Methods for transmitting a waveform having a controllable attenuation and propagation velocity

[00002] This application is a continuation-in-part of, and claims a benefit of priority under 35 U.S.C. 119(e) and/or 35 U.S.C. 120 from, U.S. Ser. No. 09/519,922, filed Mar. 7, 2000, now U.S. Pat. No. 6,441,695, the entire contents of which are hereby expressly incorporated by reference for all purposes.

[00003] 1. Field of the Invention

[00004] 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.

[00005] 2. Discussion of the Related Art

[00006] 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.

[00007] 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.

[00008] 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 2 pulse launch point to the location of the impedance change causing the pulse to be reflected and returned.

[00009] 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.

[00010] 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.

[00011] Until now, the requirements of providing a method an d/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.

[00012] There is a need for the following embodiments. Of course, the invention is not limited to these embodiments.

[00013] According to an aspect of the invent ion, a method for transmitting a waveforms having an essentially constant propagation velocity along a transmission line comprises: generating an exponential waveform, the exponential waveform (a) being characterized by the equation V_in=De^ASD(x−vSDt), where D is a magnitude, V_in is a voltage, t is time, A_SD is an attenuation coefficient, and v_SD is a propagation velocity; and (b) being truncated at a maximum value. According to another aspect of the invention, an apparatus for determining an attenuation coefficient and a propagation velocity comprises: an exponential waveform generator; an input recorder coupled to an output of the exponential waveform generator; a transmission line under test coupled to the output, of the exponential waveform generator; an output recorder coupled to the transmission line under test; an additional transmission line coupled to the transmission line under test; and a termination impedance coupled to the additional transmission line and to a ground.

[00014] 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.

[00083] 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.

[00084] 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.

[00085] The below-referenced allowed U.S. Patent Application discloses embodiments that were satisfactory for the purposes for which they are intended. The entire contents of U.S. patent application Ser. No. 09/519,922, filed Mar. 7, 2000, is hereby expressly incorporated by reference herein for all purposes.

[00086] 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.

[00087] 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.

[00088] 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.

[00089] 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, VSD, 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).

[00090] 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.

[00091] 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.

[00092] 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.

[00093] 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.

[00094] 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.

[00095] 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.

[00096] 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.

[00097] 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.

[00098] 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.

[00099] 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.

[00100] 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.

[00101] 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 a, and may be determined experimentally for a transmission line.

[00102] 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: $\frac{\partial }{\partial x}V=Z·I$ $\frac{\partial }{\partial x}I=Y·V$ $\frac{\partial }{\partial x}\left[\begin{array}{c}{V}_1\left(x,t\right)\\ V_2\left(x,t\right)\end{array}\right]=\left[\begin{array}{cc}{R}_1+L_1\frac{\partial }{\partial t}& M_{12}\frac{\partial }{\partial t}\\ M_{21}\frac{\partial }{\partial t}& R_2+L_2\frac{\partial }{\partial t}\end{array}\right]·\left[\begin{array}{c}{I}_1\left(x,t\right)\\ I_2\left(x,t\right)\end{array}\right].\frac{\partial }{\partial x}\left[\begin{array}{c}{I}_1\left(x,t\right)\\ I_2\left(x,t\right)\end{array}\right]=\left[\begin{array}{cc}{G}_1+C_1\frac{\partial }{\partial t}& C_{12}\frac{\partial }{\partial t}\\ C_{21}\frac{\partial }{\partial t}& G_2+C_2\frac{\partial }{\partial t}\end{array}\right]·\left[\begin{array}{c}{V}_1\left(x,t\right)\\ V_2\left(x,t\right)\end{array}\right]$
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. $\frac{\partial }{\partial x}\left[\begin{array}{c}{V}_1\left(x,s\right)\\ V_2\left(x,s\right)\end{array}\right]=\left[\begin{array}{cc}{\stackrel{\_}{R}}_1\left(s\right)+s{\stackrel{\_}{L}}_1\left(s\right)& s{\stackrel{\_}{M}}_{12}\left(s\right)\\ s{\stackrel{\_}{M}}_{21}\left(s\right)& {\stackrel{\_}{R}}_2\left(s\right)+s{\stackrel{\_}{L}}_2\left(s\right)\end{array}\right]·\left[\begin{array}{c}{I}_1\left(x,s\right)\\ I_2\left(x,s\right)\end{array}\right]$ $\frac{\partial }{\partial x}\left[\begin{array}{c}{I}_1\left(x,s\right)\\ I_2\left(x,s\right)\end{array}\right]=\left[\begin{array}{cc}{\stackrel{\_}{G}}_1\left(s\right)+s{\stackrel{\_}{C}}_1\left(s\right)& s{\stackrel{\_}{C}}_{12}\left(s\right)\\ s{\stackrel{\_}{C}}_{21}\left(s\right)& {\stackrel{\_}{G}}_2\left(s\right)+s{\stackrel{\_}{C}}_2\left(s\right)\end{array}\right]·\left[\begin{array}{c}{V}_1\left(x,s\right)\\ V_2\left(x,s\right)\end{array}\right]$
Taking the partial derivative of the top equation with respect to x, and substituting the bottom equation for the $\frac{\partial }{\partial x}\left[\begin{array}{c}{I}_1\left(x,s\right)\\ I_2\left(x,s\right)\end{array}\right]$
term, gives the result: $\frac{{\partial }^2}{\partial x^2}V\left(x,s\right)=\stackrel{\_}{Z}\left(s\right)\stackrel{\_}{Y}\left(s\right)·V\left(x,s\right)$ $\begin{array}{c}\frac{{\partial }^2}{\partial x^2}\left[\begin{array}{c}{V}_1\left(x,s\right)\\ V_2\left(x,s\right)\end{array}\right]=\left[\begin{array}{cc}{\stackrel{\_}{R}}_1\left(s\right)+s{\stackrel{\_}{L}}_1\left(s\right)& s{\stackrel{\_}{M}}_{12}\left(s\right)\\ s{\stackrel{\_}{M}}_{21}\left(s\right)& {\stackrel{\_}{R}}_2\left(s\right)+s{\stackrel{\_}{L}}_2\left(s\right)\end{array}\right]·\\ \left[\begin{array}{cc}{\stackrel{\_}{G}}_1\left(s\right)+s{\stackrel{\_}{C}}_1\left(s\right)& s{\stackrel{\_}{C}}_{12}\left(s\right)\\ s{\stackrel{\_}{C}}_{21}\left(s\right)& {\stackrel{\_}{G}}_2\left(s\right)+s{\stackrel{\_}{C}}_2\left(s\right)\end{array}\right]·\left[\begin{array}{c}{V}_1\left(x,s\right)\\ V_2\left(x,s\right)\end{array}\right]\end{array}$ $\mathrm{or}$ $\frac{{\partial }^2}{\partial x^2}V\left(x,s\right)={\stackrel{\_}{\Gamma }\left(s\right)}^2·V\left(x,s\right)$ $\begin{array}{c}\frac{{\partial }^2}{\partial x^2}\left[\begin{array}{c}{V}_1\left(x,s\right)\\ V_2\left(x,s\right)\end{array}\right]=\left[\begin{array}{cc}\left({\stackrel{\_}{R}}_1\left(s\right)+s{\stackrel{\_}{L}}_1\left(s\right)\right)\left({\stackrel{\_}{G}}_1\left(s\right)+s{\stackrel{\_}{C}}_1\left(s\right)\right)& s^2{\stackrel{\_}{M}}_{12}\left(s\right){\stackrel{\_}{C}}_{21}\left(s\right)\\ s^2{\stackrel{\_}{M}}_{21}\left(s\right){\stackrel{\_}{C}}_{12}\left(s\right)& \left({\stackrel{\_}{R}}_2\left(s\right)+s{\stackrel{\_}{L}}_2\left(s\right)\right)\left({\stackrel{\_}{G}}_2\left(s\right)+s{\stackrel{\_}{C}}_2\left(s\right)\right)\end{array}\right]·\\ \left[\begin{array}{c}{V}_1\left(x,s\right)\\ V_2\left(x,s\right)\end{array}\right]\end{array}$
This system of equations has the solution:
V(x,s)=e^{−{overscore (Γ)}(s)·x}·A+e^{+{overscore (Γ)}(s)·x}·B

