Methods for propagating a non sinusoidal signal without distortion in dispersive lossy media

[0001] This application is a continuation-in-part of, and claims priority to, U.S. patent application Ser. No. 10/224,541 filed Aug. 20, 2002, now U.S. Pat. No. 6,847,267, which is a continuation-in-part of U.S. patent application Ser. No. 09/519,922, filed Mar. 7, 2000, now U.S. Pat. No. 6,441,695. The entire text of each of the above related applications is incorporated by reference in its entirety.

[0002] 1. Field of the Invention

[0003] The invention relates generally to the field of signal transmission. More particularly, a representative embodiment of the invention relates to transmission line testing and communication.

[0004] 2. Discussion of the Related Art

[0005] Transmission lines, with their characteristic loss of signal as well as inherent time delay, may create problems in designing systems that employ a plurality of signals, which may be subject to delay and distortion. Typical signals used to generate inputs to transmission lines generally exhibit delay or propagation times that are not easily determinable. The propagation velocity of these waves is also variable with displacement along the transmission line.

[0006] A Time Domain Reflectometer (TDR) is a test instrument used to find faults in transmission lines and to empirically estimate transmission line lengths and other parameters characterizing the line, such as: inductance per unit length, capacitance per unit length, resistance per unit length and conductance per unit length.

[0007] An important measurement in TDR test technology is the time-of-flight (TOF) of a test pulse generated by the instrument and applied to the line. The time-of-flight may be measured by timing the passage of the pulse detected at two locations along the line. Along with a value of the propagation speed of the pulse, time-of-flight measurements can allow one to obtain the distance between measurement points or, in the case of a reflected wave, the distance from the pulse launch point to the location of the impedance change causing the pulse to be reflected and returned.

[0008] A fundamental limitation in TDR technology is the poor accuracy of TOF measurements in lossy, dispersive transmission lines. The relatively high TDR accuracy of TOF values obtainable in short low loss, low dispersion transmissions lines is possible only because the propagating test pulses keep their shape and amplitude in tact over the distances they travel during TOF measurements. By contrast, in dispersive, lossy long transmission lines the test pulses used in the art change shape, amplitude, and speed as they travel.

[0009] Further, it is difficult to provide high-speed communications in a lossy, frequency dependent transmission media. It would be advantageous to have a method to increase data transmission rates in such transmission lines.

[0010] Until now, the requirements of providing a method and/or apparatus for accurately measuring times-of-flight, estimating line lengths and other parameters characterizing lossy, dispersive transmission lines, and providing high-speed communications via such transmission media have not been met. What is needed is a solution that addresses these requirements.

[0011] Shortcomings are reduced or eliminated by the techniques disclosed here. In one respect, the disclosure involves a method for transmitting an exponential waveform. The exponential waveform, which can be characterized by the equation V_SD=De^{−ASD[x−vSDt]} and can be truncated at a maximum value and can have an essentially constant shape during transmission on a transmission line.

[0012] In another respect, the disclosure involves transmitting a waveform including a speedy delivery signal envelope modulated with a sinusoidal signal on a media. The waveform can be characterize by the equation V(z,s)=B(s)e^−zγ(s) and having a substantially constant shape during transmission on the media. In some embodiments, an exponential waveform can include a truncated speedy delivery signal envelope modulated with a sinusoidal carrier signal, in which the media can be a resistance-capacitance-inductance transmission line. In other embodiments, an electromagnetic plane waveform can include a truncated speedy delivery signal envelope modulated with a sinusoidal electromagnetic plane wave on a media, such as a dispersive plasma media.

[0013] In other respects, the disclosure involves a method for determining a temperature of a media. The method includes generating an exponential waveform on the media and determining the delay of the exponential waveform. The delay can be characterized by the equation

[0014] ${\delta }_{\mathrm{SD}}=\left(\frac{1}{\alpha }\right)·\sqrt{\begin{array}{c}\stackrel{\_}{L}\left(\alpha ,T\right)\stackrel{\_}{C}\left(\alpha ,T\right){\alpha }^2+(\stackrel{\_}{G}\left(\alpha ,T\right)\stackrel{\_}{L}\left(\alpha ,T\right)+\\ \stackrel{\_}{R}\left(\alpha ,T\right)\stackrel{\_}{C}\left(\alpha ,T\right))\alpha +\stackrel{\_}{R}\left(\alpha ,T\right)\stackrel{\_}{G}\left(\alpha ,T\right)\end{array}}$
where the temperature is proportional to the length of travel of the exponential waveform on the media.

[0015] In another respect, a method is provided for measuring a temperature of a media. The method includes generating a speedy delivery (SD) signal on a media and determining the delay of the SD signal characterized by the equation Δ_SD=l(T)·√{square root over (L(T)C(T))}{square root over (L(T)C(T))}·√{square root over (1+τ/(L(T)/R(T)))}{square root over (1+τ/(L(T)/R(T)))}. The method also includes determining the temperature of the media which can be proportional to the delay.

[0016] In yet another respect, the disclosure involves a method for reducing interconnect delays. A signal is generated on an interconnect and the delay of the signal can be characterized by a line model and the equation δ_SD=√{square root over (LC)}√{square root over (1+τ/(L/R))}. In one embodiment, the method includes reducing the delay by decreasing τ. In another embodiment, the method includes inserting a repeater on the interconnect.

[0017] In other respects, the disclosure involves a method including steps for generating an exponential waveform having an essentially constant shape during transmission on a media at a first location. A plurality of exponentials can be encoded on a leading edge of the exponential waveform, where the encoded exponential waveform is transmitted to a second location. The method also includes decoding the encoded exponential waveform at the second location.

[0018] The disclosure also involves a method for determining a length of an interconnect for a desired delay. A signal can be generated and a length of the interconnect can be characterized by the equation

[0019] $l=\frac{{\Delta }_{\mathrm{SD}}}{\sqrt{\mathrm{LC}}·\sqrt{1+\frac{\tau }{L/R}}},$
where L is the inductance per unit length of the interconnect, R resistance per unit length of the interconnect, C is the capacitance per unit length of the interconnect, and Δ_SD is the desired delay.

[0020] A thermometer is also presented. The thermometer includes a signal generator configured to generate a truncated exponential waveform characterized by the equation V_SD=De^{−ASD[x−vSDt]}, _where where D is a magnitude, V_SD is a voltage, t is time, A_SD is an attenuation coefficient, v_SD is a propagation velocity, and t is time. The thermometer also includes a processor coupled to the signal generator. The processor is configured to determine a temperature of the media, where the temperature of the media being proportional to the delay of the waveform.

[0021] In another respect, the present disclosure includes a time domain reflectometer. The time domain reflectometer includes a signal generator configured to generate a truncated exponential waveform on a media. Using the generated truncated exponential media, the time domain reflectometer is configured to determine the length of the media. In another embodiment, the time domain reflectometer is configured to detect the locations of possible faults.

[0022] These, and other, embodiments of the invention will be better appreciated and understood when considered in conjunction with the following description and the accompanying drawings. It should be understood, however, that the following description, while indicating various embodiments of the invention and numerous specific details thereof, is given by way of illustration and not of limitation. Many substitutions, modifications, additions and/or rearrangements may be made within the scope of the invention without departing from the spirit thereof, and the invention includes all such substitutions, modifications, additions and/or rearrangements.

