# I. Introduction

low probability of intercept (LPI) radar that uses frequency hopping techniques changes the transmitting frequency in time over a wide bandwidth in order to prevent an intercept receiver from intercepting the waveform. The frequency slots used are chosen from a frequency hopping sequence, and it is this unknown sequence that gives the radar the advantage over the intercept receiver in terms of processing gain. The frequency sequence appears random to the intercept receiver, and so the possibility of it following the changes in frequency is remote [PAC09]. This prevents a jammer from reactively jamming the transmitted frequency [ADA04]. Frequency hopping radar performance depends only slightly on the code used, given that certain properties are met. This allows for a larger variety of codes, making it more difficult to intercept.

Time-frequency signal analysis involves the analysis and processing of signal s with time -varying The STFT of a signal ??(??)is given in equation 1 as:
?? ?? (??, ð??"ð??"; ?) = ? ??(??)? +? ?? (?? ? ??)?? ??? 2??ð??"ð??"?? ???? (1)
Where ?(??) is a short time analysis window localized around ?? = 0 and ð??"ð??" = 0. Because multipli-cation by the relatively short window ?(?? ? ??) effectively suppresses the signal outside a neighborhood around the analysis point ?? = ??, the STFT is a 'local' spectrum of the signal ??(??) around ??. Think of the window ?(??) as sliding along the signal ??(??) and for each shift ?(?? ? ??) we compute the usual Fourier transform of the product function ??(??)?(?? ? ??). 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.

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??"ð??") = ?? ??(??) +? ?? ?(?? ? ??)?? ??? 2??ð??"ð??"?? ????? 2 (2)
The Spectrogram 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) Time and Frequency covariance -The Spectrogram preserves time and frequency shifts, thus the spectrogram is an element of the class of quadratic time-frequency distributions that are covariant by translation in time and in frequency (i.e. Cohen's class); 2) Time-Frequency Resolution -The time-frequency resolution of the Spectrogram is limited exactly as it is for the STFT; there is a trade-off between time resolution and frequency resolution. This poor resolution is the main drawback of this representation; 3) Interference Structure -As it is a quadratic (or 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. Thus, as for every quadratic distribution, the Spectrogram presents interference terms; however, those interference terms are restricted to those regions of the time-frequency plane where the signals overlap. Thus if the signal components are sufficiently distant so that their Spectrograms do not overlap significantly, then the interference term will nearly be identically zero
[ISI96], [COH95], [HLA92].
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 idea of the wavelet transform (equation (3)) is to project a signal ?? on a family of zero-mean functions (the wavelets) deduced from an elementary function (the mother wavelet) by translations and dilations:
?? ?? (??, ??; ?) = ? ??(??)? ??,?? * +? ?? (??)????(3)
Where ? ??,?? (??) = |??| 


# ??

. 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 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 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
[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 ?? 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)
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 . 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 frequency hopping 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 waveforms (4 component frequency hopping, 8 component frequency hopping). 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 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 frequency hopping (prevalent in the LPI arena [AMS09]) 4component signal whose parameters were: sampling frequency=5KHz; carrier frequencies=1KHz, 1.75KHz, 0.75KHz, 1.25KHz; modulation bandwidth=1KHz; modulation period=.025sec.

Task 2 was similar to Task 1, but for a frequency hopping 8-component signal whose parameters were: sampling frequency=5KHz; carrier frequencies=1.5 KHz, 1KHz, 1.25KHz, 1.5KHz, 1.75KHz, 1.25KHz, 0.75KHz, 1KHz; modulation bandwidth=1KHz; modulation period=.0125sec.

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.  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 98%, 78%, 75%, 63%), and the lowest of these 4 values was recorded (63%). Ten test runs were performed for both timefrequency 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).  


# 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).   ) 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.


# 7) 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.

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 7).  2dB) 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 frequency hopping signal components was visible. 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. large, the Scalogram outperformed the Spectrogram, as will be shown in the results section. From Table 1, the Scalogram outperformed the Spectrogram in average percent error: carrier frequency (0.44% vs. 0.67%), modulation bandwidth (21.62% vs. 25.70%), modulation period (10.25% vs. 11.37%), and time-frequency (y-direction) (9.44% vs. 9.77%);and in average: percent detection (80.84% vs. 69.67%), and lowest detectable SNR (-3.0db vs. -2.0db), while the Spectrogram outperformed the Scalogram in average plot time (3.43s vs. 5.62s).    


