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CLARKSON UNIVERSITY

Detection and Parameter Extraction of Low Probability of Intercept Radar Signals Using the Reassignment Method and the Hough Transform

A Dissertation By

Daniel Lee Stevens

Department of Electrical and Computer Engineering Submitted in partial fulfillment of the requirements

for the degree of

Doctor of Philosophy, Electrical and Computer Engineering

November 22, 2010

Accepted by the Graduate School

_________, _______________________ Date, Dean of the Graduate School

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UMI Number: _3437800

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The undersigned have examined the dissertation entitled ‘Detection and Parameter Extraction of Low Probability of Intercept Radar Signals Using the Reassignment Method and the Hough Transform’ presented by Daniel Lee Stevens, a candidate for the degree of Doctor of Philosophy, Electrical and Computer Engineering and hereby certify that it is worthy of acceptance.

______________________ ___________________________________ Date Dr. Stephanie A. Schuckers

______________________ ___________________________________ Date Dr. Robert J. Schilling

______________________ ___________________________________ Date Dr. Abdul J. Jerri

______________________ ___________________________________ Date Dr. Jeremiah J. Remus

______________________ ___________________________________ Date Dr. Andrew J. Noga

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Abstract

Current radar systems employ new types of signals which are difficult to detect and characterize. These signals are known as low probability of intercept radar signals, which have as their goal ‘to see and not be seen’ by intercept receivers, devices which are designed to detect and extract information from radar emissions. Digital intercept receivers are currently moving away from Fourier-based analysis and towards classical time-frequency analysis techniques, based on the Wigner-Ville distribution, Choi-Williams distribution, spectrogram, and scalogram, for the purpose of analyzing low probability of intercept radar signals (e.g. triangular modulated frequency modulated continuous wave and frequency shift keying). Although these classical time-frequency techniques are an improvement over the Fourier-based analysis, they still suffer from a lack of readability, due to poor time-frequency localization, cross-term interference, and a mediocre performance in low signal-to-noise ratio environments. This lack of readability may lead to inaccurate detection and parameter extraction of these radar signals, making for a less informed (and therefore less safe) intercept receiver environment. In this dissertation, the reassignment method, because of its ability to smooth out cross-term interference and improve time-frequency localization, is proposed as an improved signal analysis technique. In addition, the Hough transform, due to its ability to suppress cross-term interference, separate signals from cross-cross-term interference, and perform well in a low signal-to-noise ratio environment, is also proposed as an improved signal analysis technique. With these qualities, both the reassignment method and the Hough transform have the potential to produce better readability and consequently, more accurate signal detection and parameter extraction metrics. Simulations are presented that compare

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frequency representations of the classical time-frequency techniques with those of the reassignment method and the Hough transform. Two different triangular modulated frequency modulated continuous wave low probability of intercept radar signals and two frequency shift keying low probability of intercept radar signals (4-component and 8-component) were analyzed. The following metrics were used for evaluation of the analysis: percent error of: carrier frequency, modulation bandwidth, modulation period, time-frequency localization, and chirp rate. Also used were: percent detection, number of cross-term false positives, lowest signal-to-noise ratio for signal detection, and plot (processing) time. Experimental results demonstrate that the ‘squeezing’ and ‘smoothing’ qualities of the reassignment method, along with the cross-term interference suppressing/separating and low signal-to-noise ratio performance qualities of the Hough transform, do in fact lead to improved readability over the classical time-frequency analysis techniques, and consequently, provide more accurate signal detection and parameter extraction metrics than the classical time-frequency analysis techniques. In addition, the Hough transform was utilized to detect, extract parameters, and properly identify a real-world low probability of intercept radar signal in a low signal-to-noise ratio environment, where classical time-frequency analysis failed. In summary, this dissertation provides evidence that the reassignment method and the Hough transform have the potential to outperform the classical time-frequency analysis techniques, which are the current state-of-the-art, cutting edge techniques for this arena. An improvement in performance can easily translate into saved equipment and lives. Future work will include automation of metrics extraction process, analysis of additional low probability of

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intercept radar waveforms of interest, and analysis of other real-world low probability of intercept radar signals utilizing more powerful computing platforms.

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Acknowledgements

I’d like to thank the following people, who all played a part in this major accomplishment:

My ‘better-half’, Donna, my loving wife, who lives the life of a Proverbs 31 woman.

Abraham, Solomon, Joy, Judah, Christian, and Glory, my six blessings from the Lord, who continually radiate life, and who are always an inspiration to me.

My Mom, Judy, whose prayers through the years have kept me on the ‘straight and narrow’.

My sister Cathy (Snorque), and my brothers Mike and Jake, who are undoubtedly smarter than I am, but whose heads aren’t quite as large.

My Father, Roger, who was himself a brilliant man, and who is currently part of that ‘great cloud of witnesses’ that we will once again see one day.

Prof. Stephanie Schuckers, who has, in both a personal and professional way, guided me through this process, all the while believing in this ‘non-traditional’ (i.e. old  ) student. She has been as much a friend as an advisor, as we are at similar stages in our lives.

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Dr. Dick Wiley, Dr. Phil Pace, Dr. Steven Kay, and Dr. G. Faye Boudreaux-Bartels: for allowing me to bounce ideas and questions off of their sharp minds.

My committee members: Prof. Bob Schilling (who helped me get the rust out early on), Prof. Abdul Jerri (who always has a smile on his face), Prof. Jeremiah Remus (one of the nicest people around), and Dr. Andy Noga (aka Dr. Demodulation - who pushed all the right buttons to get the banana!) – Thank-you for your feedback and your helpful suggestions.

Prof. Tom Ortmeyer: for introducing me to the program.

John Grieco, Steve Johns, Ed Starczewski, and Dr. John Maher: for their in-house support.

Charlie Estrella: for his MATLAB wisdom (Go Raiders!).

Finally, I would like to thank Jesus, because apart from Him I can do nothing (John 15:5), but with Him I can do all things (Phil 4:13).

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List of Figures

Figure Page Figure 1: WVD of a triangular modulated FMCW signal at an SNR of 0 dB (256 samples) ...27 Figure 2: Same as Figure 1, but with the SNR level increased to 10 dB. Though some of the noise has disappeared, the cross-terms remain ...27 Figure 3: Same as Figure 1, but with the SNR level increased to 20 dB. Most of the noise has disappeared, but the cross-terms remain, demonstrating that cross-terms are virtually unaffected by the SNR level ...28 Figure 4: Example of cross-term false positives (XFPs). There are 4 true signals (fc1=1KHZ, fc2=1.75KHz, fc3=0.75KHz, fc4=1.25KHz) and 6 XFPs (cross-terms that were wrongly declared as signal detections) (ct1=0.875KHz, ct2=1KHz, ct3=1.125KHz, ct4=1.25KHz, ct5=1.375KHz, ct6=1.5KHz). WVD of a 4-component FSK signal at SNR=10dB (512 samples) ...29 Figure 5: RSPWVD of triangular modulated FMCW, 256 samples, SNR=10dB. Threshold of 5% of the maximum intensity (upper left) and 1% threshold (upper right). The SNR has been increased to 25dB (lower-left) and to 50dB (lower-right). The plot on the upper left gives the appearance of perfectly localized distributions for the chirp signals. However, the plot on the upper right (1% threshold) reveals artifacts that are clearly not perfectly localized. In order to determine if these artifacts are due to noise, the SNR is raised to 25dB (lower left). Many of the artifacts disappear, but some remain. The SNR is then raised to 50dB (lower right) to see if the remaining artifacts are also noise related, but most of the artifacts in the 25dB plot are also in the 50dB plot. For time-frequency reassignment, it is known that the localization is slightly degraded at those points where the IF cannot be considered as quasi-linear within a TF smoothing window (i.e. the curved areas of the signal in Figure 5) causing artifacts. In addition, when more than one component is within a TF smoothing window (i.e. the curves where the chirp signals meet in Figure 5), a beating effect occurs, resulting in interference fringes which may manifest as artifacts ...45 Figure 6: Time-frequency plot on the left and Hough transform plot on the right. A point in the TF plot maps to a sinusoidal curve in the HT plot. A line (signal) in the TF plot maps to a point in the HT plot. The rho and theta values of the point in the HT plot can be used to back-map to the TF plot in order to find the location of the line (signal) (good if time-frequency plot is cluttered with noise and/or cross-term interference and signal is not visible) ...51 Figure 7: Hough transform of the WVD of a triangular modulated FMCW signal at an SNR of 10 dB (512 samples). Each point has a unique theta and rho value which can be

