Radar signal recognition based on time-frequency ... - IEEE Xplore

Radar signal recognition based on time-frequency representations and multidimensional probability density function estimator Krzysztof Konopko, Yuri P. Grishin, Dariusz Jańczak Faculty of Electrical Engineering, Bialystok University of Technology Bialystok, Poland [email protected], [email protected], [email protected] Abstract—A radar signal recognition can be accomplished by exploiting the particular features of a radar signal observed in presence of noise. The features are the result of slight radar component variations and acts as an individual signature. The paper describes radar signal recognition algorithm based on time frequency analysis, noise reduction and statistical classification procedures. The proposed method is based on the Wigner-Ville Distribution with using a two-dimensional denoising filter which is followed by a probability density function estimator which extracts the features vector. Finally the statistical classifier is used for the radar signal recognition. The numerical simulation results for the P4-coded signals are presented. Keywords — radar signal recognition; Distribution; time-frequency analysis

Wigner-Ville

I. INTRODUCTION Joint Time-Frequency Analysis (TFA) [1-4] has been under active study in the recent years for different signal processing applications such as underwater acoustics, speech processing, medical analysis and signal recognition in telecommunication systems [3, 5-7]. It has also been investigated by radar researchers as a very useful tool for radar signal detection, analysis and recognition [5, 8-10, 12, 13]. Such an analysis provides also new identification tool for Electronic Warfare Systems [5] and for classification of the modulation type of the intercepted radar signals [14, 15]. This paper presents an application of the TFA for solving the radar signal recognition problem. The proposed method exploits the particular features of observed radar signal. These modulation features are the result of slight radar component variations and acts as an individual signature of radar. However using only the TFA tools for solving the radar signal recognition problem [5, 10] is not effective enough because of the presence of the cross terms interference and noise. Besides, there is a problem of selection a proper time-frequency distribution for parameters estimation of similar signals which are closely located in the time-frequency plane. Thus a successful radar signal recognition algorithm requires not only selection of a suitable time-frequency signal representation, but also procedures of noise reduction and extraction of a feature vector for final recognition. The objective of the paper is to present an extensive algorithm which uses the Wigner-Ville Distribution (WVD) [2-5, 8, 11] tools for a signal

representation and the procedures of denoisification followed by recognition based on distributions of specific features. II. A RADAR SIGNAL RECOGNITION ALGORITHM A. Signal Representation A conventional signal analysis is based on time or frequency domain representations. These analyses are appropriate for stationary signal that is when the signal characteristics do not vary with time. However, the radar signals have a non-stationary nature. The behaviour of radar signals cannot be sufficiently investigated by using the conventional analysis methods for stationary signals. The algorithm proposed in this paper is based on the quadratic (or bilinear) time-frequency transform [1-4] which is twodimensional function that indicates the time-varying frequency content of the one-dimensional signal. The Wigner Ville distribution is the most known example of the transform. It is defined as the Fourier transform of the time-dependent signal autocorrelation function: 



WV (t , f ) 



*



 x(t  2 ) x (t  2 )e

 j 2f

d 





where x(t) is the analysed signal. The WVD exhibits the highest signal energy concentration in the time-frequency plane for linearly modulated signals, but has drawbacks in the cases of nonlinear frequency modulated signals. The WVD also contains interfering cross terms between each element of multicomponent signal. The presence of the cross terms often makes it difficult to determine the signal modulation parameters. The influence that the cross term interference has on the WVD is analysed in [8, 9, 16]. The main efforts of researches is directed on working out new transformations which could enable to remove the cross terms. For the signal processing a windowed version of the WVD, the pseudo WVD (PWVD), and smoothed pseudo WignerVille distribution (SPWVD) are widely used. Pseudo Wigner-Ville Distribution is defined as the following:





PWV (t , f ) 









h x(t  ) x* (t  )e  j 2f d  2 2



where h(t) is a window function. Smoothed Pseudo Wigner-Ville Distribution can be written as: 

 SPWV (t , f ) 





qu  t 







 h x(t  2 ) x (t  2 )e *

 j 2f

B. Noise Reduction A time-frequency signal representation may be subject to noise and interference from several sources, including sensor noise and channel transmission errors. Generally the noise has a wide time-frequency (TF) scattering. Hence, simple 2D filtering may be effective for noise reduction in TF plane. In general, time-frequency representation TF(m,n) is convolved by a filter function F(i,j) giving the smooth representation [19]:

d du 

k





S (m, n) 

l

 TF (m  i, n  j) F (i, j) 

