Noncoherent sequential code acquisition based on ... - CiteSeerX
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Signal Processing 85 (2005) 1563–1571 www.elsevier.com/locate/sigpro
Noncoherent sequential code acquisition based on simplified maximum likelihood ratios Hyoungmoon Kwona, Iickho Songa,, Seokho Yoonb, Jumi Leea, Sun Yong Kimc, Bo-Hyun Chungd a
Department of Electrical Engineering and Computer Science, Korea Advanced Institute of Science and Technology, 373-1 Guseong Dong, Yuseong Gu, Daejeon 305-701, Republic of Korea b School of Information and Communication Engineering, Sungkyunkwan University, 300 Cheoncheon Dong, Jangan Gu, Suweon, Gyeonggi 440-746, Republic of Korea c Department of Electronic Engineering, Konkuk University, 1 Hwayang Dong, Gwangjin Gu, Seoul 143-701, Republic of Korea d Mathematics Section, College of Science and Technology, Hongik University, 300 Shinan Ri, Jochiwon, Yeongi Gun, Chungnam 339-701, Republic of Korea Received 18 June 2004; received in revised form 16 November 2004
Abstract In this paper, we consider the noncoherent code acquisition problem using sequential schemes. To alleviate the computational complexity of the maximum likelihood method, simplified schemes are proposed and analyzed for the truncated sequential probability ratio test. The performance of the simplified and original schemes are compared in additive white Gaussian noise and slowly varying fading channels. Numerical results show that the proposed schemes have essentially the same performance as the original schemes while allowing simpler structures. r 2005 Elsevier B.V. All rights reserved. Keywords: Sequential code acquisition; Maximum likelihood estimate; Bessel function
1. Introduction In direct-sequence–spread-spectrum (DS/SS) systems, the receiver is to generate a pseudonoise Corresponding author.
E-mail addresses: [email protected] (H. Kwon), [email protected] (I. Song), [email protected] (S. Yoon), [email protected] (J. Lee), [email protected] (S.Y. Kim), [email protected] (B.-H. Chung).
(PN) code which is a replica of and is in synchronization with the incoming PN code. Code synchronization is an indispensable process in any SS system in despreading and reliable data demodulation at the receiver. In the past two decades, a vast volume of research on PN code acquisitions has been carried out e.g., [1–5]. Among the various methods, sequential schemes have the potential to achieve the best performance. A sequential scheme has two
0165-1684/$ - see front matter r 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.sigpro.2005.02.009
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H. Kwon et al. / Signal Processing 85 (2005) 1563–1571
sets of thresholds in the decision process, one set for accepting phase alignment and the other for rejecting phase alignment and continuously checking alignment or nonalignment. At any time, if alignment or nonalignment is apparent from the integrator output, the inspection process terminates and the corresponding decision is made. Otherwise, it continues the test. It appears, however, that the sequential schemes are the least studied because of the difficulty associated with the design and analysis of sequential schemes. Numerical methods, computer simulations, and/or approximations are used for the performance evaluation of sequential schemes [1–3]. In [1], by viewing the truncated sequential probability ratio test (TSPRT) as a mixture of a sequential probability ratio test (SPRT) and a fixed-sample-size (FSS) test, a novel design procedure for the TSPRT is developed. In [2], a noncoherent sequential acquisition based on the noncoherent I/Q detector with continuous integration is investigated for additive white Gaussian noise (AWGN) and Rayleigh fading channels. In addition to the difficulty in the design and analysis, most of the sequential acquisition schemes using maximum likelihood estimate impose difficulty in hardware implementation because of a high level of computational complexity [3].
during acquisition, the received signal wðtÞ can be expressed as wðtÞ ¼ A0 ccðt þ iDT c Þ cosðo0 t þ yÞ þ nðtÞ.
(1)
In (1), A0 is the signal amplitude, c is the fading random variable, cðtÞ is the PN signal, i is an integral initial phase number, D is the advancing step size, T c is the chip duration, o0 is the carrier angular frequency, y is the random phase distributed uniformly over ½0; 2pÞ; and nðtÞ represents the AWGN with one-sided power spectral density N 0 : The fading variable c is assumed to have the Rician probability density function (pdf): pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 f c ðcÞ ¼ 2cð1 þ rÞerc ð1þrÞ I 0 2c rð1 þ rÞ , cX0,
ð2Þ
where r ¼ s2 =ð2s2 Þ is the ratio of the power s2 in the direct component and the power 2s2 in the diffused component with the constraint s2 þ 2s2 ¼ R 2p 1; and I 0 ðxÞ ¼ ð1=2pÞ 0 ex cos n dn is the zeroorder modified Bessel function of the first kind. The case r ! 1 clearly represents no fading or a constant value of c in (1). In addition, we have Rayleigh fading when r ¼ 0; in which case the pdf 2 (2) becomes f c ðcÞ ¼ 2cec for cX0: It is straightforward to see that the test statistic Y n ¼ X 2i;n þ X 2q;n
(3)
is a noncentral w2 random variable with the pdf [2]:
2. System model 2.1. Output statistics of noncoherent correlators Fig. 1 shows a block diagram of the acquisition system considered. The transmitted signal is affected by slowly varying fading in addition to AWGN. Assuming that no data are modulated
f Y n ðyn Þ ð1 þ rÞ 2s2n ½ð1 þ rÞ þ ln =2s2n ð1 þ rÞðyn =2s2n Þ þ rðln =2s2n Þ
exp ð1 þ rÞ þ ln =2s2n
¼
Fig. 1. A block diagram of the noncoherent correlator receiver.