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used to back-map to the time-frequency (WVD) representation in order to locate the 4 signals, as depicted in Figure 7 ...53 Figure 8: The unique theta and rho values extracted from Figure 6 are used to back-map to the time-frequency representation (WVD of a triangular modulated FMCW signal (SNR=10dB, #samples=512)) in order to locate the 4 chirp signals that make up the 4 legs of the triangular modulated FMCW signal ...55 Figure 9: Threshold percentage determination. This plot is an amplitude vs. time (x-z view) of CWD of FSK 4-component signal (512 samples, SNR= -2dB). For visually detected low SNR plots (like this one), the percent of max intensity for the peak z-value of each of the signal components was noted (here 82%, 98%, 87%, 72%), and the lowest of these 4 values was recorded (72%). Ten test runs were performed for each time-frequency analysis tool, for each of the 4 waveforms. The average of these recorded low values was determined and then assigned as the threshold for that particular time-frequency analysis tool. Note - the threshold for CWD is 70%...68 Figure 10: Percent detection (time-frequency). CWD of 4-component FSK (512 samples, SNR=10dB) with threshold value automatically set to 70%. From this threshold plot, the signal was declared a (visual) detection because at least a portion of each of the 4 FSK signal components was visible ...69 Figure 11: Percent detection (Hough transform). This plot is a theta-intensity (x-z view) of the HT of an RSPWVD (512 samples, SNR=10dB). Signal declared a (visual) detection because at least a portion of each of the signal components exceeded the noise floor threshold ...70 Figure 12: Example of cross-term false positives (XFPs). WVD of a 4-component FSK signal at SNR=10dB (512 samples). There are 4 true signals (fc1=1KHz, fc2=1.75KHz, fc3=0.75KHz, fc4=1.25KHz) and 6 XFPs (cross-terms that were wrongly declared as signal detections because they passed the signal detection criteria listed in the percent detection section above) (ct1=0.875KHz, ct2=1KHz, ct3=1.125KHz, ct4=1.25KHz, ct5=1.375KHz, ct6=1.5KHz) ...71 Figure 13: Determination of carrier frequency. CWD of a triangular modulated FMCW (256 samples, SNR=10dB). From the frequency-intensity (y-z) view, the maximum intensity value is manually determined. The frequency corresponding to the max intensity value is the carrier frequency (here fc=1062.4 Hz)...72 Figure 14: Determination of carrier frequency. CWD of a 4-component FSK (512 samples, SNR=10dB). From the frequency-intensity (y-z) view, the 4 maximum intensity values (1 for each carrier frequency) are manually determined. The frequencies corresponding to those 4 max intensity values are the 4 carrier frequency (here fc1=10001 Hz, fc2=1753Hz, fc3=757Hz, fc4=1255Hz) ...73

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Figure 15: Modulation bandwidth determination. CWD of a triangular modulated FMCW (512 samples, SNR=10dB) with threshold value automatically set to 20%. From this threshold plot, the modulation bandwidth was measured manually from the highest frequency value of the signal (top red arrow) to the lowest frequency value of the signal (bottom red arrow) in the y-direction (frequency) ...74 Figure 16: Modulation bandwidth determination. CWD of a 4-component FSK (512 samples, SNR=10dB) with threshold value automatically set to 20%. From this threshold plot, the modulation bandwidth was measured manually from the highest frequency value of the signal (top red arrow) to the lowest frequency value of the signal (bottom red arrow) in the y-direction (frequency) ...75 Figure 17: Modulation period determination. CWD of a triangular modulated FMCW (512 samples, SNR=10dB) with threshold value automatically set to 20%. From this threshold plot, the modulation period was measured manually from the highest frequency value of the signal (top red arrow) to the lowest frequency value of the signal (bottom red arrow) in the x-direction (time) ...77 Figure 18: Modulation period determination. CWD of a 4-component FSK (512 samples, SNR=10dB) with threshold value automatically set to 20%. From this threshold plot, the modulation period was measured manually from the left side of the signal (left red arrow) to the right side of the signal (right red arrow) in the x-direction (time). This was done for all 4 signal components, and the average value was determined ...78 Figure 19: Time-frequency localization determination. CWD of a triangular modulated FMCW (512 samples, SNR=10dB) with threshold value automatically set to 20%. From this threshold plot, the time-frequency localization was measured manually from the left side of the signal (left red arrow) to the right side of the signal (right red arrow) in both the x-direction (time) and the y-direction (frequency). Measurements were made at the center of each of the 4 ‘legs’, and the average values were determined. Average time and frequency ‘thickness’ values were then converted to: % of entire x-axis and % of entire y-axis ...80 Figure 20: Time-frequency localization determination for the CWD of a 4-component FSK (512 samples, SNR=10dB) with threshold value automatically set to 20%. From this threshold plot, the time-frequency localization was measured manually from the top of the signal (top red arrow) to the bottom of the signal (bottom red arrow) in the y-direction (frequency). This frequency ‘thickness’ value was then converted to: % of entire y-axis. ...81 Figure 21: Lowest detectable SNR (time-frequency). CWD of 4-component FSK (512 samples, SNR=-2dB) with threshold value automatically set to 70%. From this threshold plot, the signal was declared a (visual) detection because at least a portion of each of the 4 FSK signal components was visible. For this case, any lower SNR would have been a non-detect. Compare to Figure 10, which is the same plot, except that it has an SNR level equal to 10dB ...82

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Figure 22: Lowest detectable SNR (Hough transform). This plot is a theta-intensity (x-z view) of the HT of an RSPWVD (512 samples, SNR=-5dB). Signal declared a (visual) detection because at least a portion of each of the signal components exceeded the noise floor threshold. For this case, any lower SNR would have been a non-detect. Compare to Figure 11, which is the same plot, except that it has an SNR level equal to 10dB ...83 Figure 23: Threshold percentage determination. This plot is an amplitude vs. time (x-z view) of CWD of FSK 4-component signal (512 samples, SNR= -2dB). For visually detected low SNR plots (like this one), the percent of max intensity for the peak z-value of each of the signal components was noted (here 82%, 98%, 87%, 72%), and the lowest of these 4 values was recorded (72%). Ten test runs were performed for each time-frequency analysis tool, for each of the 4 waveforms. The average of these recorded low values was determined and then assigned as the threshold for that particular time-frequency analysis tool. Note - the threshold for CWD is 70%...95 Figure 24: Percent detection (time-frequency). CWD of 4-component FSK (512 samples, SNR=10dB) with threshold value automatically set to 70%. From this threshold plot, the signal was declared a (visual) detection because at least a portion of each of the 4 FSK signal components was visible ...96 Figure 25: Modulation bandwidth determination. CWD of a triangular modulated FMCW (512 samples, SNR=10dB) with threshold value automatically set to 20%. From this threshold plot, the modulation bandwidth was measured manually from the highest frequency value of the signal (top red arrow) to the lowest frequency value of the signal (bottom red arrow) in the y-direction (frequency) ...98 Figure 26: Modulation bandwidth determination. CWD of a 4-component FSK (512 samples, SNR=10dB) with threshold value automatically set to 20%. From this threshold plot, the modulation bandwidth was measured manually from the highest frequency value of the signal (top red arrow) to the lowest frequency value of the signal (bottom red arrow) in the y-direction (frequency) ...99 Figure 27: Lowest detectable SNR (time-frequency). CWD of 4-component FSK (512 samples, SNR=-2dB) with threshold value automatically set to 70%. From this threshold plot, the signal was declared a (visual) detection because at least a portion of each of the 4 FSK signal components was visible. For this case, any lower SNR would have been a non-detect. Compare to Figure 24, which is the same plot, except that it has an SNR level equal to 10dB ...101 Figure 28: Time-frequency localization comparison between the classical time-frequency analysis tools (left) and the reassignment method (right). The uppper two plots are the spectrogram (left) and the reassigned spectrogram (right) for a triangular modulated FMCW signal (256 samples, SNR=10dB), and the lower two plots are the scalogram (left) and the reassigned scalogram (right) for a 4-component FSK signal (512 samples,

