs different frequency components that are contiguous. Other waveforms, including pseudo random, spaced in frequency and/or time or orthogonal, etc. can be also used for as long as the ranging signal bandwidth remains narrow. In FIG. 1 the time duration T_f for every frequency component is long enough to obtain the ranging signal narrow-bandwidth property.

[0090] Another variation of a ranging signal with different frequency components is shown on FIG. 2. It includes multiple frequencies (f₁, f₂, f₃, f₄, f_n) transmitted over long period of time to make individual frequencies narrow-band. Such signal is more efficient, but it occupies in a wide bandwidth and a wide bandwidth ranging signal impacts the SNR, which, in turn, reduces the operating range. Also, such wide bandwidth ranging signal will violate FCC requirements on the VHF band or lower frequencies bands. However, in certain applications this wide-bandwidth ranging signal allows an easier integration into existing signal and transmission protocols. Also, such a signal decreases the track-locate time.

[0091] These multiple-frequency (f₁, f₂, f₃, f₄, f_n) bursts may be also contiguous and/or pseudo random, spaced in frequency and/or time or orthogonal, etc.

[0092] The narrowband ranging mode will produce the accuracy in the form of instantaneous wide band ranging while increasing the range at which this accuracy can be realized, compared to wide band ranging. This performance is achieved because at a fixed transmit power, the SNR (in the appropriate signal bandwidths) at the receiver of the narrow band ranging signal is greater than the SNR at the receiver of a wideband ranging signal. The SNR gain is on the order of the ratio of the total bandwidth of the wideband ranging signal and the bandwidth of each channel of the narrow band ranging signal. This provides a good trade-off when very rapid ranging is not required, e.g., for stationary and slow-moving targets, such as a person walking or running.

[0093] Master devices and Tag devices are identical and can operate either in Master or Transponder mode. All devices include data/remote control communication channels. The devices can exchange the information and master device(s) can remotely control tag devices. In this example depicted in FIG. 1 during an operation of a master (i.e., reader) multi-path mitigation processor originates the ranging signal to tag(s) and, after a certain delay, the master/reader receives the repeated ranging signal from the tag(s).

[0094] Thereafter, master's multi-path mitigation processor compares the received ranging signal with the one that was originally sent from the master and determines the Â(f_n) estimates in form of an amplitude and a phase for every frequency component f_n. Note that in the equation (3) Â(f_n) is defined for one-way ranging signal trip. In the exemplary embodiment the ranging signal makes a round-trip. In other words, it travels both ways: from a master/reader to a target/slave and from the target/slave back to the master/reader. Thus, this round-trip signal complex amplitude, which is received back by the master, can be calculated as follows: (4)
|Â_RT(f_n)|=|Â(f_n)|² and ∠Â_RT(f_n)=2×(∠Â(f_n))  (4)

[0095] There are many techniques available for estimating the complex amplitude and phase values, including, for example, matching filtering |Â(f_n)| and ∠Â(f_n). According to the exemplary embodiment, a complex amplitude determination is based on |Â(f_n)| values derived from the master and/or tag receiver RSSI (Received Signal Strength Indicator) values. The phase values ∠Â_RT(f_n) are obtained by comparing the received by a reader/master returned base-band ranging signal phase and the original (i.e., sent by reader/master) base band ranging signal phase. In addition, because master and tag devices have independent clock systems a detailed explanation of devices operation is augmented by analysis of the clock accuracy impact on the phase estimation error. As the above description shows, the one-way amplitude |Â(f_n)| values are directly obtainable from target/slave device. However, the one-way phase ∠Â(f_n) values cannot be measured directly.

[0096] In the exemplary embodiment, the ranging base band signal is the same as the one depicted in FIG. 1. However, for the sake of simplicity, it is assumed herein that the ranging base band signal consists of only two frequency components each containing multiple periods of cosine or sine waves of different frequency: F₁ and F₂. Note that F₁=f₁ and F₂=f₂. The number of periods in a first frequency component is L and the number of periods in a second frequency component is P. Note that L may or may not be equal to P, because for T_f=constant each frequency component can have different number of periods. Also, there is no time gap between each frequency component, and both F₁ and F₂ start from the initial phase equal to zero.

[0097] FIGS. 3A, 3B and 3C depict block diagrams of a master or a slave unit (tag) of an RF mobile tracking and locating system. F_OSC refers to the frequency of the device system clock (crystal oscillator 20 in FIG. 3A). All frequencies generated within the device are generated from this system clock crystal oscillator. The following definitions are used: M is a master device (unit); AM is a tag (target) device (unit). The tag device is operating in the transponder mode and is referred to as transponder (AM) unit.

[0098] In the preferred embodiment the device consists of the RF front-end and the RF back-end, base-band and the multi-path mitigation processor. The RF back-end, base-band and the multi-path mitigation processor are implemented in the FPGA 150 (see FIGS. 3B and 3C). The system clock generator 20 (see FIG. 3A) oscillates at: F_OSC=20 MHz; or ω_OSC=2π×20×10⁶. This is an ideal frequency because in actual devices the system clocks frequencies are not always equal to 20 MHz: F_OSC^M=F_OSCγ^M; F_OSC^AM=F_OSCγ^AM.

[0099] Note that

[0100] ${\gamma }^M=\frac{F_{\mathrm{OSC}}^M}{F_{\mathrm{OSC}}},{\gamma }^{\mathrm{AM}}=\frac{F_{\mathrm{OSC}}^{\mathrm{AM}}}{F_{\mathrm{OSC}}},\mathrm{and}{\beta }^M=\frac{1}{{\gamma }^M},{\beta }^{\mathrm{AM}}=\frac{1}{{\gamma }^{\mathrm{AM}}}$

[0101] It should be noted that other than 20 MHz F_OSC frequencies can be used without any impact on system performance.

[0102] Both units' (master and tag) electronic makeup is identical and the different modes of operations are software programmable. The base band ranging signal is generated in digital format by the master' FPGA 150, blocks 155-180 (see FIG. 2B). It consists of two frequency components each containing multiple periods of cosine or sine waves of different frequency. At the beginning, t=0, the FPGA 150 in a master device (FIG. 3B) outputs the digital base-band ranging signal to its up-converter 50 via I/Q DACs 120 and 125. The FPGA 150 starts with F₁ frequency and after time T₁ start generating F₂ frequency for time duration of T₂.

[0103] Since crystal oscillator's frequency might differ from 20 MHz the actual frequencies generated by the FPGA will be F₁γ^M and F₂γ^M. Also, time T₁ will be T₁β^M and T₂ will be T₂β^M. IT is also assumed that T₁, T₂, F₁, F₂ are such that F₁γ^M*T₁β^M=F₁T₁ and F₂γ^M*T₂β^M=F₂T₂, where both F₁T₁ & F₂T₂ are integer numbers. That means that the initial phases of F₁ and F₂ are equal to zero.

[0104] Since all frequencies are generated from the system crystal oscillator 20 clocks, the master' base-band I/Q DAC(s) 120 and 125 outputs are as follows: F₁=γ^M 20×10⁶×K_F1 and F₂=γ^M 20×10⁶ ×K_F2, where K_F1 and K_F2 are constant coefficients. Similarly, the output frequencies TX_LO and RX_LO from frequency synthesizer 25 (LO signals for mixers 50 and 85) can be expressed through constant coefficients. These constant coefficients are the same for the master (M) and the transponder (AM)—the difference is in the system crystal oscillator 20 clock frequency of each device.

[0105] The master (M) and the transponder (AM) work in a half-duplex mode. Master's RF front-end up-converts the base-band ranging signal, generated by the multi-path mitigation processor, using quadrature up-converter (i.e., mixer) 50 and transmits this up-converted signal. After the base-band signal is transmitted the master switches from TX to RX mode using RF Front-end TX/RX Switch 15. The transponder receives and down-converts the received signal back using its RF Front-end mixer 85 (producing First IF) and ADC 140 (producing Second IF).

[0106] Thereafter, this second IF signal is digitally filtered in the Transponder RF back-end processor using digital filters 190 and further down-converted to the base-band ranging signal using the RF back-end quadrature mixer 200, digital I/Q filters 210 and 230, a digital quadrature oscillator 220 and a summer 270. This base-band ranging signal is stored in the transponder's memory 170 using Ram Data Bus Controller 195 and control logic 180.