[00108] 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^{−{overscore (Γ)}(s)·x}·V(x=0,s).

[00110] Because {overscore (Γ)} is a complex symmetric matrix, it can be written {overscore (Γ)}=({overscore (ZY)})^1/2={overscore (PγP)}⁻¹, where {overscore (P)} is the eigenvector matrix of the {overscore (ZY)} product and {overscore (γ)} is the diagonal eigenvalue matrix of {overscore (Γ)}. Therefore, we can write e^{−{overscore (Γ)}(s)·x}=P(s)·e^{−{overscore (Γ)}(s)·x}·P(s)⁻¹. This gives the result:
V(x,s)=P(s)·e^{−{overscore (γ)}(s)·x}·P(s)⁻¹·V(x=0,s)

[00112] 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: $\begin{array}{c}V\left(x,t\right)=P\left({\alpha }_1\right)·e^{-\gamma \left({\alpha }_1\right)·x}·P^{-1}\left({\alpha }_1\right)·\left[\begin{array}{c}{D}_1·e^{{\alpha }_1·t}\\ 0\end{array}\right]+P\left({\alpha }_2\right)·\\ e^{-\gamma \left({\alpha }_2\right)·x}·P^{-1}\left({\alpha }_2\right)·\left[\begin{array}{c}0\\ D_2·e^{{\alpha }_2·t}\end{array}\right],\end{array}$
where P(α_i) is a real matrix. $\begin{array}{c}V\left(x,t\right)=\left[\begin{array}{cc}{P}_{11}\left({\alpha }_1\right)& P_{12}\left({\alpha }_1\right)\\ P_{21}\left({\alpha }_1\right)& P_{22}\left({\alpha }_1\right)\end{array}\right]·\left[\begin{array}{cc}{e}^{-{\gamma }_1\left({\alpha }_1\right)·x}& 0\\ 0& e^{-{\gamma }_2\left({\alpha }_1\right)·x}\end{array}\right]·\\ {\left[\begin{array}{cc}{P}_{11}\left({\alpha }_1\right)& P_{12}\left({\alpha }_1\right)\\ P_{21}\left({\alpha }_1\right)& P_{22}\left({\alpha }_1\right)\end{array}\right]}^{-1}·\left[\begin{array}{c}{D}_1·e^{{\alpha }_1·t}\\ 0\end{array}\right]+\left[\begin{array}{cc}{P}_{11}\left({\alpha }_2\right)& P_{12}\left({\alpha }_2\right)\\ P_{21}\left({\alpha }_2\right)& P_{22}\left({\alpha }_2\right)\end{array}\right]·\\ \left[\begin{array}{cc}{e}^{-{\gamma }_1\left({\alpha }_2\right)·x}& 0\\ 0& e^{-{\gamma }_2\left({\alpha }_2\right)·x}\end{array}\right]·{\left[\begin{array}{cc}{P}_{11}\left({\alpha }_2\right)& P_{12}\left({\alpha }_2\right)\\ P_{21}\left({\alpha }_2\right)& P_{22}\left({\alpha }_2\right)\end{array}\right]}^{-1}·\left[\begin{array}{c}0\\ D_2·e^{{\alpha }_2·t}\end{array}\right]\end{array}$
If we designate P⁻¹ as Q, we can write: $\begin{array}{c}V\left(x,t\right)=\left[\begin{array}{cc}{P}_{11}\left({\alpha }_1\right)& P_{12}\left({\alpha }_1\right)\\ P_{21}\left({\alpha }_1\right)& P_{22}\left({\alpha }_1\right)\end{array}\right]·\left[\begin{array}{cc}{e}^{-{\gamma }_1\left({\alpha }_1\right)·x}& 0\\ 0& e^{-{\gamma }_2\left({\alpha }_1\right)·x}\end{array}\right]·\\ \left[\begin{array}{cc}{Q}_{11}\left({\alpha }_1\right)& Q_{12}\left({\alpha }_1\right)\\ Q_{21}\left({\alpha }_1\right)& Q_{22}\left({\alpha }_1\right)\end{array}\right]·\left[\begin{array}{c}{D}_1·e^{{\alpha }_1·t}\\ 0\end{array}\right]+\\ \left[\begin{array}{cc}{P}_{11}\left({\alpha }_1\right)& P_{12}\left({\alpha }_1\right)\\ P_{21}\left({\alpha }_1\right)& P_{22}\left({\alpha }_1\right)\end{array}\right]·\left[\begin{array}{cc}{e}^{-{\gamma }_1\left({\alpha }_1\right)·x}& 0\\ 0& e^{-{\gamma }_2\left({\alpha }_1\right)·x}\end{array}\right]·\\ \left[\begin{array}{cc}{Q}_{11}\left({\alpha }_2\right)& Q_{12}\left({\alpha }_2\right)\\ Q_{21}\left({\alpha }_2\right)& Q_{22}\left({\alpha }_2\right)\end{array}\right]·\left[\begin{array}{c}0\\ D_2·e^{{\alpha }_2·t}\end{array}\right]\end{array}$ $\mathrm{or},\begin{array}{c}V\left(x,t\right)=P\left({\alpha }_1\right)·\left[\begin{array}{c}{Q}_{11}\left({\alpha }_1\right)·e^{-{\gamma }_1\left({\alpha }_1\right)·x}\\ Q_{21}\left({\alpha }_1\right)·e^{-{\gamma }_2\left({\alpha }_1\right)·x}\end{array}\right]·D_1·e^{{\alpha }_1·t}+P\left({\alpha }_2\right)·\\ \left[\begin{array}{c}{Q}_{12}\left({\alpha }_2\right)·e^{-{\gamma }_1\left({\alpha }_2\right)·x}\\ Q_{22}\left({\alpha }_2\right)·e^{-{\gamma }_2\left({\alpha }_2\right)·x}\end{array}\right]·D_2·e^{{\alpha }_2·t}\end{array}$ $\mathrm{or},\begin{array}{c}V\left(x,t\right)=\left[\begin{array}{c}{Q}_{11}\left({\alpha }_1\right)P_{11}\left({\alpha }_1\right)e^{-{\gamma }_1\left({\alpha }_1\right)·x}+Q_{21}\left({\alpha }_1\right)P_{12}\left({\alpha }_1\right)e^{-{\gamma }_2\left({\alpha }_1\right)·x}\\ Q_{11}\left({\alpha }_1\right)P_{21}\left({\alpha }_1\right)e^{-{\gamma }_1\left({\alpha }_1\right)·x}+Q_{21}\left({\alpha }_1\right)P_{22}\left({\alpha }_1\right)e^{-{\gamma }_2\left({\alpha }_1\right)·x}\end{array}\right]·\\ D_1·e^{{\alpha }_1·t}+\\ \left[\begin{array}{c}{Q}_{12}\left({\alpha }_2\right)P_{11}\left({\alpha }_2\right)e^{-{\gamma }_1\left({\alpha }_2\right)·x}+Q_{22}\left({\alpha }_2\right)P_{12}\left({\alpha }_2\right)e^{-{\gamma }_2\left({\alpha }_2\right)·x}\\ Q_{12}\left({\alpha }_2\right)P_{21}\left({\alpha }_2\right)e^{-{\gamma }_1\left({\alpha }_2\right)·x}+Q_{22}\left({\alpha }_2\right)P_{22}\left({\alpha }_2\right)e^{-{\gamma }_2\left({\alpha }_2\right)·x}\end{array}\right]·\\ D_2·e^{{\alpha }_2·t}\end{array}$

[00115] The real constants associated with the eigenvector matrix, P, its inverse, P⁻¹=Q, and the SD parameters D can be written:

• [00116] A₁₁=D₁P₁₁(α₁)Q₁₁(α₁)
• [00117] A₁₂=D₁P₁₂(α₁)Q₂₁(α₁)
• [00118] A₂₁=D₂P₁₁(α₂)Q₁₂(α₂)
• [00119] A₂₂=D₂ P₁₂(α₂)Q₂₂(α₂)
• [00120] B₁₁=D₁ P₂₁ (α₂)Q₁₁(α₁)
• [00121] B₁₂=D₁ P₂₂(α₁)Q₂₁(α₁)
• [00122] B₂₁=D₂P₂₁(α₂)Q₁₂(α₂)
• [00123] B₂₂=D₂P₂₂(α₂)Q₂₂(α₂)
  The signals on transmission lines will be:
  V₁(x,t)=A₁₁·e^{−γ1(α1)·xeα1·t}αA₂₁·e^{−γ2(α1)·xeα1·t}+A₁₂·e^{−γ1(α2)·xeα2·t}+A₂₂·e^{−γ2(α2)·xeα2·t}
  V₂(x,t)=B₁₁·e^{−γ1(α1)·xeα1·t}+B₂₁·e^{−γ1(α2)·xeα2·t}+B₁₂·e^{−γ2(α2)·xeα2·t}+B₂₂·e^{−γ2(α2)·xeα2·t}
  or,
  V₁(x,t)=A₁₁·e^−A_SD^11(x−v_SD^11t)+A₂₁·e^−A_SD^21(x−v_SD^21t)+A₁₂·e^−A_SD^12(x−v_SD12^t)+A₂₂·e^−A_SD^22(x−v_SD^22t)
  V₂(x,t)=B₁₁·e^−A_SD^11(x−v_SD^11t)+B₂₁·e^−A_SD^21(x−v_SD^21t)+B₁₂·e^−A_SD^12(x−v_SD^12t)+B₂₂·e^−A_SD^22(x−v_SD^22t)

[00130] Where:
A_SD¹¹=γ₁(α₁), A_SD¹²=γ₁(α₂), A_SD²¹=γ₂(α₁), and A_SD²²=γ₂(α₂).
$v_{{\mathrm{SD}}_{11}}=\frac{{\alpha }_1}{{\gamma }_1\left({\alpha }_1\right)},v_{{\mathrm{SD}}_{12}}=\frac{{\alpha }_2}{{\gamma }_1\left({\alpha }_2\right)},v_{{\mathrm{SD}}_{21}}=\frac{{\alpha }_1}{{\gamma }_2\left({\alpha }_1\right)},\mathrm{and}{v}_{{\mathrm{SD}}_{22}}=\frac{{\alpha }_2}{{\gamma }_2\left({\alpha }_2\right)}.$

[00133] Thus, there are SD attenuations and velocities associated with each eigenvalue, γ_i, of the product {overscore (ZY)} evaluated at each α.