[0118] The invention and the various features and advantageous details thereof are explained more fully with reference to the nonlimiting embodiments that are illustrated in the accompanying drawings and detailed in the following description. Descriptions of well known starting materials, processing techniques, components and equipment are omitted so as not to unnecessarily obscure the invention in detail. It should be understood, however, that the detailed description and the specific examples, while indicating specific embodiments of the invention, are given by way of illustration only and not by way of limitation. Various substitutions, modifications, additions and/or rearrangements within the spirit and/or scope of the underlying inventive concept will become apparent to those skilled in the art from this disclosure.

[0119] Within this application several publications are referenced by Arabic numerals within parentheses or brackets. Full citations for these, and other, publications may be found at the end of the specification immediately preceding the claims after the section heading References. The disclosures of all these publications in their entireties are hereby expressly incorporated by reference herein for the purpose of indicating the background of the invention and illustrating the state of the art.

[0120] In general, the context of the invention can include signal transmission. The context of the invention can include a method and/or apparatus for measuring and estimating transmission line parameters. The context of the invention can also include a method and/or apparatus for performing high-speed communication over a lossy, dispersive transmission line.

[0121] A practical application of the invention that has value within the technological arts is in time domain reflectometry. Further, the invention is useful in conjunction with lossy transmission lines. The invention is also useful in conjunction with frequency dependent transmission lines. Another practical application of the invention that has value within the technological arts is high-speed communications. There are virtually innumerable other uses for the invention. In fact, any application involving the transmission of a signal may benefit.

[0122] A method for transmitting a waveform having a controllable attenuation and propagation velocity, representing an embodiment of the invention, can be cost effective and advantageous. The invention improves quality and/or reduces costs compared to previous approaches.

[0123] The invention includes a method and/or apparatus for utilizing a truncated waveform, coined a Speedy Delivery (SD) waveform by the inventors. An analysis of SD propagation in coupled lossy transmission lines is presented and practical considerations associated with truncating the SD waveforms are addressed. Parameters used to describe the propagation of the SD waveform are defined and techniques for determining their values are presented. These parameters include the Speedy Delivery propagation velocity, v_SD, the Speedy Delivery attenuation coefficient, A_SD, and the Speedy Delivery impedance, Z_SD. These parameters may depend on properties of a transmission media as well as an exponential coefficient α (SD input signal parameter).

[0124] The invention describes SD signal propagation in coupled transmission lines. An embodiment involving two transmission lines is illustrated. More complex configurations that include a larger number of coupled lines may be readily developed by one of ordinary skill in the art in light of this disclosure. Additionally, the behavior of the propagation of a slowly varying envelope of an optical pulse containing a truncated SD signal in a single mode communication fiber is described.

[0125] The invention teaches a method for using SD test pulses to obtain empirical estimates of the SD signal propagation parameters, A_SD and v_SD, in lossy transmission lines, including those with frequency dependent parameters. Once the value of v_SD is determined for a calibration cable length, constant threshold time of flight measurements can be used to measure distances to discontinuities in similar cables. The calibration procedure can be repeated experimentally for each exponential coefficient α.

[0126] The invention provides a method for utilizing a wideband, non-SD waveform in order to develop an empirical transfer function for a calibration length of a given transmission media. This transfer function may be used to simulate the transmission of SD waveforms with a wide range of values for the exponential coefficient α. The SD parameters A_SD(α) and v_SD(α) can be determined from these simulated waveforms. The inclusion of a simulation step in the calibration process can significantly reduce the experimental measurements needed to determine the values of A_SD(α) and v_SD(α) for a variety of values for α. The SD behavior of a media can then be determined by simulation, without the need for an SD waveform generator.

[0127] The invention teaches a method for determining the SD impedance of a transmission line. It also illustrates the predicable variation of Z_SD as a function of α. This predictability can allow a designer to control the transmission line SD impedance by appropriately selecting the parameter α.

[0128] The invention also describes an alternate precision distance measurement method using truncated SD test pulses with high loss, long lines. As the test pulses travel down these lines, the attenuation can be so extensive that their amplitudes are too small for common threshold crossing time of flight measurements to be feasible. The invention includes a method for predicting an attenuation and overcoming this difficulty, extracting accurate length measurements for long lossy lines.

[0129] The invention demonstrates the utility of the SD test waveform in accurately determining distances to impedance discontinuities. Examples show how to use SD wave propagation as the basis for an accurate time domain reflectometry (TDR) unit. This unit may include an SD waveform generator, a means of coupling with the transmission media, a means of measuring the applied and reflected waveforms, a display, memory storage, and computational ability to analyze and interpret the SD waveforms. The invention can also include a computer. The invention can include a method which can be applied to modify currently available TDR units to increase their accuracy. This modification may be a TDR software modification resulting in a nonrecurring unit manufacturing cost. A standard TDR waveform may be used to determine an empirical transfer function for the line under test, and this transfer function may then be used to simulate the propagation of the SD waveform. This process allows all the advantages of SD to be incorporated into currently available units with only a modification of existing computational algorithms, creating a virtual SD TDR unit.

[0130] The truncated SD test pulse approach can also be applied to the propagation of acoustic waves in sonar and geophysical applications. Reflected acoustic signals can be used to accurately determine the location of underwater objects and to determine locations of geophysical strata boundaries. Empirical transfer functions can be determined and used to simulate SD pulse propagation eliminating the need to physically generate complex acoustic pulses.

[0131] The invention provides an SD waveform that has several propagation properties in lossy, frequency dependent transmission media, which make it suitable for enhancing a data transmission scheme.

[0132] The invention provides a method to increase the number of bits transmitted per pulse by varying the value of α to encode data on truncated SD waveforms. The example provided utilizes SD pulses with four different α's sequentially transmitted on a 2 kft cable. These SD pulses are received at the end of the cable with the exponential constants undistorted. The four pulses may be transmitted as positive or negative signals resulting in eight possible states. Therefore, each pulse represents three bits of transmitted information in each symbol period. This strategy can be used with short, low loss transmission lines.

[0133] The invention also teaches the use of SD propagation properties to generate a set of pulses consisting of linear combinations of truncated SD waveforms that are orthogonal at the receiver. Data can be encoded on these pulses by varying their amplitude, and these amplitude-modulated pulses may be transmitted simultaneously on a transmission line. The individual pulse amplitudes are computed at the receiver taking advantage of their orthogonality. This method provides improved noise immunity, and can be used with longer, higher loss transmission lines.

[0134] The invention also teaches the use of SD waveforms in digital circuits, specifically, on-chip clock circuits. It provides a method for utilizing truncated SD clock waveforms to reduce clock skew.

[0135] The propagation of a Speedy Delivery (SD) waveform in a transmission line can be described by the equation V(x,t)=De^{−ASD[x−vSDt]}. The associated boundary condition at the input to the line is V(x=0, t)=De^αt, wherein A_SD is the exponential coefficient describing the attenuation of the SD signal as a function of distance along the line at a fixed time instant, and v_SD is the SD signal propagation velocity. These quantities are also related by the simple equation, A_SD(α)·v_SD(α)=α. The parameters A_SD and v_SD of the propagating wave and the line boundary condition are all functions of α, and may be determined experimentally for a transmission line.