[0107] Subsequently, the transponder switches from RX to TX mode using RF front-end switch 15 and after certain delay t_RTX begins re-transmitting the stored base-band signal. Note that the delay is measured in the AM (transponder) system clock. Thus, t_RTX^AM=t_RTXβ^AM. The master receives the transponder transmission and down-converts the received signal back to the base-band signal using its RF back-end quadrature mixer 200, the digital I and Q filters 210 and 230, the digital quadrature oscillator 220 (see FIG. 3C).

[0108] Thereafter, the master calculates the phase difference between F₁ and F₂ in the received (i.e., recovered) base-band signal using multi-path mitigation processor arctan block 250 and phase compare block 255. The amplitude values are derived from the RF back-end RSSI block 240.

[0109] For improving the estimation accuracy it is always desirable to improve the SNR of the amplitude estimates from block 240 and phase difference estimates from block 255. In the preferred embodiment the multi-path mitigation processor calculates amplitude and phase difference estimates for many time instances over the ranging signal frequency component duration (T_f). These values, when averaged, improve SNR. The SNR improvement can be in an order that is proportional to √{square root over (N)}, where N is a number of instances when amplitude and phase difference values were taken (i.e., determined).

[0110] Another approach to the SNR improvement is to determine amplitude and phase difference values by applying matching filter techniques over a period of time. Yet, another approach would be to estimate the phase and the amplitude of the received (i.e., repeated) base band ranging signal frequency components by sampling them and integrating over period T≦T_f against the original (i.e., sent by the master/reader) base-band ranging signal frequency components in the I/Q form. The integration has the effect of averaging of multiple instances of the amplitude and the phase in the I/Q format. Thereafter, the phase and the amplitude values can be translated from the I/Q format to the |Â(f_n)| and ∠Â(f_n) format.

[0111] Let's assume that at t=0 under master' multi-path processor control the master base-band processor (both in FPGA 150) start the base-band ranging sequence.
φ_FPGA^M(t)=γ^M×ω_OSC×(K_F1(t)),t<T₁β^M,t<T₁β^M;
φ_FPGA^M(t)=γ^M×ω_OSC×(K_f1(T₁β^M)+K_F2(t−T₁β^M)),t>T₁β^M,
where T_f≧T₁β^M.
The phase at master's DAC(s) 120 and 125 outputs are as follows:
φ_DAC^M(t)=γ^M×ω_OSC×(K_F1(t−t_DAC^M))+φ_DAC^M(0),t<T₁β^M+t_DAC^M;
φ_DAC^M(t)=γ^M×ω_OSC×(K_F1(T₁β^M)+K_F2(t−T₁β^M−t_DAC^M))+φ_DAC^M(0),t>T₁β^M+t_DAC^M
Note that DACs 120 and 125 have internal propagation delay, t_DAC^M, that does not depend upon the system clock.

[0112] Similarly, the transmitter circuitry components 15, 30, 40 and 50 will introduce additional delay, t_TX^M, that does not depend upon the system clock.

[0113] As a result, the phase of the transmitted RF signal by the master can be calculated as follows:
φ_RF^M(t)=γ^M×ω_OSC×(K_F1(t−t_DAC^M−t_TX^M)+K_SYN—TX(t−t_TX^M))+φ_DAC^M(0)+φ_SYN—TX^M(0), t<T₁β^M+t_DAC^M+t_TX^M;
φ_RF^M(t)=γ^M×ω_OSC×(K_F1(T₁β^M)+K_F2(t−T₁β^M−t_DAC−t_TX^M)+K_SYN—TX(t−t_TX^M))+φ_DAC^M(0)+φ_SYN—TX^M(0), t>T₁β^M+t_DAC^M+t_TX^M

[0114] The RF signal from the master (M) experiences a phase shift φ^MULT that is a function of the multi-path phenomena between the master and tag.

[0115] The φ^MULT values depend upon the transmitted frequencies, e.g. F₁ and F₂. The transponder (AM) receiver' is not able to resolve each path because of limited (i.e., narrow) bandwidth of the RF portion of the receiver. Thus, after a certain time, for example, 1 microsecond (equivalent to ˜300 meters of flight), when all reflected signals have arrived at the receiver antenna, the following formulas apply:
φ_ANT^AM(t)=γ^M×ω_OSC×(K_F1(t−t_DAC^M−t_TX^M)+K_SYN—TX(t−t_TX^M))+φ_F1^MULT+φ_DAC^M(0)+φ_SYN—TX^M(0), 10⁻⁶<t<T₁β^M+t_DAC^M+t_TX^M;
φ_ANT^AM(t)=γ^M×ω_OSC×(K_F1(T₁β^M)+K_F2(t−T₁β^M−t_DAC^M−t_TX^M)+K_SYN—TX(t−t_TX^M))+φ_F2^MULT+φ_DAC^M(0)+φ_SYN—TX^M(0), t>T₁β^M+t_DAC^M+t_TX^M+10⁻⁶

[0116] In the AM (transponder) receiver at the first down converter, element 85, an output, e.g. first IF, the phase of the signal is as follows:
φ_IF—1^AM(t)=γ^M×ω_OSC×(K_F1(t−t_DAC^M−t_TX^M−t_RX^AM)+K_SYN—TX(t−t_TX^M−t_RX^AM))−γ^AM×ω_OSC×(K_SYN—RX—1t)+φ_F1^MULT+φ_SYN—^M(0)−φ_SYN—RX—1^AM(0),10⁻⁶<t<T₁β^M+t_DAC^M+t_TX^M+t_TX^AM;
φ_IF—1^AM(t)=γ^M×ω_OSC×(K_F1(T₁β^M)+K_f2(t−T₁β^M−t_DAC^M−t_TX^M−t_TX^AM)+K_SYN—TX(t−t_TX^M−t_RX^AM))−γ^AM×ω_OSC×(K_SYN—RX—1t)+φ_F2^MULT+φ_SYN—TX^M(0)−φ_SYN—RX—1^AM(0),t>T₁β^M+t_DAC^M+t_TX^M+t_RX^AM+10⁻⁶

[0117] Note that the propagation delay t in the receiver RF section (elements 15 and 60-85) does not depend upon the system clock. After passing through RF Front-end filters and amplifiers (elements 95-110 and 125) the first IF signal is sampled by the RF Back-end ADC 140. It is assumed that ADC 140 is under-sampling the input signal (e.g., first IF). Thus, the ADC also acts like a down-converter producing the second IF. The first IF filters, amplifiers and the ADC add propagation delay time. At the ADC output (second IF):