[00134] 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.

[00135] 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.

[00136] 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.

[00137] 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.

[00138] In the case of a four parameter (LCRG) transmission line, we have the expressions A_SD(α)=√{square root over (LCα²+(RC+LG)α+RG)}, and $v_{\mathrm{SD}}\left(\alpha \right)=\frac{\alpha }{\sqrt{\mathrm{LC}{\alpha }^2+\left(\mathrm{RC}+\mathrm{LG}\right)\alpha +\mathrm{RG}}}.$

[00139] 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)}.

[00140] Introducing the new coordinates:
t′=t−x√{square root over (LC)}.
x′=x

[00143] Then
V(x′,t′)=De^{−ASD(α)·[x′−vSD(α)·(t′+x′·√{square root over (LC)})]},

• • [00145] or
    V(x′,t′)=De^{α·t′e−ASD(α)·[1−vSD(α)·√{square root over (LC)}]·x′}.

[00147] 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}.

[00149] 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}

[00151] 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: $\frac{V\left(l,t_j\right)}{V\left(0,t_j\right)}=\frac{D{e}^{\alpha ·t_j}{e}^{-A_{SD}\left(\alpha \right)·\lfloor 1-v_{SD}\left(\alpha \right)·\sqrt{LC}\rfloor ·l}}{D{e}^{\alpha ·t_j}}=e^{-A_{SD}\left(\alpha \right)·\left[1-v_{SD}\left(\alpha \right)·\sqrt{LC}\right]·l}$

[00152] 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 {overscore (L)}(s), {overscore (C)}(s), {overscore (R)}(s), and {overscore (G)}(s), is: $\frac{V\left(l,t_j\right)}{V\left(0,t_j\right)}=e^{-{\stackrel{\_}{A}}_{SD}\left(\alpha \right)·\lfloor 1-{\stackrel{\_}{v}}_{SD}\left(\alpha \right)·\sqrt{\stackrel{\_}{L}\stackrel{\_}{C}\left(\alpha \right)}\rfloor ·l}$

• • [00153] where: ${\stackrel{\_}{A}}_{SD}\left(\alpha \right)=\sqrt{\stackrel{\_}{LC}\left(\alpha \right)·{\alpha }^2+\left(\stackrel{\_}{RC}\left(\alpha \right)+\stackrel{\_}{LG}\left(\alpha \right)\right)·\alpha +\stackrel{\_}{RG}\left(\alpha \right)},\mathrm{and}$ ${\stackrel{\_}{v}}_{SD}\left(\alpha \right)=\frac{\alpha }{\sqrt{\stackrel{\_}{LC}\left(\alpha \right)·{\alpha }^2+\left(\stackrel{\_}{RC}\left(\alpha \right)+\stackrel{\_}{LG}\left(\alpha \right)\right)·\alpha +\stackrel{\_}{RG}\left(\alpha \right)}}.$

[00154] 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.

[00155] Coupled transmission lines can be treated in a similar fashion. The equations governing SD propagation on one of two coupled transmission lines is: $\begin{array}{c}{V}_1\left(x,t\right)=A_{11}·e^{-A_{S{D}_{11}}\left(x-v_{S{D}_{11}}t\right)}+A_{21}·e^{-A_{S{D}_{21}}\left(x-v_{S{D}_{21}}t\right)}+\\ A_{12}·e^{-A_{S{D}_{12}}\left(x-v_{S\mathrm{D2}}t\right)}+A_{22}·e^{-A_{S{D}_{22}}\left(x-v_{S{D}_{22}}t\right)}.\end{array}$

[00156] Applying the substitution: t′=t−x√{square root over (LC)}, x′=x, gives the result: $\begin{array}{c}\frac{V_1\left(x^{\prime }=l,t^{\prime }=t_i\right)}{V_1\left(x=0,t=t_i\right)}=\frac{\begin{array}{c}{A}_{11}·e^{{\alpha }_1{t}_i}·e^{-A_{S{D}_{11}}\left(1-v_{S{D}_{11}}\sqrt{\stackrel{\_}{LC}}\right)l}+\\ A_{21}·e^{{\alpha }_1{t}_i}·e^{-A_{S{D}_{21}}\left(1-v_{S{D}_{21}}\sqrt{\stackrel{\_}{LC}}\right)l}\end{array}}{\begin{array}{c}{A}_{11}·e^{{\alpha }_1{t}_i}+A_{21}·e^{{\alpha }_1{t}_i}+\\ A_{12}·e^{{\alpha }_2{t}_i}+A_{22}·e^{{\alpha }_2{t}_i}\end{array}}+\\ \frac{\begin{array}{c}{A}_{12}·e^{{\alpha }_2{t}_i}·e^{-A_{S{D}_{12}}\left(1-v_{S{D}_{12}}\sqrt{\stackrel{\_}{LC}}\right)l}+\\ A_{22}·e^{{\alpha }_2{t}_i}·e^{-A_{S{D}_{22}}\left(1-v_{S{D}_{22}}\sqrt{\stackrel{\_}{LC}}\right)l}\end{array}}{\begin{array}{c}{A}_{11}·e^{{\alpha }_1{t}_i}+A_{21}·e^{{\alpha }_1{t}_i}+\\ A_{12}·e^{{\alpha }_2{t}_i}+A_{22}·e^{{\alpha }_2{t}_i}\end{array}}\end{array}$

