Paper Title (Use Style: Paper Title) s35

Characteristic Analysis of White Gaussian Noise in S-Tran sformation Domain

Xinliang Zhang1, and Yue Qi1, Mingzhe Zhu1,2

1: Department of Electronic Engineering, Xidian University, Xi’an, 710071 China; 2: The Ministry Key Laboratory of Electronic Information Countermeasure and Simulation. Email: [email protected]

ABSTRACT The characteristic property of white Gaussian noise (WGN) is derived in S-transformation domain. The results show tha t the distribution of normalized S-spectrum of WGN follows c 2 distribution with two degree of freedom. The conclusi on has been confirmed through both theoretical derivations and numerical simulations. Combined with different criteria, an effective signal detection in S-transformation can be realized.

Keywords: signal detection, S-transform; white Gaussian noise; c 2 distribution

1. Introduction rifies the proposition by Monte Carlo method. In Sect ion 4, the performance of signal detection is illustrate THE S-transform(ST) has been well studied since it w d by detecting transient sinusoid signals under the co as proposed by R. G. Stockwell in 1996 [1] and has alr nstant-false-alarm-rate (CFAR) criterion. Finally, co eady found applications in many fields such as geoph nclusions are drawn in Section 5. ysics [2], mechanical systems [3] and medical signal a nalyses [4]. The signal processing framework in S-tra nsformation domain is under building and more and 2. Characteristic Property of WGN in ST Domai more ST based methods are proposed. The most cont n ributive works among them were conducted by C. R. Pinnegar who extended the original ST to generalized 2.1. Proposition ST [2] and presented several kinds of useful STs for s x( t ) pecific conditions, such as the asymmetric ST [2], the The S-transform of a signal is defined by [1] bi-Gaussian ST [5] and the complex ST [6]. Other out 1 Sx ( t , f )= x (t ) exp( - j 2 p f t ) w ( t - t , f ) d t (1) standing works were concentrated on ST based time-f - 2p requency filtering [7], the side effects of inverse ST where the window function is [8], ST based instantaneous frequency estimation [9] 2 2 and ST based pattern recognition [10-11]. However, t w(t , f )= f exp( - t f / 2) (2) he noise analysis in S-transformation domain, which i And the characteristic and the advantage of ST has bee s the essential problem for signal detection, has not be n sufficiently described by previous literature [2-6]. en well studied yet. [11] presented an illuminating ide 2 Let Sx ( t , f ) be the S-spectrum of x( t ) and white Ga a for noise distribution in a specific generalized ST do ussian noise n( t ): N (0,s 2 ) ; then the normalized S-s main but didn’t prove it theoretically. In this paper, t n( t ) 2S ( t , f )2 / E轾 S ( t , f ) 2 pectrum of , n臌 n (denoted he characteristic property of WGN in original ST do 2 2 as NSn ( t , f ) ), follows c distribution with two degr main is derived theoretically and the conclusion is fur ee of freedom. ther verified by Monte Carlo method. Its application i n nonstationary signal detection is also illustrated. 2.2. Proof

The rest of this paper is organized into as follows: In Step.1: Derive the distribution of Re[Sn ( t , f )] and