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I 0
2
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi! rð1 þ rÞðln =2s2n Þðyn =2s2n Þ , ð1 þ rÞ þ ln =2s2n
and
yn X0,
ð4Þ
where X i;n and X q;n are the outputs (ignoring the double-frequency terms) of the in-phase and quadrature branches of the noncoherent correlator and ln ¼ A20 T 2c S2n =4 is a measure of the signal energy contained in Y n : The cumulative distribution function (cdf) of Y n is sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rðln =s2n Þ ; F Y n ðyn Þ ¼ 1 Q ð1 þ rÞ þ ðln =2s2n Þ sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi! ð1 þ rÞðyn =s2n Þ ; yn X0, ð1 þ rÞ þ ðln =2s2n Þ
ð5Þ
where Q( ; ) is the Marcum’s Q function given by Z Qða; bÞ ¼ b
1
ðn2 þ a2 Þ n exp I 0 ðanÞ dn [6]. 2
For the normalized random variable Z n ¼ Y n =ð2s2n Þ; let en ¼ ln =ð2s2n Þ; zn ¼ yn =ð2s2n Þ; and qn ¼ ð1 þ rÞ þ en : Then, from (4) and (5), the pdf and cdf of Z n are
f Zn ðzn Þ ¼
ð1 þ rÞ ð1 þ rÞzn þ ren exp qn qn pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi! 2 rð1 þ rÞen zn
I 0 ; zn X0 qn
ð6Þ
and sffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi! 2ren 2ð1 þ rÞzn F Zn ðzn Þ ¼ 1 Q ; ; qn qn
zn X0,
respectively. When r ¼ 0 (the Rayleigh fading case), we have qn ¼ 1 þ e n , f Zn ðzn Þ ¼ exp
zn 1 ,
1 þ en 1 þ en
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sffiffiffiffiffiffiffiffiffiffiffiffiffi! 2zn F Zn ðzn Þ ¼ 1 Q 0; 1 þ en
zn ¼ 1 exp 1 þ en
R1 since I 0 ð0Þ ¼ 1 and Qð0; bÞ ¼ b n expððn2 = 2ÞÞ dn ¼ expððb2 =2ÞÞ: On the other hand, when there is no fading (r ! 1), we have ð1 þ rÞ=qn ! 1; r=qn ! 1; and rð1 þ rÞen zn =q2n ! en zn : Thus, Zn is a w2 random variable with pdf: pffiffiffiffiffiffiffiffi (7) f Zn ðzn Þ ¼ eðzn þen Þ I 0 2 en zn ; zn X0, pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi and cdf F Zn ðzn Þ ¼ 1 Qð 2en ; 2zn Þ; zn X0: It is noteworthy that en is a measure of the signal-to-noise ratio (half of the effective signal-tonoise ratio) since the variance of Y n can be written as (
2 A0 4 2 cT c Sn cos y VarðY n Þ ¼ 4sn þ 4sn 2
2 ) A0 þ cT c S n sin y 2 ¼ 4s4n þ 4s2n ln c2 ,
ð8Þ
of which the first and second terms on the righthand side are the noise and signal power, respectively. Consequently, the effective signalto-noise ratio without fading ðc ¼ 1Þ is 4s2n ln =4s4n ¼ ln =s2n : Comparing (6) and (7), we can observe that qn is a measure of the effect of fading relative to the signal-to-noise ratio en (which would eventually reduce the effective signal-to-noise ratio). 2.2. Maximum likelihood estimate Let the phase of the local PN signal be ðj þ gÞDT c ; where j is an integer and g 2 ð12; 12 is the residual code phase offset. Based on Z n ; the sequential decision processor decides if (1) the phases of the incoming and local PN signals are aligned to be within DT c =2 of each other (i.e., j ¼ i), (2) they differ by at least one chip (i.e., jðj þ gÞ ijDT c XT c ), or (3) neither of the first two. Here, without loss of generality, we assume
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i ¼ 0 so that the phase difference becomes j þ g: Equivalently, the decision processor attempts to test the following hypotheses: 1 , D 1 H1 ðalignmentÞ: jj þ gjp , 2 1 1 H2 ðdecision deferredÞ: ojj þ gjo . 2 D H0 ðnon-alignmentÞ: jj þ gjX
ð9Þ
Since g 2 ð 12 ; 12 ; both H0 and H1 are composite hypotheses and the value of ln generally differs depending on the value of g and the hypothesis. Thus, assuming the worst-case scenario, let ln;0 denote the largest value of ln under H0 and ln;1 the smallest value of ln under H1 : clearly, ln;1 4ln;0 : Note that ln;0 and ln;1 are the worst-case values. We use one additional subscript 0 or 1 to denote the parameters en and qn under H0 or H1 ; respectively: for example, en;0 denotes the parameter en under H0 : Then, the likelihood ratio can be expressed as Ln ðzn Þ qn;0 ð1 þ rÞzn þ ren;1 ð1 þ rÞzn þ ren;0 ¼ exp þ qn;1 qn;1 qn;0 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi!, pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi! 2 rð1 þ rÞen;1 zn 2 rð1 þ rÞen;0 zn I0 .