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SNR=10dB). The reassignment method gives a much more concentrated time-frequency localization than does its classical time-frequency analysis counterparts ...104 Figure 29: Cross-term interference of the WVD (left) and ability of the RSPWVD to reduce cross-term interference (right). The uppper two plots are the WVD (left) and the RSPWVD (right) for a triangular modulated FMCW signal (512 samples, SNR=10dB), and the lower two plots are the WVD (left) and the RSPWVD (right) for a 8-component FSK signal (512 samples, SNR=10dB). The WVD displays a lot of cross-term interference. There appears to be one additional triangle signal in the middle of the two-triangle signal in the upper-left plot, and there appears to be 9 additional signals to go along with the 8-component signal in the lower-left plot. The RSPWVD has drastically reduced the cross-term interference found in the WVD plots, making for more readable presentations. Notice also that the RSPWVD is more concentrated than its WVD counterpart (as per Table 1) ...105 Figure 30: Readability degradation due to reduction in SNR. Spectrogram, triangular modulated FMCW, modulation bandwidth=500Hz, 512 samples. SNR=10dB (left), 0dB (center), -4dB (left). Readability degrades as SNR decreases, negatively affecting the accuracy of the metrics extracted, as per Table 2. ...107 Figure 31: Comparison between Task 1 (left) and Task 2 (right). The uppper two plots are both spectrogram plots of a triangular modulated FMCW signal (512 samples, SNR=10dB), Task 1 (modulation bandwidth=500Hz) is on the left and Task 2 (modulation bandwidth=2400Hz) is on the right. The lower two plots are CWD plots of the same signals. The Task 2 plots (right) have a larger modulation bandwidth than Task 1 plots, therefore the signals appear taller and more upright than the Task 1 signals ...109 Figure 32: Percent detection (Hough transform). This plot is a theta-intensity (x-z view) of the HT of an RSPWVD (512 samples, SNR=10dB). Signal declared a (visual) detection because at least a portion of each of the signal components exceeded the noise floor threshold ...132 Figure 33: Threshold percentage determination. This plot is an amplitude vs. time (x-z view) of CWD of FSK 4-component signal (512 samples, SNR= -2dB). For visually detected low SNR plots (like this one), the percent of max intensity for the peak z-value of each of the signal components was noted (here 82%, 98%, 87%, 72%), and the lowest of these 4 values was recorded (72%). Ten test runs were performed for each time-frequency analysis tool, for each of the 4 waveforms. The average of these recorded low values was determined and then assigned as the threshold for that particular time-frequency analysis tool. Note - the threshold for CWD is 70%...133 Figure 34: Percent detection (time-frequency). CWD of 4-component FSK (512 samples, SNR=10dB) with threshold value automatically set to 70%. From this threshold plot, the signal was declared a (visual) detection because at least a portion of each of the 4 FSK signal components was visible ...134

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Figure 35: S Example of cross-term false positives (XFPs). WVD of a 4-component FSK signal at SNR=10dB (512 samples). There are 4 true signals (fc1=1KHz, fc2=1.75KHz, fc3=0.75KHz, fc4=1.25KHz) and 6 XFPs (cross-terms that were wrongly declared as signal detections because they passed the signal detection criteria listed in the percent detection section above) (ct1=0.875KHz, ct2=1KHz, ct3=1.125KHz, ct4=1.25KHz, ct5=1.375KHz, ct6=1.5KHz) ...135 Figure 36: Lowest detectable SNR (Hough transform). This plot is a theta-intensity (x-z view) of the HT of an RSPWVD (512 samples, SNR=-5dB). Signal declared a (visual) detection because at least a portion of each of the signal components exceeded the noise floor threshold. For this case, any lower SNR would have been a non-detect. Compare to Figure 32, which is the same plot, except that it has an SNR level equal to 10dB ...136 Figure 37: Lowest detectable SNR (time-frequency). CWD of 4-component FSK (512 samples, SNR=-2dB) with threshold value automatically set to 70%. From this threshold plot, the signal was declared a (visual) detection because at least a portion of each of the 4 FSK signal components was visible. For this case, any lower SNR would have been a non-detect. Compare to Figure 34, which is the same plot, except that it has an SNR level equal to 10dB ...137 Figure 38: Cross-term comparison between classical time-frequency analysis techniques (left) and the Hough transform (right). Top left: WVD of a triangular modulated FMCW signal (512 samples, SNR=10dB) (left). Top right: the Hough transform of the WVD of a triangular modulated FMCW signal (512 samples, SNR=10dB). Bottom left: WVD of an FSK (8-component) signal (512 samples, SNR=10dB). Bottom right: the Hough transform of the WVD of an FSK (8-component) signal (512 samples, SNR=10dB). Upper 2 plots – the Hough transform (right) has eliminated the cross-term interference that the WVD (left) displays, making it easier to see the signal (better readability) in the Hough transform plot (the four bright spots which represent the four legs of the triangular modulated FMCW signal). The WVD appears to have another triangle signal between the outer two triangle signals. Lower 2 plots - In the WVD plot (left) there are 8 signal components and 9 cross-term components that appear to be signal components, all melded in together with one another. The 8 signal components are located at 5 distinct frequencies (one at 0.75KHz, two at 1KHz, two at 1.25KHz, two at 1.5KHz, and one at 1.75KHz). In the Hough transform plot (right) there are 8 signal components (located at 5 different frequencies (3 on the left-hand side of the plot and 2 in the middle of the plot)) plus 2 cross-term components, clearly separated from the signal components. The Hough transform plot makes it easier to see the signal components (better readability) as compared to the WVD plot ...141 Figure 39: Low SNR comparison between classical time-frequency analysis techniques (left) and the Hough transform (right). Top left: CWD of a triangular modulated FMCW signal (512 samples, SNR=-6dB) (left). Top right: the Hough transform of the CWD of a triangular modulated FMCW signal (512 samples, SNR=-6dB). Bottom left: CWD of an FSK (8-component) signal (512 samples, SNR=-3dB). Bottom right: the Hough transform of the CWD of an FSK (8-component) signal (512 samples, SNR=-3dB).