[00157] Which can be written: $\begin{array}{c}\frac{V_1\left(l,t_j\right)}{V_1\left(0t_j\right)}=\frac{\begin{array}{c}(A_{11}·e^{-A_{S{D}_{11}}\left(1-v_{S{D}_{11}}\sqrt{\stackrel{\_}{L}\stackrel{\_}{C}}\right)l}+\\ A_{21}·e^{-A_{S{D}_{21}}\left(1-v_{S{D}_{21}}\sqrt{\stackrel{\_}{L}\stackrel{\_}{C}}\right)l})\end{array}}{\left(A_{11}+A_{21}\right)+\left(A_{12}+A_{22}\right)·e^{\left({\alpha }_2-{\alpha }_1\right)t_i}}+\\ \frac{\begin{array}{c}(A_{12}·e^{-A_{S{D}_{12}}\left(1-v_{S{D}_{12}}\sqrt{\stackrel{\_}{L}\stackrel{\_}{C}}\right)l}+\\ A_{22}·e^{-A_{S{D}_{22}}\left(1-v_{S{D}_{22}}\sqrt{\stackrel{\_}{L}\stackrel{\_}{C}}\right)l})\end{array}}{\left(A_{11}+A_{21}\right)·e^{\left({\alpha }_1-{\alpha }_2\right)t_i}+\left(A_{12}+A_{22}\right)}\end{array}$

[00158] 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 Schrodinger partial differential equation [3], $\frac{\partial E}{\partial x}+{\beta }_1\frac{\partial E}{\partial t}-\frac{j}{2}{\beta }_2\frac{{\partial }^{2}E}{\partial t^2}-\frac{1}{6}{\beta }_3\frac{{\partial }^{3}E}{\partial t^3}=-\frac{\gamma }{2}E,$

• • [00159] 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 ${\beta }_n={\left[\frac{{\partial }^n\beta }{\partial {\omega }^n}\right]}_{\omega ={\omega }_o}$
  • [00160] and ω_o is the frequency of the light being modulated in the fiber.

[00161] 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]

[00162] 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.

[00163] The SD solution of the linear Schrodinger equation with loss (γ≠0) is: $E\left(x,t\right)={\mathrm{De}}^{{\left[\frac{t}{T_o}-\left(\frac{{\beta }_1}{T_o}-\frac{{\beta }_3}{6T_o^3}+\frac{\gamma }{2}\right)·x\right]}_{e}j\left(\frac{{\beta }_2·x}{2T_o^2}\right)}.$

[00164] The boundary condition at the input to the fiber at x=0 is $E\left(0,t\right)={\mathrm{De}}^{\frac{t}{\mathrm{To}}},\left(\mathrm{see}\mathrm{FIG}\mathrm{.1}A\mathrm{with}\alpha =\frac{1}{T_o}\right).$

[00165] 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]},

• • [00167] where the velocity of propagation of a truncated SD leading edge of the envelope is $v_{\mathrm{sd}}\left(T_o\right)=\frac{1}{\left({\beta }_1-\frac{{\beta }_3}{6T_o^2}+\frac{T_o\gamma }{2}\right)},\mathrm{and}A\left(T_o\right)=\left(\frac{{\beta }_1}{T_o}-\frac{{\beta }_3}{6T_o^3}+\frac{\gamma }{2}\right).$

[00168] Note that $v_{\mathrm{sd}}\left(T_o\right)·A\left(T_o\right)=\frac{1}{T_o}.$

[00169] Transforming to a moving reference frame traveling at the group velocity 1/β₁,
t′=t−β₁x.
x′=x

[00172] The SD propagation speed in this moving reference frame is ${{\stackrel{\_}{v}}_{\mathrm{sd}}\left(T_o\right)}_{\mathrm{in}{x}^{\prime },t^{\prime }\mathrm{frame}}=\frac{1}{T_o\left(\frac{\gamma }{2}-\frac{{\beta }_3}{6T_o^3}\right)},$

[00173] The attenuation of the truncated SD portion of the leading edge of the pulse envelope is $\frac{{E\left(l,t_j\right)}^2}{{E\left(0,t_j\right)}^2}=e^{-\left(\frac{\gamma }{2}-\frac{{\beta }_1}{6T_o^3}\right)l}.$

[00174] 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 becomes smaller.

[00175] 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.

[00176] 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 a can be determined empirically.

[00177] 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.

[00178] 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.

[00179] 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 $v_{\mathrm{SD}}=\frac{d}{{\mathrm{TOF}}_{\mathrm{AVG}}}=\frac{1002\mathrm{ft}}{1\text{,}716\times {10}^{-9}\sec }=5\text{.}839\times {10}^8\frac{\mathrm{ft}}{\sec }.$

[00180] 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 $A_{\mathrm{SD}}=\frac{\alpha }{v_{\mathrm{SD}}}=\frac{3\text{.}0\times {10}^6\frac{1}{\sec }}{5\text{.}839\times {10}^8\frac{\mathrm{ft}}{\sec }}=5.138\times {10}^{-3}\frac{1}{\mathrm{ft}}.$

[00181] 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.

[00182] Referring to FIG. 5A, the measured input voltage of the two twisted wire pairs in a T1line 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).

[00183] 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.

[00184] 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.

[00185] 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 sect 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: $H_{\mathrm{est}}\left(j\omega \right)=\frac{FFT\left(V_{\mathrm{Measured}\mathrm{at}l}\right)}{FFT\left(V_{\mathrm{Applied}}\right)}.$
Once the transfer function is determined, the propagation of SD waveforms with a series of different values for α can be simulated and the times of flight and SD parameters, A_SD and v_SD, determined.

[00188] 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 (R₁ 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.

[00189] Once the transfer function is known, a waveform with any a 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.