[0136] For the case of two balanced twisted wire pair transmission lines, the differential voltage can be equally divided between the two wires and the current on each wire in the pair is equal in magnitude but opposite in direction. In this case, the telegrapher's equations can be written:

[0137] $\frac{\partial }{\partial x}V=Z·I$ $\frac{\partial }{\partial x}I=Y·V$ $\frac{\partial }{\partial x}\left[\begin{array}{c}{V}_1\left(x,t\right)\\ V_2\left(x,t\right)\end{array}\right]=\left[\begin{array}{cc}{R}_1+L_1\frac{\partial }{\partial t}& M_{12}\frac{\partial }{\partial t}\\ M_{21}\frac{\partial }{\partial t}& R_2+L_2\frac{\partial }{\partial t}\end{array}\right]·\left[\begin{array}{c}{I}_1\left(x,t\right)\\ I_2\left(x,t\right)\end{array}\right]·\frac{\partial }{\partial x}\left[\begin{array}{c}{I}_1\left(x,t\right)\\ I_2\left(x,t\right)\end{array}\right]=\left[\begin{array}{cc}{G}_1+C_1\frac{\partial }{\partial t}& C_{12}\frac{\partial }{\partial t}\\ C_{21}\frac{\partial }{\partial t}& G_2+C_2\frac{\partial }{\partial t}\end{array}\right]·\left[\begin{array}{c}{V}_1\left(x,t\right)\\ V_2\left(x,t\right)\end{array}\right]$
This analysis includes the net cross capacitance, C_ij, and the net cross inductances, M_ij. The cross resistance and conductance terms are neglected. These equations are then Laplace transformed into the complex frequency domain.

[0138] $\frac{\partial }{\partial x}\left[\begin{array}{c}{V}_1\left(x,s\right)\\ V_2\left(x,s\right)\end{array}\right]=\left[\begin{array}{cc}{\stackrel{\_}{R}}_1\left(s\right)+s{\stackrel{\_}{L}}_1\left(s\right)& s{\stackrel{\_}{M}}_{12}\left(s\right)\\ s{\stackrel{\_}{M}}_{21}\left(s\right)& {\stackrel{\_}{R}}_2\left(s\right)+s{\stackrel{\_}{L}}_2\left(s\right)\end{array}\right]·\left[\begin{array}{c}{I}_1\left(x,s\right)\\ I_2\left(x,s\right)\end{array}\right]$ $\frac{\partial }{\partial x}\left[\begin{array}{c}{I}_1\left(x,s\right)\\ I_2\left(x,s\right)\end{array}\right]=\left[\begin{array}{cc}{\stackrel{\_}{G}}_1\left(s\right)+s{\stackrel{\_}{C}}_1\left(s\right)& s{\stackrel{\_}{C}}_{12}\left(s\right)\\ s{\stackrel{\_}{C}}_{21}\left(s\right)& {\stackrel{\_}{G}}_2\left(s\right)+s{\stackrel{\_}{C}}_2\left(s\right)\end{array}\right]·\left[\begin{array}{c}{V}_1\left(x,s\right)\\ V_2\left(x,s\right)\end{array}\right]$
Taking the partial derivative of the top equation with respect to x, and substituting the bottom equation for the

[0139] $\frac{\partial }{\partial x}\left[\begin{array}{c}{I}_1\left(x,s\right)\\ I_2\left(x,s\right)\end{array}\right]$
term, gives the result:

[0140] $\frac{{\partial }^2}{\partial x^2}V\left(x,s\right)=\stackrel{\_}{Z}\left(s\right)\stackrel{\_}{Y}\left(s\right)·V\text{(}\text{x,s}\text{)}$ $\begin{array}{c}\frac{{\partial }^2}{\partial x^2}\left[\begin{array}{c}{V}_1\left(x,s\right)\\ V_2\left(x,s\right)\end{array}\right]=\left[\begin{array}{cc}{\stackrel{\_}{R}}_1\left(s\right)+s{\stackrel{\_}{L}}_1\left(s\right)& s{\stackrel{\_}{M}}_{12}\left(s\right)\\ s{\stackrel{\_}{M}}_{21}\left(s\right)& {\stackrel{\_}{R}}_2\left(s\right)+s{\stackrel{\_}{L}}_2\left(s\right)\end{array}\right]·\\ \left[\begin{array}{cc}{\stackrel{\_}{G}}_1\left(s\right)+s{\stackrel{\_}{C}}_1\left(s\right)& s{\stackrel{\_}{C}}_{12}\left(s\right)\\ s{\stackrel{\_}{C}}_{21}\left(s\right)& {\stackrel{\_}{G}}_2\left(s\right)+s{\stackrel{\_}{C}}_2\left(s\right)\end{array}\right]·\left[\begin{array}{c}{V}_1\left(x,s\right)\\ V_2\left(x,s\right)\end{array}\right]\end{array}$ $\mathrm{or},\frac{{\partial }^2}{\partial x^2}V\left(x,s\right)={\stackrel{\_}{\Gamma }\left(s\right)}^2·V\text{(}\text{x,s}\text{)}$ $\frac{{\partial }^2}{\partial x^2}\left[\begin{array}{c}{V}_1\left(x,s\right)\\ V_2\left(x,s\right)\end{array}\right]=\left[\begin{array}{cc}\left({\stackrel{\_}{R}}_1\left(s\right)+s{\stackrel{\_}{L}}_1\left(s\right)\right)\left({\stackrel{\_}{G}}_1\left(s\right)+s{\stackrel{\_}{C}}_1\left(s\right)\right)& s^2{\stackrel{\_}{M}}_{12}\left(s\right){\stackrel{\_}{C}}_{21}\left(s\right)\\ s^2{\stackrel{\_}{M}}_{21}\left(s\right){\stackrel{\_}{C}}_{12}\left(s\right)& \left({\stackrel{\_}{R}}_2\left(s\right)+s{\stackrel{\_}{L}}_2\left(s\right)\right)\left({\stackrel{\_}{G}}_2\left(s\right)+s{\stackrel{\_}{C}}_2\left(s\right)\right)\end{array}\right]·\left[\begin{array}{c}{V}_1\left(x,s\right)\\ V_2\left(x,s\right)\end{array}\right]$
This system of equations has the solution:
V(x,s)=e^{− Γ(s)·x}·A+e^{+ Γ(s)·x}·B

[0141] Applying the boundary conditions for a semi-infinite pair of transmission lines,
V(x=0,s) at x=0, and V(x→∞,s)=0 gives:
V(x,s)=e^{− Γ(s)·x}·V(x=0,s).

[0142] Because Γ is a complex symmetric matrix, it can be written Γ=( ZY)^1/2= PγP⁻¹, where P is the eigenvector matrix of the ZY product and γ is the diagonal eigenvalue matrix of Γ. Therefore, we can write e^{− Γ(s)·x}=P(s)·e^{− γ(s)·x}·P(s)⁻¹. This gives the result:
V(x,s)=P(s)·e^{− γ(s)·x}·P(s)⁻¹·V(x=0,s)

[0143] For the case where the applied boundary conditions at x=0 are SD waveforms, V₁(x=0,t)=D₁e^α1t, V₂(x=0,t)=D₂e^α2t, the SD solution is:

[0144] $V\left(x,t\right)=P\left({\alpha }_1\right)·e^{-\gamma \left({\alpha }_1\right)·x}·P^{-1}\left({\alpha }_1\right)·\left[\begin{array}{c}{D}_1·e^{{\alpha }_1·t}\\ 0\end{array}\right]+P\left({\alpha }_2\right)·e^{-\gamma \left({\alpha }_2\right)·x}·P^{-1}\left({\alpha }_2\right)·\left[\begin{array}{c}0\\ D_2·e^{{\alpha }_2·t}\end{array}\right],$
where P(α_i) is a real matrix.