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Upper 2 plots - Though the signal is not visible in the CWD plot (left) (due to the low SNR (-6dB)), the four bright spots that represent the four legs of the triangular modulated FMCW signal are clearly seen in the Hough transform of the CWD plot (right). Each bright spot has a unique rho and theta value that can be used to back-map to the time-frequency representation (here CWD) and find the location of the 4 (non-visible) chirps that make up the triangular modulated FMCW signal. Lower 2 plots - Though the signal components are not visible in the CWD plot (due to the low SNR (-3dB)), the 5 bright spots (3 on the left and 2 in the middle) corresponding to the 5 different frequencies of the 8 FSK components are clearly visible in the Hough transform of the CWD plot (as are 2 cross-term components). The Hough transform does a good job of detecting the signal components and separating the signal components from the cross-term components, all in a low SNR environment ...142 Figure 40: Readability degradation due to reduction in SNR. Spectrogram, triangular modulated FMCW, modulation bandwidth=500Hz, 512 samples. SNR=10dB (left), 0dB (center), -4dB (left). Readability degrades as SNR decreases, negatively affecting the accuracy of the metrics extracted, as per Table 5 ...144 Figure 41: Comparison between Task 1 (left) and Task 2 (right). The uppper two plots are both spectrogram plots of a triangular modulated FMCW signal (512 samples, SNR=10dB), Task 1 (modulation bandwidth=500Hz) is on the left and Task 2 (modulation bandwidth=2400Hz) is on the right. The lower two plots are CWD plots of the same signals. The Task 2 plots (right) have a larger modulation bandwidth than Task 1 plots, therefore the signals appear taller and more upright than the Task 1 signals ...145 Figure 42: Spectrogram (left) and Hough transform of Spectrogram (right) of area where Task 5 real-world signal was located. Signal was not visible in the spectrogram, but was visible in the Hough transform, due to the Hough transform’s ability to extract signals from low SNR environments. ...146 Figure 43: Hough transform of Spectrogram (zoomed-in on receiver IF bandwidth (~750MHz to 1250MHz)) of area where Task 5 real-world signal was located. Signal is clearly visible at theta=2.872 and rho=228.8 ...147 Figure 44: Spectrogram (zoomed-in on receiver IF bandwidth (~750MHz to 1250MHz)). Shows how the Hough transform theta and rho values back-map to the time-frequency representation for signal location ...148 TASK 1

Figure 45: Test 1 Plot – WVD, 256, 10 ...184 Figure 46: Test 2 Plot – WVD, 256, 0 ...185 Figure 47: Test 3 Plot – WVD, 256, -2 ...186

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Figure 48: Test 4 Plot – WVD, 512, 10 ...187

Figure 49: Test 7 Plot – CWD, 256, 10 ...189

Figure 51: Test 9 Plot – CWD, 256, -3 ...191

Figure 55: Test 13 Plot – Spectrogram, 256, 10 ...195

Figure 57: Test 15 Plot – Spectrogram, 256, -3 ...197

Figure 61: Test 19 Plot – Scalogram, 256, 10...201

Figure 63: Test 21 Plot – Scalogram, 256, -2 ...203

Figure 67: Test 25 Plot – Reassigned Spectrogram, 256, 10 ...207

Figure 69: Test 27 Plot – Reassigned Spectrogram, 256, -2 ...209

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Figure 73: Test 31 Plot – Reassigned Scalogram, 256, 10 ...213

Figure 75: Test 33 Plot – Reassigned Scalogram, 256, -2 ...215

Figure 78: Test 36 Plot – Reassigned Scalogram, 512, -3 ...218

Figure 79: Test 37 Plot – RSPWVD, 256, 10 ...219

Figure 81: Test 39 Plot – RSPWVD, 256, -3 ...221

Figure 84: Test 42 Plot – RSPWVD, 512, -3 ...224

Figure 85: Test 43 Plot – HT of WVD, 256, 10 ...225

Figure 87: Test 45 Plot – HT of WVD, 256, -3 ...227

Figure 89: Test 49 Plot – HT of CWD, 256, 10 ...230

Figure 91: Test 51 Plot – HT of CWD, 256, -4 ...232

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Figure 94: Test 54 Plot – HT of CWD, 512, -6 ...235

Figure 95: Test 55 Plot – HT of RSPWVD, 256, 10 ...236

Figure 97: Test 57 Plot – HT of RSPWVD, 256, -4 ...238

Figure 100: Test 60 Plot – HT of RSPWVD, 512, -5 ...241

TASK 2 Figure 101: Test 61 Plot – WVD, 512, 10 ...242

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Figure 116: Test 78 Plot – Reassigned Scalogram, 512, -3 ...258 Figure 117: Test 79 Plot – RSPWVD, 512, 10 ...259 Figure 118: Test 80 Plot – RSPWVD, 512, 0 ...260 Figure 119: Test 81 Plot – RSPWVD, 512, -3 ...261 Figure 120: Test 82 Plot – HT of WVD, 512, 10 ...263 Figure 121: Test 85 Plot – HT of CWD, 512, 10 ...264 Figure 122: Test 86 Plot – HT of CWD, 512, 0 ...265 Figure 123: Test 87 Plot – HT of CWD, 512, -6 ...266 Figure 124: Test 88 Plot – HT of RSPWVD, 512, 10 ...267 Figure 125: Test 89 Plot – HT of RSPWVD, 512, 0 ...268 Figure 126: Test 90 Plot – HT of RSPWVD, 512, -6 ...269 TASK 3

Figure 127: Test 91 Plot – WVD, 512, 10 ...270 Figure 128: Test 94 Plot – CWD, 512, 10 ...272 Figure 129: Test 95 Plot – CWD, 512, 0 ...273 Figure 130: Test 96 Plot – CWD, 512, -2 ...274 Figure 131: Test 97 Plot – Spectrogram, 512, 10 ...275 Figure 132: Test 98 Plot – Spectrogram, 512, 0 ...276 Figure 133: Test 99 Plot – Spectrogram, 512, -3 ...277 Figure 134: Test 100 Plot – Scalogram, 512, 10...278 Figure 135: Test 101 Plot – Scalogram, 512, 0...279 Figure 136: Test 102 Plot – Scalogram, 512, -3 ...280 Figure 137: Test 103 Plot – Reassigned Spectrogram, 512, 10 ...281

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Figure 138: Test 104 Plot – Reassigned Spectrogram, 512, 0 ...282 Figure 139: Test 105 Plot – Reassigned Spectrogram, 512, -3 ...283 Figure 140: Test 106 Plot – Reassigned Scalogram, 512, 10 ...284 Figure 141: Test 107 Plot – Reassigned Scalogram, 512, 0 ...285 Figure 142: Test 108 Plot – Reassigned Scalogram, 512, -4 ...286 Figure 143: Test 109 Plot – RSPWVD, 512, 10 ...287 Figure 144: Test 110 Plot – RSPWVD, 512, 0 ...288 Figure 145: Test 111 Plot – RSPWVD, 512, -3 ...289 Figure 146: Test 112 Plot – HT of WVD, 512, 10 ...290 Figure 147: Test 112 Plot – X-Y View - HT of WVD, 512, 10 ...290 Figure 148: Test 113 Plot – HT of CWD, 512, -3 ...292 Figure 149: Test 113 Plot – X-Y View - HT of CWD, 512, -3 ...292 Figure 150: Test 114 Plot – HT of RSPWVD, 512, -4 ...294 Figure 151: Test 114 Plot – X-Y View - HT of RSPWVD, 512, -4 ...294 TASK 4

Figure 152: Test 115 Plot – WVD, 512, 10 ...296 Figure 153: Test 118 Plot – CWD, 512, 10 ...298 Figure 154: Test 119 Plot – CWD, 512, 0 ...299 Figure 155: Test 120 Plot – CWD, 512, -1 ...300 Figure 156: Test 121 Plot – Spectrogram, 512, 10 ...301 Figure 157: Test 122 Plot – Spectrogram, 512, 0 ...302 Figure 158: Test 123 Plot – Spectrogram, 512, -1 ...303 Figure 159: Test 124 Plot – Scalogram, 512, 10...304