(5)

i   k j  l

where q(u-t) and h() are window functions. In addition to the WVD a number of bilinear distributions have been proposed [1-4, 12] for removing the cross terms. L. Cohen [16] has shown that most of the time-frequency distributions can be written in a generalized form

S (t , f ;  )    







     , ; t, f x(s  2 ) x (s  2 )e *

j 2 [ ( t  s )  f ]

dsd d



  

which makes it possible to obtain new time-frequency distributions. This class of the time-frequency distributions is known as Cohen. Any distribution of the Cohen class can be thought of as two dimensional linear filtering the WVD. The most popular kernel functions   , ; t , f  are shown in Tab. I. However taking into consideration high computational burden necessary for implementing the latter transformations, in practice transformations (1)-(3) are usually used [5, 6, 17, 18]. These transformations are also employed in the numerical simulations of this paper. TABLE I.

THE POPULAR KERNEL FUNCTION   , ; t , f

The TFA Wigner-Ville Pseudo WignerVille Smooth Pseudo Wigner-Ville

C. Features Extraction General problem of radar signal recognition based on timefrequency distribution is a dimensionality issue. Thus feature vector extraction is required for dimensionality reduction. As the elements of feature vector, the join time-frequency moments, Kolmogorov and Kullback-Leibler distance [19] can be chosen. The two-dimensional moments of order (p+q) of a time-frequency distribution TFD(t, f) are defined as:  



 t

p

f qTFD (t , f )dtdf 

where p, q=0,1,… , . The central moments of a time-frequency distribution TFD(t, f) are defined as:  



 pq 

  (t  t )

p

( f  f ) q TFD (t , f )dtdf

sin( )

  (2 t ) 2  exp     

Choi-Williams

where t and f are defined as t 

e j 





 h  t  2  h



*

   2 t dt t  e  2

(7)

M10 M , f  01 M 00 M 00

The second order moments (02, 11, 20) may be used to determine an significant image feature orientation. In general the orientation describes the directions of the principal axes. In terms of moments the orientations may be given by:

e  j 

Page Rihaczek



 



    h  t   h*  t   G     2  2

(6)

 

1    h  t   h*  t    2  2

M pq 



the kernel function   , ; t , f 

Born-Jordan

Spectrogram

Next the noise and interference reduction by means of thresholding is performed. After this step the output representation values are zero for all points where the inputs are below the threshold level.



 2 11  1    tan 1  2   20  02 



where  is the angle of the principal axis nearest to the x-axis which takes on values in the range: –/4  /4.

The Kolmogorov distance is defined as:

The test function Tk (F ) reduces n-dimensional stochastic process to one-dimensional process.

DK ( NTFD1 , NTFD2 )   







NTFD1 (t , f )  NTFD2 (t , f ) dtdf





Small values of the test function lead to acceptance of hypothesis H0, so the critical set is given by:

Ak  0, ak  



 

where NTFD1, NTFD2 are normalized time-frequency distributions.

where ak can be given by:

The Kullback-Leibler distance is defined as:

ak  maxak i 

DKL ( NTFD1 , NTFD2 )    

 NTFD1 (t , f )  NTFD1 (t , f ) ln  NTFD (t , f )  2     0 



 dtdf NTFD2  0    NTFD2  0

D. Statistical Classifier The main task of the radar signal recognition is classification. The algorithm proposed in this paper uses statistical classifier for solving the problem. The recognition is carried out by sequential testing of the hypotheses H1, H2, …, Hk which denote the k-th radar signal against one alternative hypothesis H0 which represents other type of signal source. In general statistical decision theory [19] there are the following fundamental entities: the observation space, that is the set of values of  which parameterizes the possible distributions of the observations, the decision space D of all decisions d which can be taken, and the loss function l(,d). In the proposed algorithm the set  has only two elements which represent k-th radar signal and other type of signal source. The decision space D has also only two possible decisions dk and d0 which represent acceptance of hypotheses Hk and H0 respectively. The loss function l(,d) has the following form:



0  l ( , d )  l1 l 2

correct decision missed decision 





i  1, N  1 

The threshold value ak(i) can be computed on the basis of Bayes test. In the case the risk function is given in the form:

l  r (d )   1 l2 



missed detection false decision





where

  Pk Tk F   ak i     Pi Tk F   ak i 





where Pk and Pi are probabilities of k-th and i-th radar signals respectively. Because of: 

