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\title{Low Probability of Intercept Triangular Modulated Frequency Modulated Continuous Wave Signal Characterization Comparison using the Wigner Ville Distribution and the Choi Williams Distribution}
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             \author[1]{Daniel L.  Stevens}

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\date{\small \em Received: 8 December 2018 Accepted: 1 January 2019 Published: 15 January 2019}

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\begin{abstract}
        


Digital intercept receivers are currently moving away from Fourier-based analysis and towards classical timefrequency analysis techniques for the purpose of analyzing low probability of intercept radar signals. This paper presents the novel approach of characterizing low probability of intercept triangular modulated frequency modulated continuous wave radar signals through utilization and direct comparison of the Wigner Ville Distribution versus the Choi Williams Distribution. The following metrics were used for evaluation: percent error of: carrier frequency, modulation bandwidth, modulation period, chirp rate, and time-frequency localization (x and y direction). Also used were: percent detection, lowest signal-tonoise ratio for signal detection, and plot (processing) time. Experimental results demonstrate that overall, the Wigner Ville Distribution produced more accurate characterization metrics than the Choi Williams Distribution. An improvement in performance may well translate into an increase in personnel safety.

\end{abstract}


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\let\tabcellsep& 	 	 		 \par
The frequency modulation spreads the transmitted energy over a large modulation bandwidth Î?"??, providing good range resolution that is critical for discriminating targets from clutter. The power spectrum of the FMCW signal is nearly rectangular over the modulation bandwidth, so non-cooperative interception is difficult. Since the transmit waveform is deterministic, the form of the return signals can be predicted. This gives it the added advantage of being resistant to interference (such as jamming), since any signal not matching this form can be suppressed  {\ref [WIL06]}. Consequently, it is difficult for an intercept receiver to detect the FMCW waveform and measure the parameters accurately enough to match the jammer waveform to the radar waveform  {\ref [PAC09]}.\par
The most popular linear modulation utilized is the triangular FMCW emitter  
\section[{Wigner Ville Distribution (WVD)}]{Wigner Ville Distribution (WVD)}\par
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 realvalued, preserves time and frequency shifts, and satisfies marginal properties [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  {\ref [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 hard to read, 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 decreased signal detection and parameter extraction metrics, potentially placing the intercept receiver signal analyst's platform in harm's way.\par
The WVD of a signal ??(??) is given in equation (1) as:?? ?? (??, ð??"ð??") = ? ??(?? + ?? 2 +? ?? )?? * ??? ? ?? 2 ? ?? ??? 2??ð??"ð??"?? ???? (1)\par
or equivalently in equation (2) as:?? ?? (??, ð??"ð??") = ? ??(ð??"ð??" + ?? 2 +? ?? )?? * ?ð??"ð??" ? ?? 2 ? ?? ?? 2?????? ???? (2) Choi Williams Distribution (CWD)\par
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 timefrequency localization. Because of this, the signal detection and 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).\par
The CWD of a signal ??(??)is given in equation (3) as:???? ?? (??, ð??"ð??") = ? 2 ?? ? ?? |??| +? ?? ?? ?2?? 2 (?????) 2 /?? 2 ?? ??? + ?? 2 ? ?? * ??? ? ?? 2 ? ?? ??? 2??ð??"ð??"?? ???? ????\textbf{(3)}\par
As can be seen from equation (3), the CWD uses an exponential kernel in the generalized class of bilinear time-frequency distributions. Choi and Williams introduced one of the earliest 'new' distributions [CHO89], which they called the Exponential Distribution or ED.\par
This new distribution overcomes several drawbacks of the Spectrogram and the WVD, providing a good trade-off between localization and suppressed interferences [WIL92], [GUL07], [UPP08]. Interference terms tend to lie away from the axes in the ambiguity plane, while auto terms (signals) tend to lie on the axes. The Spectrogram kernel attenuates everything away from the (0,0) point, the WVD kernel passes everything, and the CWD kernel passes everything on the axes and attenuates away from the axes. Thus, the CWD generally attenuates interference terms[PAC09], [HLA92].\par
This provides its reduced interference characteristic. The Spectrogram reduces interference also, but at a cost to the signal concentration. 