I 0 qn;1 qn;0 ð10Þ When there is no fading (r ! 1), (10) reduces to Ln ðzn Þ ¼ e
pffiffiffiffiffiffiffiffiffiffiffi en;1 zn Þ pffiffiffiffiffiffiffiffiffiffiffi . I 0 ð2 en;0 zn Þ
ðen;0 en;1 Þ I 0 ð2
(11)
It is noteworthy that, due to the computational complexities of the Bessel and exponential functions, (10) and (11) are quite difficult to implement for real-time processing [7]. Specifically, for the Bessel function, a great quantity of complicated computations is required. In addition, since there is no closed form expression for the Bessel function, one must resort to numerical integration techniques to provide quantitative answers for a given argument [8].
3. Proposed schemes The modified Bessel function I 0 ðxÞ of order zero can be approximated as [6] 2k 1 X 1 1 x I 0 ðxÞ ¼ 2 2 k¼0 ðk!Þ 8 1 2 > > 0pjxj51; > > p ffiffiffiffiffiffiffiffi ; jxjb1: > : 2px With the two approximated expressions in (12), we propose simplifications of (10) and (11) for Ln ðzn Þ in fading and AWGN channels. 3.1. AWGN (no fading) channel Let us first focus on the ratio containing I 0 ð Þ in (11): pffiffiffiffiffiffiffiffiffiffiffi I 0 ð2 en;1 zn Þ (13) pffiffiffiffiffiffiffiffiffiffiffi . I 0 ð2 en;0 zn Þ Approximation 1 for AWGN. Using I 0 ðxÞ pffiffiffiffiffiffiffiffi ex = 2px; (13) can be written as rffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffi pffiffiffi pffiffiffiffiffi pffiffiffiffiffi I 0 ð2 en;1 zn Þ en;0 2 z ð e e Þ . (14) pffiffiffiffiffiffiffiffiffiffiffi e n n;1 n;0 4 I 0 ð2 en;0 zn Þ en;1 Then, the likelihood ratio (11) becomes rffiffiffiffiffiffiffi pffiffiffi pffiffiffiffiffi pffiffiffiffiffi en;0 Ln ðzn Þ efen;0 en;1 þ2 zn ð en;1 en;0 Þg 4 en;1 e pffiffiffiffiffi ¼ zn ,
ð15Þ
e
where ‘¼’ means ‘equals equivalently to’: that is, the likelihood ratios using the quantities before e and after ¼ result in the equivalent detector. Approximation 2 for AWGN. We apply I 0 ðxÞ pffiffiffiffiffiffiffiffi ex = 2px and I 0 ðxÞ 1 þ 14 x2 to the numerator and denominator of (13), respectively, and then we can rewrite Ln ðzn Þ in (11) as pffiffiffiffiffiffiffiffi eðen;0 en;1 þ2 en;1 zn Þ ffi Ln ðzn Þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffi 4p en;1 zn ð1 þ en;0 zn Þ 1 e pffiffiffiffiffiffiffiffiffiffiffi ¼ 2 en;1 zn ln zn lnð1 þ en;0 zn Þ. ð16Þ 4
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Approximation 3 for AWGN. Applying I 0 ðxÞ 1 þ 14x2 to (13), it is easy to see that (11) can be expressed as Ln ðzn Þ eðen;0 en;1 Þ
e
¼
1 þ en;1 zn 1 þ en;0 zn
1 þ en;1 zn . 1 þ en;0 zn
ð17Þ
Note that it is meaningless pffiffiffiffiffiffiffiffi to apply I 0 ðxÞ 1 þ 14x2 and I 0 ðxÞ ex = 2px to the numerator and denominator of (13), respectively, since en;1 4en;0 : 3.2. Fading channel We now focus on the ratio of the zeroth-order modified Bessel functions pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi!, pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi! 2 rð1 þ rÞen;1 zn 2 rð1 þ rÞen;0 zn I0 I0 qn;1 qn;0 (18) in the maximum likelihood ratio (10) for fading channels. Approximation 1 for fading. With I 0 ðxÞ pffiffiffiffiffiffiffiffi ex = 2px and some manipulations as we have done for the AWGN channel case, the likelihood ratio (10) can be written as rffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi qn;1 en;0 qn;0 en;1 pffiffiffiffiffi r Ln ðzn Þ zn 2 zn . qn;1 qn;0 1þr (19) Approximation 2 p forffiffiffiffiffiffiffiffi fading. Using the approximations I 0 ðxÞ ex = 2px and I 0 ðxÞ 1 þ 14 x2 in the numerator and denominator of (18), respectively, the likelihood ratio (10) becomes
1 1 Ln ðzn Þ ð1 þ rÞ zn qn;0 qn;1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 rð1 þ rÞen;1 pffiffiffiffiffi þ zn qn;1 1 ln zn ln½q2n;0 þ rð1 þ rÞen;0 zn . ð20Þ 4 Approximation 3 for fading. With thepapproximaffiffiffiffiffiffiffiffi tions I 0 ðxÞ 1 þ 14 x2 and I 0 ðxÞ ex = 2px in the numerator and denominator of (18), respectively,
