Improved Adaptive Time Delay Estimation Algorithm Based on Fourth-order Cumulants

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1 Available online Journal of hemical and Pharmaceutical Reearch, 016, 8(5): Reearch Article ISSN : ODEN(USA) : JPR5 Imroved Adative Time Delay Etimation Algorithm Baed on Fourth-order umulant Ling Tang and Yumei hen ollege of Automation and Electronic Information, Sichuan Univerity of Science & Engineering, Zigong 64000, hina ABSTRAT Time-delay etimation i widely ued in onar and radar alication. In order to etimate time delay accurately in variable ignal to noie ratio(snr), an imroved algorithm of adative time delay etimation() baed on fourth-order cumulant ha been rooed in thi aer. The ingle adative filter in the conventional model i earated into two adative ubunit connected in cacade, one i to track the delay, the other to track the SNR, o a to obtain the accurate ignal time delay etimation in the low and variable SNR. The theoretical analyi and imulation reult confirm the validity and feaibility of the rooed algorithm. Keyword: fourth-order cumulant, time delay, adative filter, variable SNR INTRODUTION Time Delay Etimation() a an active reearch in ignal roceing i widely ued in the field of radar, onar, communication and o on. The common method are hae method, double ectrum method, correlation method, adative filter arameter model method and o on. With the continuou develoment of ignal roceing method, a variety of algorithm are introduced into time delay etimation, which imrove the reciion and convergence, reduce the comutation. Becaue of blind Gau roerty high order accumulation i widely ued in time delay etimation. Liu Ying et al.[1] rooed an adative time delay etimation algorithm baed on four-order cumulant(fo-lms), which can effectively ure the influence of correlated Gau noie, but in the cae of low SNR the etimated reult are not atifactory. Aiming at thi roblem, a new algorithm of time delay etimation algorithm FO-E i rooed in the literature[], which can ure the imact of related or not related to Gau noie, and obtain the accurate time delay etimation of non Gau ignal in lower SNR. However when SNR change, the erformance of the ytem i greatly reduced. In thi aer, an imroved adative time delay etimation algorithm i rooed. The ingle adative filter i earated into two adative ubunit connected in cacade, one i to track the delay, the other to track the SNR, o a to imrove the erformance in the variable SNR. MATHEMATIAL MODEL onider one ignal beamed from a ditant ource, with interference and noie in the channel. Thu the ignal arameter model received on two atially earated enor[] i: x ( ( + w1 ( y ( a( k D) + w ( = (1) = () 889

2 Ling Tang and Yumei hen J. hem. Pharm. Re., 016, 8(5): In the formula ( i non Gau and zero mean tationary random ignal, ( k D) i the time delay of (, a i the fading factor that atified a (0,1], uually a = 1. w1 ( and w ( are the correlative zero mean Gau noie which i indeendent with (. By convolution theory: n= k D) = in n D) ( k n) ( () Where in v) = in(π v), then relace with a large oitive integer ( > D πv truncation error, we can get the amling value of the function. ), and ignoring the Inert the formula () into (): ( = a in ( k i) + n ( k = a in [ k i) n1 ( k i) ] + n ( y ) (4) The uroe of time delay etimation i to ure the Gau noie effectively by uing the limited obervation ignal ) k and y (, o a to etimate the time delay D. ALGORITHM ANALYSIS For tationary random roce with zero mean ( defined a follow: ( τ,0,0) cum[ k + τ ), ] x and y (, the FO of the random variable can be = (5) Define cro four order cumulant(fo) of the random variable and y( ( τ,0,0) cum[ y( k + τ ), ] = (6) Inert the formula (4) into (6), we get ( τ,0,0) = cum a in [ k i + τ ) w1 ( k i + τ )] + w ( k + τ ), (7) According to the roertie that the cumulant can be added and the FO of Gau noie i zero, the equation(7) can be orted out: [ ( k i τ ), )] ( τ,0,0) = a in cum x + k a: (8) That i: ( τ,0,0) = ( τ,0,0) (9) (,0,0) = a in ( τ i,0,0) = a in ( τ i,0,0) τ (10) Aarently the Gau noie contained in ( τ,0,0), ( τ,0,0) i ureed. The formula (10) can be exreed a a vector: = a4 S (11) xy x 890

3 Ling Tang and Yumei hen J. hem. Pharm. Re., 016, 8(5): Where, a i the contant, and S = [in D),in + 1 D), L,in D)] xy [ (,0,0), ( + 1,0,0), L, (,0,0 ] T = ) 4x = (0,0,0) (1,0,0) M (,0,0) ( 1,0,0) (0,0,0) M ( 1,0,0) L L O L T (,0,0) ( + 1,0,0) M (0,0,0) In the ue of adative time delay etimation method to get the time delay, the otimal weight coefficient hould be [4]: a σ T [in, i =, + 1, L,0, L, 1, ] σ + σ w (1) Baed on the maximum value of the weight coefficient, i obtained. However thi method can get the maximum error in the cae of low noie, which lead to inaccurate etimation. In thi cae, the original adative filter coefficient can be decouled into two art of g and in c, with uing in filter, at the ame time adding the gain g for the iteration to track the factor ( a σ ) ( σ + σ ) w o a to obtain the otimal weight coefficient. In thi way the etimation can be obtained directly without the incorrect maximum value. The ytem model i howed in Figure 1. y( FO calculation FO calculation (τ,0,0) g (τ,0,0) FIR filter with inc amling xzxx (τ,0,0) + e( + - Figure 1: Sytem model (τ,0,0 In Figure 1 ignal ) (τ,0,0 ignal ) : xzxx xzxx after an adative ower factor g ) and an adative FIR filter into the outut ) ) ( τ,0,0) = g in ( τ i,0,0) Adative error function i defined a: If = [ ( τ,0,0) ( τ,0,0)] xzxx J (14) τ =, + 1, L, τ, the error function i (1) 891