[00190] 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):

[00002]

TABLE I
vSD and ASD Determined
By Simulation with Empirical Transfer Function
	Alpha	1002 Simulation	Simulation vSD	Simulation ASD
	(sec−1)	TOF (μsec)	(×108 ft/sec)	(×10−3 1/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	5.849	5.129
		(1.716*)	(5.839*)	(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

[00191] 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 {overscore (R)}(s), {overscore (L)}(s), C(s), and {overscore (G)}(s), are known, then Z_SD(α) may be computed from $Z_{SD}=\sqrt{\frac{\alpha \stackrel{\_}{L}\left(\alpha \right)+\stackrel{\_}{R}\left(\alpha \right)}{\alpha \stackrel{\_}{C}\left(\alpha \right)+\stackrel{\_}{G}\left(\alpha \right)}}.$

[00192] 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 $V_{SD}\left(R_t\right)=\left(1+{\Gamma }_t\right)·V_{SD}^{+}=\frac{2·R_t}{R_t+Z_{SD}}·V_{SD}^{+}.$
Note that all lumped and distributed impedance values are real for SD signals.

[00194] 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 a of 1.5×10⁶ l/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, ${\Gamma }_t=\frac{V_{SD}\left(R_t\right)}{V_{SD}^{+}}-1.$
FIG. 15B shows the measured ratios for this cable and α. Using the relationship for the transmission coefficient, $1+{\Gamma }_t=\frac{2·R_t}{R_t+Z_{SD}},\mathrm{gives}{Z}_{SD}=\frac{R_t}{1+{\Gamma }_t}\left(1-{\Gamma }_t\right).$

[00196] Using the two finite values for R_t, 98.6 Ω and 49.5 Ω, yields two estimates for Zsd: $Z_{SD}\left(\alpha =1.5\times {10}^6\frac{1}{\sec }\right)=\frac{98.6}{1+\left(0.919-1\right)}\left(1-\left(0.919-1\right)\right)=116\Omega ,\mathrm{or}$ $Z_{SD}\left(\alpha =1.5\times {10}^6\frac{1}{\sec }\right)=\frac{49.5}{1+\left(0.598-1\right)}\left(1-\left(0.598-1\right)\right)=116\Omega ,\mathrm{for}\mathrm{this}\mathrm{T1}\mathrm{cable}.$

[00197] 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.

[00198] 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.

[00199] 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 a D of 0.19 volts and an α of 3.5×10⁵ /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=8.279×10⁻⁴ l/ft.

[00200] Even though the signal was detectable at much greater distances than 6 kft, the constant threshold time of flight could not be measured. FIG. 17 shows the voltage traces measured at the input, ˜6 kft, ˜12 kft, and ˜18 kft distances, and it also shows the preservation of shape of the SD region of the highly attenuated propagating waveform. The ˜18 kft trace is corrected for the reflection from the open termination by dividing the measured waveform in half. The last two traces show that even with this smaller α, beyond 12,000 ft there is little or no SD signal above 0.19 volts. This is the value of the input SD waveform parameter D, thus there is little or no SD voltage threshold common to the input SD waveform and the SD waveforms at these longer distances.

[00201] The first step in precisely measuring highly attenuated SD pulse time of flight is to detect the SD region of the propagating pulse, V(t), measured and recorded at the distance l along the cable. This is done by computing the ratio $\frac{\frac{dV\left(t\right)}{dt}}{V\left(t\right)}$
of the measured pulse waveform. In the SD region of the pulse, this ratio is α. The end of the SD region is then found by detecting the time that this ratio diverges from α. The ratio $\frac{\frac{d^{2}V\left(t\right)}{d{t}^2}}{\frac{dV\left(t\right)}{dt}}$
can also be used to detect the end of the SD region. In the SD region, this ratio is also α. The later ratio of the second and first derivative responds more quickly, but is also more susceptible to noise than the ratio of the first derivative and the signal. These ratios as a function of time are plotted in FIG. 18 for the SD waveform measured at 12-kft. In general, the ratio of $\frac{\frac{{\partial }^{n}V\left(x,t\right)}{{\partial }^{n}t}}{\frac{{\partial }^{n-1}V\left(x,t\right)}{{\partial }^{n-1}t}}$
for any positive n will have this property. Additionally, the ratio of the signal and its integral can be used to locate the SD region. In the SD region, we have the relationship: $\frac{D{e}^{\alpha t}}{{\int }_0^{t}D{e}^{\alpha t}d\tau }=\frac{D{e}^{\alpha t}}{\frac{1}{\alpha }\left(D{e}^{\alpha t}-D\right)}=\alpha \frac{1}{1-e^{-\alpha t}}.$
Thus, the ratio will converge on α as time progresses. This ratio diverges at the end of the SD region. In a lossy transmission medium with constant transmission line parameters, the end of the estimated SD region can provide a good marker to determine the speed of light in the medium. The truncated SD leading edge of the pulse does not disperse, even with frequency dependent parameters. The high frequency components of the closing pulse, however, do undergo dispersion. These fast, high frequency components tend to erode the end of the truncated SD region as the pulse propagates. At long distances, these high frequency components are also more highly attenuated so the amount of SD region erosion is not proportional to the distance traveled. In this case, the detected end of the SD region serves as an initial estimate of the pulse time of flight or the distance the wave has traveled and serves to define the region examined for more precise time of flight measurements in the next step.

[00206] The attenuation of the truncated SD signal propagating a distance d was shown to be e^{−Asd·(l−v·√{square root over (LC)})d}. In the application of time domain reflectometry (TDR) to these lines, the parameter 1/√{square root over (LC )} is commonly referred to as Vp and is given as a fraction of the speed of light in a vacuum. The standard value of Vp used in TDR for this air core poly 24 AWG cable is 0.67. However, the value of the ratio used in this analysis is 0.59. The reason for this can be seen in FIG. 19. The value of 0.67 is appropriate for the high frequencies that are found in a standard TDR pulse used to test these telephone lines, but the α selected here for low truncated SD signal attenuation in these long lines is equivalent to a low frequency and propagates at a slower speed.

