the detailed description of a preferred embodiment of the present invention is described with reference to the figures . fig1 provides a system overview . fig2 a - 2d , 3a - 3d and 4a - 4i are graphs illustrating characteristics of the data samples before and after processing according to the present invention . fig5 is a flowchart illustrating the processing according to the present invention . the present invention has been incorporated into a network analyzer called the wiltron 360 , manufactured by wiltron , 490 jarvis drive , morgan hill , calif . 95037 . this system is described in detail in the co - pending u . s . patent application entitled microwave measurement system and associated method referred to above . fig1 schematically illustrates a system of the present invention set up for a typical reflection characteristic measurement . the system includes a network analyzer 10 and a network 11 coupled to the analyzer across input lines 12 . the network analyzer 10 includes measurement apparatus 13 , a storage facility 14 , a display 15 , a user input interface 16 , a central processing unit 17 and a transform engine 18 , all coupled to bus 19 for communication . the measurement apparatus 13 samples the reflection or transmission characteristics under control of the cpu 17 across a frequency spectrum from f1 to f2 and stores the measured data samples into the storage 14 . the cpu processes the stored data for display according to a variety of formats as desired by the user . the user communicates with the cpu 17 through a user input interface 16 consisting of a plurality of control switches and a keyboard . associated with the cpu is a discrete fourier transform ( dft ) engine 18 which is utilized for transforming the information taken in the frequency domain to the time domain . in the preferred system , the transform engine is a software utility which performs the chirp z - transform as described below with respect to the preferred embodiment . the chirp z - transform is described in detail in l . r . rabiner , &# 34 ; chirp z - transform algorithm program &# 34 ;, programs for digital signal processing , ieee press , new york , n . y ., 1979 , pp . 1 . 6 - 1 to 1 . 6 - 13 . the network 11 in fig1 consists of a transmission line 20 , having a characteristic impedance of 50 ohms . along the transmission line 20 , a discontinuity 21 exists . the nature of the discontinuity can be a step - up in impedance , a step - down in impedance , capacitive , inductive , or a combination of these characteristics . the user of the network analyzer 10 desires to determine the location of the discontinuity 21 along the transmission line 20 as well as its nature . thus , the user sets up the measurement apparatus 13 through the cpu 17 to take measurements over a range of equally spaced frequencies from frequency f1 to frequency f2 . the analyzer samples the real and imaginary components of the reflection coefficient at each frequency . results can be displayed in a real and imaginary format , or they can be converted to magnitude and phase for user convenience . fig2 a - 2d and 3a - 3d illustrate the stages of data processing according to the present invention for a simple short circuit discontinuity with a rectangular window ( fig2 a - 2d ), or a two - term hamming window ( fig3 a - 3d ). each of fig2 a - 2d and 3a - 3d includes plots of the real and imaginary components of the data and a polar plot with the same data . fig2 a illustrates the windowed frequency domain data with a rectangular window . the real component is illustrated in chart 300 , the imaginary component in chart 301 and the polar graph is shown in chart 302 . as can be seen , it is simply a constant magnitude signal with a phase rotating around 360 degrees . after transforming the single - sided spectrum , the data from point t1 to point t2 takes the form shown in fig2 b , with the real component shown in chart 303 , the imaginary component shown in chart 304 and the polar magnitude plot shown in chart 305 . the portion of the time domain data shown in fig2 b is those data points taken around the main lobe of the discontinuity of interest as selected by a user through the keyboard interface . this does not illustrate the entire range from 0 to t which could be generated by the transform . as is illustrated in the chart 305 , the phase information is quite difficult to decipher . fig2 c illustrates the partially processed time domain data from point t1 to point t2 after removing the portion of the phase error that is not dependent on the distance to the discontinuity through the network . the real component is shown in chart 306 , the imaginary component in chart 307 and the polar plot in chart 308 . as can be seen , the phase information in the data is now normalized to a simple line in the polar plot . however , this includes an error that is dependent on the particular location of the discontinuity in the network . in the final unwinding step , that error is removed and the resulting data is shown in fig2 d . the real component of the resulting data is shown in chart 309 , the imaginary component in chart 310 and the polar plot in chart 311 . this takes the form that would be expected for a simple short circuit discontinuity in the network . the imaginary part is zero everywhere as shown at chart 310 . the polar plot takes the form of a vector with a phase of minus 180 degrees illustrating a step - down in impedance with no reactive component . fig3 a - 3d illustrate the same data using a two - term hamming window in order to improve the displayed characteristics for the purposes of analysis . accordingly , fig3 a shows the real and imaginary components of the frequency domain data in charts 312 and 313 , respectively . the