\section[{II.}]{II.} 
\section[{Methodology}]{Methodology}\par
The methodologies detailed in this section describe the processes involved in obtaining and comparing metrics between the classical time-frequency analysis techniques of the Wigner Ville Distribution and the Choi Williams Distribution for the detection and characterization of low probability of intercept triangular modulated FMC Wradar signals.\par
The tools used for this testing were: MATLAB (version 8.3), Signal Processing Toolbox (version 6.21), Wavelet Toolbox (version 4.13), Image Processing Toolbox (version 9.0), Time-Frequency Toolbox (version 1.0) (http://tftb.nongnu.org/).\par
All testing was accomplished on a desktop computer (Dell Precision T1700; Processor -Intel Xeon CPU E3-1226 v3 3.30GHz; Installed RAM -32.0GB; System type -64-bit operating system, x64-based processor).\par
Testing was performed for the triangular modulated FMCW waveform, whose parameters were chosen for academic validation of signal processing techniques. Due to computer processing resources they were not meant to represent real-world values. The number of samples was chosen to be either 256 or 512, which seemed to be the optimum size for the desktop computer. Testing was performed at three different SNR levels: 10dB, 0dB, and the lowest SNR at which the signal could be detected. The noise added was white Gaussian noise, which best reflects the thermal noise present in the IF section of an intercept receiver [PAC09]. Kaiser windowing was used, when windowing was applicable. 100 runs were performed for each test, for statistical purposes. The plots included in this paper were done at a threshold of 5\% of the maximum intensity and were linear scale (not dB) of analytic (complex) signals; the color bar represented intensity. 
\section[{Global Journal of Researches in Engineering ( ) Volume XIX Issue IV Version I}]{Global Journal of Researches in Engineering ( ) Volume XIX Issue IV Version I}\par
The triangular modulated FMCW signal (most prevalent LPI radar waveform [LIA09]) used had the following parameters: sampling frequency = 4 KHz; carrier frequency = 1 KHz; modulation bandwidth = 500Hz; modulation period =. 02 sec.\par
After each particular run of each test, metrics were extracted from the time-frequency representation. The different metrics extracted were as follows: a) Plot (processing) time Time required for plot to be displayed. 
\section[{b) Percent detection}]{b) Percent detection}\par
Percent of time signal was detected -signal was declared a detection if any portion of each of the signal components (4 chirp components for triangular modulated FMCW) exceeded a set threshold (a certain percentage of the maximum intensity of the timefrequency representation).\par
Threshold percentages were determined based on visual detections of low SNR signals (lowest SNR at which the signal could be visually detected in the timefrequency representation) (see Figure  {\ref 1}).\par
Figure  {\ref 1}: Threshold percentage determination. This plot is a time vs. amplitude (x-z view) of the CWD of a triangular modulated FMCW signal (256 samples, with SNR= -3dB). 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 (the 2 legs for each of the 2 triangles of the triangular modulated FMCW) was noted (here 61\%, 91\%, 98\%, 61\%), and the lowest of these 4 values was recorded (61\%). Ten test runs were performed for both time-frequency analysis tools (CWD and WVD) for this waveform. 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 the CWD is 60\%. 
\section[{Thresholds were assigned as follows: CWD (60\%); WVD (50\%).}]{Thresholds were assigned as follows: CWD (60\%); WVD (50\%).}\par
For percent detection determination, these threshold values were included in the time-frequency plot algorithms so that the thresholds could be applied automatically during the plotting process. From the threshold plot, the signal was declared a detection if any portion of each of the signal components was visible (see Figure  {\ref 2}).\par
The signal processing tools used for each task were the Wigner Ville Distribution and the Choi Williams Distribution. 
\section[{Global Journal of Researches in Engineering ( ) Volume XIX Issue IV Version I}]{Global Journal of Researches in Engineering ( ) Volume XIX Issue IV Version I}\par
Figure  {\ref 2}: Percent detection (time-frequency). This plot is a time vs. frequency (x-y view) of the CWD of a triangular modulated FMCW signal (256 samples, with SNR= 10dB) with threshold value automatically set to 60\%. From this threshold plot, the signal was declared a (visual) detection because at least a portion of each of the 4 signal components (the 2 legs for each of the 2 triangles of the triangular modulated FMCW) was visible. 
\section[{c) Carrier frequency}]{c) Carrier frequency}\par
The frequency corresponding to the maximum intensity of the time -frequency representation (see Figure \hyperref[fig_2]{3}). ). From the frequency vs. amplitude (y-z view), the maximum intensity value is manually determined. The frequency corresponding to the max intensity value is the carrier frequency (here fc=976.1 Hz). 
\section[{d) Modulation bandwidth}]{d) Modulation bandwidth}\par
Distance from highest frequency value of signal (at a threshold of 20\% maximum intensity) to lowest frequency value of signal (at same threshold) in Ydirection (frequency).\par
The threshold percentage was determined based on manual measurement of the modulation bandwidth of the signal in the time-frequency representation. This was accomplished for ten test runs of each time-frequency analysis tool (Wigner Ville Distribution and Choi Williams Distribution), for the triangular modulated FMCW waveform. During each manual measurement, the max intensity of the high and low measuring points was recorded. The average of the max intensity values for these test runs was 20\%. This was adopted as the threshold value, and is representative of what is obtained when performing manual measurements. This 20\% threshold was also adapted for determining the modulation period and the time-frequency localization (both are described below).\par
For modulation bandwidth determination, the 20\% threshold value was included in the time-frequency plot algorithms so that the threshold could be applied  For modulation period determination, the 20\% threshold value was included in the time-frequency plot algorithms so that the threshold could be applied automatically during the plotting process. From the threshold plot, the modulation period was manually measured (see Figure \hyperref[fig_5]{5}).  