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we can rewrite (10) as
1 1 zn Ln ðzn Þ ð1 þ rÞ qn;0 qn;1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 rð1 þ rÞen;0 pffiffiffiffiffi zn qn;0 1 þ ln zn þ ln½q2n;1 þ rð1 þ rÞen;1 zn . ð21Þ 4 Approximation 4 for fading. Using I 0 ðxÞ 1 þ 14x2 ; we obtain the likelihood ratio Ln ðyn Þ in (10) as
1 1 Ln ðzn Þ ð1 þ rÞ zn qn;0 qn;1 " # q2n;1 þ rð1 þ rÞen;1 zn þ ln 2 . ð22Þ qn;0 þ rð1 þ rÞen;0 zn Unlike in the case of AWGN channel, we have also considered the papproximations I 0 ðxÞ 1 þ ffiffiffiffiffiffiffiffi 1 2 x x and I ðxÞ e = 2px to the numerator and 0 4 denominator of (18), respectively, in the fading channel. This is due to the fact that we pffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffi could have both e z =q 4 en;0 zn =qn;0 and pffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffi n;1 n n;1 en;1 zn =qn;1 o en;0 zn =qn;0 depending on the specific values of the four parameters en;0 ; en;1 ; qn;0 ; and qn;1 :
4. Performance evaluation We have so far obtained simplifications to the maximum likelihood ratios using the two approximations in (12). We now investigate the performance of the proposed and original acquisition schemes with the TSPRT. 4.1. Decision processor with TSPRT In the SPRT scheme, we compare the likelihood ratio Ln ðzn Þ with two thresholds A and B, where A4B40; for n ¼ 1; 2; 3; . . . ; until one of the thresholds is reached. When the detected value corresponds to neither H0 nor H1 ; the SPRT may not end for a long period of time. To avoid such an excessive delay associated with the SPRT, the TSPRT [2] puts an upper bound n^ on the
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test length:
ASN function
350
No approximation Approximation 1 for AWGN Approximation 2 for AWGN Approximation 3 for AWGN
say H1 ; 300
say H0 ; continue; H1 ; H0 :
ð23Þ
250
ASN
8 XA^ > > < ^ Ln ðzn Þ pB^ TSPRT : if non; > > : otherwise ( X^t say ^ Ln^ ðzn^ Þ if n ¼ n; o^t say
200
SNR=-10dB
Therefore, the test is truncated at n ¼ n^ if it has not been previously terminated.
=βdTSPRT=0.005 αTSPRT d
150
p0=p1=0.5
4.2. Simulation results and discussions
100
(a)
(1) PN code sequence of 1023 chips generated from the primitive polynomial 1 þ x2 þ x5 þ x6 þ x10 ; (2) chip SNR ¼ 10 dB; (3) advancing step size D ¼ 1=2; (4) residual code phase offset g ¼ 1=2: Fig. 2 shows the performance of the original and proposed schemes for AWGN, obtained from (11), (15)–(17). In this figure, the TSPRT is designed with ad ¼ bd ¼ 0:005 and p0 ¼ p1 ¼ 0:5; where p0 and p1 are the design constants of the TSPRT [2]. If p0 ¼ p1 ¼ 0; the TSPRT becomes the FSS test and if p0 ¼ p1 ¼ 1; the TSPRT becomes the SPRT. For values of p0 and p1 in between, the TSPRT can
1
1.5
2
|j+γ| Power function
1
No approximation Approximation 1 for AWGN Approximation 2 for AWGN Approximation 3 for AWGN
0.9 0.8
Power function
Let us employ the average sample number (ASN) and power function as the performance measures. The ASN is the average number of chips needed for a test to terminate. The power function is the probability of accepting H1 as a function of the absolute phase difference jj þ gj: If jj þ gj is consistent with the hypothesis H1 ; then the power function value is the detection probability. Similarly, if jj þ gj is consistent with the hypothesis H0 ; the power function value is the false alarm probability. Briefly, the ASN represents the price we have to pay, in terms of the number of observations required for the sequential test, and the power function describes how well the sequential test procedure achieves its objective of making correct decisions. In evaluating the performance, we use the following parameters:
0.5
0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0.5
(b)
1
1.5
2
|j+γ|
Fig. 2. ASN and power functions of the TSPRT in AWGN channel. (a) ASN function and (b) Power function.