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Figure 160: Test 125 Plot – Scalogram, 512, 0...305 Figure 161: Test 126 Plot – Scalogram, 512, -3 ...306 Figure 162: Test 127 Plot – Reassigned Spectrogram, 512, 10 ...307 Figure 163: Test 128 Plot – Reassigned Spectrogram, 512, 0 ...308 Figure 164: Test 129 Plot – Reassigned Spectrogram, 512, -2 ...309 Figure 165: Test 130 Plot – Reassigned Scalogram, 512, 10 ...310 Figure 166: Test 131 Plot – Reassigned Scalogram, 512, 0 ...311 Figure 167: Test 132 Plot – Reassigned Scalogram, 512, -2 ...312 Figure 168: Test 133 Plot – RSPWVD, 512, 10 ...313 Figure 169: Test 134 Plot – RSPWVD, 512, 0 ...314 Figure 170: Test 135 Plot – RSPWVD, 512, -3 ...315 Figure 171: Test 136 Plot – HT of WVD, 512, 10 ...316 Figure 172: Test 136 Plot – X-Y View - HT of WVD, 512, 10 ...316 Figure 173: Test 137 Plot – HT of CWD, 512, -3 ...318 Figure 174: Test 137 Plot – X-Y View - HT of CWD, 512, -3 ...318 Figure 175: Test 138 Plot – HT of RSPWVD, 512, -4 ...320 Figure 176: Test 138 Plot – X-Y View - HT of RSPWVD, 512, -4 ...320

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xxiv List of Tables

Table 1: Signal Processing Tool viewpoint of the overall test metrics (average percent error) for the 4 classical time-frequency analysis techniques (WVD, CWD, spectrogram, scalogram), along with their combined average (TF) and for the 3 reassignment methods (reassigned spectrogram, reassigned scalogram, RSPWVD), along with their combined average (RM). The parameters extracted are listed in the left-hand column: carrier frequency (fc), modulation bandwidth (modbw), modulation period (modper), time-frequency localization in the x-direction (tf res-x), time-time-frequency localization in the y-direction (tf res-y), chirp rate (cr), percent detection (% det), number of cross-term false positives (#XFP), lowest detectable SNR (low snr), plot time (plottime) ...103 Table 2: SNR viewpoint of overall test metrics (average percent error) for the classical time-frequency analysis techniques (TF) and for the reassignment method (RM) for SNR=10dB, 0dB, and lowest detectable SNR (low snr). The parameters extracted are listed in the left-hand column: carrier frequency (fc), modulation bandwidth (modbw), modulation period (modper), time-frequency localization in the x-direction (tf res-x), time-frequency localization in the y-direction (tf res-y), chirp rate (cr), percent detection (% det), number of cross-term false positives (#XFP), plot time (plottime) ...106 Table 3: Task 1, 2, 3, and 4 viewpoint of overall test metrics (average percent error) for the classical time-frequency analysis techniques (TF) and for the reassignment method (RM). Task1=triangular modulated FMCW signal (modulation bandwidth=500Hz), Task2=triangular modulated FMCW signal (modulation bandwidth=2400Hz), Task 3=FSK (4-component) signal, Task 4=FSK (8-component) signal. The parameters extracted are listed in the left-hand column: carrier frequency (fc), modulation bandwidth (modbw), modulation period (modper), time-frequency localization in the x-direction (tf res-x), time-frequency localization in the y-direction (tf res-y), chirp rate (cr), percent detection (% det), number of cross-term false positives (#XFP), lowest detectable SNR (low snr), plot time (plottime) ...108 Table 4: Signal Processing Tool viewpoint of the overall test metrics (average percent error) for the 4 classical time-frequency analysis techniques (WVD, CWD, spectrogram, scalogram) along with their combined average (TF) and for the 2 Hough transform methods (WVD + HT, CWD + HT), along with their combined average (HT). The parameters extracted are listed in the left-hand column: chirp rate (cr), percent detection (% det), # of cross-term false positives (#XFP), lowest detectable SNR (low snr) ...139 Table 5: SNR viewpoint of overall test metrics (average percent error) for the classical time-frequency analysis techniques (TF) and for the Hough transform methods (HT) for SNR=10dB, 0dB, and lowest detectable SNR (low SNR). The parameters extracted are listed in the left-hand column: chirp rate (cr), percent detection (% det), number of cross-term false positives (#XFP). ...143 Table 6: Task 1, 2, 3, and 4 viewpoint of overall test metrics (average percent error) for the classical time-frequency analysis techniques (TF) and for the Hough transform (HT).

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Task1=triangular modulated FMCW signal (modulation bandwidth=500Hz), Task2=triangular modulated FMCW signal (modulation bandwidth=2400Hz), Task 3=FSK (4-component) signal, Task 4=FSK (8-component) signal. The parameters extracted are listed in the left-hand column: chirp rate (cr), percent detection (% det), number of cross-term false positives (#XFP), lowest detectable SNR (low snr) ...144 Table 7: Overall test metrics for the classical time-frequency analysis techniques (Classical TF), reassignment method (RM), Hough transform (HT), and reassignment method plus the Hough transform (RM + HT). The parameters extracted are listed in the left-hand column: chirp rate, lowest detectable SNR (Low SNR), and percent detection (% Detection) ...159 Table 8: Test Matrix for Task 1 through Task 5 ...176 TASK 1

Table 9: Test 1 Metrics – WVD, 256, 10 ...184 Table 10: Test 2 Metrics – WVD, 256, 0 ...185 Table 11: Test 3 Metrics – WVD, 256, -2 ...186 Table 12: Test 4 Metrics – WVD, 512, 10 ...187 Table 13: Test 7 Metrics – CWD, 256, 10 ...189 Table 14: Test 8 Metrics – CWD, 256, 0 ...190 Table 15: Test 9 Metrics – CWD, 256, -3...191 Table 16: Test 10 Metrics – CWD, 512, 10 ...192 Table 17: Test 11 Metrics – CWD, 512, 0 ...193 Table 18: Test 12 Metrics – CWD, 512, -3...194 Table 19: Test 13 Metrics – Spectrogram, 256, 10 ...195 Table 20: Test 14 Metrics – Spectrogram, 256, 0 ...196 Table 21: Test 15 Metrics – Spectrogram, 256, -3 ...197 Table 22: Test 16 Metrics – Spectrogram, 512, 10 ...198 Table 23: Test 17 Metrics – Spectrogram, 512, 0 ...199

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Table 24: Test 18 Metrics – Spectrogram, 512, -4 ...200 Table 25: Test 19 Metrics – Scalogram, 256, 10 ...201 Table 26: Test 20 Metrics – Scalogram, 256, 0 ...202 Table 27: Test 21 Metrics – Scalogram, 256, -2 ...203 Table 28: Test 22 Metrics – Scalogram, 512, 10 ...204 Table 29: Test 23 Metrics – Scalogram, 512, 0 ...205 Table 30: Test 24 Metrics – Scalogram, 512, -3 ...206 Table 31: Test 25 Metrics – Reassigned Spectrogram, 256, 10 ...207 Table 32: Test 26 Metrics – Reassigned Spectrogram, 256, 0 ...208 Table 33: Test 27 Metrics – Reassigned Spectrogram, 256, -2 ...209 Table 34: Test 28 Metrics – Reassigned Spectrogram, 512, 10 ...210 Table 35: Test 29 Metrics – Reassigned Spectrogram, 512, 0 ...211 Table 36: Test 30 Metrics – Reassigned Spectrogram, 512, -3 ...212 Table 37: Test 31 Metrics – Reassigned Scalogram, 256, 10 ...213 Table 38: Test 32 Metrics – Reassigned Scalogram, 256, 0 ...214 Table 39: Test 33 Metrics – Reassigned Scalogram, 256, -2 ...215 Table 40: Test 34 Metrics – Reassigned Scalogram, 512, 10 ...216 Table 41: Test 35 Metrics – Reassigned Scalogram, 512, 0 ...217 Table 42: Test 36 Metrics – Reassigned Scalogram, 512, -3 ...218 Table 43: Test 37 Metrics – RSPWVD, 256, 10...219 Table 44: Test 38 Metrics – RSPWVD, 256, 0...220 Table 45: Test 39 Metrics – RSPWVD, 256, -3 ...221 Table 46: Test 40 Metrics – RSPWVD, 512, 10...222