polar plot is shown in chart 314 . the raw time domain data real and imaginary components between points t1 and t2 which illustrate the main lobe of the discontinuity are shown in charts 315 and 316 of fig3 b . the polar plot of the raw time domain data is shown in chart 317 of fig3 b . after removing a portion of the phase error that is independent of the distance to the discontinuity , the time domain data as partially processed , is shown in fig3 c . the real component of the partially processed data is shown in chart 318 , the imaginary component in chart 319 and the polar plot in chart 320 . after the final unwinding , removing the portion of the phase error that is dependent upon the location of the discontinuity in the network , the low pass equivalent data according to the present invention , is shown in fig3 d , with the real component shown in chart 321 , the imaginary component in chart 322 and the polar plot shown in chart 323 . as can be seen , this is an improved version of the processed data over that shown in fig2 d because of the hamming window . the real component is below zero across the entire discontinuity , the imaginary component is zero everywhere . thus , the polar plot is the form of a vector from zero to minus one as illustrated in chart 323 . again , this is the form of a polar plot that would be expected for a simple short circuit discontinuity in the network . the expected characteristics of time domain data for various discontinuities are illustrated in fig4 a - 4i . fig4 a - 4i are plots illustrating time domain data after correction according to the present invention . graphs 4a , 4b and 4c are plots in the time domain of corrected data for a discontinuity which is a step - down in impedance ( r = 49 . 0099 ohms ), no discontinuity ( r = 50 . 00 ohms ), and for a discontinuity which is a step - up in impedance ( r = 50 . 0101 ohms ) with no reactive components ( l = 0 , c = 0 ). fig4 d , 4e and 4f are plots of the real and imaginary components of the corrected time domain data for a capacitive discontinuity ( c = 0 . 0032 pf ) across a frequency range of 18 to 26 . 5 ghz with step - down in impedance , no impedance change , and a step - up in impedance , respectively . fig4 g - 4i illustrate the real and imaginary components of the corrected time domain data for an inductive discontinuity ( l = 0 . 0080 mh ) across a frequency range of 18 to 26 . 5 ghz with step - down , no change , and a step - up in impedance , respectively . using the location of the main lobe of the discontinuity as well as the plots as illustrated in fig4 a - 4i , a great deal of information about the nature of the discontinuity and its location can be determined . of course for complex discontinuities that involve combinations of reactive and non - reactive components , the shapes of these plots may vary widely . also , the plots shown in fig4 a - 4i were generated using a hamming window with two terms as a bias to the time domain data . the hamming window is used to enhance the characteristics of the discontinuity so that plots as illustrated in fig4 a - 4i are more readily readable . the present invention will work with or without biasing windows . let us diagram the frequency domain measurement , the transformed time domain data and their relationships . the frequency domain is shown below : ______________________________________0 f1 f2 frequency domain x1 x2 x3 . . . xn ( real part of data ) y1 y2 y3 . . . yn ( imaginary part ) ______________________________________ the inverse fourier transform can be used to obtain n complex points of independent data in the time domain , as shown below : ______________________________________0 t1 t2 t time domainx1 x2 x2 . . . xn ( real part of data ) y1 y2 y3 . . . yn ( imaginary part ) ______________________________________ the time domain covers a range from 0 to t , after which it repeats indefinitely . this behavior is called aliasing . the range t is the inverse of the frequency domain step size : the time domain step size t /( ndata - 1 ) is the inverse of the frequency domain range : in the preferred embodiment , the chirp z - transform is used to display the arbitrary time range from t1 ( starttime ) to t2 ( endtime ) indicated in the last figure . other dft algorithms can be employed if desired . the number of equally spaced points calculated for this range t1 to t2 is equal to the number of points in the frequency domain , although the chirp z - transform does allow the number of input and output points to differ . note that the chirp z - transform performs a band limited interpolation because t1 to t2 is less than the unaliased time range 0 to t . the chirp z - transform as published by the ieee uses certain conventions which will be adopted here , for the parameters used in calling the transform . the published algorithm is nominally a forward transform . converting it to an inverse transform can be accomplished by reversing the rotation of the output data , which can be done by using negative values for the parameters dltomg and ome0 . in addition , ndata and nopts are set to a common integer value and fs is set to the same value , using a floating point variable to satisfy the variable type requirements of fortran , the language in which the chirp z - transform is written . phase rotation of complex data is accomplished by doing a complex multiplication of the data by a unit vector with an angle equal to the desired change in radians . where array index values are specified , they fall in the range of 1 to ndata . this is the fortran indexing convention . when the foregoing conventions are followed , the low pass equivalent algorithm can be expressed as shown in fig5 . first , the system generates the ndata data samples in a frequency range from f1 to f2 ( block 500 ). the routine processes the samples as follows : 1 . apply a suitable window to the frequency domain data with nterms terms ( block 501 ). a hamming raised cosine window with two terms ( nterm = 2 ) is found to provide useful results for many measurements . also , 3 or 4 term raised cosine windows , known as blackman windows , can be used . for information concerning selection of windows , see f . j . harris , &# 34 ; on the use of windows for harmonic analysis with the discrete fourier transform &# 34 ;, proc . ieee , vol 66 , no . 