[0145] $V\left(x,t\right)=\left[\begin{array}{cc}{P}_{11}\left({\alpha }_1\right)& P_{12}\left({\alpha }_1\right)\\ P_{21}\left({\alpha }_1\right)& P_{22}\left({\alpha }_1\right)\end{array}\right]·\left[\begin{array}{cc}{e}^{-{\gamma }_1\left({\alpha }_1\right)·x}& 0\\ 0& e^{-{\gamma }_2\left({\alpha }_1\right)·x}\end{array}\right]·{\left[\begin{array}{cc}{P}_{11}\left({\alpha }_1\right)& P_{12}\left({\alpha }_1\right)\\ P_{21}\left({\alpha }_1\right)& P_{22}\left({\alpha }_1\right)\end{array}\right]}^{-1}·\left[\begin{array}{c}{D}_1·e^{{\alpha }_1·t}\\ 0\end{array}\right]+\left[\begin{array}{cc}{P}_{11}\left({\alpha }_2\right)& P_{12}\left({\alpha }_2\right)\\ P_{21}\left({\alpha }_2\right)& P_{22}\left({\alpha }_2\right)\end{array}\right]·\left[\begin{array}{cc}{e}^{-{\gamma }_1\left({\alpha }_2\right)·x}& 0\\ 0& e^{-{\gamma }_2\left({\alpha }_2\right)·x}\end{array}\right]·{\left[\begin{array}{cc}{P}_{11}\left({\alpha }_2\right)& P_{12}\left({\alpha }_2\right)\\ P_{21}\left({\alpha }_2\right)& P_{22}\left({\alpha }_2\right)\end{array}\right]}^{-1}·\left[\begin{array}{c}0\\ D_2·e^{{\alpha }_2·t}\end{array}\right]$
If we designate P⁻¹ as Q, we can write:

[0146] $V\left(x,t\right)=\left[\begin{array}{cc}{P}_{11}\left({\alpha }_1\right)& P_{12}\left({\alpha }_1\right)\\ P_{21}\left({\alpha }_1\right)& P_{22}\left({\alpha }_1\right)\end{array}\right]·\left[\begin{array}{cc}{e}^{-{\gamma }_1\left({\alpha }_1\right)·x}& 0\\ 0& e^{-{\gamma }_2\left({\alpha }_1\right)·x}\end{array}\right]·\left[\begin{array}{cc}{Q}_{11}\left({\alpha }_1\right)& Q_{12}\left({\alpha }_1\right)\\ Q_{21}\left({\alpha }_1\right)& Q_{22}\left({\alpha }_1\right)\end{array}\right]·\left[\begin{array}{c}{D}_1·e^{{\alpha }_1·t}\\ 0\end{array}\right]+\left[\begin{array}{cc}{P}_{11}\left({\alpha }_2\right)& P_{12}\left({\alpha }_2\right)\\ P_{21}\left({\alpha }_2\right)& P_{22}\left({\alpha }_2\right)\end{array}\right]·\left[\begin{array}{cc}{e}^{-{\gamma }_1\left({\alpha }_2\right)·x}& 0\\ 0& e^{-{\gamma }_2\left({\alpha }_2\right)·x}\end{array}\right]·\left[\begin{array}{cc}{Q}_{11}\left({\alpha }_2\right)& Q_{12}\left({\alpha }_2\right)\\ Q_{21}\left({\alpha }_2\right)& Q_{22}\left({\alpha }_2\right)\end{array}\right]·\left[\begin{array}{c}0\\ D_2·e^{{\alpha }_2·t}\end{array}\right]$ $\mathrm{or},V\left(x,t\right)=P\left({\alpha }_1\right)·\left[\begin{array}{c}{Q}_{11}\left({\alpha }_1\right)·e^{-{\gamma }_1\left({\alpha }_1\right)·x}\\ Q_{21}\left({\alpha }_1\right)·e^{-{\gamma }_2\left({\alpha }_1\right)·x}\end{array}\right]·D_1·e^{{\alpha }_1·t}+P\left({\alpha }_2\right)·\left[\begin{array}{c}{Q}_{12}\left({\alpha }_2\right)·e^{-{\gamma }_1\left({\alpha }_2\right)·x}\\ Q_{22}\left({\alpha }_2\right)·e^{-{\gamma }_2\left({\alpha }_2\right)·x}\end{array}\right]·D_2·e^{{\alpha }_2·t}$ $\mathrm{or},V\left(x,t\right)=\hspace{1em}\left[\begin{array}{c}{Q}_{11}\left({\alpha }_1\right)P_{11}\left({\alpha }_1\right)e^{-{\gamma }_1\left({\alpha }_1\right)·x}+Q_{21}\left({\alpha }_1\right)P_{12}\left({\alpha }_1\right)e^{-{\gamma }_2\left({\alpha }_1\right)·x}\\ Q_{11}\left({\alpha }_1\right)P_{21}\left({\alpha }_1\right)e^{-{\gamma }_1\left({\alpha }_1\right)·x}+Q_{21}\left({\alpha }_1\right)P_{22}\left({\alpha }_1\right)e^{-{\gamma }_2\left({\alpha }_1\right)·x}\end{array}\right]·\hspace{1em}{D}_1·e^{{\alpha }_1·t}+\hspace{1em}\left[\begin{array}{c}{Q}_{12}\left({\alpha }_2\right)P_{11}\left({\alpha }_2\right)e^{-{\gamma }_1\left({\alpha }_2\right)·x}+Q_{22}\left({\alpha }_2\right)P_{12}\left({\alpha }_2\right)e^{-{\gamma }_2\left({\alpha }_2\right)·x}\\ Q_{12}\left({\alpha }_2\right)P_{21}\left({\alpha }_2\right)e^{-{\gamma }_1\left({\alpha }_2\right)·x}+Q_{22}\left({\alpha }_2\right)P_{22}\left({\alpha }_2\right)e^{-{\gamma }_2\left({\alpha }_2\right)·x}\end{array}\right]·D_2·e^{{\alpha }_2·t}$

[0147] The real constants associated with the eigenvector matrix, P, its inverse, P⁻¹=Q, and the SD parameters D can be written:
A₁₁=D₁P₁₁(α₁)Q₁₁(α₁)
A₁₂=D₁P₁₂(α₁)Q₂₁(α₁)
A₂₁=D₂P₁₁(α₂)Q₁₂(α₂)
A₂₂=D₂P₁₂(α₂)Q₂₂(α₂)
B₁₁=D₁P₂₁(α₁)Q₁₁(α₁)
B₁₂=D₁P₂₂(α₁)Q₂₁(α₁)
B₂₁=D₂P₂₁(α₂)Q₁₂(α₂)
B₂₂=D₂P₂₂(α₂)Q₂₂(α₂)
The signals on transmission lines will be:

[0148] $\begin{array}{c}{V}_1\left(x,t\right)=A_{11}·e^{-{\gamma }_1\left({\alpha }_1\right)·x}{e}^{{\alpha }_1·t}+A_{21}·e^{-{\gamma }_2\left({\alpha }_1\right)·x}{e}^{{\alpha }_1·t}+\\ A_{12}·e^{-{\gamma }_1\left({\alpha }_2\right)·x}{e}^{{\alpha }_2·t}+A_{22}·e^{-{\gamma }_2\left({\alpha }_2\right)·x}{e}^{{\alpha }_2·t}\\ V_2\left(x,t\right)=B_{11}·e^{-{\gamma }_1\left({\alpha }_1\right)·x}{e}^{{\alpha }_1·t}+B_{21}·e^{-{\gamma }_2\left({\alpha }_1\right)·x}{e}^{{\alpha }_1·t}+\\ B_{12}·e^{-{\gamma }_1\left({\alpha }_2\right)·x}{e}^{{\alpha }_2·t}+B_{22}·e^{-{\gamma }_2\left({\alpha }_2\right)·x}{e}^{{\alpha }_2·t}\end{array}$ $\mathrm{or},\text{}\begin{array}{c}{V}_1\left(x,t\right)=A_{11}·e^{-A_{{\mathrm{SD}}_{11}}\left(x-v_{{\mathrm{SD}}_{11}}t\right)}+A_{21}·e^{-A_{{\mathrm{SD}}_{21}}\left(x-v_{{\mathrm{SD}}_{21}}t\right)}+\\ A_{12}·e^{-A_{{\mathrm{SD}}_{12}}\left(x-v_{\mathrm{SD}12}t\right)}+A_{22}·e^{-A_{{\mathrm{SD}}_{22}}\left(x-v_{{\mathrm{SD}}_{22}}t\right)}\\ V_2\left(x,t\right)=B_{11}·e^{-A_{{\mathrm{SD}}_{11}}\left(x-v_{{\mathrm{SD}}_{11}}t\right)}+B_{21}·e^{-A_{{\mathrm{SD}}_{21}}\left(x-v_{{\mathrm{SD}}_{21}}t\right)}+\\ B_{12}·e^{-A_{{\mathrm{SD}}_{12}}\left(x-v_{{\mathrm{SD}}_{12}}t\right)}+B_{22}·e^{-A_{{\mathrm{SD}}_{22}}\left(x-v_{{\mathrm{SD}}_{22}}t\right)}\end{array}$

[0149] Where:

[0150] $A_{{\mathrm{SD}}_{11}}={\gamma }_1\left({\alpha }_1\right),A_{{\mathrm{SD}}_{12}}={\gamma }_1\left({\alpha }_2\right),A_{{\mathrm{SD}}_{21}}={\gamma }_2\left({\alpha }_1\right),\mathrm{and}{A}_{{\mathrm{SD}}_{22}}={\gamma }_2\left({\alpha }_2\right).v_{{\mathrm{SD}}_{11}}=\frac{{\alpha }_1}{{\gamma }_1\left({\alpha }_1\right)},v_{{\mathrm{SD}}_{12}}=\frac{{\alpha }_2}{{\gamma }_1\left({\alpha }_2\right)},v_{{\mathrm{SD}}_{21}}=\frac{{\alpha }_1}{{\gamma }_2\left({\alpha }_1\right)},\mathrm{and}{v}_{{\mathrm{SD}}_{22}}=\frac{{\alpha }_2}{{\gamma }_2\left({\alpha }_2\right)}.$

[0151] Thus, there are SD attenuations and velocities associated with each eigenvalue, γ_i, of the product ZY evaluated at each α.

[0152] Because positive exponential waveforms continually increase, practical considerations may create the need to truncate or limit the waveform at some level determined by the specific application. Truncation may be accomplished by several methods, as is known in the art.

[0153] The result of this truncation of the input of a transmission media is that the propagating exponential (SD) signal is also limited in magnitude as it travels in the media, and in lossy media, this maximum amplitude decreases with distance traveled exhibiting an attenuating behavior.

[0154] The propagating (SD) signal is described in the coordinate frame, (x, t), by V(x,t)=De^{−ASD(α)[x−vSD(α)·t]}=De^α·te^{−ASD(α)·x}, where A_SD(α)v_SD(α)=α.

[0155] Referring to FIG. 1A, a graph of an SD waveform for (x=0,t) is depicted. The truncated input waveform at x=0 reaching a limit at t=t_i is shown.

[0156] In the case of a four parameter (LCRG) transmission line, we have the expressions

[0157] $\begin{array}{c}{A}_{\mathrm{SD}}\left(\alpha \right)=\sqrt{\mathrm{LC}{\alpha }^2+\left(\mathrm{RC}+\mathrm{LG}\right)\alpha +\mathrm{RG}},\mathrm{and}\\ v_{\mathrm{SD}}\left(\alpha \right)=\frac{\alpha }{\sqrt{\mathrm{LC}{\alpha }^2+\left(\mathrm{RC}+\mathrm{LG}\right)\alpha +\mathrm{RG}}}.\end{array}$

[0158] The attenuation of the traveling SD wave may be viewed from the perspective of a moving reference frame traveling at the speed 1/√{square root over (LC)}.

[0159] Introducing the new coordinates:
t′=t−x√{square root over (LC)}
x′=x
Then
V(x′,t′)=De^{−ASD(α)·[x′−vSD(α)·(t′+x′·√{square root over (LC)})]},
or
V(x′,t′)=De^α·t′e^{−ASD(α)·[1−vSD(α)·√{square root over (LC)}]·x′}.

[0160] If we view the traveling wave after traveling a distance, x=x′=l, in the transmission media, then
V(l,t′)=De^α·t′e^{−ASD(α)·[1−vSD(α)·√{square root over (LC)}]·l}.

[0161] Setting t′=t_i, yields the maximum value of the truncated SD waveform at x=l, and for any value 0≦t_j≦t_i, the value of the signal at the associated point t_j in the moving reference frame at x=l is:
V(l,t_j)=De^α·tje^{−ASD(α)·[1−vSD(α)·√{square root over (LC)}]·l}

[0162] Referring to FIG. 1B, a graph of an SD waveform in moving frame at x′=l (x=l,t′) is depicted. The attenuation of this truncated waveform is defined by the ratio:

[0163] $\begin{array}{c}\frac{V\left(l,t_j\right)}{V\left(0,t_j\right)}=\frac{D{e}^{\alpha ·t_j}{e}^{-A_{\mathrm{SD}}\left(\alpha \right)·\lfloor 1-v_{\mathrm{SD}}\left(\alpha \right)·\sqrt{\mathrm{LC}}\rfloor ·l}}{D{e}^{\alpha ·t_j}}\\ =e^{-A_{\mathrm{SD}}\left(\alpha \right)·\left[1-v_{\mathrm{SD}}\left(\alpha \right)·\sqrt{\mathrm{LC}}\right]·l}\end{array}$

[0164] This attenuation of the signal at a point t_j of a truncated SD waveform traveling in a transmission line with frequency dependent parameters whose Laplace transforms are L(s), C(s), R(s), and G(s), is:

[0165] $\frac{V\left(l,t_j\right)}{V\left(0,t_j\right)}=e^{-{\stackrel{\_}{A}}_{\mathrm{SD}}\left(\alpha \right)·\lfloor 1-{\stackrel{\_}{v}}_{\mathrm{SD}}\left(\alpha \right)·\sqrt{\stackrel{\_}{\mathrm{LC}}\left(\alpha \right)}\rfloor ·l}\mathrm{where}\text{:}$ ${\stackrel{\_}{A}}_{\mathrm{SD}}\left(\alpha \right)=\sqrt{\stackrel{\_}{\mathrm{LC}}\left(\alpha \right)·{\alpha }^2+\left(\stackrel{\_}{\mathrm{RC}}\left(\alpha \right)+\stackrel{\_}{\mathrm{LG}}\left(\alpha \right)\right)·\alpha +\stackrel{\_}{\mathrm{RG}}\left(\alpha \right)},\mathrm{and}$ ${\stackrel{\_}{v}}_{\mathrm{SD}}\left(\alpha \right)=\frac{\alpha }{\sqrt{\stackrel{\_}{\mathrm{LC}}\left(\alpha \right)·{\alpha }^2+\left(\stackrel{\_}{\mathrm{RC}}\left(\alpha \right)+\stackrel{\_}{\mathrm{LG}}\left(\alpha \right)\right)·\alpha +\stackrel{\_}{\mathrm{RG}}\left(\alpha \right)}}.$

[0166] Furthermore, in this transmission media with frequency dependent parameters, if the input exponential pulse is rapidly closed when the exponential amplitude truncation limit is reached, then the end of the exponential region of the leading edge of the transmitted pulse is further reduced in amplitude and rounded by the chromatic dispersion.