\section[{f) Time-frequency localization}]{f) Time-frequency localization}\par
Measure of the thickness of a signal component (at a threshold of 20\% maximum intensity on each side of the component) -converted to \% of entire X-Axis, and \% of entire Y-Axis.\par
For time-frequency localization determination, the 20\% threshold value was included in the time-frequency plot algorithms so that the threshold could be applied automatically during the plotting process. From the threshold plot, the time-frequency localization was manually measured (see Figure \hyperref[fig_6]{6}). 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. 
\section[{g) Chirp rate (modulation bandwidth)/(modulation period) h) Lowest detectable SNR}]{g) Chirp rate (modulation bandwidth)/(modulation period) h) Lowest detectable SNR}\par
The lowest SNR level at which at least a portion of each of the signal components exceeded the set threshold listed in the percent detection section above.\par
For lowest detectable SNR determination, these threshold values were included in the time-frequency plot algorithms so that the thresholds could be applied automatically during the plotting process. From the threshold plot, the signal was declared a detection if any portion of each of the signal components was visible. The lowest SNR level for which the signal was declared a detection is the lowest detectable SNR (see Figure  {\ref 7}).\par
Figure  {\ref 7}: Lowest detectable SNR. This plot is a time vs. frequency (x-y view) of the CWD of a triangular modulated FMCW signal (256 samples, with SNR= -3dB) with threshold value automatically set to 60\%. From this threshold plot, the signal was declared a (visual) detection because at least a portion of each of the 4 signal components (the 2 legs for each of the 2 triangles of the triangular modulated FMCW) was visible. Note that the signal portion for the two 61\% max intensities are barely visible, because the threshold for the CWD is 60\%. For this case, any lower SNR than -3dB would have been a non-detect. The data from all 100 runs for each test was used to produce the actual, error, and percent error for each of these metrics listed above.\par
The metrics from the Wigner Ville Distribution were then compared to the metrics from the Choi Williams Distribution. By and large, the Wigner Ville Distribution outperformed the Choi Williams Distribution, as will be shown in the results section.  \hyperref[tab_0]{1}, the WVD outperformed the CWD in average percent error: carrier frequency (2.23\% vs. 5.22\%), modulation bandwidth (5.53\% vs. 9.61\%), modulation period (0.48\% vs. 0.49\%), and chirp rate (5.28\% vs. 9.67\%). The WVD also outperformed the CWD in average: percent detection (77.6\% vs. 69.6\%), time-frequency localization (x-direction) (0.62\% vs. 1.89\%), and time-frequency localization (y-direction) (1.28\% vs. 3.52\%). The CWD outperformed the WVD in ). The SNR for the top row is 10dB, for the middle row is 0dB, and for the bottom row is -3dB.In general, the WVD signalappears more localized ('thinner') than does the CWD signal, however, the cross-term interference in the WVDplots makes it more difficult to differentiate betweeen the signal and the cross-term interference. 
\section[{III.}]{III.} 
\section[{Results}]{Results} 
\section[{IV. Discussion}]{IV. Discussion}\par
This section will elaborate on the results from the previous section.\par
From Table \hyperref[tab_0]{1}, the WVD outperformed the CWD in average percent error: carrier frequency (2.23\% vs. 5.22\%), modulation bandwidth (5.53\% vs. 9.61\%), modulation period (0.48\% vs. 0.49\%), and chirp rate (5.28\% vs. 9.67\%); and in average: percent detection (77.6\% vs. 69.6\%), time-frequency localization (xdirection) (0.62\% vs. 1.89\%), and time-frequency localization (y-direction) (1.28\% vs. 3.52\%). These results are by and large a result of the WVD signal being much more localized signal than the CWD signal. The CWD's 'thicker' signal is a result of its cross-term reduction -at the expense of signal localization.\par
The CWD outperformed the WVD in average: lowest detectable SNR (-3.0db vs. -2.3db) and average plot time (10.44s vs. 1341.84s). The combination of the CWD's reduction of cross-term interference along with the WVD being very computationally complex [MIL02] are the grounds for the CWD's better plot time. In addition, lowest detectable SNR is based on visual detection in the Time-Frequency representation. Figure \hyperref[fig_8]{8} shows that, for the WVD plots, as the SNR gets lower, it becomes more difficult to distinguish between the actual signal and the cross-term interference. However, for the CWD plots there is no cross-term interference to confuse with the actual signals, making the CWD signal, though not as localized, more easily detected than the WVD signal -at these lower SNRs.\par
The WVD might be used in a scenario where you need good signal localization in a fairly low SNR environment, without tight time constraints. The CWD might be used in a scenario where a short plot time is necessary, and where signal localization is not an issue. Such a scenario might be a 'quick and dirty' check to see if a signal is present, without precise extraction of its parameters. 