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Table 47: Test 41 Metrics – RSPWVD, 512, 0...223 Table 48: Test 42 Metrics – RSPWVD, 512, -3 ...224 Table 49: Test 43 Metrics – HT of WVD, 256, 10 ...225 Table 50: Test 44 Metrics – HT of WVD, 256, 0 ...226 Table 51: Test 45 Metrics – HT of WVD, 256, -3 ...227 Table 52: Test 46 Metrics – HT of WVD, 512, 10 ...228 Table 53: Test 49 Metrics – HT of CWD, 256, 10 ...230 Table 54: Test 50 Metrics – HT of CWD, 256, 0 ...231 Table 55: Test 51 Metrics – HT of CWD, 256, -4 ...232 Table 56: Test 52 Metrics – HT of CWD, 512, 10 ...233 Table 57: Test 53 Metrics – HT of CWD, 512, 0 ...234 Table 58: Test 54 Metrics – HT of CWD, 512, -6 ...235 Table 59: Test 55 Metrics – HT of RSPWVD, 256, 10 ...236 Table 60: Test 56 Metrics – HT of RSPWVD, 256, 0 ...237 Table 61: Test 57 Metrics – HT of RSPWVD, 256, -4 ...238 Table 62: Test 58 Metrics – HT of RSPWVD, 512, 10 ...239 Table 63: Test 59 Metrics – HT of RSPWVD, 512, 0 ...240 Table 64: Test 60 Metrics – HT of RSPWVD, 512, -5 ...241 TASK 2

Table 65: Test 61 Metrics – WVD, 512, 10 ...242 Table 66: Test 64 Metrics – CWD, 512, 10 ...244 Table 67: Test 65 Metrics – CWD, 512, 0 ...245 Table 68: Test 66 Metrics – CWD, 512, -3...246

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Table 69: Test 67 Metrics – Spectrogram, 512, 10 ...247 Table 70: Test 68 Metrics – Spectrogram, 512, 0 ...248 Table 71: Test 69 Metrics – Spectrogram, 512, -4 ...249 Table 72: Test 70 Metrics – Scalogram, 512, 10 ...250 Table 73: Test 71 Metrics – Scalogram, 512, 0 ...251 Table 74: Test 72 Metrics – Scalogram, 512, -3 ...252 Table 75: Test 73 Metrics – Reassigned Spectrogram, 512, 10 ...253 Table 76: Test 74 Metrics – Reassigned Spectrogram, 512, 0 ...254 Table 77: Test 75 Metrics – Reassigned Spectrogram, 512, -3 ...255 Table 78: Test 76 Metrics – Reassigned Scalogram, 512, 10 ...256 Table 79: Test 77 Metrics – Reassigned Scalogram, 512, 0 ...257 Table 80: Test 78 Metrics – Reassigned Scalogram, 512, -3 ...258 Table 81: Test 79 Metrics – RSPWVD, 512, 10...259 Table 82: Test 80 Metrics – RSPWVD, 512, 0...260 Table 83: Test 81 Metrics – RSPWVD, 512, -3 ...261 Table 84: Test 82 Metrics – HT of WVD, 512, 10 ...262 Table 85: Test 85 Metrics – HT of CWD, 512, 10 ...264 Table 86: Test 86 Metrics – HT of CWD, 512, 0 ...265 Table 87: Test 87 Metrics – HT of CWD, 512, -6 ...266 Table 88: Test 88 Metrics – HT of RSPWVD, 512, 10 ...267 Table 89: Test 89 Metrics – HT of RSPWVD, 512, 0 ...268 Table 90: Test 90 Metrics – HT of RSPWVD, 512, -6 ...269 TASK 3

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Table 91: Test 91 Metrics – WVD, 512, 10 ...270 Table 92: Test 94 Metrics – CWD, 512, 10 ...272 Table 93: Test 95 Metrics – CWD, 512, 0 ...273 Table 94: Test 96 Metrics – CWD, 512, -2...274 Table 95: Test 97 Metrics – Spectrogram, 512, 10 ...275 Table 96: Test 98 Metrics – Spectrogram, 512, 0 ...276 Table 97: Test 99 Metrics – Spectrogram, 512, -3 ...277 Table 98: Test 100 Metrics – Scalogram, 512, 10 ...278 Table 99: Test 101 Metrics – Scalogram, 512, 0 ...279 Table 100: Test 102 Metrics – Scalogram, 512, -3 ...280 Table 101: Test 103 Metrics – Reassigned Spectrogram, 512, 10...281 Table 102: Test 104 Metrics – Reassigned Spectrogram, 512, 0...282 Table 103: Test 105 Metrics – Reassigned Spectrogram, 512, -3 ...283 Table 104: Test 106 Metrics – Reassigned Scalogram, 512, 10 ...284 Table 105: Test 107 Metrics – Reassigned Scalogram, 512, 0 ...285 Table 106: Test 108 Metrics – Reassigned Scalogram, 512, -4...286 Table 107: Test 109 Metrics – RSPWVD, 512, 10...287 Table 108: Test 110 Metrics – RSPWVD, 512, 0...288 Table 109: Test 111 Metrics – RSPWVD, 512, -3 ...289 Table 110: Test 112 Metrics – HT of WVD, 512, 10 ...291 Table 111: Test 113 Metrics – HT of CWD, 512, -3 ...293 Table 112: Test 114 Metrics – HT of RSPWVD, 512, -4...295 TASK 4

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Table 113: Test 115 Metrics – WVD, 512, 10 ...296 Table 114: Test 118 Metrics – CWD, 512, 10 ...298 Table 115: Test 119 Metrics – CWD, 512, 0 ...299 Table 116: Test 120 Metrics – CWD, 512, -1...300 Table 117: Test 121 Metrics – Spectrogram, 512, 10 ...301 Table 118: Test 122 Metrics – Spectrogram, 512, 0 ...302 Table 119: Test 123 Metrics – Spectrogram, 512, -1 ...303 Table 120: Test 124 Metrics – Scalogram, 512, 10 ...304 Table 121: Test 125 Metrics – Scalogram, 512, 0 ...305 Table 122: Test 126 Metrics – Scalogram, 512, -3 ...306 Table 123: Test 127 Metrics – Reassigned Spectrogram, 512, 10...307 Table 124: Test 128 Metrics – Reassigned Spectrogram, 512, 0...308 Table 125: Test 129 Metrics – Reassigned Spectrogram, 512, -2 ...309 Table 126: Test 130 Metrics – Reassigned Scalogram, 512, 10 ...310 Table 127: Test 131 Metrics – Reassigned Scalogram, 512, 0 ...311 Table 128: Test 132 Metrics – Reassigned Scalogram, 512, -2...312 Table 129: Test 133 Metrics – RSPWVD, 512, 10...313 Table 130: Test 134 Metrics – RSPWVD, 512, 0...314 Table 131: Test 135 Metrics – RSPWVD, 512, -3 ...315 Table 132: Test 136 Metrics – HT of WVD, 512, 10 ...317 Table 133: Test 137 Metrics – HT of CWD, 512, -3 ...319 Table 134: Test 138 Metrics – HT of RSPWVD, 512, -4...321

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1

1. Introduction

Today, radar systems face serious threats from anti-radiation systems [FAU02] and from electronic attack [GAU02]. In order to be able to perform their functions properly, they must be able to ‘see without being seen’ [PAC09], [WIL06]. This necessitates that they be low probability of intercept (LPI) radars. These radars typically have very low peak power, wide bandwidth, high duty cycle, and power management, making them difficult to be detected and characterized by intercept receivers. In the radar arena, there is a current trend towards radars becoming LPI radars.