Section 2, the proposition of WGN distribution follow Im[Sn ( t , f )] and prove that they follow the identical Ga ed by its theoretical derivation are given. Section 3 ve ussian distribution. 2 Paper Title By definition, the S-spectrum of n( t ) can be expressed a Step.2: Derive the expression of the average S-spectrum s n( t ) E轾 S( t , f ) 2 Re[NS ( t , f )] of , 臌 n , and Prove that n a 2 2 2 Sn( t, f) = Re [ S n ( t , f )] + Im [ S n ( t , f )] (3) nd Im[NSn ( t , f )] follow identical standard normal distri Before further proceeding, let us analyze the distribution bution. of Re[Sn ( t , f )] and Im[Sn ( t , f )] , respectively. The real The normalized S-spectrum of n( t ) can be written as part of S( t , f ) is given by n 骣 2 2 2 2 2 琪Re [Sn ( t , f ) ] Im [ S n ( t , f ) ] f -(t -t ) f NS( t , f )= 2� n 2 2 Re[Sn ( t , f )]= n (t )exp( )cos(2 p f t ) d t 琪轾轾琪E Sn( t , f ) E S n ( t , f ) - 2p 2 桫臌臌 (4) 2 2 =Re [NSn ( t , f )] + Im [ NS n ( t , f )] Re[S ( t , f )] n( t ) n is the integral of multiplied by dete (7) rministic signals, thus the distribution of Re[S ( t , f )] is 2 n Since the average S-spectrum of n( t ) , E轾 S( t , f ) , is 2 臌 n the same as n( t ) with zero mean and its variance s R (t ) given by expressed as 2 * f 2 2 E轾 S( t , f )= E [ n (t ) n ( t )] w ( t - t , f ) 2 -(t -t1 ) f 臌 n 蝌 1 2 1 sR (t )= E n ( t1 ) exp( )cos(2 p f t 1 ) d t 1 - 2p 2 �w(t2- t , f )exp( - 2 p fj ( t 1 t 2 )) d t 1 d t 2 f -(t -t )2 f 2 n* (t )exp(2 )cos(2 p f t ) d t 2 2 2 f 2 - 2p 2 = s 2 2 p 骣 f ゥ -(t -t )2 f 2 = E轾 n(t ) n* ( t ) exp(1 ) (8) 琪蝌臌 1 2 桫 2p -� 2 then the distribution of Re[NSn ( t , f )] and 2 2 -(t -t 2 ) f 鬃exp( ) cos(2pf t ) cos(2 p f t ) d t d t Im[NSn ( t , f )] can be obtained as 2 1 2 1 2 2 Re[NSn ( t , f )] ~ N (0,1) 2 骣 f 2 2 2 (9) = s 琪 exp(- (t -t ) f )cos (2 p f t ) d t Im[NSn ( t , f )] ~ N (0,1) 桫 2p - 2 2骣 f 2 2 骣cos(4pf t )+ 1 Step.3: Prove that Re[NS ( t , f )] and Im[NS ( t , f )] are =s琪 exp( - (t - t ) f ) 琪 d t n n 桫 2p - 桫 2 independent. 2 2 1 骣 f To determine the distribution of NS( t , f ) , the corre =s2�琪 exp( - (t t ) 2 f 2 ) cos(4 p f t ) d t n 2 桫 2p - lation of Re[NSn ( t , f )] and Im[NSn ( t , f )] need to be an alyzed. The correlation function is +exp( - (t -t )2 f 2 ) d t RRI( t1 , t 2 )= E{ Re[ NS n ( t 1 , f )] Im[ NS n ( t 2 , f )]} - f ゥ骣-f2( t -t ) 2 f = 2E [ n (t ) n* ( t )]exp 1 1 1 2 2 2 蝌 1 2 琪 =s� (exp( + 4 p )cos(4 p ft ) 1) s p -� 桫 2 2 2 p 2 2 骣-f( t2 -t 2 ) (5) 鬃exp琪 cos(2pf t1 ) sin(2 p f t 2 ) d t 1 d t 2 2 Im[S ( t , f )] 桫 Similarly, the imaginary part n follows the 2 2 2 2 f 骣-f( t1 -t ) 骣 - f ( t 2 - t ) Gaussian distribution with Im[S ( t , f )]: N (0,s 2 ( t )) , w = 2鬃 exp琪 exp 琪 n I p - 桫2 桫 2 here 鬃cos(2pf t ) sin(2pf t ) d t 2 2 2 2 21 2f 2 f 骣-f( t -t ) 骣 - f ( t - t ) s(t )= s� 1 - exp( 4 p )cos(4 p ft ) = exp琪1鬃 exp 琪 2 sin( 4pf t) d t I ( ) (6) 2 2 2 2 p p - 桫桫 In equation (5) and equation (6), the constant term, exp(- 4p 2 ) 7.1572e-18 , is of very small value which c an approximately equal to zero, thus Sn ( t , f ) can be rega rded as having identically distributed real and imaginary parts, that is, Re[Sn ( t , f )] and Im[Sn ( t , f )] 1 f : N(0,s 2 ) . 2 2 p