be viewed as a mixture of the SPRT and FSS test. The hypotheses H1 and H0 are represented by jj þ gj ¼ 0:5 and jj þ gj ¼ 2:0; respectively, in the figure. From this figure, we can see that Approximation 2 (or the approximation (16)) is better than the other approximations and has some performance improvements over the original scheme at the cost of false alarm probability. Compared with the original scheme, the approximation (16) has no larger ASN at all phase difference. Approximation (16) also exhibits a similarly decreasing characteristic of power function compared with the original scheme. Note that the approximation (16) has a
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The ratio of the zeroth– order modified Bessel functions
The ratio of the zeroth – order modified Bessel functions
1070 1060 1050
O
SNR=0dB
1040 1030
No Approximation (SNR=0dB) Approximation 1 for AWGN (SNR=0dB) Approximation 2 for AWGN (SNR=0dB) Approximation 3 for AWGN (SNR=0dB) No Approximation (SNR=20dB) Approximation 1 for AWGN (SNR=20dB) Approximation 2 for AWGN (SNR=20dB) Approximation 3 for AWGN (SNR=20dB)
O
1020 1010
O
O
100 SNR=20dB 10-10
0
20
40
60
: SNR=0dB : SNR=20dB
80
100
zn pffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffi Fig. 3. The ratio I 0 ð2 en;1 zn Þ=I 0 ð2 en;0 zn Þ with the proposed approximations for the AWGN channel.
small loss in false alarm probability in comparison with the original scheme. To account for these results, let us consider pffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffi Fig. 3 showing the ratio I 0 ð2 en;1 zn Þ=I 0 ð2 en;0 zn Þ for the AWGN channel. In this figure, defining the signal-to-noise ratio (SNR) per chip as A20 T c =ð2N 0 Þ; we have assumed the approximations en;1 ¼ 1n ðSNRÞS 2n;1 nðSNRÞð1 jgjDÞ2 and en;0 ¼ 2 1 n ðSNRÞS n;0 SNR which have been proposed and proved very effective in [2]. In this figure, n ¼ 100; g ¼ 12 (the worst case value of the residual code phase offset), and D ¼ 12 are used. Since it has been found empirically that the general tendency does not depend on the value of n, we have chosen n ¼ 100: We can see that all the proposed approximations exhibit a similarly increasing characteristic in comparison with the original scheme at the lower value 20 dB of SNR. At the higher value of SNR, the original scheme, Approximation 1, and Approximation 2 show an increasing characteristic, while Approximation 3 does not follow the original scheme, being stationary at a relatively small value.
Note that the ratios in Approximation 2 grow up faster than the others regardless of the SNR. This result means that Approximation 2 generates a larger value of Ln ðzn Þ than the others for a given value of zn : Thus, Ln ðzn Þ of Approximation 2 is more likely to be larger than a certain threshold, and would consequently accept H1 more frequently. Another consequence is that using Approximation 2 may produce some adverse effects when Ln ðzn Þ is desired to be smaller than a certain threshold (that is, when it is desired to accept H0 ). In short, we find that approximation (16) yields a higher value than the original ratio (11), and thus is more likely to be larger than the upper threshold. Consequently, the ASN decreases and the power function value increases when the absolute phase difference is within a half of the chip duration ðjj þ gjo1:25Þ: On the other hand, when the absolute phase difference is larger than a half of the chip duration ðjj þ gj41:25Þ; the ASN still decreases and the power function tends to be higher (than that for the original scheme) since the event fLn ðzn Þ4upper thresholdg occurs more
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ASN function 700 SNR=10dB, r=10 TSPRT
αd
600
TSPRT
=βd
=0.01
p0=p1=0.5
ASN
500
400
300 No approximation Approximation 1 for Fading Approximation 2 for Fading Approximation 3 for Fading Approximation 4 for Fading
200
100 0.5
1
1.5
2
|j+γ|
(a)
Power function 1 No approximation Approximation 1 for Fading Approximation 2 for Fading Approximation 3 for Fading Approximation 4 for Fading
0.9 0.8
Power function
0.7 0.6 0.5 0.4
ability when jj þ gjo1:25: Similarly, approximation (19) (Approximation 1) has the best performance among the approximations and improved ASN characteristic over the original scheme at the expense of false alarm probability when jj þ gj41:25 in fading channels. These observations can be similarly explained with a rationale similar to that described for AWGN channels. Fig. 5 shows the ratio (18) with the proposed approximations for the fading channel when r ¼ 10: When the SNR is low, it is observed that the ratios of I 0 ð Þ in the original scheme, Approximations 1 and 2 have a similarly increasing characteristic, but those of Approximations 3 and 4 do not have such a characteristic. At the higher value of SNR, the ratios in the original scheme, Approximations 1 and 2 still show a similar pattern, although the ratios in Approximation 2 become somewhat larger than the others. On the other hand, the ratios of Approximations 3 and 4 again do not follow that of the original scheme. We can thus anticipate that Approximations 1 and 2 will have a better performance than Approximations 3 and 4, as is confirmed in Fig. 4.