On the other side of the fence, the intercept receiver that intercepts these LPI radar signals faces threats as well. It must be able to detect and characterize the signals from these LPI radars in order to provide timely information about threatening systems, in addition to providing information about defensive systems - which is important in maintaining a credible deterrent force to penetrate those defenses. In this case, the knowledge provided by the intercept receiver provides insight for determining the details of the threat so that an effective response can be prepared [WIL06].

LPI radars attempt to detect specific targets at a longer range than the intercept receiver can detect the LPI radar. This means that the success of the LPI radar is measured by how difficult it is for them to be detected by intercept receivers. Consequently, the

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properties of LPI radar, such as high duty cycle, low power, wide bandwidth, power management, make the LPI radar signals difficult to be detected and characterized by intercept receivers. Examples of intercept receivers are Electronic Support receivers, Electronic Intelligence receivers, and Radar Warning Receivers.

Different from the older pulsed radar systems, which were relatively easy for an intercept receiver to detect due to their high peak power, the LPI radar transmitter makes use of sophisticated phase and frequency modulation to spread the signal bandwidth, making the signal difficult to intercept. The LPI radar receiver is able to make use of the appropriate matched filter so that the radar’s performance is similar to that of traditional pulsed radar radiating the same amount of average power [GAU02].

To make matters even more challenging for intercept receivers, most of the intercept receivers currently in the fleet are analog intercept receivers that were designed to intercept ‘older’ (non-LPI) radar signals, and which perform poorly when faced with LPI radar signals. Digital receivers, which are becoming a growing trend, are seen as the eventual solution to the intercept receiver’s LPI challenge [PAC09].

In this dissertation, the use of the reassignment method and the Hough transform are explored as methods for improving the readability of classical time-frequency analysis representations of LPI radar signals collected by intercept receivers. This improved readability may lead to more accurate signal detection and parameter extraction metrics of these signals.

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The following sections give a brief overview of the current intercept receiver signal analysis techniques, the classical time-frequency analysis techniques, the reassignment method, and the Hough transform. Each of these will be explained in more detail in the background section of this dissertation.

1.1. Intercept Receiver Signal Analysis Techniques

Currently, most of the intercept receivers in the fleet are analog, and since analog intercept receivers perform poorly when faced with LPI radar signals, digital intercept receivers represent the growing trend for LPI signal analysis.

To identify the emitter parameters, Fourier analysis techniques using the FFT have been used as the basic tool of the digital intercept receiver, and make up a majority of the digital intercept receiver techniques that are currently in the fleet [PAC09]. From a Fourier Transform representation, the evolution of the frequency content in time cannot be determined. This is due to the fact that the Fourier Transform is a decomposition of complex exponentials, which are of infinite duration and completely un-localized in time. Time information is in fact encoded in the phase of the Fourier Transform (which is simply ignored by the energy spectrum), but its interpretation is not straightforward and its direct extraction is faced with a number of difficulties such as phase unwrapping [ISI96]. Consequently, when a practical non-stationary signal (such as an LPI radar signal) is processed, the Fourier transform cannot efficiently analyze and process the time-varying characteristics of the signal’s frequency spectrum [XIE08], [STE96]. In

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order to have a more informative description of such signals, it would be better to directly represent their frequency content while still keeping the time description parameter – which is what time-frequency analysis accomplishes [ISI96], [SOL00].

1.2. Classical Time-Frequency Analysis Techniques

From the basic tool of the FFT, more complex signal processing techniques have evolved, such as time-frequency analysis techniques. In many operational radar environments, the non-stationary nature of the received radar signal mandates the use of some form of time-frequency analysis technique, which can clearly bring out the non-stationary behavior of the signal. The important virtue of time-frequency analysis is that it provides an identification of the specific times during which certain spectral components of the signal are observed. Time-frequency techniques can also be used to detect and extract the parameters from LPI modulations [MIL02]. For digital intercept receivers, classical time-frequency analysis techniques are currently at the lab phase, being analyzed by industry in an attempt to port the time-frequency analysis algorithms that have been developed to hardware (FPGAs, DSPs) for the purpose of running them in real-time on realistic IF bandwidths [PAC09].

Four of the classical time-frequency analysis techniques (Wigner-Ville Distribution (WVD), Choi-Williams Distribution (CWD), Spectrogram, Scalogram) will be described briefly in this section, and then in more detail in the background section of this dissertation.

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WVD: One of the most prominent members of the time-frequency analysis techniques

family is the WVD. The WVD satisfies a large number of desirable mathematical properties. In particular, it is always real-valued, preserves time and frequency shifts, and satisfies marginal properties [ISI96], [QIA02]. The WVD, which is a transformation of a continuous time signal into the time-frequency domain, is computed by correlating the signal with a time and frequency translated version of itself, making it bilinear. The WVD exhibits the highest signal energy concentration in the time-frequency plane [WIL06]. By using the WVD, an intercept receiver can come close to having a processing gain near the LPI radar’s matched filter processing gain [PAC09]. The WVD also contains cross term interference between every pair of signal components, which may limit its applications [GUL07], [STE96], and which can make the WVD time-frequency representation very unreadable, especially if the components are numerous or close to each other, and the more so in the presence of noise [BOA03]. This lack of readability can in turn translate into poor signal detection and parameter extraction metrics, potentially placing the intercept receiver signal analyst in harm’s way.

CWD: The CWD is a member of the Cohen’s class of time-frequency distributions

which use smoothing kernels [GUL07] to help reduce cross-term interference so prevalent in the WVD [BOA03], [PAC09], [UPP08]. The reduction in cross-term interference can make the time-frequency representation more readable and can make signal detection and parameter extraction more accurate. The down-side is that the CWD, like all members of Cohen’s class, is faced with an inevitable trade-off between cross-term reduction and time-frequency localization. So the signal detection and

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parameter extraction benefits gained by the cross-term reduction may be offset by the decrease in time-frequency localization (smearing or widening of the signal).

Spectrogram: The Short-Time Fourier Transform (STFT) will be examined first, and

then connected to the Spectrogram. The Spectrogram is defined as the magnitude squared of the STFT [HIP00], [HLA92], [MIT01]. The STFT was the first tool devised for analyzing a signal simultaneously in both time and frequency. The basic idea of the STFT is to analyze a small part of the signal around the time of interest based on Fourier methods in order to determine the frequencies at that time. Since the time interval is short compared to the whole signal, the process is called taking the Short-Time Fourier Transform [STE96]. The observation window allows localization of the spectrum in time, but also smears the spectrum in frequency in accordance with the uncertainty principle, leading to a trade-off between time resolution and frequency resolution. In general, if the window is short, the time resolution is good, but the frequency resolution is poor, and if the window is long, the frequency resolution is good, but the time resolution is poor. For the Spectrogram, its robustness and ease of implementation have made the spectrogram a popular tool for speech analysis [BOA03], [COH95]. Since the Spectrogram is a quadratic (bilinear) representation, the Spectrogram of the sum of two signals is not the sum of the two Spectrograms (quadratic superposition principle); there is a cross-Spectrogram part and a real part. Therefore, as for every quadratic distribution, the Spectrogram presents interference terms, however, the interference terms are restricted to those regions of the time-frequency plane where the signals overlap. If the

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signal components are sufficiently far apart so that their spectrograms do not overlap significantly, then the interference terms will be nearly identically zero [ISI96].