[0167] Coupled transmission lines can be treated in a similar fashion. The equations governing SD propagation on one of two coupled transmission lines is:

[0168] $V_1\left(x,t\right)=A_{11}·e^{-A_{{\mathrm{SD}}_{11}}\left(x-v_{{\mathrm{SD}}_{11}}t\right)}+A_{21}·e^{-A_{{\mathrm{SD}}_{21}}\left(x-v_{{\mathrm{SD}}_{21}}t\right)}+A_{12}·e^{-A_{{\mathrm{SD}}_{12}}\left(x-v_{\mathrm{SD}2}t\right)}+A_{22}·e^{-A_{{\mathrm{SD}}_{22}}\left(x-v_{{\mathrm{SD}}_{22}}t\right)}.$

[0169] Applying the substitution: t′=t−x√{square root over ( LC, x′=x, gives the result:

[0170] $\frac{V_1\left(x^{\prime }=l,t^{\prime }=t_i\right)}{V_1\left(x=0,t=t_i\right)}=\frac{\begin{array}{c}{A}_{11}·e^{{\alpha }_1{t}_i}·e^{-A_{{\mathrm{SD}}_{11}}\left(1-v_{{\mathrm{SD}}_{11}}\sqrt{\stackrel{\_}{\mathrm{LC}}}\right)l}+\\ A_{21}·e^{{\alpha }_1{t}_i}·e^{-A_{{\mathrm{SD}}_{21}}\left(1-v_{{\mathrm{SD}}_{21}}\sqrt{\stackrel{\_}{\mathrm{LC}}}\right)l}\end{array}}{A_{11}·e^{{\alpha }_1{t}_i}+A_{21}·e^{{\alpha }_1{t}_i}+A_{12}·e^{{\alpha }_2{t}_i}+A_{22}·e^{{\alpha }_2{t}_i}}+\frac{\begin{array}{c}{A}_{12}·e^{{\alpha }_2{t}_i}·e^{-A_{{\mathrm{SD}}_{12}}\left(1-v_{{\mathrm{SD}}_{12}}\sqrt{\stackrel{\_}{\mathrm{LC}}}\right)l}+\\ A_{22}·e^{{\alpha }_2{t}_i}·e^{-A_{{\mathrm{SD}}_{22}}\left(1-v_{{\mathrm{SD}}_{22}}\sqrt{\stackrel{\_}{\mathrm{LC}}}\right)l}\end{array}}{A_{11}·e^{{\alpha }_1{t}_i}+A_{21}·e^{{\alpha }_1{t}_i}+A_{12}·e^{{\alpha }_2{t}_i}+A_{22}·e^{{\alpha }_2{t}_i}}$

[0171] Which can be written:

[0172] $\frac{V_1\left(l,t_j\right)}{V_1\left(0t_j\right)}=\frac{\left(A_{11}·e^{-A_{{\mathrm{SD}}_{11}}\left(1-v_{{\mathrm{SD}}_{11}}\sqrt{\stackrel{\_}{\mathrm{LC}}}\right)l}+A_{21}·e^{-A_{{\mathrm{SD}}_{21}}\left(1-v_{{\mathrm{SD}}_{21}}\sqrt{\stackrel{\_}{\mathrm{LC}}}\right)l}\right)}{\left(A_{11}+A_{21}\right)+\left(A_{12}+A_{22}\right)·e^{\left({\alpha }_2-{\alpha }_1\right)t_i}}+\frac{\left(A_{12}·e^{-A_{{\mathrm{SD}}_{12}}\left(1-v_{{\mathrm{SD}}_{12}}\sqrt{\stackrel{\_}{\mathrm{LC}}}\right)l}+A_{22}·e^{-A_{{\mathrm{SD}}_{22}}\left(1-v_{{\mathrm{SD}}_{22}}\sqrt{\stackrel{\_}{\mathrm{LC}}}\right)l}\right)}{\left(A_{11}+A_{21}\right)·e^{\left({\alpha }_1-{\alpha }_2\right)t_i}+\left(A_{12}+A_{22}\right)}.$

[0173] Another example includes of the evolution of a slowly varying envelope, E(x,t), of an optical pulse in a single mode communication fiber. The Schrödinger partial differential equation [3],

[0174] $\frac{\partial E}{\partial x}+{\beta }_1\frac{\partial E}{\partial t}-\frac{j}{2}{\beta }_2\frac{{\partial }^{2}E}{\partial t^2}-\frac{1}{6}{\beta }_3\frac{{\partial }^{3}E}{\partial t^3}=-\frac{\gamma }{2}E,$
describes the evolution of the shape of the propagating pulse envelope undergoing chromatic dispersion in fiber. Chromatic dispersion occurs because the mode-propagation constant, β(ω), is a nonlinear function of the angular frequency ω), where

[0175] ${\beta }_n={\left[\frac{{\partial }^n\beta }{\partial {\omega }^n}\right]}_{\omega ={\omega }_o}$
and ω_o is the frequency of the light being modulated in the fiber.

[0176] ω_o/β_o is the phase velocity of the pulse. 1/β₁ is the group velocity, β₂ is the group velocity dispersion (GVD) parameter which causes the pulse to broaden as it propagates in the fiber. β₃ is the third-order dispersion (TOD) parameter [4]

[0177] The higher order derivatives of β with respect to ω are assumed negligible. However, they may be added in these analyses by anyone familiar with the art. The fiber attenuation is represented by the parameter γ.

[0178] The SD solution of the linear Schrödinger equation with loss (γ≠0) is:

[0179] $E\left(x,t\right)={\mathrm{De}}^{\left[\frac{t}{T_o}-\left(\frac{{\beta }_1}{T_o}-\frac{{\beta }_3}{6T_o^3}+\frac{\gamma }{2}\right)·x\right]}{e}^{j\left(\frac{{\beta }_2·x}{2T_o^2}\right)}.$

[0180] The boundary condition at the input to the fiber at x=0 is

[0181] $E\left(0,t\right)={\mathrm{De}}^{\frac{t}{\mathrm{To}}},$
(see FIG. 1A with

[0182] $\alpha =\frac{1}{T_o}).$
The receiver detector in a fiber optic communication network responds to the square of the magnitude |E(x,t)|² of the propagating pulse envelope. In this case
|E(x,t)|²=D²e^{−2·A(To)[x−vsd(To)·t]},

[0183] where the velocity of propagation of a truncated SD leading edge of the envelope is

[0184] $v_{\mathrm{sd}}\left(T_o\right)=\frac{1}{\left({\beta }_1-\frac{{\beta }_3}{6T_o^2}+\frac{T_o\gamma }{2}\right)},\mathrm{and}A\left(T_o\right)=\left(\frac{{\beta }_1}{T_o}-\frac{{\beta }_3}{6T_o^3}+\frac{\gamma }{2}\right).\mathrm{Note}\mathrm{that}{v}_{\mathrm{sd}}\left(T_o\right)·A\left(T_o\right)=\frac{1}{T_o}.$