1 , pp . 51 - 83 , january 1978 . biasing with hamming windows is not necessary for all measurements , but can aid interpretation of the data . a &# 34 ; rectangular &# 34 ; window ( nterms = 1 ) is suitable for many measurements . 2 . convert data to the time domain with the chirp z - transform , selecting dltomg and ome0 as shown , so that the main lobe of the discontinuity from t1 to t2 ( block 502 ), fills the display of time domain data . 3 . unwind the phase of the time domain data by an amount given by where j is the index into the time domain data ( block 503 ). this corrects the portion of the phase error which is not dependent on the distance to the discontinuity . 4 . find the magnitude peak jmax of the displayed time range t1 to t2 ( block 504 ). find the phase angles of the data at two points removed from the magnitude peak by a number of data points equal to the integer part of where nterms is the number of terms in the window used ( block 505 ). nterms reflects the width of the main lobe relative to its width for a rectangular window . determine the difference in phase angle at these two data points to identify the phase slope through jmax . 5 . compare this phase difference to a threshold value determined by experiment for the particular instrument configuration ( block 506 ). if the difference , or phase slope , is less than the threshold value , the discontinuity is not reactive , branch to block 507 . 6 . if the phase slope exceeds the threshold value , jmax is not the best estimate of the location of the discontinuity , branch to block 508 . an improved estimate is generated by comparing the phase at the middle of the spectrum of the discontinuity with the phase of the time domain data over which the phase slope was determined in block 505 . the processing performed in step 3 ( block 503 ) is a heterodyne operation in the time domain . it is the equivalent of a shift in the frequency domain of the middle of the spectrum of the discontinuity to zero frequency . the spectral component of interest can be computed with a discrete fourier transform . since the component is at zero frequency , the transform degenerates into a simple summation of the real and imaginary parts of the data . the inverse tangent operation is then used to find the angle of interest . once the angle has been found , it is compared point by point with the angles of the time domain data between t1 and t2 until the best match is found . linear interpolation can be used to find the point of match between the closest two data points , if desired . this matching point is an improved estimate of the location of the discontinuity . alternative methods for estimating the location may be derived for a particular application . 7 . the estimate of location determined in block 505 or block 508 is set as variable z with a value between 1 and ndata in blocks 507 and 509 , respectively . z is used to do a final unwrapping of the phase of the time domain data ( block 510 ), using a phase value given by 8 . finally , the resulting data is displayed ( block 511 ). the data will now have consistent real and imaginary parts which can be interpreted in terms of the nature of the discontinuity as illustrated in fig4 a - 4i . the interpretation will be somewhat different from that for true low pass data . the real part will be positive or negative if there is an increase or decrease in impedance . the imaginary part will be positive or negative if there is an inductive or capacitive reactance . combined discontinuities will exhibit combined effects . source code in fortran implementing a preferred embodiment of the present invention is provided as an appendix to the present application . the present invention provides low pass equivalent processing useful in analyzing data sampled from a complex network . users of the present invention will be able to better identify discontinuities in networks for design and trouble shooting tasks . it has been established that according to the present invention , frequency domain data taking over a range of frequencies from f1 to f2 , where f1 is not equal to zero , can be transformed to the time domain and the phase information for a particular point in that time domain data can be reconstructed with an accuracy that is useful for analyzing characteristics of the network from which the frequency domain samples were taken . results may not be obtainable for all networks . the foregoing description of the preferred embodiment of the present invention has been presented for purposes of illustration and description . it is not intended to be exhaustive or to limit the invention to the precise form disclosed . obviously , many modifications and variations will be apparent to practitioners skilled in this art . the embodiment was chosen and described in order to best explain the principles of the invention and its practical application , thereby enabling others skilled in the art to understand the invention for various embodiments and with various modifications as are suited to the particular use contemplated . it is intended that the scope of the invention be defined by the following claims and their equivalents .