2 f 禳- f 2 2 2 5 4 3 2 =exp睚 ( 2t - 2 t (t1 + t 2) +( t 1 + t 2 )) j2 p - 铪 2 �(exp( j 4p f t) - exp( j 4 p f t)) d t A good agreement between the theoretical values and t 1 骣-f2( t - t ) 2 he simulation results can be observed. And as the times o =exp( 鬃 4p2 ) exp1 2 + expj 2 p f ( t t ) � 琪 ( 1 2 ) f Monte Carlo experiments increase, the simulation result 2j 桫 4 s become closer to their theoretical values. Hence, we ha 禳 2 f 镲 2 骣骣t+ t 2p 鬃 exp 睚-f琪t -琪1 2 + j d t ve proved our proposition and obtained the distribution o p 2 f - 铪镲桫桫 f WGN in S transformation domain. 2 2 1 2 骣-f( t1 - t 2 ) -�exp( 鬃 4p ) exp琪 - exp( + j 2 p f ( t1 t 2 )) 2j 桫 4 4. Signal Detection in ST Domain 禳 2 In this section, the task of ST based detector is to provide f 镲 2 骣骣t1+ t 2 2p 鬃 exp 睚-f琪t -琪 - j dt time-varying information in addition to detecting whethe - p 桫桫 2 f 铪 r a transient signal is present or absent. We use the conce 2 2 2 骣-f( t1 - t 2 ) =exp( - 4p )鬃 exp琪 sin( 2 p f ( t1 + t 2 )) pt of frame detection which is similar to the common met 桫 4 hod in voice activity detection (VAD) and no cumulation 0 operator is used. The time slice of S-spectrum is employe (10) d as the frame and the decision function is Since both Re[NS ( t , f )] and Im[NS ( t , f )] follow n n H0 : x ( t )= n ( t ) l < G (11) Gaussian distribution, equation (10) means they are inde H: x ( t )= s ( t ) + n ( t ) l 矴 pendent. 1 where x( t ) is the observed signal, s( t ) is the signal to be This concludes the proof that NSn ( t , f ) has independent and identical distributed real and imaginary parts of stan detected and n( t ) is WGN at the time t (also t the fram 2 2 e index). G is the detection threshold and its value can b dard normal distribution, thus NS( t , f ) follows c di n e calculated as stribution with two degree of freedom. 1 g 2 G = E Sn ( t , f ) 2 f { } (12) 3.Numerical Simulations where the value of g can be determined by c 2 distributi In this section, Monte Carlo methods are used to demons on table according to the specific false alarm probability trate the rationalities of above derivations. The variance 2 P and the value of E S( t , f ) can be estimated un of WGN is set to be zero mean and s 2 = 0.2 with 512 d FA { n } ata points. The results are shown in Table 1, where the th der the assumption that the initialization of x( t ) is the no 2 2 eoretical values ca (n ) are obtained by c distribution t ise only. The detection statistic can be expressed as 2 2 2 n l= max S ( f ) f able according to P{c( n )> ca ( n )} = a , is the degre ( x t ) (13) e of freedom, k is the times of Monte Carlo methods an 2 2 where Sx ( f ) is the time slice of Sx ( t , f ) at the time d the simulation results are obtained by k realizations. t t . The constant-false-alarm-rate(CFAR) criterion is emp loyed in our detector. The probability of detection P is Table 1. Simulation results of c 2 (n ) according to D a used as the performance variable and its value can be nu 2 2 P{c( n )> ca ( n )} = a by k realizations merically expressed as k P= (1/ k ) ( N N ) D i=1 H1|H1 H1 (14) a 0.025 0.05 0.1 0.25 0.75 where k is the time of Monte Carlo realizations, N is t theoretical 7.378 5.991 4.605 2.773 0.575 H1 6.000 4.605 2.751 0.564 he number of the transient signal frames and NH1|H1 is th k = 1000 7.4969 5 8 4 0 at of the correct detection frames. 5.991 4.603 2.762 0.573 In the analysis, a transient sinusoid signal under WGN k = 10000 7.3757 5 6 8 2 environment is used to illustrate the performance of sign al detection in ST domain. The simulation signal is of 50 a 0.90 0.95 0.975 0.99 0.995 0 data points with 150 points pure noise at the beginning theoretical 0.211 0.103 0.051 0.020 0.010 segment and the end segment, respectively. 0.100 0.050 0.021 0.010 Figure 1 shows an example of frame detection at k = 1000 0.2079 6 0 8 8 NH1|H1 . In this simulation, 4 different frames are selected k = 10000 0.2096 0.102 0.050 0.020 0.010 and it is obvious that the frame at t = 175 is the signal fr