[0118] ${\phi }_{\mathrm{ADC}}^{\mathrm{AM}}\left(t\right)={\gamma }^M\times {\omega }_{\mathrm{OSC}}\times \left(K_{F_1}\left(t-t_{\mathrm{DAC}}^M-t_{\mathrm{TX}}^M-t_{\mathrm{RX}}^{\mathrm{AM}}-t_{\mathrm{IF\_}1}^{\mathrm{AM}}-t_{\mathrm{ADC}}^{\mathrm{AM}}\right)+K_{\mathrm{SYN\_TX}}\left(t-t_{\mathrm{TX}}^M-t_{\mathrm{RX}}^{\mathrm{AM}}-t_{\mathrm{IF\_}1}^{\mathrm{AM}}-t_{\mathrm{ADC}}^{\mathrm{AM}}\right)\right)-{\gamma }^{\mathrm{AM}}\times {\omega }_{\mathrm{OSC}}\times \left(K_{\mathrm{SYN\_RX}\_1}\left(t-t_{\mathrm{IF\_}1}^{\mathrm{AM}}-t_{\mathrm{ADC}}^{\mathrm{AM}}\right)+K_{\mathrm{ADC}}\left(t\right)\right)+{\phi }_{F_1}^{\mathrm{MULT}}+{\phi }_{\mathrm{SYN\_TX}}^M\left(0\right)-{\phi }_{\mathrm{SYN\_RX}\_1}^{\mathrm{AM}}\left(0\right)-{\phi }_{\mathrm{ADC\_CLK}}^{\mathrm{AM}}\left(0\right),{10}^{-6}<t<T_1{\beta }^M+t_{\mathrm{DAC}}^M+t_{\mathrm{TX}}^M+t_{\mathrm{RX}}^{\mathrm{AM}}+t_{\mathrm{IF\_}1}^{\mathrm{AM}}+t_{\mathrm{ADC}}^{\mathrm{AM}};$ ${\phi }_{\mathrm{ADC}}^{\mathrm{AM}}\left(t\right)={\gamma }^M\times {\omega }_{\mathrm{OSC}}\times \left(K_{F_1}\left(T_1{\beta }^M\right)+K_{F_2}\left(t-T_1{\beta }^M-t_{\mathrm{DAC}}^M-t_{\mathrm{TX}}^M-t_{\mathrm{RX}}^{\mathrm{AM}}-t_{\mathrm{IF\_}1}^{\mathrm{AM}}-t_{\mathrm{ADC}}^{\mathrm{AM}}\right)+K_{\mathrm{SYN\_TX}}\left(t-t_{\mathrm{TX}}^M-t_{\mathrm{RX}}^{\mathrm{AM}}-t_{\mathrm{IF\_}1}^{\mathrm{AM}}-t_{\mathrm{ADC}}^{\mathrm{AM}}\right)\right)-{\gamma }^{\mathrm{AM}}\times {\omega }_{\mathrm{OSC}}\times \left(K_{\mathrm{SYN\_RX}\_1}\left(t-t_{\mathrm{IF\_}1}^{\mathrm{AM}}-t_{\mathrm{ADC}}^{\mathrm{AM}}\right)+K_{\mathrm{ADC}}\left(t\right)\right)+{\phi }_{F2}^{\mathrm{MULT}}+{\phi }_{\mathrm{SYN\_TX}}^M\left(0\right)-{\phi }_{\mathrm{SYN\_RX}\_1}^{\mathrm{AM}}\left(0\right)-{\phi }_{\mathrm{ADC\_CLK}}^{\mathrm{AM}}\left(0\right),t>T_1{\beta }^M+t_{\mathrm{DAC}}^M+t_{\mathrm{TX}}^M+t_{\mathrm{RX}}^{\mathrm{AM}}+t_{\mathrm{IF\_}1}^{\mathrm{AM}}+t_{\mathrm{ADC}}^{\mathrm{AM}}+{10}^{-6}$

[0119] In the FPGA 150 the second IF signal (from the ADC output) is filtered by the RF Back-end digital filters 190 and further down-converted back to base-band ranging signal by the third down-converter (i.e., quadrature mixer 200, digital filters 230 and 210 and digital quadrature oscillator 220), summed in the summer 270 and is stored in the memory 170. At the third down-converter output (i.e., quadrature mixer):

[0120] ${\phi }_{\mathrm{BB}}^{\mathrm{AM}}\left(t\right)={\gamma }^M\times {\omega }_{\mathrm{OSC}}\times \left(K_{F_1}\left(t-t_{\mathrm{DAC}}^M-t_{\mathrm{TX}}^M-t_{\mathrm{RX}}^{\mathrm{AM}}+t_{\mathrm{IF}\_1}^{\mathrm{AM}}+t_{\mathrm{ADC}}^{\mathrm{AM}}-t_{\mathrm{FIR}}{\beta }^{\mathrm{AM}}\right)+K_{\mathrm{SYN\_TX}}\left(t-t_{\mathrm{TX}}^M-t_{\mathrm{RX}}^{\mathrm{AM}}-t_{\mathrm{IF}\_1}^{\mathrm{AM}}-t_{\mathrm{ADC}}^{\mathrm{AM}}-t_{\mathrm{FIR}}{\beta }^{\mathrm{AM}}\right)\right)-{\gamma }^{\mathrm{AM}}\times {\omega }_{\mathrm{OSC}}\times \left(K_{\mathrm{SYN\_RX}\_1}\left(t-t_{\mathrm{IF}\_1}^{\mathrm{AM}}-t_{\mathrm{ADC}}^{\mathrm{AM}}-t_{\mathrm{FIR}}{\beta }^{\mathrm{AM}}\right)+K_{\mathrm{ADC}}\left(t-t_{\mathrm{FIR}}{\beta }^{\mathrm{AM}}\right)+K_{\mathrm{SYN\_RX}\_2}t\right)+{\phi }_{F_1}^{\mathrm{MULT}}+{\phi }_{\mathrm{SYN\_TX}}^M\left(0\right)-{\phi }_{\mathrm{SYN\_RX}\_1}^{\mathrm{AM}}\left(0\right)-{\phi }_{\mathrm{ADC\_CLK}}^{\mathrm{AM}}\left(0\right)-{\phi }_{\mathrm{SYN\_RX}\_2}^{\mathrm{AM}}\left(0\right),{10}^{-6}<t<T_1{\beta }^M+t_{\mathrm{DAC}}^M+t_{\mathrm{TX}}^M+t_{\mathrm{IF}\_1}^{\mathrm{AM}}+t_{\mathrm{ADC}}^{\mathrm{AM}}+t_{\mathrm{FIR}}{\beta }^{\mathrm{AM}};$ ${\phi }_{\mathrm{BB}}^{\mathrm{AM}}\left(t\right)={\gamma }^M\times {\omega }_{\mathrm{OSC}}\times \left(K_{F_1}\left(T_1{\beta }^M\right)+K_{F_2}\left(t-T_1{\beta }^M-t_{\mathrm{DAC}}^M-t_{\mathrm{TX}}^M-t_{\mathrm{RX}}^{\mathrm{AM}}-t_{\mathrm{IF}\_1}^{\mathrm{AM}}-t_{\mathrm{ADC}}^{\mathrm{AM}}-t_{\mathrm{FIR}}{\beta }^{\mathrm{AM}}\right)+K_{\mathrm{SYN\_TX}}\left(t-t_{\mathrm{TX}}^M-t_{\mathrm{RX}}^{\mathrm{AM}}-t_{\mathrm{IF}\_1}^{\mathrm{AM}}-t_{\mathrm{ADC}}^{\mathrm{AM}}-t_{\mathrm{FIR}}{\beta }^{\mathrm{AM}}\right)\right)-{\gamma }^{\mathrm{AM}}\times {\omega }_{\mathrm{OSC}}\times \left(K_{\mathrm{SYN\_RX}\_1}\left(t-t_{\mathrm{IF}\_1}^{\mathrm{AM}}-t_{\mathrm{ADC}}^{\mathrm{AM}}-t_{\mathrm{FIR}}{\beta }^{\mathrm{AM}}\right)+K_{\mathrm{ADC}}\left(t-t_{\mathrm{FIR}}{\beta }^{\mathrm{AM}}\right)+K_{\mathrm{SYN\_RX}\_2}t\right)+{\phi }_{F_2}^{\mathrm{MULT}}+{\phi }_{\mathrm{SYN\_TX}}^M\left(0\right)-{\phi }_{\mathrm{SYN\_RX}\_1}^{\mathrm{AM}}\left(0\right)-{\phi }_{\mathrm{ADC\_CLK}}^{\mathrm{AM}}\left(0\right)-{\phi }_{\mathrm{SYN\_RX}\_2}^{\mathrm{AM}}\left(0\right),t>T_1{\beta }^M+t_{\mathrm{DAC}}^M+t_{\mathrm{TX}}^M+t_{\mathrm{RX}}^{\mathrm{AM}}+t_{\mathrm{IF}\_1}^{\mathrm{AM}}+t_{\mathrm{ADC}}^{\mathrm{AM}}+t_{\mathrm{FIR}}{\beta }^{\mathrm{AM}}+{10}^{-6}$

[0121] Note that propagation delay t_FIR^AM=t_FIRβ^AM in the FIR section 190 does not depend upon the system clock.

[0122] After RX->TX delay the stored (in memory 170) base-band ranging signal from the master (M) is retransmitted. Note that RX->TX delay t_RTX^AM=t_RTXβ^AM.

