# Introduction

requency Modulated Continuous Wave (FMCW) signals are frequently encountered in modern radar systems [WAN10], [WON09], [WAJ08]. The frequency modulation spreads the transmitted energy over a large modulation bandwidth Î?"F, 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 [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 
? ? (?, ð??"; ?) = ? ?(?)? +? ?? (? ? ?)? ??2?ð??"? ?? (1)
Where h(t) is a short time analysis window localized around t=0 and f=0. Because multiplication by the relatively short window h(u-t) effectively suppresses the signal outside a neighborhood around the analysis point u=t, the STFT is a 'local' spectrum of the signal x(u) around t. Think of the window h(t) as sliding along the signal x(u) and for each shift h(u-t) we compute the usual Fourier transform of the product function x(u)h(u-t). The observation window allows The STFT was the first tool devised for analyzing a signal in both time and frequency simultaneously. For analysis of human speech, the main method was, and still is, the STFT. In general, the STFT is still the most widely used method for studying non-stationary signals [COH95].

The Spectrogram (the squared modulus of the STFT) is given by equation 2 as:
? ? (?, ð??") = ?? ?(?) +? ?? ?(? ? ?)? ??2?ð??"? ??? 2 (2)
The Spectrog ram is a real-valued and nonnegative distribution. Since the window h of the STFT is assumed of unit energy, the Spectrogram satisfies the global energy distribution property. Thus we can interpret the Spectrogram as a measure of the energy of the signal contained in the time-frequency domain centered on the point (t, f) and whose shape is independent of this localization.

Here are some properties of the Spectrogram: 1) The idea of the wavelet transform (equation (3)) is to project a signal x on a family of zero-mean functions (the wavelets) deduced from an elementary function (the mother wavelet) by translations and dilations:
? ? (?, ?; ?) = ? ?(?)? ?,? * +? ?? (?)?? (3)
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 translations 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 temporal localization of a signal's features [BOA03]. Unlike many of the quadratic functions such as the Wigner-Ville Distribution (WVD) and Choi-Williams Distribution (CWD), the wavelet transform is a linear transformation, therefore cross-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 The variable a corresponds to a scale factor, in the sense that taking |a|>1 dilates the wavelet ? and taking |a|<1 compresses ?. By definition, the wavelet transform is more a time-scale than a time-frequency representation. However, for wavelets which are well localized around a non-zero frequency ?_0 at a scale =1 , a time-frequency interpretation is possible thanks to the formal identification ? = ? 0 ? . localization of the spectrum in time, but also smears the contractions) and translations (moving window) of an analyzing wavelet.


# Global

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
[GRI08], [FAR96], [SAR98], [SAT98].
Like the design of conventional digital filters, the design of a wavelet filter can be accomplished by using a number of methods including weighted least squares [ALN00], [GOH00], orthogonal matrix methods [ZAH99], nonlinear optimization, optimization of a single parameter (e.g. the passband edge) [ZHA00], and a method that minimizes an objective function that bounds the out-of-tile energy [FAR99].

Here are some properties of the wavelet transform: 1) The wavelet transform is covariant by translation in time and scaling. The corresponding group of transforms is called the Affine group; 2) The signal x can be recovered from its wavelet transform via the synthesis wavelet; 3) Time and frequency resolutions, like in the STFT case, are related via the Heisenberg-Gabor inequality. However in the wavelet transform case, these two resolutions depend on the frequency: the frequency resolution becomes poorer and the time resolution becomes better as the analysis frequency grows; 4) Because the wavelet transform is a linear transform, it does not contain cross-term interferences
[GRI07], [LAR92].
A similar distribution to the Spectrogram can be defined in the wavelet case. Since the wavelet transform behaves like an orthonormal basis decomposition, it can be shown that it preserves energy:
? |? ? (?, ?; ?)| 2 +? ?? ?? ?? ? 2 = ? ? (4)
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].

For this paper, the Morlet Scalogram will be used. The Morlet wavelet is obtained by taking a complex sine wave and by localizing it with a Gaussian envelope. The Mexican hat wavelet isolates a single bump of the Morlet wavelet. The Morlet wavelet has good focusing in both time and frequency [CHE09].


# II.


# Methodology

The methodologies detailed in this section describe the processes involved in obtaining and comparing metrics between the classical time-frequency analysis techniques of the Spectrogram and the Scalogram for the detection and characterization of low probability of intercept triangular modulated FMCW radar signals.