[00207] Once the initial TOF estimate of end of the SD region in the leading edge of the propagating pulse is obtained at the distance l, the input waveform is attenuated and time shifted assuming the end of the SD region of the traveling wave corresponds to the point of truncation of the initial SD waveform, (FIG. 20, top). The input waveform is then incrementally translated in time and attenuated based on each new estimated distance traveled. At each distance estimate, this shifted and attenuated input waveform is subtracted from the measured waveform. If there is no dc voltage offset and no noise, this difference will be zero in the SD region when the estimated distance is equal to the actual distance. Because there is always noise and there is frequently a varying differential, dc offset at different points along the cable, the variance of this difference is evaluated at each estimated distance. This correlation has a minimum when the input waveform has been translated to the actual distance traveled, (FIG. 20, middle). If there is a dc differential offset between the lines, this minimum will be non-zero. At this estimated distance, the erosion of the end of the truncated SD region due to the dispersion from the closing pulse is obvious. If the waveform is translated too far, this deviation error measurement increases, (FIG. 20, bottom). FIG. 21 shows the calculated standard deviation of the difference of the shifted and attenuated input waveform and the SD waveform measured at the splice between the second and third twisted wire pairs as a function of estimated distance. The minimum occurs at an estimated distance of 12,216 ft. FIG. 22A shows the difference of the two waveforms and the variance of the difference of the waveforms at the minimum. FIG. 22A indicates that there is a five to ten millivolt differential dc offset between the two measuring points. FIG. 22B is presented to show how the variance has increased when the estimated distance is 38 ft longer than the distance associated with the minimum.

[00208] This process was repeated eight times giving average distances of 6,113 ft, 12,216 ft, and 18,327 ft for the best-fit shift locations of the attenuated input waveform to the measured waveforms. The standard deviations of the three lengths were 1 ft, 1 ft, and 5 ft respectively. The fits associated with one of these data sets are shown in FIG. 23. The actual length of the cable was not exactly known, however, each pair in the cable is parallel and the only differences in length are small and due to different twist rates. This process yields the distance of two lengths within 10 ft of two times the distance of one length. The distance of three lengths is within 11 ft of three times the distance of one length. Using the ˜6 kft distance as a standard gives a discrepancy of −0.08% for the ˜12 kft length and −0.06% for the ˜18 kft length.

[00209] 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. A fundamental measurement in TDR test technology is the time of flight (TOF) of a test pulse generated by the instrument and applied to the line. This time of flight may be measured by timing the passage of the pulse detected at two locations along the line, referred to as Time Domain Transmission measurements (TDT). Or by Time Domain Reflection measurements which estimate the launch time of a pulse at one position and the subsequent return time of the pulse back to the same location after the pulse has been reflected from a fault or other impedance change some distance along the line. These measured TOF values, along with a value of the propagation speed of the pulse, allows one to obtain the distance between measurement points or in the case of the reflected wave, the distance from the pulse launch point to the location of the impedance change causing the pulse to be reflected and returned to the launch point.

[00210] A fundamental limitation in TDR technology is the poor accuracy of these 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, change amplitude and speed as they travel. The TOF measurements used in the art under these circumstances focus on estimating the emergence time of the leading profile of the test pulses. This part of the signature of test pulses used in the art has characteristically a low signal level and low signal slope making an accurate pulse emergence time measurement difficult to obtain.

[00211] Several advantages can be obtained in TDR technology for lossy, dispersive transmission lines by using a test pulse that contains a truncated SD signal in its leading edge. This truncated SD leading edge travels at a constant speed along these transmission lines without changing shape. The speed of propagation of this SD edge is a function of the line parameters and the SD signal parameter alpha and is controllable by changing alpha. The truncated SD leading edge of the test pulse will be attenuated as it travels along a lossy line. However, this rate of attenuation is also a function of the line parameters and the SD signal parameter alpha and is controllable by changing alpha.

[00212] The same principles used for TDR can be applied to waveforms other than electromagnetic waves in a transmission line. The reflections of acoustic waves in SONAR or geophysical applications can be analyzed using the techniques discussed in this section to provide accurate time of flight and distance estimates as well as be used to characterize the transmission media.

[00213] The accuracy of the SD test waveform in a TDR application can be demonstrated by an example using the T1 cable, as discussed in examples 1 and 2. The two twisted wire pairs inside the T1 cable are connected in series to form a 2004 ft cable and the truncated SD signal is applied to the input. The voltage trace is measured at the splice of the two lines at 1002 ft. This can eliminate the need to correct for any line impedance mismatch at the point of measurement. The exponential coefficient a of the applied wave is 6.7×10⁶ sec⁻¹. The SD parameters, A_SD and v_SD, are obtained from the empirical transfer function analysis presented in example 2. Interpolating from the data of table I gives an A_SD of 10.703×10⁻³ ft⁻¹ and a V_SD of 6.276×10⁸ ft/sec for this α.

[00214] The first demonstration terminates the cable with an open circuit. The reflected coefficient of an open circuit is +1, which results in a positive reflection. FIG. 24 shows the voltage waveform measured at the 1002 ft point showing the forward traveling applied wave and the wave reflected from an open termination at the end of the cable. FIG. 25 shows the result of the measured reflected wave together with the applied wave time shifted and attenuated according to the procedure of example 4. The total distance traveled by the reflected wave is measured to be 2004 ft by this process, or exactly two times the 1002 ft length of this twisted wire pair section. FIGS. 26A-B show the constant threshold timing method result for TOF discussed in example 1. This method results in a time of flight of 3,203 nsec, which gives an estimated total distance traveled of 2010 ft. This results in an estimated twisted wire pair section length of 1005 ft, or 3 ft in error (+0.3%). This is an illustration of reduced accuracy using the constant threshold TOF measurements when the pulse travels far enough to experience significant attenuation. The accuracy of this method may decline more in cases where the test pulse experiences even more attenuation. This level of attenuation may not generally occur in digital circuits. The maximum attenuation of digital signals in interconnects is generally less than one-half of the driving signal. In such cases, the constant threshold TOF measurements with the SD wave should be preferably more accurate than current TDR methods using estimated time of emergence of the reflected pulses.