[0123] ${\phi }_{\mathrm{RF}}^{\mathrm{AM}}\left(t\right)={\gamma }^M\times {\omega }_{\mathrm{OSC}}\times \left(K_{F_1}\left(t-t_{\mathrm{DAC}}^M-t_{\mathrm{TX}}^M-t_{\mathrm{RX}}^{\mathrm{AM}}-t_{\mathrm{IF}\_1}^{\mathrm{AM}}-t_{\mathrm{ADC}}^{\mathrm{AM}}-t_{\mathrm{FIR}}{\beta }^{\mathrm{AM}}-t_{\mathrm{RTX}}{\beta }^{\mathrm{AM}}-t_{\mathrm{RTX}}{\beta }^{\mathrm{AM}}-t_{\mathrm{DAC}}^{\mathrm{AM}}-t_{\mathrm{TX}}^{\mathrm{AM}}\right)+K_{\mathrm{SYN\_TX}}\left(t-t_{\mathrm{TX}}^M-t_{\mathrm{RX}}^{\mathrm{AM}}-t_{\mathrm{IF}\_1}^{\mathrm{AM}}-t_{\mathrm{ADC}}^{\mathrm{AM}}-t_{\mathrm{FIR}}{\beta }^{\mathrm{AM}}-t_{\mathrm{RTX}}{\beta }^{\mathrm{AM}}-t_{\mathrm{DAC}}^{\mathrm{AM}}-t_{\mathrm{TX}}^{\mathrm{AM}}\right)\right)-{\gamma }^{\mathrm{AM}}\times {\omega }_{\mathrm{OSC}}\times \left(K_{\mathrm{SYN\_RX}\_1}\left(t-t_{\mathrm{IF}\_1}^{\mathrm{AM}}-t_{\mathrm{ADC}}^{\mathrm{AM}}-t_{\mathrm{FIR}}{\beta }^{\mathrm{AM}}-t_{\mathrm{RTX}}{\beta }^{\mathrm{AM}}-t_{\mathrm{DAC}}^{\mathrm{AM}}-t_{\mathrm{TX}}^{\mathrm{AM}}\right)+K_{\mathrm{ADC}}\left(t-t_{\mathrm{FIR}}{\beta }^{\mathrm{AM}}-t_{\mathrm{RTX}}{\beta }^{\mathrm{AM}}-t_{\mathrm{DAC}}^{\mathrm{AM}}-t_{\mathrm{TX}}^{\mathrm{AM}}\right)+K_{\mathrm{SYN\_RX}\_2}\left(t-t_{\mathrm{RTX}}{\beta }^{\mathrm{AM}}-t_{\mathrm{DAC}}^{\mathrm{AM}}-t_{\mathrm{TX}}^{\mathrm{AM}}\right)-K_{\mathrm{SYN\_TX}}\left(t-t_{\mathrm{TX}}^{\mathrm{AM}}\right)\right)+{\phi }_{F_1}^{\mathrm{MULT}}+{\phi }_{\mathrm{SYN\_TX}}^M\left(0\right)-{\phi }_{\mathrm{SYN\_RX}\_1}^{\mathrm{AM}}\left(0\right)-{\phi }_{\mathrm{ADC\_CLK}}^{\mathrm{AM}}\left(0\right)-{\phi }_{\mathrm{SYN\_RX}\_2}^{\mathrm{AM}}\left(0\right)+{\phi }_{\mathrm{SYN\_TX}}^{\mathrm{AM}}\left(0\right),{10}^{-6}<t<T_1{\beta }^M+t_{\mathrm{DAC}}^M+t_{\mathrm{TX}}^M+t_{\mathrm{RX}}^{\mathrm{AM}}+t_{\mathrm{IF}\_1}^{\mathrm{AM}}+t_{\mathrm{ADC}}^{\mathrm{AM}}+t_{\mathrm{FIR}}{\beta }^{\mathrm{AM}}+t_{\mathrm{RTX}}{\beta }^{\mathrm{AM}}+t_{\mathrm{DAC}}^{\mathrm{AM}}+t_{\mathrm{TX}}^{\mathrm{AM}};$ ${\phi }_{\mathrm{RF}}^{\mathrm{AM}}\left(t\right)={\gamma }^M\times {\omega }_{\mathrm{OSC}}\times \left(K_{F_1}\left(T_1{\beta }^M\right)+K_{F_2}\left(t-T_1{\beta }^M-t_{\mathrm{DAC}}^M-t_{\mathrm{TX}}^M-t_{\mathrm{RX}}^{\mathrm{AM}}-t_{\mathrm{IF}\_1}^{\mathrm{AM}}-t_{\mathrm{ADC}}^{\mathrm{AM}}-t_{\mathrm{FIR}}{\beta }^{\mathrm{AM}}-t_{\mathrm{RTX}}{\beta }^{\mathrm{AM}}-t_{\mathrm{DAC}}^{\mathrm{AM}}-t_{\mathrm{TX}}^{\mathrm{AM}}\right)+K_{\mathrm{SYN\_TX}}\left(t-t_{\mathrm{TX}}^M-t_{\mathrm{RX}}^{\mathrm{AM}}-t_{\mathrm{IF}\_1}^{\mathrm{AM}}-t_{\mathrm{ADC}}^{\mathrm{AM}}-t_{\mathrm{FIR}}{\beta }^{\mathrm{AM}}-t_{\mathrm{RTX}}{\beta }^{\mathrm{AM}}-t_{\mathrm{DAC}}^{\mathrm{AM}}-t_{\mathrm{TX}}^{\mathrm{AM}}\right)\right)-{\gamma }^{\mathrm{AM}}\times {\omega }_{\mathrm{OSC}}\times \left(K_{\mathrm{SYN\_RX}\_1}\left(t-t_{\mathrm{IF}\_1}^{\mathrm{AM}}-t_{\mathrm{ADC}}^{\mathrm{AM}}-t_{\mathrm{FIR}}{\beta }^{\mathrm{AM}}-t_{\mathrm{RTX}}{\beta }^{\mathrm{AM}}-t_{\mathrm{DAC}}^{\mathrm{AM}}-t_{\mathrm{TX}}^{\mathrm{AM}}\right)+K_{\mathrm{ADC}}\left(t-t_{\mathrm{FIR}}{\beta }^{\mathrm{AM}}-t_{\mathrm{RTX}}{\beta }^{\mathrm{AM}}-t_{\mathrm{DAC}}^{\mathrm{AM}}-t_{\mathrm{TX}}^{\mathrm{AM}}\right)+K_{\mathrm{SYN\_RX}\_2}\left(t-t_{\mathrm{RTX}}{\beta }^{\mathrm{AM}}-t_{\mathrm{DAC}}^{\mathrm{AM}}-t_{\mathrm{TX}}^{\mathrm{AM}}\right)-K_{\mathrm{SYN\_TX}}\left(t-t_{\mathrm{TX}}^{\mathrm{AM}}\right)\right)+{\phi }_{F_2}^{\mathrm{MULT}}+{\phi }_{\mathrm{SYN\_TX}}^M\left(0\right)-{\phi }_{\mathrm{SYN\_RX}\_1}^{\mathrm{AM}}\left(0\right)-{\phi }_{\mathrm{ADC\_CLK}}^{\mathrm{AM}}\left(0\right)-{\phi }_{\mathrm{SYN\_RX}\_2}^{\mathrm{AM}}\left(0\right)+{\phi }_{\mathrm{SYN\_TX}}^{\mathrm{AM}}\left(0\right),t>T_1{\beta }^M+t_{\mathrm{DAC}}^M+t_{\mathrm{TX}}^M+t_{\mathrm{RX}}^{\mathrm{AM}}+t_{\mathrm{IF}\_1}^{\mathrm{AM}}+t_{\mathrm{ADC}}^{\mathrm{AM}}+t_{\mathrm{FIR}}{\beta }^{\mathrm{AM}}+t_{\mathrm{RTX}}{\beta }^{\mathrm{AM}}+t_{\mathrm{DAC}}^{\mathrm{AM}}+t_{\mathrm{TX}}^{\mathrm{AM}}+{10}^{-6}$

[0124] By the time the signal from the transponder reaches the master' (M) receiver antenna MULT the RF signal from transponder (AM) experiences another phase shift φ^MULT that is a function of the multi-path. As discussed above, this phase shift happens after a certain time period when all reflected signals have arrived at the master' receiver antenna:

[0125] ${\phi }_{\mathrm{ANT}}^M\left(t\right)={\gamma }^M\times {\omega }_{\mathrm{OSC}}\times \left(K_{F_1}\left(t-t_{\mathrm{DAC}}^M-t_{\mathrm{TX}}^M-t_{\mathrm{RX}}^{\mathrm{AM}}-t_{\mathrm{IF\_}1}^{\mathrm{AM}}-t_{\mathrm{ADC}}^{\mathrm{AM}}-t_{\mathrm{FIR}}{\beta }^{\mathrm{AM}}-t_{\mathrm{RTX}}{\beta }^{\mathrm{AM}}-t_{\mathrm{DAC}}^{\mathrm{AM}}-t_{\mathrm{TX}}^{\mathrm{AM}}\right)+K_{\mathrm{SYN\_TX}}\left(t-t_{\mathrm{TX}}^M-t_{\mathrm{RX}}^{\mathrm{AM}}-t_{\mathrm{IF\_}1}^{\mathrm{AM}}-t_{\mathrm{ADC}}^{\mathrm{AM}}-t_{\mathrm{FIR}}{\beta }^{\mathrm{AM}}-t_{\mathrm{RTX}}{\beta }^{\mathrm{AM}}-t_{\mathrm{DAC}}^{\mathrm{AM}}-t_{\mathrm{TX}}^{\mathrm{AM}}\right)\right)-{\gamma }^{\mathrm{AM}}\times {\omega }_{\mathrm{OSC}}\times \left(K_{\mathrm{SYN\_RX}\_1}\left(t-t_{\mathrm{IF\_}1}^{\mathrm{AM}}-t_{\mathrm{ADC}}^{\mathrm{AM}}-t_{\mathrm{FIR}}{\beta }^{\mathrm{AM}}-t_{\mathrm{DAC}}^{\mathrm{AM}}-t_{\mathrm{TX}}^{\mathrm{AM}}\right)+K_{\mathrm{ADC}}\left(t-t_{\mathrm{FIR}}{\beta }^{\mathrm{AM}}-t_{\mathrm{RTX}}{\beta }^{\mathrm{AM}}-t_{\mathrm{DAC}}^{\mathrm{AM}}-t_{\mathrm{TX}}^{\mathrm{AM}}\right)+K_{\mathrm{SYN\_RX}\_2}\left(t-t_{\mathrm{RTX}}{\beta }^{\mathrm{AM}}-t_{\mathrm{DAC}}^{\mathrm{AM}}-t_{\mathrm{TX}}^{\mathrm{AM}}\right)-K_{\mathrm{SYN\_TX}}\left(t-t_{\mathrm{TX}}^{\mathrm{AM}}\right)\right)+2\times {\phi }_{F_1}^{\mathrm{MULT}}+{\phi }_{\mathrm{SYN\_TX}}^M\left(0\right)-{\phi }_{\mathrm{SYN\_RX}\_1}^{\mathrm{AM}}\left(0\right)-{\phi }_{\mathrm{ADC\_CLK}}^{\mathrm{AM}}\left(0\right)-{\phi }_{\mathrm{SYN\_RX}\_2}^{\mathrm{AM}}\left(0\right)+{\phi }_{\mathrm{SYN\_TX}}^{\mathrm{AM}}\left(0\right),2\times {10}^{-6}<t<T_1{\beta }^M+t_{\mathrm{DAC}}^M+t_{\mathrm{TX}}^M+t_{\mathrm{RX}}^{\mathrm{AM}}+t_{\mathrm{IF\_}1}^{\mathrm{AM}}+t_{\mathrm{ADC}}^{\mathrm{AM}}+t_{\mathrm{FIR}}{\beta }^{\mathrm{AM}}+t_{\mathrm{RTX}}{\beta }^{\mathrm{AM}}+t_{\mathrm{DAC}}^{\mathrm{AM}}+t_{\mathrm{TX}}^{\mathrm{AM}};$ ${\phi }_{\mathrm{ANT}}^M\left(t\right)={\gamma }^M\times {\omega }_{\mathrm{OSC}}\times \left(K_{F_1}\left(T_1{\beta }^M\right)+K_{F_2}\left(t-T_1{\beta }^M-t_{\mathrm{DAC}}^M-t_{\mathrm{TX}}^M-t_{\mathrm{RX}}^{\mathrm{AM}}-t_{\mathrm{IF\_}1}^{\mathrm{AM}}-t_{\mathrm{ADC}}^{\mathrm{AM}}-t_{\mathrm{FIR}}{\beta }^{\mathrm{AM}}-t_{\mathrm{RTX}}{\beta }^{\mathrm{AM}}-t_{\mathrm{DAC}}^{\mathrm{AM}}-t_{\mathrm{TX}}^{\mathrm{AM}}\right)+K_{\mathrm{SYN\_TX}}\left(t-t_{\mathrm{TX}}^M-t_{\mathrm{IF\_}1}^{\mathrm{AM}}-t_{\mathrm{ADC}}^{\mathrm{AM}}-t_{\mathrm{FIR}}{\beta }^{\mathrm{AM}}-t_{\mathrm{RTX}}{\beta }^{\mathrm{AM}}-t_{\mathrm{DAC}}^{\mathrm{AM}}-t_{\mathrm{TX}}^{\mathrm{AM}}\right)\right)-{\gamma }^{\mathrm{AM}}\times {\omega }_{\mathrm{OSC}}\times \left(K_{\mathrm{SYN\_RX}\_1}\left(t-t_{\mathrm{IF\_}1}^{\mathrm{AM}}-t_{\mathrm{ADC}}^{\mathrm{AM}}-t_{\mathrm{FIR}}{\beta }^{\mathrm{AM}}-t_{\mathrm{RTX}}{\beta }^{\mathrm{AM}}-t_{\mathrm{DAC}}^{\mathrm{AM}}-t_{\mathrm{TX}}^{\mathrm{AM}}\right)+K_{\mathrm{ADC}}\left(t-t_{\mathrm{FIR}}{\beta }^{\mathrm{AM}}-t_{\mathrm{RTX}}{\beta }^{\mathrm{AM}}-t_{\mathrm{DAC}}^{\mathrm{AM}}-t_{\mathrm{TX}}^{\mathrm{AM}}\right)+K_{\mathrm{SYN\_RX}\_2}\left(t-t_{\mathrm{RTX}}{\beta }^{\mathrm{AM}}-t_{\mathrm{DAC}}^{\mathrm{AM}}-t_{\mathrm{TX}}^{\mathrm{AM}}\right)-K_{\mathrm{SYN\_TX}}\left(t-t_{\mathrm{TX}}^{\mathrm{AM}}\right)\right)+2\times {\phi }_{F_2}^{\mathrm{MULT}}+{\phi }_{\mathrm{SYN\_TX}}^M\left(0\right)-{\phi }_{\mathrm{SYN\_RX}\_1}^{\mathrm{AM}}\left(0\right)-{\phi }_{\mathrm{ADC\_CLK}}^{\mathrm{AM}}\left(0\right)-{\phi }_{\mathrm{SYN\_RX}\_2}^{\mathrm{AM}}\left(0\right)+{\phi }_{\mathrm{SYN\_TX}}^{\mathrm{AM}}\left(0\right),t>T_1{\beta }^M+t_{\mathrm{DAC}}^M+t_{\mathrm{TX}}^M+t_{\mathrm{RX}}^{\mathrm{AM}}+t_{\mathrm{IF\_}1}^{\mathrm{AM}}+t_{\mathrm{ADC}}^{\mathrm{AM}}+t_{\mathrm{FIR}}{\beta }^{\mathrm{AM}}+t_{\mathrm{RTX}}{\beta }^{\mathrm{AM}}+t_{\mathrm{DAC}}^{\mathrm{AM}}+t_{\mathrm{TX}}^{\mathrm{AM}}+2\times {10}^{-6}$

[0126] In the master receiver the signal from transponder goes through the same down-conversion process as in the transponder receiver. The result is the recovered base-band ranging signal that was originally sent by the master.