# III. Results


# IV. Discussion

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

From Table 1, the performance of the Spectrogram and the Scalogram will be summarized, including strengths, weaknesses, and generic scenarios in which each particular signal analysis tool might be used. Spectrogram: The Spectrogram outperformed the Scalogram in average plot time (3.43s vs 5.62s). However, the Spectrogram was outperformed by the Scalogram in every other category. The Spectrogram's extreme reduction of cross-term interference is grounds for its good plot time, but at the expense of signal localization (i.e. it produces a 'thicker' signal (as is seen in Figure 8 and Figure 9) -due to the trade-off between cross-term interference and signal localization). This poor signal localization ('thicker' signals) can account for the Spectrogram being outperformed in the areas of average percent error of modulation bandwidth, modulation period, and time-frequency localization (ydirection). The spectrogram 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. Scalogram: The Scalogram outperformed the Spectrogram in every category but plot time. Because of the Spectrogram's extreme reduction of cross-terms at the expense of signal localization (i.e. it produces a 'thicker' signal), the Scalogram was more localized than the Spectrogram, accounting for its better performance in the areas of average percent error of modulation bandwidth, modulation period, and time-frequency localization (y-direction). In addition, since the compactly supported nature of thewavelet (basis of Scalogram) enables temporal localization of a signal's features, this may also have contributed to the the Scalogram's better average percent error of modulation period. Average percent detection and lowest detectable SNR are both based on visual detection in the Time-Frequency representation. Figures 8 and 9 clearly show that the signals in the Scalogram plots are more readable than those in the Spectrogram plots, which accounts for the Scalogram's better average percent detection and lowest detectable SNR. Since the irregularity of the wavelet basis (basis of Scalogram) lends itself to analysis of signals with discontinuities (such as the frequency hopping signals used in this testing), this may have been a contributing factor to the Scalogram's better overall performance versus the Spectrogram. Also, since the wavelet is irregular in shape and compactly supported, it makes it an ideal tool for analyzing signals of transient nature (such as the frequency hopping signals used in this testing), which may also have been a contributing factor to the Future plans include analysis of an additional low probability of intercept radar waveform (triangular modulated FMCW), again using the Spectrogram and the Scalogram as time-frequency analysis techniques.
![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].](image-2.png "")
2![Percent detection: Percent of time signal was detected -signal was declared a detection if any portion of each of the signal components (4 or 8 signal components for frequency hopping) exceeded a set threshold (a certain percentage of Global Journal of Researches in Engineering ( ) Volume XVI Issue II Version I 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 Figure1).](image-3.png "2 )")
1![Figure1: Threshold percentage determination. This plot is an amplitude vs. time (x-z view) of the Spectrogram of a frequency hopping 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 98%, 78%, 75%, 63%), and the lowest of these 4 values was recorded (63%). Ten test runs were performed for both timefrequency 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%.](image-4.png "Figure 1 :")
2![Figure 2 : Percent detection (time-frequency). Spectrogram of 4-component frequency hopping signal (512 samples, 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 FSK signal components was visible. 3) Carrier frequency: The frequency corresponding to the maximum intensity of the time-frequency representation (there are multiple carrier frequencies (4 or 8) for the frequency hopping waveforms).](image-5.png "Figure 2 :")
3![Figure 3: Determination of carrier frequency. Spectrogram of a 4-component frequency hopping signal (512 samples, SNR=10dB). From the frequency-intensity (y-z) view, the 4 maximum intensity values (1 for each carrier](image-6.png "Figure 3 :")
4![Figure 4 : Modulation bandwidth determination. Spectrogram of a 4-component frequency hopping signal (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). 5) Modulation period: From Figure 5 (which is at a threshold of 20% maximum intensity), the modulation period is the manual measurement of the width of each of the 4 frequency hopping signals in the x-direction (time), and then the average of the 4 signals is calculated.](image-7.png "Figure 4 :")
5![Figure 5 : Modulation period determination. Spectrogram of a 4-component frequency hopping signal (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. 6) Time-frequency localization: From Figure 6, the time-frequency localization is a manual measurement (at a threshold of 20% maximum intensity) of the 'thickness' (in the y-direction) of the](image-8.png "Figure 5 :")
6![Figure 6 : Time-frequency localization determination for the Spectrogram of a 4-component frequency hopping signal (512 samples, SNR=10dB) with threshold value automatically set to 20%. From this threshold plot, the time-](image-9.png "Figure 6 :")
7![Figure 7 : Lowest detectable SNR. Spectrogram of 4-component frequency hopping signal (512 samples, SNR=-2dB) 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 frequency hopping signal components was visible. 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.](image-10.png "Figure 7 :")
8![Figure 8 shows comparative plots of the Spectrogram vs. the Scalogram (4 component frequency hopping) at SNRs of 10dB (top), 0dB (middle), and -3dB (bottom).](image-11.png "Figure 8")
8![Figure 8 : Comparative plots of the 4-component frequency hopping 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](image-12.png "Figure 8 :")
9![Figure 9 shows comparative plots of the Spectrogram vs. the Scalogram (8 component frequency hopping) at SNRs of 10dB (top), 0dB (middle), and -3dB (bottom).](image-13.png "Figure 9")
9![Figure 9 : Comparative plots of the 8-component frequency hopping low probability of intercept radar signals (Spectrogram (left-hand side) vs. the Scalogram (right-hand side)). The SNR for the top row plots is 10dB, for the middle row plots is 0dB, and for the bottom row plots is -3dB (which is a non-detect for the Spectrogram).In general, the Scalogram signals appear more localized ('thinner') than do the Spectrogram signals. In addition, the Scalogram signals appear more readable than the Spectrogram signals at every SNR level.](image-14.png "Figure 9 :")
![Scalogram's better overall perfromance. The scalogram might be used in a scenario where you need good signal localization in a fairly low SNR environment, without tight time constraints. V. Conclusions 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 frequency hopping radar signals) it was shown that the Scalogram by-and-large outperforms the Spectrogram 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-15.png "")
1The metrics from the Spectrogram were thencompared to the metrics from the Scalogram. By andparametersSpectrogramScalogramcarrier frequency0.67%0.44%modulation bandwidth25.70%21.62%modulation period11.37%10.25%time-frequency localization-y9.77%9.44%
			© 2016 Global Journals Inc. (US)
		
