The Bootstrapped Matched Filter and Its Accuracy

(1) where Pr3 fT 3 < xg is the distribution of T under H0 . If st cos(!t + ) where ! 6= 0 mod 2 , ! 6= 0 mod 2=3 and [0; ), then
IHale Waihona Puke . THEORETICAL RESULTS
Manuscript received June 18, 1999. The associate editor coordinating the review of this manuscript and approving it for publication was Prof. A. Nehorai. H.-T. Ong is with the School of Electrical and Computer Engineering and the Australian Telecommunications Research Institute, Curtin University of Technology, Perth 6845, Western Australia, Australia (e-mail: h.ong@). A. M. Zoubir is with the Australian Telecommunications Research Institute and the School of Electrical and Computer Engineering, Curtin University of Technology, Perth 6845, Western Australia, Australia. Publisher Item Identifier S 1070-9908(00)00272-8.
^ t 0 n01 u W ^ t = Xt 0 ~t = W ^u Theorem 1: Let ^st , W PW 2 0 1 2 3 ~ and ^ = n . Let the bootstrap sample be X W t = t t 3 ^st + Wt3 , t = 0; 1 1 1 ; n 0 1, and T 3 = (^ 3 0 ^)= , where ^ Wt3 , t = 0; 1 1 1 ; n 0 1, are i.i.d. random variables from the ~ t, t = ^ that places a probability mass 1=n on W distribution F 2 = 2 =(s0s). If 0; 1 1 1 ; n 0 1; ^3 = (s 0s)01s0X 3 , and ^ 3 ^ st < 1, t = 0; 1 1 1 ; n 0 1, then Pr3 fT 3 < xg = PrfT < xg + Op (n01)
Abstract—We use Edgeworth expansions to show the higher-order accuracy of the bootstrapped matched filter (MF). Specifically, the bootstrapped MF has a probability of false alarm of order n01 distant from the preset level, where n is the sample size. We also show that the order is n03=2 for sinusoidal signals. Simulation results compare the bootstrapped MF with the classical MF. We briefly discuss the bootstrapping approach when the noise variance is unknown and when signal parameters are unknown. Index Terms—Bootstrap, bootstrapped matched filter, Edgeworth expansions, higher-order accuracy.
IEEE SIGNAL PROCESSING LETTERS, VOL. 7, NO. 1, JANUARY 2000
11
The Bootstrapped Matched Filter and Its Accuracy
Hwa-Tung Ong, Member, IEEE, and Abdelhak M. Zoubir, Senior Member, IEEE
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=
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Pr3 fT 3 < xg = Pr fT < xg + Op (n03=2)
(2)
^. where Pr3 means probability with respect to F Proof: Under the condition of finite variance and higherorder cumulants, k , k = 3; 4; 1 1 1, the Edgeworth expansion (3), shown at the bottom of the next page, applies (see Lemma 1, Appendix), where is the standard Gaussian density; Hr (x) is the Hermite polynomial degree r ; k P = k =k , k = P k of 0 1 k 2 0 1 3 3; 4; 1 1 1; k = n st =s , and s = n s2 t . Pr3 fT < k xg is given by (3) with k replaced by ^k = ^k = ^ where ^k, ^ . The difference between k = 3; 4; 1 1 1, are the cumulants of F Pr fT < xg and Pr3 fT 3 < xg is Pr3 fT 3 < xg 0 PrfT < xg 3 0 (^ 0 ) 3 ) 3 p H2(x) + 4 4 4 H3(x) = 0(x) (^ 24n 6 n 2 ) 2 0 (^ 2 + 3 72n3 3 H5 (x) + Op (n03=2) : (4) ^k 0 k = Op(n01=2 ), k = 3; 4; 1 1 1. It can be shown [4] that Also, it can be shown that if st < 1, t = 0; 1 1 1 ; n 0 1, then 3 = O(1) and 4 = O(1). Thus, the difference in (4) is seen to be Op(n01 ) and (1) follows. If st = cos (!t + ), then the stronger result 3 = O(n01 ) holds while 4 remains O(1) (see Lemma 2, Appendix). The first term in (4) becomes Op (n02 ) while the second term remains Op (n03=2), and (2) follows. From (3), it can be seen that 8(x) = Pr fT < xg + O(n01=2), when st < 1, and 8(x) = Pr fT < xg + O(n01 ), when st is sinusoidal. The bootstrap distribution Pr3 fT 3 < xg approximates Pr fT < xg by an order n01=2 better than 8. If the threshold is computed from Pr3 fT 3 < xg, then the actual PFA is + Op (n01) for st < 1 and + Op(n03=2 )
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I. INTRODUCTION IGNAL detection is used in radar and other engineering fields. Under the Neyman–Pearson criterion, the probability of false alarm (PFA) is constrained to not exceed a specific level and the probability of detection is maximized under this constraint. However, this requires the distribution of the noise to be known, which is not practical in many applications. If the noise distribution is unknown, methods based on the Gaussian assumption may still work but are no longer optimal. The PFA may be much higher than the preset level. We provide support for the use of the bootstrap [1] to improve the threshold so that the PFA is closer to the level. Let the discrete-time model be described by Xt = st + Wt , t = 0; 61; 1 1 1, where fst g is a bounded, deterministic signal sequence to be detected, and fWtg is an independent and identically-distributed (i.i.d.) noise process with zero mean and variance 2 . The signal detection problem is formulated as a test of the hypotheses H0 : = 0 against H1 : > 0, given observations xt = st + wt , t = 0; 1 1 1 ; n 0 1, or, in vector s + w . Under the Neyman–Pearson criterion, the form, x = s matched filter [2] is optimal for Gaussian noise. For a given level , H0 is rejected if T > , where T = ^= ^ = (s0s )01s 0x , ^, 2 2 0 0 1 ^ = =(s s), = 8 (10 ), and 8 is the standard Gaussian distribution. The bootstrapped MF [3] carries out the same test but with calculated from the bootstrap distribution of T 3 = (^ 3 0 ^)= ^3 instead of from 8. The bootstrap distribution is based on the data x itself and does not need the noise distribution to be known.