Copyright © 2013 SciRes. JCC 4 Paper Title ame while others are noise ones. In Figure 2, the detecto 1 r performance is shown in terms of PD versus SNR curve P s according to three different FA under CFAR criteria. T 0 . 9 he results show that the proposed detector can achieve rel iable detection at the noise level higher than 0dB . 0 . 8

t = 1 0 0 d 0 . 7 t = 1 2 5 P t = 1 7 5 2 0 t = 4 0 0 0 . 6 T h r e s h o l d

S T F T 1 5

e 0 . 5

d S T u t i l p m

A 1 0 0 . 4 - 5 0 5 S N R ( d B )

5 Figure 3. Detection performance of frame detector -2 based on ST and STFT at PFA = 10 .

0 0 . 2 0 . 4 0 . 6 0 . 8 1 N o r m a l i z e d F r e q u e n c y 5. Conclusions In this paper, the characteristic analysis of white Gaussia Figure 1. Frame detection results at four different ind n noise in S transformation domain has been performed. exes under the noise level of SNR= 0dB Accurate distribution of WGN has been analytically deri 1 ved and been further verified through numerical analysis.

0 . 9 This result could be useful in signal detection and ST-bas ed signal processing scheme. 0 . 8

0 . 7 1

0 . 6 0 . 9

0 . 8 d 0 . 5 P

0 . 4 0 . 7

0 . 3 0 . 6 P = 1 0 - 2 F A d 0 . 5 0 . 2 P P = 1 0 - 3 F A 0 . 4 0 . 1 P = 1 0 - 4 F A 0 . 3 0 - 1 5 - 1 0 - 5 0 5 S N R ( d B ) 0 . 2

0 . 1 Figure 2. SNR - PD curve for presented detector unde 0 r three different false alarm probabilities - 1 5 - 1 0 - 5 0 5 S N R ( d B ) Figure 3 depicts the performance comparison of propo Figure 4. Initiation time detection results of transient sed detector with the same frame detector which uses ST sinusoid signals using frame detection FT instead of ST. The threshold corresponding to specifi c PFA of STFT based detector is obtained by Monte Carl 6. Acknowledgments o method of 10000 realizations. The superiority of frame This research was supported in part by a National Natura detection is to provide time-varying information. Thus in l Science Foundation of China (61301286, 61201287) an Figure 4, the initiation time detection result of the transi d a Fundamental Research Funds for the Central Univers ent sinusoid signal is shown under the error tolerance of ities (K50511020022). 1 frame. The simulation result essentially means that this linear time-frequency representation could be potentially used for accurate estimation of the time-varying paramet REFERENCES ers for nonstationary signals. [1] R. G. Stockwell, L. Mansinha and R. P. Lowe. Lo-

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