Pi Tk F   ak i   Pi Tk F   ak i   1 



one can write 

  1  Pi Tk F   ak i  



The decision dk is accepted by the Bayes rule when: l1  l2 and decision d0 when: l1  l2  Thus threshold value ak can be computed on the basis of equation:

false decision

A procedure which makes decision dk or d0 and leads to acceptance of hypothesis Hk or H0, is based on a test function Tk (F ) [17]:



l1  l2  



The equation (17) can be presented in the form: 

Tk ( F ) 

f k (F )

ik

N

f k (F ) 





fi (F )

i 1

where N is a number of known radar signals, F is n-dimensional feature vector, fk specifies the probability density function for the feature vector of k-th radar signal.



Pi Tk F   ak i  

l1 Pk Tk F   ak i   1  l2



In order to solve this equation the values of probabilities must be given in the form:

Pi Tk F   ak i   D f k ( Fi ) ak i 



Pk Tk F   ak i   D f k ( Fk ) ak i 





Waveform of the S1 P4-coded signal and its spectrum (SNR=0dB) are shown in Fig. 1 and 2 respectively.

where D f k ( Fk ) a k i  and D f k ( Fi ) a k i  are distributions of test function Tk F  of k-th and i-th radar signals. The threshold value ak(i) can be computed in a recursive way:

aki 0   a0 ,



aki n   aki n  1  c1  g k aki n  1,

where g k  D f k Fi  ak i  





l1 D f F  ak i  and c are constant l2 k k

values. The statistical classifier proposed in this paper can use various pdf estimators. The pdf estimators can be classified as parametric, non-parametric and semi-parametric. In parametric methods the pdf is assumed to be of a standard form. The nonparametric methods include histogram, the kernel based methods and the K-nearest neighbour methods. In semiparametric methods the given density can be modelled as a combination of known densities. These methods combine the flexibility of nonparametric methods and the evaluation efficiency of parametric methods.

Fig. 1. A waveform realization of the S1 signal (SNR=0dB)

III. SIMULATION RESULTS For analysis of the proposed recognition algorithm efficiency the P4-coded radar signal [5, 11, 20] has been used. A high efficiency of the algorithm in case of a signal with linear frequency modulation under relatively small values of the SNR (about 0dB  -4dB) was shown in [17]. The P4-code belongs to a general class of polyphase codes, where the discrete phases of the linear chirp waveform are taken at specific time intervals. The code exhibits the same rangeDoppler properties as the chirp waveform. At the same time the peak sidelobe levels of the P4-code are lower than those of the unweighted chirp waveform and as a consequence it possesses better resolution capabilities.

Fig. 2. The spectrum of the S1 signal realization (SNR=0dB) Each signal was sampled in 512 points. The WVD, PWVD and SPWVD were chosen as signal representation for a comparison. The WVD of S1, S2, S3 signals (SNR=0dB) are presented in Fig. 3-5 respectively.

The phase sequence of the P4 signal is described by:



n 

 N

(n  1) 2   (n  1) n  1, ..., N 



where N is the pulse compression ratio [11, 20]. The following three P4 code signals have been tested:  S1 - frequency = 1kHz, number of sub-pulses N=64, number of cycles per phase = 1;  S2 - frequency = 1kHz, number of sub-pulses N=68, number of cycles per phase = 1;  S3 - frequency = 1.01kHz, number of sub-pulses N=64, number of cycles per phase = 1.

Fig. 3. The WVD of the S1 signal realization (SNR=0dB)

The signal recognition was carried out by the proposed statistical classifier. The density function of the feature vector was estimated by using HRBF neural network [17]. The recognition performance of the algorithm for the P4-coded signals with different representation and different SNR has been tested on the basis Monte Carlo simulations. The probabilities of the lack of recognition (LR) and false recognition (FR) for algorithm with using the Gaussian smoothing filter are presented in Table II. The corresponding probabilities for algorithm which for a noise reduction uses only thresholding are presented in Table III. TABLE II. PROBABILITIES OF FALSE RECOGNITION AND LACK OF RECOGNITION WITH USING THE 2D GAUSSIAN FILTER AS NOISE REDUCTION PROCEDURE

TFD,

Fig.4. The WVD of the S2 signal realization (SNR=0dB)

S/N ratio

S1 recognition S2 recognition S3 recognition FR

LR

FR

LR

FR

LR

WVD 4 dB