\section[{V. onclusions}]{V. onclusions}\par
Digital intercept receivers, whose main job is to detect and extract parameters from low probability of intercept radar signals, are currently moving away from Fourier-based analysis and towards classical timefrequency analysis techniques, such as the Wigner Ville Distribution, and the Choi Williams Distribution, for the purpose of analyzing low probability of intercept radar signals. Based on the research performed for this paper (the novel direct comparison of the Wigner Ville Distribution versus the Choi Williams Distribution for the signal analysis of low probability of intercept triangular modulated FMCW radar signals) it was shown that the Wigner Ville Distribution by-and-large outperformed the Choi Williams Distribution for analyzing these low probability of intercept radar signals -for reasons brought out in the discussion section above. More accurate characterization metrics could well translate into saved equipment and lives.\par
Future plans include continuing to analyze low probability of intercept radar waveforms (such as the frequency hopping and the triangular modulated FMCW), using additional time-frequency analysis techniques.\begin{figure}[htbp]
\noindent\textbf{}\includegraphics[]{image-2.png}
\caption{\label{fig_0}}\end{figure}
 \begin{figure}[htbp]
\noindent\textbf{}\includegraphics[]{image-3.png}
\caption{\label{fig_1}}\end{figure}
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\noindent\textbf{3}\includegraphics[]{image-4.png}
\caption{\label{fig_2}Figure 3 :}\end{figure}
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\noindent\textbf{}\includegraphics[]{image-5.png}
\caption{\label{fig_3}}\end{figure}
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\noindent\textbf{4}\includegraphics[]{image-6.png}
\caption{\label{fig_4}Figure 4 :}\end{figure}
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\noindent\textbf{5}\includegraphics[]{image-7.png}
\caption{\label{fig_5}Figure 5 :}\end{figure}
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\noindent\textbf{6}\includegraphics[]{image-8.png}
\caption{\label{fig_6}Figure 6 :}\end{figure}
 \begin{figure}[htbp]
\noindent\textbf{}\includegraphics[]{image-9.png}
\caption{\label{fig_7}}\end{figure}
 \begin{figure}[htbp]
\noindent\textbf{8}\includegraphics[]{image-10.png}
\caption{\label{fig_8}Figure 8 :}\end{figure}
 \begin{figure}[htbp]
\noindent\textbf{}\includegraphics[]{image-11.png}
\caption{\label{figure11}}\end{figure}
 \begin{figure}[htbp]
\noindent\textbf{1} \par 
\begin{longtable}{P{0.20957446808510638\textwidth}P{0.4819148936170213\textwidth}P{0.06382978723404256\textwidth}P{0.09468085106382979\textwidth}}
Parameters carrier frequency\tabcellsep Choi Williams Distribution 5.22\%\tabcellsep Wigner Ville Distribution 2.23\%\tabcellsep Year 2019\\
modulation bandwidth\tabcellsep 9.61\%\tabcellsep 5.53\%\tabcellsep 31\\
modulation period chirp rate percent detection lowest detectable snr plot time time-frequency localization-x time-frequency localization-y From Table\tabcellsep \multicolumn{2}{l}{0.49\% 9.67\% 69.6\% -3.0db 10.44s 1.89\% 3.52\% average: lowest detectable SNR (-3.0db vs. -2.3db) and 0.48\% 5.28\% 77.6\% -2.3db 1341.84s 0.62\% 1.28\% average plot time (10.44s vs. 1341.84s). Figure 8 shows comparative plots of the Choi Williams Distribution (left) vs. the Wigner Ville Distribution (right) (triangular modulated FMCW signal) at SNRs of 10dB (top row), 0dB (middle row), and -3dB (bottom row).}\tabcellsep of Researches in Engineering F ( ) Volume XIX Issue IV Version I\\
\tabcellsep \tabcellsep \tabcellsep Global Journal\\
\tabcellsep \tabcellsep © 2019 Global Journals\tabcellsep \end{longtable} \par
 
\caption{\label{tab_0}Table 1 :}\end{figure}
 			\footnote{F © 2019 Global Journals Low Probability of Intercept Triangular Modulated Frequency Modulated Continuous Wave Signal Characterization Comparison Using the Wigner Ville Distribution and the Choi Williams Distribution} 		 		\backmatter  			  				\begin{bibitemlist}{1}
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