		
			

				


* 
	
		EW 102: A Second Course in Electronic Warfare
		
			DAdamy
		
		
			2004
			Artech House
			Norwood, MA
		
	




* 
	
		On the Design of 2-Band FIR QMF Filter Banks Using WLS Techniques
		
			FAl-Namiy
		
		
			MNigam
		
	
	
		Proceedings of the Fourth IEEE International Conference on High Performance Computing in the Asia-Pacific Region
				the Fourth IEEE International Conference on High Performance Computing in the Asia-Pacific Region
		
			May 2000
			2
			
		
	




* 
	
		Identification of LPI Radar Signal Modulation using Bi-coherence Analysis and Artificial Neural Networks Techniques
		
			LAnjaneyulu
		
		
			NMurthy
		
		
			N;Sarma
		
		
			Iit Guwahati
		
		
			2009. January 16-18, 2009
			
		
	




* 
	
		A Novel Method for Recognition of Modulation Code of LPI Radar Signals
		
			LAnjaneyulu
		
		
			NMurthy
		
		
			NSarma
		
	
	
		International Journal of Recent Trends in Engineering
		
			1
			3
			
			May 2009
		
	




* 
	
		Time Frequency Signal Analysis and Processing: A Comprehensive Reference
		
			BBoashash
		
		
			2003
			Elsevier
			Oxford, England
		
	




* 
	
		A Novel Method for Extraction of In-Pulse Feature of Multi-Component LFM Signal
		
			CChen
		
		
			MHe
		
		
			HJin
		
		
			HLi
		
		
			ZShen
		
		
			2009. 2009
			
		
	




* 
	
		The Wavelet Transform and Time-Frequency Analysis
		
			LCohen
		
	
	
		Debnath, L., Wavelet Transforms and Signal Processing
				
			Birkhauser
			2002
			
		
	




* 
	
		Time-Frequency Analysis
		
			LCohen
		
		
			1995
			Prentice Hall
			Upper Saddle River, NJ
		
	




* 
	
		A Low Probability of Intercept Signal Detection Receiver Using Quadrature Mirror Filter Bank Trees
		
			TCFarrell
		
		
			GPrescott
		
		
			1996
			IEEE Proceedings
			
		
	




* 
	
		A Method for Finding Orthogonal Wavelet Filters with Good Energy Tiling Characteristics
		
			TFarrell
		
		
			GPrescott
		
	
	
		IEEE Transactions on Signal Processing
		
			47
			1
			
			Jan. 1999
		
	




* 
	
		A Time-Frequency Approach to the Adjustable Bandwidth Concept
		
			[Gal05] Galleani
		
		
			LCohen
		
		
			LNoga
		
		
			A
		
	
	
		Digital Signal Processing
		
			16
			5
			
			Sept. 2006
		
	




* 
	
		A WLS Algorithm for the Design of Low-Delay Quadrature Mirror Filter Banks
		
			[goh00]Goh
		
		
			CLim
		
		
			T
		
	
	
		Proceedings of the IEEE International Symposium on Circuits and Systems
				the IEEE International Symposium on Circuits and Systems
		