For the first frequency component F₁:

[0127] ${\phi }_{\mathrm{BB\_RECOV}}^M\left(t\right)={\gamma }^M\times {\omega }_{\mathrm{OSC}}\times (K_{F_1}\left(t-t_{\mathrm{DAC}}^M-t_{\mathrm{TX}}^M-t_{\mathrm{RX}}^{\mathrm{AM}}-t_{\mathrm{IF\_}1}^{\mathrm{AM}}-t_{\mathrm{ADC}}^{\mathrm{AM}}-t_{\mathrm{FIR}}{\beta }^{\mathrm{AM}}-t_{\mathrm{RTX}}{\beta }^{\mathrm{AM}}-t_{\mathrm{DAC}}^{\mathrm{AM}}-t_{\mathrm{TX}}^{\mathrm{AM}}-t_{\mathrm{RX}}^M-t_{\mathrm{IF\_}1}^M-t_{\mathrm{ADC}}^M-t_{\mathrm{FIR}}{\beta }^M\right)+K_{\mathrm{SYN\_TX}}\left(t-t_{\mathrm{TX}}^M-t_{\mathrm{RX}}^{\mathrm{AM}}-t_{\mathrm{IF\_}1}^{\mathrm{AM}}-t_{\mathrm{ADC}}^{\mathrm{AM}}-t_{\mathrm{FIR}}{\beta }^{\mathrm{AM}}-t_{\mathrm{RTX}}{\beta }^{\mathrm{AM}}-t_{\mathrm{DAC}}^{\mathrm{AM}}-t_{\mathrm{TX}}^{\mathrm{AM}}-t_{\mathrm{RX}}^M-t_{\mathrm{IF\_}1}^M-t_{\mathrm{ADC}}^M-t_{\mathrm{FIR}}{\beta }^M\right)-K_{\mathrm{SYN\_RX}\_1\hspace{1em}}\hspace{1em}\left(t-t_{\mathrm{IF\_}1}^M-t_{\mathrm{ADC}}^M-t_{\mathrm{FIR}}{\beta }^M\right)-K_{\mathrm{ADC}}\left(t-t_{\mathrm{FIR}}{\beta }^M\right)-K_{\mathrm{SYN\_RX}\_2}\left(t\right))-{\gamma }^{\mathrm{AM}}\times {\omega }_{\mathrm{OSC}}\times \left(K_{\mathrm{SYN\_RX}\_1}\left(t-t_{\mathrm{IF\_}1}^{\mathrm{AM}}-t_{\mathrm{ADC}}^{\mathrm{AM}}-t_{\mathrm{FIR}}{\beta }^{\mathrm{AM}}-t_{\mathrm{RTX}}{\beta }^{\mathrm{AM}}-t_{\mathrm{DAC}}^{\mathrm{AM}}-t_{\mathrm{TX}}^{\mathrm{AM}}-t_{\mathrm{RX}}^M-t_{\mathrm{IF\_}1}^M-t_{\mathrm{ADC}}^M-t_{\mathrm{FIR}}{\beta }^M\right)+K_{\mathrm{ADC}}\left(t-t_{\mathrm{FIR}}{\beta }^{\mathrm{AM}}-t_{\mathrm{RTX}}{\beta }^{\mathrm{AM}}-t_{\mathrm{DAC}}^{\mathrm{AM}}-t_{\mathrm{TX}}^{\mathrm{AM}}-t_{\mathrm{RX}}^M-t_{\mathrm{IF\_}1}^M-t_{\mathrm{ADC}}^M-t_{\mathrm{FIR}}{\beta }^M\right)+K_{\mathrm{SYN\_RX}\_2}(t-t_{\mathrm{RTX}}{\beta }^{\mathrm{AM}}-t_{\mathrm{DAC}}^{\mathrm{AM}}-t_{\mathrm{TX}}^{\mathrm{AM}}-t_{\mathrm{RX}}^M-t_{\mathrm{IF\_}1}^M-t_{\mathrm{ADC}}^M-t_{\mathrm{FIR}}{\beta }^M)-K_{\mathrm{SYN\_TX}}\left(t-t_{\mathrm{TX}}^{\mathrm{AM}}-t_{\mathrm{RX}}^M-t_{\mathrm{IF\_}1}^M-t_{\mathrm{ADC}}^M-t_{\mathrm{FIR}}{\beta }^M\right)\right)+2\times {\phi }_{F_1}^{\mathrm{MULT}}+{\phi }_{\mathrm{SYN\_TX}}^M\left(0\right)-{\phi }_{\mathrm{SYN\_RX}\_1}^{\mathrm{AM}}\left(0\right)-{\phi }_{\mathrm{ADC\_CLK}}^{\mathrm{AM}}\left(0\right)-{\phi }_{\mathrm{SYN\_RX}\_2}^{\mathrm{AM}}\left(0\right)+{\phi }_{\mathrm{SYN\_TX}}^{\mathrm{AM}}-{\phi }_{\mathrm{SYN\_RX}\_1}^M\left(0\right)-{\phi }_{\mathrm{ADC\_CLK}}^M\left(0\right)-{\phi }_{\mathrm{SYN\_RX}\_2}^M\left(0\right),2\times {10}^{-6}<t<T_1{\beta }^M+t_{\mathrm{DAC}}^M+t_{\mathrm{TX}}^M+t_{\mathrm{RX}}^{\mathrm{AM}}+t_{\mathrm{IF\_}1}^{\mathrm{AM}}+t_{\mathrm{ADC}}^{\mathrm{AM}}+t_{\mathrm{FIR}}{\beta }^{\mathrm{AM}}+t_{\mathrm{RTX}}{\beta }^{\mathrm{AM}}+t_{\mathrm{DAC}}^{\mathrm{AM}}+t_{\mathrm{TX}}^{\mathrm{AM}}+t_{\mathrm{RX}}^M+t_{\mathrm{IF\_}1}^M+t_{\mathrm{ADC}}^M+t_{\mathrm{FIR}}{\beta }^M;$

[0128] For the second frequency component F₂:

[0129] ${\phi }_{\mathrm{BB\_RECOV}}^M\left(t\right)={\gamma }^M\times {\omega }_{\mathrm{OSC}}\times \left(K_{F_1}\left(T_1{\beta }^M\right)+K_{F_2}\left(t-T_1{\beta }^M-t_{\mathrm{DAC}}^M-t_{\mathrm{TX}}^M-t_{\mathrm{RX}}^{\mathrm{AM}}-t_{\mathrm{IF\_}1}^{\mathrm{AM}}-t_{\mathrm{ADC}}^{\mathrm{AM}}-t_{\mathrm{FIR}}{\beta }^{\mathrm{AM}}-t_{\mathrm{RTX}}{\beta }^{\mathrm{AM}}-t_{\mathrm{DAC}}^{\mathrm{AM}}-t_{\mathrm{TX}}^{\mathrm{AM}}-t_{\mathrm{RX}}^M-t_{\mathrm{IF\_}1}^M-t_{\mathrm{ADC}}^M-t_{\mathrm{FIR}}{\beta }^M\right)+K_{\mathrm{SYN\_TX}}\left(t-t_{\mathrm{TX}}^M-t_{\mathrm{RX}}^{\mathrm{AM}}-t_{\mathrm{IF\_}1}^{\mathrm{AM}}-t_{\mathrm{ADC}}^{\mathrm{AM}}-t_{\mathrm{FIR}}{\beta }^{\mathrm{AM}}-t_{\mathrm{RTX}}{\beta }^{\mathrm{AM}}-t_{\mathrm{DAC}}^{\mathrm{AM}}-t_{\mathrm{TX}}^{\mathrm{AM}}-t_{\mathrm{RX}}^M-t_{\mathrm{IF\_}1}^M-t_{\mathrm{ADC}}^M-t_{\mathrm{FIR}}{\beta }^M\right)-K_{\mathrm{SYN\_RX}\_1}\left(t-t_{\mathrm{IF\_}1}^M-t_{\mathrm{ADC}}^M-t_{\mathrm{FIR}}{\beta }^M\right)-K_{\mathrm{ADC}}\left(t-t_{\mathrm{FIR}}{\beta }^M\right)-K_{\mathrm{SYN\_RX}\_2}\left(t\right)\right)-{\gamma }^{\mathrm{AM}}\times {\omega }_{\mathrm{OSC}}\times \left(K_{\mathrm{SYN\_RX}\_1}\left(t-t_{\mathrm{IF\_}1}^{\mathrm{AM}}-t_{\mathrm{ADC}}^{\mathrm{AM}}-t_{\mathrm{FIR}}{\beta }^{\mathrm{AM}}-t_{\mathrm{DAC}}^{\mathrm{AM}}-t_{\mathrm{TX}}^{\mathrm{AM}}-t_{\mathrm{RX}}^M-t_{\mathrm{IF\_}1}^M-t_{\mathrm{ADC}}^M-t_{\mathrm{FIR}}{\beta }^M\right)+K_{\mathrm{ADC}}\left(t-t_{\mathrm{FIR}}{\beta }^{\mathrm{AM}}-t_{\mathrm{RTX}}{\beta }^{\mathrm{AM}}-t_{\mathrm{DAC}}^{\mathrm{AM}}-t_{\mathrm{TX}}^{\mathrm{AM}}-t_{\mathrm{RX}}^M-t_{\mathrm{IF\_}1}^M-t_{\mathrm{ADC}}^M-t_{\mathrm{FIR}}{\beta }^M\right)+K_{\mathrm{SYN\_RX}\_2}(t-t_{\mathrm{RTX}}{\beta }^{\mathrm{AM}}-t_{\mathrm{DAC}}^{\mathrm{AM}}-t_{\mathrm{TX}}^{\mathrm{AM}}-t_{\mathrm{RX}}^M-t_{\mathrm{IF\_}1}^M-t_{\mathrm{ADC}}^M-t_{\mathrm{FIR}}{\beta }^M)-K_{\mathrm{SYN\_TX}}\left(t-t_{\mathrm{TX}}^{\mathrm{AM}}-t_{\mathrm{RX}}^M-t_{\mathrm{IF\_}1}^M-t_{\mathrm{ADC}}^M-t_{\mathrm{FIR}}{\beta }^M\right)\right)+2\times {\phi }_{F_2}^{\mathrm{MULT}}+{\phi }_{\mathrm{SYN\_TX}}^M\left(0\right)-{\phi }_{\mathrm{SYN\_RX}\_1}^{\mathrm{AM}}\left(0\right)-{\phi }_{\mathrm{ADC\_CLK}}^{\mathrm{AM}}\left(0\right)-{\phi }_{\mathrm{SYN\_RX}\_2}^{\mathrm{AM}}\left(0\right)+{\phi }_{\mathrm{SYN\_TX}}^{\mathrm{AM}}-{\phi }_{\mathrm{SYN\_RX}\_1}^M\left(0\right)-{\phi }_{\mathrm{ADC\_CLK}}^M\left(0\right)-{\phi }_{\mathrm{SYN\_RX}\_2}^M\left(0\right),t>T_1{\beta }^M+t_{\mathrm{TX}}^M+t_{\mathrm{RX}}^{\mathrm{AM}}+t_{\mathrm{IF\_}1}^{\mathrm{AM}}+t_{\mathrm{ADC}}^{\mathrm{AM}}+t_{\mathrm{FIR}}{\beta }^{\mathrm{AM}}+t_{\mathrm{RTX}}{\beta }^{\mathrm{AM}}+t_{\mathrm{DAC}}^{\mathrm{AM}}+t_{\mathrm{TX}}^{\mathrm{AM}}+t_{\mathrm{RX}}^M+t_{\mathrm{IF\_}1}^M+t_{\mathrm{ADC}}^M+t_{\mathrm{FIR}}{\beta }^M+2\times {10}^{-6}$

[0130] Substitutions:
T_D—M-AM=t_DAC^M+t_DAC^M+t_TX^M+t_RX^AM+t_IF—1^AM+t_ADC^AM+t_FIRβ^AM+t_RTXβ^AM+t_DAC^AM+t_TX^AM+t_RX^M+t_IF—1^M+t_ADC^M+t_FIRβ^M;
where T_D—M-AM is the propagation delay through master (M) and transponder (AM) circuitry.
φ_BB—M-AM(0)=φ_SYN—TX^M(0)−φ_SYN—RX—1^AM(0)−φ_ADC—CLK^AM(0)−φ_SYN—RX—2^AM(0)+φ_SYN—TX^AM−φ_SYN—RX—1^M(0)−φ_ADC—CLK^M(0)−φ_SYN—RX—2^M(0)=Const;
where: φ_BB—M-AM(0) is the LO phase shift, at time t=0, from master (M) and transponder (AM) frequency mixers, including ADC(s).
Also: K_SYN—TX=K_SYN—RX—1+K_ADC+K_SYN—RX—2

[0131] First frequency component F₁:

[0132] ${\phi }_{\mathrm{BB\_RECOV}}^M\left(t\right)={\gamma }^M\times {\omega }_{\mathrm{OSC}}\times \left(K_{F_1}\left(t-T_{\mathrm{D\_M}-\mathrm{AM}}\right)-K_{\mathrm{SYN\_TX}}\left(t\right)+K_{\mathrm{SYN\_RX}\_1}\left(t\right)-K_{\mathrm{ADC}}\left(t\right)-K_{\mathrm{SYN\_RX}\_2}\left(t\right)+K_{\mathrm{SYN\_TX}}(-t_{\mathrm{TX}}^M-t_{\mathrm{RX}}^{\mathrm{AM}}-t_{\mathrm{IF\_}1}^{\mathrm{AM}}-t_{\mathrm{ADC}}^{\mathrm{AM}}-t_{\mathrm{FIR}}{\beta }^{\mathrm{AM}}-t_{\mathrm{RTX}}{\beta }^{\mathrm{AM}}-t_{\mathrm{DAC}}^{\mathrm{AM}}-t_{\mathrm{TX}}^{\mathrm{AM}}-t_{\mathrm{RX}}^M-t_{\mathrm{IF\_}1}^M-t_{\mathrm{ADC}}^M-t_{\mathrm{FIR}}{\beta }^M)-K_{\mathrm{SYN\_RX}\_1}\left(-t_{\mathrm{IF\_}1}^M-t_{\mathrm{ADC}}^M-t_{\mathrm{FIR}}{\beta }^M\right)-K_{\mathrm{ADC}}\left(-t_{\mathrm{FIR}}{\beta }^M\right)\right)-{\gamma }^{\mathrm{AM}}\times {\omega }_{\mathrm{OSC}}\times \left(K_{\mathrm{SYN\_RX}\_1}\left(t\right)+K_{\mathrm{ADC}}\left(t\right)+K_{\mathrm{SYN\_RX}\_2}\left(t\right)-K_{\mathrm{SYN\_TX}}\left(t\right)+K_{\mathrm{SYN\_RX}\_1}\left(t-t_{\mathrm{IF\_}1}^{\mathrm{AM}}-t_{\mathrm{ADC}}^{\mathrm{AM}}-t_{\mathrm{FIR}}{\beta }^{\mathrm{AM}}-t_{\mathrm{RTX}}{\beta }^{\mathrm{AM}}-t_{\mathrm{DAC}}^{\mathrm{AM}}\right)+K_{\mathrm{ADC}}\left(-t_{\mathrm{FIR}}{\beta }^{\mathrm{AM}}-t_{\mathrm{RTX}}{\beta }^{\mathrm{AM}}-t_{\mathrm{DAC}}^{\mathrm{AM}}\right)+K_{\mathrm{SYN\_RX}\_2}\left(-t_{\mathrm{RTX}}{\beta }^{\mathrm{AM}}-t_{\mathrm{DAC}}^{\mathrm{AM}}\right)K_{\mathrm{SYN\_RX}\_1}\left(-t_{\mathrm{TX}}^{\mathrm{AM}}-t_{\mathrm{RX}}^M-t_{\mathrm{IF\_}1}^M-t_{\mathrm{ADC}}^M-t_{\mathrm{FIR}}{\beta }^M\right)+K_{\mathrm{ADC}}\left(-t_{\mathrm{TX}}^{\mathrm{AM}}-t_{\mathrm{RX}}^M-t_{\mathrm{IF\_}1}^M-t_{\mathrm{ADC}}^M-t_{\mathrm{FIR}}{\beta }^M\right)+K_{\mathrm{SYN\_RX}\_2}\left(-t_{\mathrm{TX}}^{\mathrm{AM}}-t_{\mathrm{RX}}^M-t_{\mathrm{IF\_}1}^M-t_{\mathrm{ADC}}^M-t_{\mathrm{FIR}}{\beta }^M\right)-K_{\mathrm{SYN\_TX}}\left(-t_{\mathrm{TX}}^{\mathrm{AM}}-t_{\mathrm{RX}}^M-t_{\mathrm{IF\_}1}^M-t_{\mathrm{ADC}}^M-t_{\mathrm{FIR}}{\beta }^M\right)\right)+2\times {\phi }_{F_1}^{\mathrm{MULT}}+{\phi }_{\mathrm{BB\_M}-\mathrm{AM}}\left(0\right),2\times {10}^{-6}<t<T_1{\beta }^M+T_{\mathrm{D\_M}-\mathrm{AM};}$