[00215] The next demonstration terminates the second twisted wire pair with a short circuit. The reflected coefficient at a short circuit is −1, which results in a negative reflection. FIG. 27 shows the voltage waveform measured at the 1002 ft point showing the forward traveling applied wave and the wave reflected from a short circuit termination at the end of the second twisted wire pair. FIG. 28 shows the result of the inverted reflected wave and the applied wave time shifted and attenuated according to the procedure of example 4. The total distance traveled is measured to be 2004 ft by this process, or exactly two times the 1002 ft twisted wire pair length. FIGS. 29A-B show the constant threshold timing method of estimating TOF discussed in example 1. This method results in a time of flight of 3,200 nsec, which gives an estimated total distance traveled of 2008 ft. This results in an estimated twisted wire pair section length of 1004 ft, or 2 ft in error (+0.2%).

[00216] The utility of the SD waveform to the TDR process can be increased by using an empirical transfer function obtained with the use of a non-SD pulse applied to the line in a way similar to the process discussed in example 2. As an example with this cable, an experimental voltage pulse is generated by an arbitrary function generator connected to a single unterminated twisted wire pair from the T1 cable. This waveform is shown in FIGS. 30-31 (dotted). The same waveform can be applied by the signal generator to a variable resistor load substituted for the 1002 ft twisted wire pair. The resistor is adjusted until the waveform approximates the shape and magnitude of the waveform applied by the arbitrary waveform generator to the twisted wire pair, see FIGS. 30-31 (solid). The waveform match is not exact because the resistor does not have the exact same frequency response as that of the twisted wire pair. A load resistance equal to the characteristic impedance, Z_o, of the twisted wire pair results in a waveform that is closely matched to the waveform applied to the cable and has similar frequency content. The difference between the signal measured with the resistive load and the signal measured with the transmission line consists of the response of the transmission line to the applied signal and contains any reflected waves from impedance mismatches on the transmission line. The transfer function is then calculated from the ratio of the fast Fourier transforms (FFT) of the difference of the two signals, and the signal of the resistor alone: $H_{est}\left(j\omega \right)=\frac{FFT\left(V_{\mathrm{Measured}\mathrm{with}\mathrm{Cable}}-V_{\mathrm{Measured}\mathrm{with}\mathrm{Load}\mathrm{Resistance}{Z}_o}\right)}{FFT\left(V_{\mathrm{Measured}\mathrm{with}\mathrm{Load}\mathrm{Resistance}{Z}_o}\right)}.$
This function, shown in FIG. 32, can be used to evaluate the reflected transmission line response to a SD pulse.

[00218] As a test of this process, a SD pulse with an α equal to the α used in the previous two experiments was simulated. The transfer function determined for the open terminated T1 cable, (FIG. 32), was used to calculate the expected response, see FIG. 33. There is a small pulse at the time of the applied pulse due to the resistor and cable having different high frequency impedance values. The influence of this pulse was eliminated with appropriate time windowing. The simulated response was then used to estimate the distance to the open termination using the methods of example 4 (FIG. 34), and example 1 (FIGS. 35A-B). The result of the slide and attenuate method resulted in an estimated total round trip distance the pulse traveled of 2004, or double the exact distance of 1002 ft to the open. The constant threshold method resulted in time of flight of 3,196 nsec for a estimated distance traveled of 2006 ft, or two times an estimated distance of 1003 ft to the open (0.1% Error).

[00219] No actual SD signal was experimentally applied to the twisted wire pair to obtain these results. The test signal applied to the line (FIG. 30) was similar to the type obtained form a commercial TDR instrument for testing communication cables with lengths ranging up to 5-kft. The measurements resulting from this test line voltage were used to obtain an empirical line transfer function. The SD TOF and the line distance estimates were all obtained computationally by simulating the SD propagation along the line utilizing this empirical line transfer function estimated with non-SD signals.

[00220] The SD waveform has several properties that make it suitable as the basis for a data transmission scheme. For example: the exponential shape is maintained as it propagates at constant speed in a uniform cable; the attenuation of the SD waveform is adjustable by changing the exponential waveform parameter a; the propagating speed of the SD waveform is adjustable by changing the exponential waveform parameter α; and the SD waveforms with different α's are linearly independent.

[00221] This example teaches using the first property as the basis for a method of transmitting data using SD waveforms. The signals are transmitted 2004 ft along a transmission line consisting of the two twisted wire pairs in a 1002 ft T1 cable connected in series. The second twisted wire pair is terminated with a 100-ohm resistor. The data is encoded on four SD waveforms, each having a distinct α. Each waveform can be transmitted with a positive or a negative D, resulting in eight total states for three bits per symbol. One symbol is transmitted every three microseconds, giving a data rate of 1 Mbps. As one of ordinary skill in the art will recognize with the benefit of this disclosure, several other communication schemes can be readily derived from the teachings contained herein.

[00222] The symbols can consist of a two-microsecond truncated SD leading edge period followed by a one-microsecond recovery period. This recovery period includes a variable width half-sine compensating pulse adjusted to shorten the time required for the transmission line to return to a zero voltage. The need for this recovery period and compensating pulse is illustrated in FIGS. 36A-B and FIGS. 37A-B. FIG. 36A shows the transmitted pulse without the compensating pulse, and FIG. 36B shows the received pulse without the compensating pulse. The truncated SD waveform in the leading edge of the input pulse is closed with a simple ramp signal, which is rounded due to the line response. Although the SD portion of the propagating pulse does not undergo deformation, the closing ramp does disperse into the region beyond the three-microsecond symbol period. FIGS. 37A-B show the effect of the compensating pulse in eliminating this inter-symbol interference. Each truncated SD waveform may requires a different amount of compensation so the width of the half-sine pulse can be varied based on observing the pulse at the terminated end of the second twisted wire pair.