Scalogram: The wavelet transform will be examined first, and then connected to the

Scalogram. The Scalogram is defined as the magnitude squared of the wavelet transform, and can be used as a time-frequency distribution [COH02], [GAL05], [BOA03]. The wavelet transform is of interest for the analysis of non-stationary signals, because it provides still another alternative to the STFT and to many of the quadratic time-frequency distributions. The basic difference between the STFT and the wavelet transform is that the STFT uses a fixed signal analysis window, whereas the wavelet transform uses short windows at high frequencies and long windows at low frequencies. This helps to diffuse the effect of the uncertainty principle by providing good time resolution at high frequencies and good frequency resolution at low frequencies. This approach makes sense especially when the signal at hand has high frequency components for short durations and low frequency components for long durations. The signals encountered in practical applications are often of this type. The wavelet transform allows localization in both the time domain via translation of the mother wavelet, and in the scale (frequency) domain via dilations. The wavelet is irregular in shape and compactly supported, thus making it an ideal tool for analyzing signals of a transient nature; the irregularity of the wavelet basis lends itself to analysis of signals with discontinuities or sharp changes, while the compactly supported nature of wavelets enables time localization of a signal’s features [BOA03]. Unlike many of the quadratic functions such as the WVD and CWD, the wavelet transform is a linear transformation; therefore

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term interference is not generated. There is another major difference between the STFT and the wavelet transform; the STFT uses sines and cosines as an orthogonal basis set to which the signal of interest is effectively correlated against, whereas the wavelet transform uses special ‘wavelets’ which usually comprise an orthogonal basis set. The wavelet transform then computes coefficients, which represents a measure of the similarities, or correlation, of the signal with respect to the set of wavelets. In other words, the wavelet transform of a signal corresponds to its decomposition with respect to a family of functions obtained by dilations (or contractions) and translations (moving window) of an analyzing wavelet.

A filter bank concept is often used to describe the wavelet transform. The wavelet transform can be interpreted as the result of filtering the signal with a set of bandpass filters, each with a different center frequency.

The Scalogram (the Affine Class counterpart of the Spectrogram) can be considered as an energy distribution of the signal in the time-scale (frequency) plane. As is the case for the wavelet transform, the time and frequency resolutions of the Scalogram are related via the Heisenberg-Gabor principle.

The interference terms of the Scalogram, as for the spectrogram, are also restricted to those regions of the time-frequency plane where the corresponding signals overlap. Therefore, if two signal components are sufficiently far apart in the time-frequency plane, their cross-Scalogram will be essentially zero [ISI96], [HLA92].

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Though classical time-frequency analysis techniques, such as those described above, are a great improvement over Fourier analysis techniques, they suffer in general from poor time-frequency localization, cross-term interference, and mediocre performance in low SNR environments. In this dissertation, two signal processing analysis techniques, the reassignment method and the Hough transform are introduced as potential solutions for these deficiencies. The next sections provide a brief overview of these techniques.

1.3. The Reassignment Method

A brief overview of the reassignment method is given in this section, while a more detailed description is provided in the background section of this dissertation.

Much work is underway to develop more advanced signal processing techniques which will more effectively and efficiently exploit modern radar signals. These techniques are directed at improving the tasks of detection, classification, and identification, meaning a readability that is based on a good concentration of the signal components and few misleading interference terms. This is what the reassignment method has been devised for. It’s scheme assumes that the energy distribution in the time-frequency plane resembles a mass distribution and moves each value of the time-frequency plane located at a point (t, f) to another point (t’, f’), which is the center of gravity of the energy distribution in the area of (t, f). The result is a more focused representation with high intensity, since the value at the point (t’, f’) is the sum of all the neighboring values. Reasoning with the mechanical analogy, the situation is as if the total mass of an object

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was assigned to its geometrical center, an arbitrary point which except in the very specific case of a homogeneous distribution, has no reason to suit the actual distribution. A much more meaningful choice is to assign the total mass of an object - as well as the time-frequency value – to the center of gravity of their respective distribution. This is what the reassignment method performs [BOA03].

The reassignment method can be applied to most energy distributions as well as to the Spectrogram [HIP00]. One of the most important properties of the reassignment method is that the application of the method theoretically yields perfectly localized distributions for chirp signals, frequency tones and impulses [BOA03] making it a good candidate for analyzing triangular modulated frequency modulated continuous wave (FMCW) and frequency shift keying (FSK) LPI radar signals. A potential drawback of the reassignment method is that there is a chance it could generate spurious artifacts due to the reassignment process randomly clustering noise (vice signals). Because of this, its performance may suffer in low SNR scenarios.

The reassignment method can be viewed as a step in the process whose goal is to build a readable time-frequency representation, which consists of: a smoothing, whose main purpose is to rub out oscillatory interferences, but whose drawback is to smear localized components; and a squeezing, whose effect is to refocus the contributions which survived the smoothing [BOA03]. Experimental results have shown for other applications that this method provides, without a drastic increase in computational complexity – enhanced

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contrast (when compared to smoothed distributions) with a much reduced level of interference (when compared to the WVD) [BOA03], [ISI96].

In this dissertation, the reassignment method, due to its smoothing and squeezing qualities, has been put forth as a potential solution for intercept receivers to the classical time-frequency analysis techniques deficiencies of poor time-frequency localization and cross-term interference.

1.4. The Hough Transform

The Hough transform is briefly described in this section, and in more detail in the background section of this dissertation.

The Hough transform (which is very similar to the Radon transform) is used for the detection of straight lines and other curves [BAR92], [BEN05], [ZAI99], [INC07], [BAR95]. The Hough transform of a particular time-frequency representation is found by computing the integral of the time-frequency representation along straight lines at different angles. The presence of a ‘spike’ in the Hough transform representation reveals the presence of high positive values concentrated along a line in the time-frequency representation – whose parameters (such as chirp rate) correspond to the coordinates of the spike (theta and rho values) [BAR92], [YAS06], [BAR95]. Detection can be achieved by establishing a threshold value for the amplitude of the Hough transform spike. Therefore the Hough transform can be used to convert a difficult global detection

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problem in the time-frequency representation into a more easily solved local peak detection problem in the Hough transform representation.

As shown in this dissertation, the Hough transform is especially suited for detecting and analyzing low SNR signals, due to its robustness in the presence of noise [SHA07], [DAH08], where the integration carried out by the Hough transform produces an improvement in the SNR. This integration also allows the Hough transform to work well in the presence of multi-component signals which produce cross-terms. This is because the cross-terms have an amplitude modulation which is reduced by the Hough transform integration, while the useful contributions, which are always positive, are correctly integrated [BAR95].

In this dissertation, the Hough transform, because of its ability to suppress cross-term interference, separate signals from cross-term interference, and perform well in a low SNR environment, has been put forth as a potential solution to the classical time-frequency analysis techniques deficiencies of cross-term interference and mediocre performance in low SNR environments.

For digital intercept receivers, the evolution from Fourier-based analysis, to classical time-frequency analysis techniques to the reassignment method and the Hough transform will produce enhanced readability, and as a result, more accurate signal detection and parameter extraction. When the intercept receiver signal analyst correlates these more accurate parameters with existing signals in databases, he will be able to more precisely

Out