[0133] First frequency component F₁ continued:

[0134] ${\phi }_{\mathrm{BB\_RECOV}}^M\left(t\right)={\gamma }^M\times {\omega }_{\mathrm{OSC}}\times (K_{F_1}\left(t-T_{\mathrm{D\_M}-\mathrm{AM}}\right)+K_{\mathrm{SYN\_TX}}\left(-t_{\mathrm{TX}}^M-t_{\mathrm{RX}}^{\mathrm{AM}}-t_{\mathrm{IF\_}1}^{\mathrm{AM}}-t_{\mathrm{ADC}}^{\mathrm{AM}}-t_{\mathrm{FIR}}{\beta }^{\mathrm{AM}}-t_{\mathrm{RTX}}{\beta }^{\mathrm{AM}}-t_{\mathrm{DAC}}^{\mathrm{AM}}-t_{\mathrm{TX}}^{\mathrm{AM}}-t_{\mathrm{RX}}^M-t_{\mathrm{IF\_}1}^M-t_{\mathrm{ADC}}^M-t_{\mathrm{FIR}}{\beta }^M\right)-K_{\mathrm{SYN\_RX}\_1}\left(-t_{\mathrm{IF\_}1}^M-t_{\mathrm{ADC}}^M-t_{\mathrm{FIR}}{\beta }^M\right)-K_{\mathrm{ADC}}\left(-t_{\mathrm{FIR}}{\beta }^M\right){\gamma }^{\mathrm{AM}}\times {\omega }_{\mathrm{OSC}}\times \left(K_{\mathrm{SYN\_RX}\_1}\left(-t_{\mathrm{IF\_}1}^{\mathrm{AM}}-t_{\mathrm{ADC}}^{\mathrm{AM}}-t_{\mathrm{FIR}}{\beta }^{\mathrm{AM}}-t_{\mathrm{RTX}}{\beta }^{\mathrm{AM}}-t_{\mathrm{DAC}}^{\mathrm{AM}}\right)+K_{\mathrm{ADC}}\left(-t_{\mathrm{FIR}}{\beta }^{\mathrm{AM}}-t_{\mathrm{RTX}}{\beta }^{\mathrm{AM}}-t_{\mathrm{DAC}}^{\mathrm{AM}}\right)+K_{\mathrm{SYN\_RX}\_2}\left(-t_{\mathrm{RTX}}{\beta }^{\mathrm{AM}}-t_{\mathrm{DAC}}^{\mathrm{AM}}\right)\right)+2\times {\phi }_{F_1}^{\mathrm{MULT}}+{\phi }_{\mathrm{BB\_M}-\mathrm{AM}}\left(0\right),2\times {10}^{-6}<t<T_1{\beta }^M+T_{\mathrm{D\_M}-\mathrm{AM}};$

[0135] Second frequency component F₂:

[0136] ${\phi }_{\mathrm{BB\_RECOV}}^M\left(t\right)={\gamma }^M\times {\omega }_{\mathrm{OSC}}\times \left(K_{F_1}\left(T_1{\beta }^M\right)+K_{F_2}\left(t-T_1{\beta }^M-T_{\mathrm{D\_M}-\mathrm{AM}}\right)-K_{\mathrm{SYN\_TX}}\left(t\right)+K_{\mathrm{SYN\_RX}\_1}\left(t\right)-K_{\mathrm{ADC}}\left(t\right)-K_{\mathrm{SYN\_RX}\_2}\left(t\right)+K_{\mathrm{SYN\_TX}}\left(-t_{\mathrm{TX}}^M-t_{\mathrm{RX}}^{\mathrm{AM}}-t_{\mathrm{IF\_}1}^{\mathrm{AM}}-t_{\mathrm{ADC}}^{\mathrm{AM}}-t_{\mathrm{FIR}}{\beta }^{\mathrm{AM}}-t_{\mathrm{RTX}}{\beta }^{\mathrm{AM}}-t_{\mathrm{DAC}}^{\mathrm{AM}}-t_{\mathrm{TX}}^{\mathrm{AM}}-t_{\mathrm{RX}}^M-t_{\mathrm{IF\_}1}^M-t_{\mathrm{ADC}}^M-t_{\mathrm{FIR}}{\beta }^M\right)-K_{\mathrm{SYN\_RX}\_1}\left(-t_{\mathrm{IF\_}1}^M-t_{\mathrm{ADC}}^M-t_{\mathrm{FIR}}{\beta }^M\right)-K_{\mathrm{ADC}}\left(-t_{\mathrm{FIR}}{\beta }^M\right)\right)-{\gamma }^{\mathrm{AM}}\times {\omega }_{\mathrm{OSC}}\times \left(K_{\mathrm{SYN\_RX}\_1}\left(t\right)+K_{\mathrm{ADC}}\left(t\right)+K_{\mathrm{SYN\_RX}\_2}\left(t\right)-K_{\mathrm{SYN\_TX}}\left(t\right)+K_{\mathrm{SYN\_RX}\_1}\left(-t_{\mathrm{IF\_}1}^{\mathrm{AM}}-t_{\mathrm{ADC}}^{\mathrm{AM}}-t_{\mathrm{FIR}}{\beta }^{\mathrm{AM}}-t_{\mathrm{RTX}}{\beta }^{\mathrm{AM}}\right)+K_{\mathrm{ADC}}(-t_{\mathrm{FIR}}{\beta }^{\mathrm{AM}}-t_{\mathrm{RTX}}{\beta }^{\mathrm{AM}}-t_{\mathrm{DAC}}^{\mathrm{AM}})+K_{\mathrm{SYN\_RX}\_2}\left(-t_{\mathrm{RTX}}{\beta }^{\mathrm{AM}}-t_{\mathrm{DAC}}^{\mathrm{AM}}\right)K_{\mathrm{SYN\_RX}\_1}\left(-t_{\mathrm{TX}}^{\mathrm{AM}}-t_{\mathrm{RX}}^M-t_{\mathrm{IF\_}1}^M-t_{\mathrm{ADC}}^M-t_{\mathrm{FIR}}{\beta }^M\right)+K_{\mathrm{ADC}}\left(-t_{\mathrm{TX}}^{\mathrm{AM}}-t_{\mathrm{RX}}^M-t_{\mathrm{IF\_}1}^M-t_{\mathrm{ADC}}^M-t_{\mathrm{FIR}}{\beta }^M\right)+K_{\mathrm{SYN\_RX}\_2}\left(-t_{\mathrm{TX}}^{\mathrm{AM}}-t_{\mathrm{RX}}^M-t_{\mathrm{IF\_}1}^M-t_{\mathrm{ADC}}^M-t_{\mathrm{FIR}}{\beta }^M\right)-K_{\mathrm{SYN\_TX}}\left(-t_{\mathrm{TX}}^{\mathrm{AM}}-t_{\mathrm{RX}}^M-t_{\mathrm{IF\_}1}^M-t_{\mathrm{ADC}}^M-t_{\mathrm{FIR}}{\beta }^M\right)\right)+2\times {\phi }_{F_2}^{\mathrm{MULT}}+{\phi }_{\mathrm{BB\_M}-\mathrm{AM}}\left(0\right),t>T_1{\beta }^M+T_{\mathrm{D\_M}-\mathrm{AM}}+2\times {10}^{-6}$