0.3 0.2
5. Conclusion
0.1 0 0.5
(b)
1
1.5
2
|j+γ|
Fig. 4. ASN and power functions of the TSPRT in fading channel. (a) ASN function and (b) Power function.
frequently under approximation (16) than under the original ratio (11). Next, we consider the performance of code acquisition schemes under fading environment assuming that the fading pdf f c ðcÞ is known. Fig. 4 shows the ASN and power function of the TSPRT designed for the known fading with r ¼ 10; ad ¼ bd ¼ 0:01; and p0 ¼ p1 ¼ 0:5: We can observe that approximation (20) (Approximation 2) outperforms the other approximations and has a performance improvement in ASN over the original scheme at the cost of false alarm prob-
We have considered noncoherent sequential code acquisition with the TSPRT. As the code acquisition schemes using the maximum likelihood estimate are known to be impractical to implement, we have derived simple approximations to the original schemes for use in channels with and without channel fading. We have compared the performance of the proposed and original schemes. It has been found that Approximation 2 for AWGN channel and Approximations 1 and 2 for fading channel outperform the other approximations and have some performance improvements over the corresponding original schemes at the cost of false alarm probability.
Acknowledgements This research was supported by Korea Science and Engineering Foundation (KOSEF) under
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The ratio of the zeroth– order modified Bessel functions (r =10) The ratio of the zeroth – order modified Bessel functions
108
O: D:
106
SNR = 0dB SNR = 20dB
O
104
D O
102
O
100 No approximation (SNR=0dB) Approximation 1 for Fading (SNR=0dB) Approximation 2 for Fading (SNR=0dB) Approximation 3 for Fading (SNR=0dB) Approximation 4 for Fading (SNR=0dB) No approximation (SNR=20dB) Approximation 1 for Fading (SNR=20dB) Approximation 2 for Fading (SNR=20dB) Approximation 3 for Fading (SNR=20dB) Approximation 4 for Fading (SNR=20dB)
10-2
10-4
10-6
D
0
20
40
O 60
80
100
zn pffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffi Fig. 5. The ratio I 0 ð2 en;1 zn Þ=I 0 ð2 en;0 zn Þ with the proposed approximations for fading channel (r ¼ 10).
Grant R01-2004-000-10019-0, for which the authors would like to express their thanks. The authors also wish to thank the anonymous reviewers for their helpful and constructive comments and suggestions. References [1] S. Tantaratana, H.V. Poor, Asymptotic efficiencies of truncated sequential tests, IEEE Trans. Inform. Theory 28 (November 1982) 911–923. [2] S. Tantaratana, A.W. Lam, P.J. Vincent, Noncoherent sequential acquisition of PN sequences for DS/SS communications with/without channel fading, IEEE Trans. Comm. 43 (February/March/April 1995) 1738–1745. [3] W.-H. Sheen, H.-C. Wang, Performance analysis of the biased square-law sequential detection with signal present, IEEE Trans. Inform. Theory 43 (July 1997) 1268–1273.
[4] S. Yoon, I. Song, S.Y. Kim, S.R. Park, DS/SS code acquisition with joint detection of multiple correct cells using locally optimum test statistics in Rayleigh fading channels, Signal Process. 82 (November 2001) 609–623. [5] H.G. Kim, I. Song, S. Yoon, S.R. Park, Nonparametric PN code acquisition using the signed-rank statistic for DS/ CDMA systems in frequency-selective Rician fading channels, IEEE Trans. Veh. Technol. 51 (September 2002) 1138–1144. [6] R.N. McDonough, A.D. Whalen, Detection of Signals in Noise, Academic Press, CA, 1995. [7] C.F. du Toit, The numerical computation of Bessel functions of the first and second kind for integer orders and complex arguments, IEEE Trans. Antennas and Propagation 38 (September 1990) 1341–1349. [8] F.B. Gross, New approximations to J 0 and J 1 Bessel functions, IEEE Trans. Antennas and Propagation 43 (August 1995) 904–907.