[0185] Transforming to a moving reference frame traveling at the group velocity 1/β₁,
t′=t−β₁x
x′=x

[0186] The SD propagation speed in this moving reference frame is

[0187] ${{\stackrel{\_}{v}}_{\mathrm{sd}}\left(T_o\right)}_{\mathrm{in}{x}^{\prime },t^{\prime }\mathrm{frame}}=\frac{1}{T_o\left(\frac{\gamma }{2}-\frac{{\beta }_3}{6T_o^3}\right)},$

[0188] The attenuation of the truncated SD portion of the leading edge of the pulse envelope is

[0189] $\frac{{E\left(l,t_j\right)}^2}{{E\left(0,t_j\right)}^2}=e^{-\left(\frac{\gamma }{2}-\frac{{\beta }_3}{6T_o^3}\right)l}.$

[0190] This result implies that if T_o can be made small enough, this attenuation may be reduced. Higher order terms of β(ω), which were not included is this particular analysis, may become more significant as T_o become smaller.

[0191] Specific embodiments of the invention will now be further described by the following, nonlimiting examples which will serve to illustrate in some detail various features. The following examples are included to facilitate an understanding of ways in which the invention may be practiced. It should be appreciated that the examples which follow represent embodiments discovered to function well in the practice of the invention, and thus can be considered to constitute preferred modes for the practice of the invention. However, it should be appreciated that many changes can be made in the exemplary embodiments which are disclosed while still obtaining like or similar result without departing from the spirit and scope of the invention. Accordingly, the examples should not be construed as limiting the scope of the invention.

[0192] This example teaches how SD test pulses may be used to obtain empirical estimates of the transmission line parameters, A_SD and v_SD, that describe the propagation of SD waveforms in lossy, dispersive lines, including those with frequency dependent parameters. The numerical values of A_SD and v_SD as a function of α can be determined empirically.

[0193] Referring to FIG. 2, an experimental set up that may be used to determine these parameters for a copper, twisted wire pair transmission line is depicted. The test length of the transmission line, d, and the value of a for the applied SD input signal are measured. An additional length of transmission line that is identical to the line under test is attached as shown and terminated with an open circuit. In this example the cables are two 24 AWG, individually shielded, twisted wire pairs in a 1002 ft coil of T1 cable.

[0194] The values of D, α and the duration of the SD signal were chosen to prevent the occurrence of reflections at the measurement point, d, during the initial time of flight of the SD signal propagating in the test line. The additional length of transmission line also ensures that all reflections occurring in the test line are delayed until well after the propagating SD waveform measurement at d is complete.

[0195] The SD signal time of flight can be directly measured by timing the propagating SD waveform crossing of constant voltage thresholds (FIG. 3). The value of the SD signal propagation velocity in the line, v_SD, can be calculated using the relationship, TOF=(t_f−t_o)=v_sdd. The times of flight for these waveforms shown again in FIG. 4A at thresholds from 1.2 volts to 4.8 volts are plotted in FIG. 4B. The average TOF for the threshold overlap region shown in FIG. 4.3b is 1,716 ns. This TOF results in

[0196] $v_{\mathrm{SD}}=\frac{d}{{\mathrm{TOF}}_{\mathrm{AVG}}}=\frac{1002\mathrm{ft}}{1\text{,}716\times {10}^{-9}\sec }=5.839\times {10}^8\frac{\mathrm{ft}}{\sec }.$

[0197] The SD attenuation coefficient, A_SD, can be calculated from α and v_sd using the relationship A_SD·v_SD=α. The calculated A_SD obtained from the waveforms shown in FIG. 4.3a is

[0198] $A_{\mathrm{SD}}=\frac{\alpha }{v_{\mathrm{SD}}}=\frac{3.0\times {10}^{6}1\text{/}\sec }{5.839\times {10}^8\mathrm{ft}\text{/}\sec }=5.138\times {10}^{-3}1\text{/}\mathrm{ft}.$

[0199] Once this velocity is known for a calibrated length of the cable, this type of threshold crossing TOF measurement can be used to determine the unknown length of another sample of the same transmission line.

[0200] Referring to FIG. 5A, the measured input voltage of the two twisted wire pairs in a T1 line connected in series is depicted (see FIG. 2) and one-half of the voltage measured at the end of the open terminated second twisted wire pair in the T1 line (R_t=∞ in FIG. 2). The average TOF for the threshold overlap region shown in FIG. 5B is 3,418 ns. This time of flight together with the previously calibrated v_SD, results in an estimated distance of 1,996 ft, or 99.6% of the actual length (2004 ft).

[0201] FIG. 5A also indicates a potential difficulty in longer lines caused by the attenuation of the truncated SD pulse. The region of SD threshold overlap grows smaller as the pulse propagates, and is attenuated over longer distances. The accuracy of this measurement can be improved, and the difficulty caused by the reduced region of threshold overlap can be overcome for very long transmission lines using the TOF measurement technique discussed in example 4.

[0202] The high accuracy of this method results from the use of an average of TOF measurements taken over a range of signal threshold amplitudes. It is feasible to use the average of these values to improve the accuracy of the measurement of TOF since all points on the leading SD edge of the waveform travel at the same speed. This is in contrast with the current time domain reflectometry art in which a measurement of the time of emergence of the dispersing test pulse is attempted. This is equivalent to attempting to accurately measure the threshold crossing time of the pulse at a zero threshold level when the slope of the pulse is also nearly zero.

[0203] The method described in example 1 can be limiting in that it should be experimentally repeated for each exponential coefficient α. This may require a lengthy laboratory calibration time for each cable type. Simulation can be included in the calibration process to significantly reduce the experimental measurements needed to determine the values of A_SD and V_SD. The use of simulation in this process requires a transfer function that describes the response of the specific type of cable. The α's analyzed for the T1 cable range from 1×10⁵ sec⁻¹ to 1×10⁷ sec⁻¹. This range of α's corresponds to the frequency band of 16 kHz to 1.6 MHz, so the transfer function must be accurate for this range. The transfer function of the cable may available from the manufacturer or can be determined empirically by applying a known pulse and measuring the applied cable input waveform and the response waveform at a known (calibration) distance along the cable. The transfer function is then calculated from the ratio of the fast Fourier transforms (FFT) of the input-output pulses:

[0204] $H_{\mathrm{est}}\left(\mathrm{j\omega }\right)=\frac{\mathrm{FFT}\left(V_{\mathrm{Measured}\mathrm{at}l}\right)}{\mathrm{FFT}\left(V_{\mathrm{Applied}}\right)}.$
Once the transfer function is determined, the propagation of SD waveforms with a series of different values for a can be simulated and the times of flight and SD parameters, A_SD and v_SD, determined.

[0205] This method can be demonstrated for the T1 cable used in example 1. The end of the total 2004 length has been terminated with a 100-Ohm resistor (R_t in FIG. 4.1) to minimize reflections at the 1002 ft measurement point. FIG. 6A shows a measured input pulse applied to the T1. FIG. 6B shows the output pulse measured at 1002 ft. FIGS. 7A-B show the Power Spectral Density of the measured pulses. The range of α's is indicated to show that there is sufficient energy in the frequency band of interest to empirically estimate an accurate transfer function. The magnitude and the phase of the transfer function determined from the measured signals are shown in FIG. 8.