The tools used for this testing were: MATLAB (version 7.12), Signal Processing Toolbox (version 6.15), Wavelet Toolbox (version 4.7), Image Processing Toolbox (version 7.2), Time-Frequency Toolbox (version 1.0) (http://tftb.nongnu.org/).

All testing was accomplished on a desktop computer (HP Compaq, 2.5GHz processor, AMD Athlon 64X2 Dual Core Processor 4800+, 2.00GB Memory (RAM), 32 Bit Operating System).

Testing was performed for 2 different triangular modulated FMCW waveforms. For each waveform, 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 for each test 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. 50 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. The signal processing tools used for each task were the Spectrogram and the Scalogram.

Task 1 consisted of analyzing a triangular modulated FMCW signal (most prevalent LPI radar waveform [LIA09]) whose parameters were: sampling frequency=4KHz; carrier frequency=1KHz; modulation bandwidth=500Hz; modulation period=.02sec.

Task 2 was similar to Task 1, but with different parameters: sampling frequency=6KHz; carrier frequency=1.5KHz; modulation bandwidth=2400Hz; modulation period=.15sec. The different parameters were chosen to see how the different shapes/heights of the triangles of the triangular modulated FMCW would affect the metrics.

After each particular run of each test, metrics were extracted from the time-frequency representation.

The different metrics extracted were as follows: 1) Plot (processing) time: Time required for plot to be displayed.

where ? ? is the energy of ?. This leads us to define the Scalogram (equation (4)) of ? as the squared modulus of the wavelet transform. It is an energy distribution of the signal in the time-scale plane, associated with the measure
?? ? 2 .
2) Percent detection: 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 time-frequency representation).

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 1).

Figure 1: Threshold percentage determination. This plot is an amplitude vs. time (x-z view) of the Spectrogram 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 98%, 60%, 95%, 63%), and the lowest of these 4 values was recorded (60%). Ten test runs were performed for both time-frequency analysis tools (Spectrogram and Scalogram) 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 Spectrogram is 60%.

Thresholds were assigned as follows: Spectrogram (60%); Scalogram (50%).

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 2).  ). 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=984.4 Hz).


# 4) Modulation bandwidth:

Distance from highest frequency value of signal (at a threshold of 20% maximum intensity) to lowest frequency value of signal (at same threshold) in Y-direction (frequency).

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 (Spectrogram and Scalogram), for each of the 2 waveforms. 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).

For modulation bandwidth 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 bandwidth was manually measured (see Figure 4). Distance from highest frequency value of signal (at a threshold of 20% maximum intensity) to lowest frequency value of signal (at same threshold) in X-direction (time). automatically during the plotting process. From the threshold plot, the modulation period was manually measured (see Figure 5).

For modulation period determination, the 20% threshold value was included in the time-frequency plot algorithms so that the threshold could be applied 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.

For time-frequency localization determination, the 20% threshold value was included in the timefrequency 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 6). 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 7). Year 2017


# F

For lowest detectable SNR determination, these threshold values were included in the time-frequency   7: Lowest detectable SNR. This plot is an frequency vs. time (x-y view) of the Spectrogram 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 60% max intensity (just above the 'x' in 'max') is barely visible, because the threshold for the Spectrogram is 60%. For this case, any lower SNR would have been a non-detect. Compare to Figure 2, which is the same plot, except that it has an SNR level equal to 10dB.

The data from all 50 runs for each test was used to produce the actual, error, and percent error for each of these metrics listed above.

The metrics from the Spectrogram were then compared to the metrics from the Scalogram. By and large, the Spectrogram outperformed the Scalogram, as will be shown in the results section.


# III. Results

Table 1 presents the overall test metrics for the two classical time-frequency analysis techniques used in this testing (Spectrogram versus Scalogram).     


# Discussion

This section will elaborate on the results from the previous section.