Signal Processing 85 (2005) 1563–1571 www.elsevier.com/locate/sigpro
Noncoherent sequential code acquisition based on simplified maximum likelihood ratios Hyoungmoon Kwona, Iickho Songa,, Seokho Yoonb, Jumi Leea, Sun Yong Kimc, Bo-Hyun Chungd a
Department of Electrical Engineering and Computer Science, Korea Advanced Institute of Science and Technology, 373-1 Guseong Dong, Yuseong Gu, Daejeon 305-701, Republic of Korea b School of Information and Communication Engineering, Sungkyunkwan University, 300 Cheoncheon Dong, Jangan Gu, Suweon, Gyeonggi 440-746, Republic of Korea c Department of Electronic Engineering, Konkuk University, 1 Hwayang Dong, Gwangjin Gu, Seoul 143-701, Republic of Korea d Mathematics Section, College of Science and Technology, Hongik University, 300 Shinan Ri, Jochiwon, Yeongi Gun, Chungnam 339-701, Republic of Korea Received 18 June 2004; received in revised form 16 November 2004
Abstract In this paper, we consider the noncoherent code acquisition problem using sequential schemes. To alleviate the computational complexity of the maximum likelihood method, simplified schemes are proposed and analyzed for the truncated sequential probability ratio test. The performance of the simplified and original schemes are compared in additive white Gaussian noise and slowly varying fading channels. Numerical results show that the proposed schemes have essentially the same performance as the original schemes while allowing simpler structures. r 2005 Elsevier B.V. All rights reserved. Keywords: Sequential code acquisition; Maximum likelihood estimate; Bessel function
1. Introduction In direct-sequence–spread-spectrum (DS/SS) systems, the receiver is to generate a pseudonoise Corresponding author.
E-mail addresses: [email protected] (H. Kwon), [email protected] (I. Song), [email protected] (S. Yoon), [email protected] (J. Lee), [email protected] (S.Y. Kim), [email protected] (B.-H. Chung).
(PN) code which is a replica of and is in synchronization with the incoming PN code. Code synchronization is an indispensable process in any SS system in despreading and reliable data demodulation at the receiver. In the past two decades, a vast volume of research on PN code acquisitions has been carried out e.g., [1–5]. Among the various methods, sequential schemes have the potential to achieve the best performance. A sequential scheme has two
0165-1684/$ - see front matter r 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.sigpro.2005.02.009
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sets of thresholds in the decision process, one set for accepting phase alignment and the other for rejecting phase alignment and continuously checking alignment or nonalignment. At any time, if alignment or nonalignment is apparent from the integrator output, the inspection process terminates and the corresponding decision is made. Otherwise, it continues the test. It appears, however, that the sequential schemes are the least studied because of the difficulty associated with the design and analysis of sequential schemes. Numerical methods, computer simulations, and/or approximations are used for the performance evaluation of sequential schemes [1–3]. In [1], by viewing the truncated sequential probability ratio test (TSPRT) as a mixture of a sequential probability ratio test (SPRT) and a fixed-sample-size (FSS) test, a novel design procedure for the TSPRT is developed. In [2], a noncoherent sequential acquisition based on the noncoherent I/Q detector with continuous integration is investigated for additive white Gaussian noise (AWGN) and Rayleigh fading channels. In addition to the difficulty in the design and analysis, most of the sequential acquisition schemes using maximum likelihood estimate impose difficulty in hardware implementation because of a high level of computational complexity [3].
during acquisition, the received signal wðtÞ can be expressed as wðtÞ ¼ A0 ccðt þ iDT c Þ cosðo0 t þ yÞ þ nðtÞ.
(1)
In (1), A0 is the signal amplitude, c is the fading random variable, cðtÞ is the PN signal, i is an integral initial phase number, D is the advancing step size, T c is the chip duration, o0 is the carrier angular frequency, y is the random phase distributed uniformly over ½0; 2pÞ; and nðtÞ represents the AWGN with one-sided power spectral density N 0 : The fading variable c is assumed to have the Rician probability density function (pdf): pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 f c ðcÞ ¼ 2cð1 þ rÞerc ð1þrÞ I 0 2c rð1 þ rÞ , cX0,
ð2Þ
where r ¼ s2 =ð2s2 Þ is the ratio of the power s2 in the direct component and the power 2s2 in the diffused component with the constraint s2 þ 2s2 ¼ R 2p 1; and I 0 ðxÞ ¼ ð1=2pÞ 0 ex cos n dn is the zeroorder modified Bessel function of the first kind. The case r ! 1 clearly represents no fading or a constant value of c in (1). In addition, we have Rayleigh fading when r ¼ 0; in which case the pdf 2 (2) becomes f c ðcÞ ¼ 2cec for cX0: It is straightforward to see that the test statistic Y n ¼ X 2i;n þ X 2q;n
(3)
is a noncentral w2 random variable with the pdf [2]:
2. System model 2.1. Output statistics of noncoherent correlators Fig. 1 shows a block diagram of the acquisition system considered. The transmitted signal is affected by slowly varying fading in addition to AWGN. Assuming that no data are modulated
f Y n ðyn Þ ð1 þ rÞ 2s2n ½ð1 þ rÞ þ ln =2s2n ð1 þ rÞðyn =2s2n Þ þ rðln =2s2n Þ
exp ð1 þ rÞ þ ln =2s2n
¼
Fig. 1. A block diagram of the noncoherent correlator receiver.