			May 2000
			1
			
		
	




* 
	
		Computer-Aided Methods of the LPI Radar Signal Detection and Classification
		
			YGrishin
		
		
			DJanczak
		
	
	
		Proc. of SPIE
				of SPIE
		
			2007
			6937
			
		
	




* 
	
		Interferences Excision Via Time-Frequency Distribution in Radio Communication Systems
		
			YGrishin
		
	
	
		XVIII-th International Conference on Electromagnetic Disturbances
				Vilnius, Lithuania
		
			2008. September 2008
			
		
	




* 
	
		Extraction of Polyphase Radar Modulation Parameters Using a Wigner-Ville Distribution-Radon Transform
		
			TGulum
		
		
			PPace
		
		
			RCristi
		
	
	
		IEEE International Conference on Acoustics, Speech, and Signal Processing
				Las Vegas, NV
		
			April 2008
		
	




* 
	
		Target Position Extraction Based on Instantaneous Frequency Estimation in a Fixed-Reticle Seeker
		
			SHan
		
		
			HHong
		
		
			DSeo
		
		
			JChoi
		
	
	
		Opt. Eng
		
			39
			
			September 2000
		
	




* 
	
		Detection and Parameter Estimation of Chirped Radar Signals. Final Report, Naval Postgraduate School
		
			RHippenstiel
		
		
			MFargues
		
		
			IMoraitakis
		
		
			CWilliams
		
		
			Jan. 10, 2000
			Monterey, CA
		
	




* 
	
		Linear and Quadratic Time-Frequency Signal Representations
		
			FHlawatsch
		
		
			GFBoudreaux-Bartels
		
	
	
		IEEE Signal Processing Mag
		
			9
			2
			
			April 1992
		
	




* 
	
		Time-Frequency Toolbox Users Manual
		
			FAuger
		
		
			PFlandrin
		
		
			PGoncalves
		
		
			OLemoine
		
		
			1996
		
		
			Centre National de la Recherche Scientifique and Rice University
		
	




* 
	
		Automatic Classification of Active Sonar Data Using Time-Frequency Transforms
		
			FLari
		
		
			AZakhor
		
	
	
		Proceedings of IEEE-SP International Symposium on Time-Frequency and Time-Scale Analysis
				IEEE-SP International Symposium on Time-Frequency and Time-Scale AnalysisVictoria, BC
		
			Oct. 4-6, 1992
			
		
	




* 
	
		
			XLi
		
		
			GBi
		
		A New Reassigned Time-Frequency Representation. 16 th European Signal Processing Conference
				Lausanne, Switzerland
		
			August 25-29, 2008
			
		
	




* 
	
		Recursive Filtering Radon-Ambiguity Transform Algorithm for Detecting Multi-LFM Signals
		
			YLi
		
		
			XXiao
		
	
	
		Journal of Electronics (China)
		
			20
			3
			
			May 2003
		
	




* 
	
		Digital Signal Processing, A Computer-Based Approach, Second Edition
		
			SMitra
		
		
			2001
			McGraw-Hill
			Boston, MA
		
	




* 
	
		Time-Frequency Component Analyzer. Dissertation
		
			AOzdemir
		
		
			Sept. 2003
			Ankara, Turkey
		
		
			Bilkent University
		
	




* 
	
		Detecting and Classifying Low Probability of Intercept Radar
		
			PPace
		
		
			2009
			Artech House
			Norwood, MA
		
	




* 
	
		Detection and Estimation of Generalized Chirps Using Time-Frequency Representations
		
			APapandreou
		
		
			GFBoudreaux-Bartels
		
		
			SKay
		
		
	




* 
	
		Conference Record of the Twenty-Eighth Asilomar Conference on Signals, Systems and Computers
				
			1994
			
		
	




* 
	
		Feature Identification in the Time-Frequency Plane by Using the Hough-Radon Transform
		
			RRangayyan
		
		
			SKrishnan
		
	
	
		Pattern Recognition
		
			34
			
			2001
		
	




* 
	
		
			[Sar98] Sarkar
		
		
			TSu
		
		
			C
		
	
	
		Discrete Wavelet Techniques. IEEE Antennas and Propagation Magazine
				
			October 1998
			1
			
		
	




* 
	
		A Tutorial on Wavelets from an Electrical Engineering Perspective, Part 2: The Continuous Case
		
			[Sat98] Sarkar
		
		
			TSu
		
		
			C
		
	
	
		IEEE Antennas and Propagation Magazine
		
			40
			6
			
			December 1998
		
	




* 
	
		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
		
	




* 
	
		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