4 Ling Tang and Yumei hen J. hem. Pharm. Re., 016, 8(5): ) ) J = g in ( τ i,0,0) ( τ,0,0) τ = ) ) = ( g S ) ( g S ) T 4 x xy 4 x xy (15) Make the gradient of the error function to the arameter D ) and g ) are zero, the otimal arameter vector can be olved. The iterative roce of the otimal arameter vector i realized by uing the teeet decent method in the adative roce[5]. Each time the value of the current vector i changed by a negative gradient vector: ) ) µ d J D( k + 1) = D( ) ) g( D (16) µ d and g tability and eed. ) ) J g k + 1) = g( µ g ) g ( (17) µ i reectively the convergence factor of D ) ( and g ) (, which i ued to adjut the adative We can ee that the initial value of delay time and gain in the iteration are needed, initial value (0) [6] mut D ˆ (0), g ) (0) atifie 1.45 D D delay time [7]. hould be between 0 and 1, o a to be converged to the real SIMULATIONS AND ANALYSIS The ignal ource i a tationary ignal with zero mean value generated by a non Gau random ignal generator. The real time delay i D=, initial time delay etimation D=, fading factor a i 1. The oitive order of the filter i =10, the amling oint are 000, and the noie i the correlated Gau white noie. The erformance of FO-E, FO-LMS and the imroved algorithm are comared uing MATLAB in the imulation. Exeriment 1 Time delay etimation under different SNR Aume SNR i 5dB or -5dB, the reult of different algorithm are hown in Figure. In the high SNR (SNR=5dB), three algorithm can quickly converge to the true value of time delay. But with SNR decreaing, the detection ability of the algorithm have been reduced. FO-LMS can not accurately etimate the time delay, comared with FO-E algorithm the rooed method obtain more accurate etimation ˆD loo loo loo (a) FO-LMS(SNR=5dB) (b) FO-E(SNR=5dB) (c) The rooed algorithm(snr=5db) 89

5 Ling Tang and Yumei hen J. hem. Pharm. Re., 016, 8(5): loo loo loo (d) FO-LMS(SNR=-5dB) (e) FO-E(SNR=-5dB) (f) The rooed algorithm(snr=-5db) Figure : Performance comarion under correlated noie Exeriment Time delay etimation invariable SNR When SNR reduced from 0dB to -10dB, comare the delay etimation. The firt 1000 data with SNR=0dB, the lat 1000 data with SNR=-10dB.The otimal olution of the gain factor in FO-ETDGE method i g * = 0. 09, when SNR=-10dB the otimal olution i g * = In FO-E method, g i fixed to 1. In Figure we comare their erformance in variable SNR. Obviouly, FO-LMS algorithm get incorrect, the effect of FO EE i not good, which i becaue that any change in the filter arameter i conidered to be a change in the time delay, o can not be tracked in time. In contrat, the effect of the imroved algorithm i the bet, even in the cae of a large change to SNR, it can alo be converged to the etimated value loo loo (a) FO-LMS (b) FO-E (c) The rooed algorithm Figure : in variable SNR ONLUSION loo In thi aer an imroved adative time delay etimation algorithm baed on four order cumulant i rooed. The ingle adative filter i earated into two adative ubunit connected in cacade, which making the adative roce of time delay and ignal to noie ratio to be earated, avoiding the oor erformance of delay etimation in variable SNR. Simulation reult demontrate the effectivene of the rooed algorithm. Acknowledgment Thi work wa uorted by roject of Sichuan Provincial Deartment of Education(1ZB018) and roject of Artificial Intelligence Key Laboratory Of Sichuan Province(01RYY0). 89

6 Ling Tang and Yumei hen J. hem. Pharm. Re., 016, 8(5): REFERENES [1] LIU Ying, WANG Shu-xun, WANG Ben-ing, Journal of Sytem Simulation, 00, 14(6), []LI ong-ying, WANG Jian-ying, YIN Zhong-ke, ZHANG Jiang-li, Journal of the hina railway ociety, 006, 8(6), [] ZHANG Duan-jin ZHANG Zhong-hua GUO Jian-jun ZHANG De-jing, Journal Of Zhengzhou Univerity(Engineeringcience), 010,1(1), [4]H..So and P.. hing. A novel contrained time delay etimator. Proc. Of ISP 9. Beijing hina. 199, [5]XIONG Qiu-ben, WANG Jiang, YANG Jing-hu, Electronic Warfare Technology, 011, 6(), [6]Faramarz Fekri, Mina Sartiii and Reell M. Merereau. IEEE Tranaction on Signal Proceing. May 005, 5(5) [7]H..So and P..hing. Performance analyi of ETDGE an efficient and unbiaed TDOA etimator. IEEE Proc.-Radar Navig. December 1998, 145(6),