From Table 1, the Spectrogram outperformed the Scalogram in every category. The Spectrogram's reduction of cross-term interference is grounds for its better plot time. Average percent detection and lowest the Time-Frequency representation. Figure 8 and Figure 9 show clearly that the signals in the Spectrogram plots are more readable than those in the Scalogram plots, which account for the Spectrogram's better average percent detection and lowest detectable SNR. At relatively low frequencies (as in this paper), wavelets (Scalograms), because of their multi-resolution analysis basis, are better resolved (localized) in frequency and more poorly resolved (localized) in time. Therefore for relatively low frequencies, the best waveforms to be analyzed by wavelets (Scalograms) are tonals. In addition, the irregularity of the wavelet (Scalogram) basis lends itself to analysis of signals with discontinuities (such as frequency hopping signals (tonals)). Also, since the wavelet is irregular shape and compactly supported, it makes it an ideal tool for analyzing signals of transient nature (such as the frequency hopping signals (tonals)). Therefore as the signal goes from being 'flat' (i.e. a tonal) signal, to more 'upright' (i.e. a triangular modulated FMCW) signal, the Scalogram of this signal becomes more poorly resolved (localized), i.e. 'fatter', accounting for the Scalogram's poorer metrics in the categories of modulation bandwidth, modulation period, chirp rate, carrier frequency, time-frequency localization (x), and timefrequency localization (y). 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.
![[PAC09]. The most popular linear modulation utilized is the triangular FMCW emitter [LIA09], since it can measure the target's range and Doppler [MIL02], [LIW08]. Triangular modulated FMCW is the waveform that is employed in this paper. Time-frequency signal analysis involves the analysis and processing of signals with time-varying frequency content. Such signals are best represented by a time-frequency distribution [PAP95], [HAN00], which is intended to show how the energy of the signal is distributed over the two-dimensional time-frequency plane [WEI03], [LIX08], [OZD03]. Processing of the signal may then exploit the features produced by the concentration of signal energy in two dimensions (time and frequency), instead of only one dimension (time or frequency) [BOA03], [LIY03]. Since noise tends to spread out evenly over the time-frequency domain, while signals concentrate their energies within limited time intervals and frequency bands; the local SNR of a noisy signal can be improved simply by using time-frequency analysis [XIA99]. Also, the intercept receiver can increase its processing gain by implementing timefrequency signal analysis [GUL08]. Time-frequency distributions are useful for the visual interpretation of signal dynamics [RAN01]. An experienced operator can quickly detect a signal and extract the signal parameters by analyzing the timefrequency distribution [ANJ09]. The Spectrogram is defined as the magnitude squared of the Short-Time Fourier Transform (STFT) [HIP00], [HLA92], [MIT01], [PAC09], [BOA03]. For nonstationary signals, the STFT is usually in the form of the Spectrogram [GRI08]. The STFT of a signal x(u) is given in equation 1 as:](image-2.png "")
2![Figure 2: Percent detection (time-frequency). This plot is an frequency vs. time (x-y view) of the Spectrogram 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.](image-3.png "Figure 2 :")
3![Figure 3: Determination of carrier frequency. Spectrogram of a triangular modulated FMCW signal (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=984.4 Hz).](image-4.png "Figure 3 :")
4![Figure 4: Modulation bandwidth determination. Spectrogram of a triangular modulated FMCW signal (256 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 white arrow) to the lowest frequency value of the signal (bottom white arrow) in the y-direction (frequency) 5) Modulation period:Distance from highest frequency value of signal (at a threshold of 20% maximum intensity) to lowest frequency value of signal (at same threshold) in X-direction (time).](image-5.png "Figure 4 :")
![Journal of Researches in Engineering ( ) Volume XVII Issue II Version I Modulation period determination. Spectrogram of a triangular modulated FMCW signal (256 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 white arrow) to the lowest frequency value of the signal (bottom white arrow) in the x-direction (time) 6) Time-frequency localization:](image-6.png "GlobalF")
6![Figure 6: Time-frequency localization determination. Spectrogram of a triangular modulated FMCW signal (256 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 white arrow) to the right side of the signal (right white 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. 7) Chirp rate: (modulation bandwidth)/(modulation period) 8) Lowest detectable SNR: 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.](image-7.png "Figure 6 :")
![Figure 5:](image-8.png "")
![Figure7: Lowest detectable SNR. This plot is an frequency vs. time (x-y view) of the Spectrogram 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 60% max intensity (just above the 'x' in 'max') is barely visible, because the threshold for the Spectrogram is 60%. For this case, any lower SNR would have been a non-detect. Compare to Figure2, which is the same plot, except that it has an SNR level equal to 10dB.](image-9.png "Figure")
8![Figure 8 shows comparative plots of the Spectrogram (left) vs. the Scalogram (right) (triangular](image-10.png "Figure 8")
8![Figure 8: Comparative plots of the triangular modulated FMCW (task 1) low probability of intercept radar signals (Spectrogram (left-hand side) vs. the Scalogram (right-hand side)). The SNR for the top row is 10dB, for the middle row is 0dB, and for the bottom row is -3dB. In general, the Spectrogram signals appear more localized ('thinner') than do the Scalogram signals. In addition, the Spectrogram signals appear more readable than the Scalogram signals at every SNR level.](image-11.png "Figure 8 :")
9![Figure 9 shows comparative plots of the Spectrogram (left) vs. the Scalogram (right) (triangular modulated FMCW signal -Task 2) at SNRs of 10dB (top), 0dB (middle), and -3dB (bottom).](image-12.png "Figure 9")
9![Figure 9: Comparative plots of the triangular modulated FMCW (task 2) low probability of intercept radar signals (Spectrogram (left-hand side) vs. the Scalogram (right-hand side)). The SNR for the top row is 10dB, for the middle row is 0dB, and for the bottom row is -3dB. In general, the Spectrogram signals appear more localized ('thinner') than do the Scalogram signals. In addition, the Spectrogram signals appear more readable than the Scalogram signals at every SNR level.IV.](image-13.png "Figure 9 :")
![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 Spectrogram, and Scalogram, 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 Spectrogram versus the Scalogram for the signal analysis of low probability of intercept triangular modulated FMCW radar signals) it was shown that the Spectrogram by-and-large outperformed the Scalogram 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.](image-14.png "Conclusions")
1 Approved for Public Release; Distribution Unlimited: CaseNumber: 88ABW-2017-1873, 25-Apr-2017
1parametersSpectrogramScalogramcarrier frequency6.83%8.26%modulation bandwidth16.60%28.17%modulation period0.68%0.72%chirp rate16.25%28.47%percent detection70.0%62.22%lowest detectable snr-3.67db-2.67dbplot time3.28s4.16stime-frequency localization-x2.88%4.51%time-frequency localization-y5.75%9.0%From Table 1, the Spectrogram outperformedthe Scalogram in every metrics category.
			Year 2017 © 2017 Global Journals Inc. (US)
			© 2017 Global Journals Inc. (US)
			F © 2017 Global Journals Inc. (US) detectable SNR are both based on visual detection in
		