[0206] Once the transfer function is known, a waveform with any ax in the indicated range of interest can be simulated and the value of A_SD and V_SD determined. The values of A_SD and v_SD of the measured SD waveforms of section three provide a test of the transfer function for an α of 3×10⁶ sec⁻¹. FIG. 9 shows the time of flight plots determined using the input signal measured in section two, and the 1002 ft waveform predicted by simulation using the empirical transfer function. The simulation TOF result (FIG. 9) of 1,714 nsec is almost identical to the experimentally measured (FIG. 5B) TOF of 1,716 nsec. FIG. 10 shows the difference between the measured waveform at 1002 ft and the simulated waveform at 1002 ft (FIG. 9, Top). During the SD propagation region the difference is on the order of ±20 mV.

[0207] This process is performed for the range of α's by simulating the initial SD waveforms and using the transfer function to predict the waveforms at 1002 ft. In all cases, the simulated initial waveforms were closed with a 1 μsec ramp to reduce their high frequency components. FIGS. 11-12 show the simulated input and 1002 ft curves for the endpoint α's of 1×10⁵ sec⁻¹ and 1×10⁷ sec⁻¹ along with their respective constant threshold times of flight curves. The values of v_SD and A_SD from simulation are presented in FIGS. 13A-B, illustrated in Table I for the range of α's, as set forth below (* values in parenthesis obtained using methods of example 1, with measured input and output SD signals along actual T1 Line):

[0208]

TABLE I
vSD and ASD Determined By Simulation with Empirical
Transfer Function
Alpha	1002 Simulation TOF	Simulation vSD	Simulation ASD
(sec−1)	(μsec)	(×108 ft/sec)	(×10−3 l/ft)
1 × 105	4.487	2.233	0.448
2 × 105	3.068	3.266	0.612
3 × 105	2.608	3.842	0.781
4 × 105	2.382	4.207	0.951
5 × 105	2.271	4.412	1.133
6 × 105	2.172	4.613	1.301
7 × 105	2.103	4.765	1.469
8 × 105	2.040	4.912	1.629
9 × 105	1.999	5.013	1.796
1 × 106	1.973	5.079	1.969
2 × 106	1.792	5.592	3.577
3 × 106	1.714 (1.716*)	5.849 (5.839*)	5.129 (5.138*)
4 × 106	1.666	6.014	6.651
5 × 106	1.634	6.132	8.154
6 × 106	1.610	6.224	9.641
7 × 106	1.592	6.294	11.122
8 × 106	1.577	6.354	12.591
9 × 106	1.565	6.403	14.057
1 × 107	1.555	6.444	15.519

[0209] The SD line impedance, Z_SD(α), is a real number which may be determined for the α's of interest. If the Laplace transform of the frequency dependent transmission line parameters R(s), L(s), C(s), and G(s), are known, then Z_SD(α) may be computed from,

[0210] $Z_{\mathrm{SD}}=\sqrt{\frac{\alpha \stackrel{\_}{L}\left(\alpha \right)+\stackrel{\_}{R}\left(\alpha \right)}{\alpha \stackrel{\_}{C}\left(\alpha \right)+\stackrel{\_}{G}\left(\alpha \right)}}.$

[0211] The SD line impedance, Z_SD(α), may also be experimentally determined for a test transmission line by measuring the SD portion of the voltage waveform across various known termination impedances and computing the value Z_SD(α) from the measurements. The measured SD waveform at a termination, V_SD(R_t), consists of the sum of the incident SD waveform, V_SD⁺, and a reflected SD wave. The reflected wave is the product of the SD reflection coefficient, Γ, and the incident wave. The measured SD signal at the termination will be

[0212] $V_{\mathrm{SD}}\left(R_t\right)=\left(1+{\Gamma }_t\right)·V_{\mathrm{SD}}^{+}=\frac{2·R_t}{R_t+Z_{\mathrm{SD}}}·V_{\mathrm{SD}}^{+}.$
Note that all lumped and distributed impedance values are real for SD signals.

[0213] The experimental setup used to determine Z_SD from line measurements is shown in FIG. 14. The measured input and termination waveforms of the voltages for a 1002 ft of T1 line, are shown in FIG. 15A. These measurements were made with an α of 1.5×10⁶ 1/sec and termination resistances of 49.5 ohms, 98.6 ohms and an open circuit. The magnitude of the waveform with no reflection, (the incident waveform V_SD⁺), is determined by dividing the signal measured with an open termination, (+R_t=∞, Γ_∞=1), in half. The ratio of the waveforms V_SD(R_t)/V_SD⁺, with known terminations and the incident waveform gives a direct measurement of the reflection coefficient, Γ_t, for each termination resistance,

[0214] ${\Gamma }_t=\frac{V_{\mathrm{SD}}\left(R_t\right)}{V_{\mathrm{SD}}^{+}}-1.$
FIG. 15B shows the measured ratios for this cable and α. Using the relationship for the transmission coefficient,

[0215] $1+{\Gamma }_t=\frac{2·R_t}{R_t+Z_{\mathrm{SD}}},\mathrm{gives}{Z}_{\mathrm{SD}}=\frac{R_t}{1+{\Gamma }_t}\left(1-{\Gamma }_t\right).$

[0216] Using the two finite values for R_t, 98.6 Ω and 49.5 Ω, yields two estimates for Zsd:

[0217] $\begin{array}{c}{Z}_{\mathrm{SD}}\left(\alpha =1.5\times {10}^6\frac{1}{\sec }\right)=\frac{98.6}{1+\left(0.919-1\right)}\left(1-\left(0.919-1\right)\right)\\ =116\Omega ,\end{array}$ $\mathrm{or}$ $\begin{array}{c}{Z}_{\mathrm{SD}}\left(\alpha =1.5\times {10}^6\frac{1}{\sec }\right)=\frac{49.5}{1+\left(0.598-1\right)}\left(1-\left(0.598-1\right)\right)\\ =116\Omega ,\mathrm{for}\mathrm{this}\mathrm{T1}\mathrm{cable}.\end{array}$

[0218] This process was repeated for α's from 5×10⁵ sec⁻¹ to 10×10⁶ sec⁻¹ and the results are shown in FIG. 16.

[0219] This example describes an alternate precision TOF measurement method of SD test pulses for higher loss, longer lines whose attenuation of the test pulses is so extensive that their amplitudes become too small as they travel down the line for a common threshold crossing measurement to be feasible.

[0220] The time of flight measured at a constant voltage threshold becomes more difficult to estimate as the waveform is attenuated. This can be seen by examining the voltage traces along an approximately 6-kft, 24 AWG, fifty twisted wire pair, telephone cable (See FIG. 17). Three pairs of the fifty have been connected in series to generate an approximately 18-kft cable with three equal length sections. The applied SD input waveform is De^αt, with D of 0.19 volts and an α of 3.5×10⁵ 1/sec. This α is approximately one tenth of the α used for the 2-kft T1 cable. The smaller α is required here to reduce the attenuation of the truncated SD leading edge of the propagating pulse over these longer distances as discussed in section two. This allows signal detection over longer distances. The value of A_SD and v_SD for this cable and α were determined using the procedure of example 1. These measured values are based on assuming a 6000 ft calibration length for each twisted pair length inside the telephone cable. The results are v_SD=4.348×10⁸ ft/sec and A_SD=8.279×10^{31 4} 1/ft.