		
			

				


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		The frequency corresponding to the maximum intensity of the time-frequency representation
				
	




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		EURASIP Journal on Advances in Signal Processing
		
			
			2008. 2008
		
	




* 
	
		Integrated Cubic Phase Function for Linear FM Signal Analysis
		
			PWang
		
		
			HLi
		
		
			IDjurovic
		
		
			BHimed
		
	
	
		IEEE Transactions on Aerospace and Electronic Systems
		
			2010
		
	
	to appear




* 
	
		Analysis of Multicomponent LFM Signals Using Time-Frequency and The Gray-Scale Inverse Hough Transform
		
			GWei
		
		
			SWu
		
		
			EMao
		
	
	
		IEEE Workshop on Statistical Signal Processing
		
			
			September 28 -October 1, 2003
		
	




* 
	
		ELINT: The Interception and Analysis of Radar Signals
		
			RWiley
		
		
			2006
			Artech House
			Norwood, MA
		
	




* 
	
		Detection of Low Probability of Intercept Radar Signals. Defence R&D Canada -Ottawa Contract Report
		
			KWong
		
		
			TDavidson
		
		
			SAbelkader
		
		
			2009. September 2009
		
	




* 
	
		A Quantitative SNR Analysis for the Pseudo Wigner-Ville Distribution
		
			XXia
		
		
			VChen
		
	
	
		IEEE Transactions on Signal Processing
		
			47
			10
			
			October, 1999
		
	




* 
	
		Design of Selective M-Channel Perfect Reconstruction FIR Filter Banks
		
			AZahhad
		
		
			MSabah
		
	
	
		IEE Electronics Letters
		
			35
			15
			
			1999
		
	




* 
	
		A Simple Method for Designing Pseudo QMF Banks
		
			ZZhang
		
		
			LJiao
		
	
	
		Proceedings of the IEEE International Conference on Communication Technology
				the IEEE International Conference on Communication Technology
		
			Aug. 2000
			2