ARTICLE IN PRESS H. Kwon et al. / Signal Processing 85 (2005) 1563–1571
I 0
2
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi! rð1 þ rÞðln =2s2n Þðyn =2s2n Þ , ð1 þ rÞ þ ln =2s2n
and
yn X0,
ð4Þ
where X i;n and X q;n are the outputs (ignoring the double-frequency terms) of the in-phase and quadrature branches of the noncoherent correlator and ln ¼ A20 T 2c S2n =4 is a measure of the signal energy contained in Y n : The cumulative distribution function (cdf) of Y n is sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rðln =s2n Þ ; F Y n ðyn Þ ¼ 1 Q ð1 þ rÞ þ ðln =2s2n Þ sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi! ð1 þ rÞðyn =s2n Þ ; yn X0, ð1 þ rÞ þ ðln =2s2n Þ
ð5Þ
where Q( ; ) is the Marcum’s Q function given by Z Qða; bÞ ¼ b
1
ðn2 þ a2 Þ n exp I 0 ðanÞ dn [6]. 2
For the normalized random variable Z n ¼ Y n =ð2s2n Þ; let en ¼ ln =ð2s2n Þ; zn ¼ yn =ð2s2n Þ; and qn ¼ ð1 þ rÞ þ en : Then, from (4) and (5), the pdf and cdf of Z n are
f Zn ðzn Þ ¼
ð1 þ rÞ ð1 þ rÞzn þ ren exp qn qn pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi! 2 rð1 þ rÞen zn
I 0 ; zn X0 qn
ð6Þ
and sffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi! 2ren 2ð1 þ rÞzn F Zn ðzn Þ ¼ 1 Q ; ; qn qn
zn X0,
respectively. When r ¼ 0 (the Rayleigh fading case), we have qn ¼ 1 þ e n , f Zn ðzn Þ ¼ exp
zn 1 ,
1 þ en 1 þ en
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sffiffiffiffiffiffiffiffiffiffiffiffiffi! 2zn F Zn ðzn Þ ¼ 1 Q 0; 1 þ en
zn ¼ 1 exp 1 þ en
R1 since I 0 ð0Þ ¼ 1 and Qð0; bÞ ¼ b n expððn2 = 2ÞÞ dn ¼ expððb2 =2ÞÞ: On the other hand, when there is no fading (r ! 1), we have ð1 þ rÞ=qn ! 1; r=qn ! 1; and rð1 þ rÞen zn =q2n ! en zn : Thus, Zn is a w2 random variable with pdf: pffiffiffiffiffiffiffiffi (7) f Zn ðzn Þ ¼ eðzn þen Þ I 0 2 en zn ; zn X0, pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi and cdf F Zn ðzn Þ ¼ 1 Qð 2en ; 2zn Þ; zn X0: It is noteworthy that en is a measure of the signal-to-noise ratio (half of the effective signal-tonoise ratio) since the variance of Y n can be written as (
2 A0 4 2 cT c Sn cos y VarðY n Þ ¼ 4sn þ 4sn 2
2 ) A0 þ cT c S n sin y 2 ¼ 4s4n þ 4s2n ln c2 ,
ð8Þ
of which the first and second terms on the righthand side are the noise and signal power, respectively. Consequently, the effective signalto-noise ratio without fading ðc ¼ 1Þ is 4s2n ln =4s4n ¼ ln =s2n : Comparing (6) and (7), we can observe that qn is a measure of the effect of fading relative to the signal-to-noise ratio en (which would eventually reduce the effective signal-to-noise ratio). 2.2. Maximum likelihood estimate Let the phase of the local PN signal be ðj þ gÞDT c ; where j is an integer and g 2 ð12; 12 is the residual code phase offset. Based on Z n ; the sequential decision processor decides if (1) the phases of the incoming and local PN signals are aligned to be within DT c =2 of each other (i.e., j ¼ i), (2) they differ by at least one chip (i.e., jðj þ gÞ ijDT c XT c ), or (3) neither of the first two. Here, without loss of generality, we assume
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i ¼ 0 so that the phase difference becomes j þ g: Equivalently, the decision processor attempts to test the following hypotheses: 1 , D 1 H1 ðalignmentÞ: jj þ gjp , 2 1 1 H2 ðdecision deferredÞ: ojj þ gjo . 2 D H0 ðnon-alignmentÞ: jj þ gjX
ð9Þ
Since g 2 ð 12 ; 12 ; both H0 and H1 are composite hypotheses and the value of ln generally differs depending on the value of g and the hypothesis. Thus, assuming the worst-case scenario, let ln;0 denote the largest value of ln under H0 and ln;1 the smallest value of ln under H1 : clearly, ln;1 4ln;0 : Note that ln;0 and ln;1 are the worst-case values. We use one additional subscript 0 or 1 to denote the parameters en and qn under H0 or H1 ; respectively: for example, en;0 denotes the parameter en under H0 : Then, the likelihood ratio can be expressed as Ln ðzn Þ qn;0 ð1 þ rÞzn þ ren;1 ð1 þ rÞzn þ ren;0 ¼ exp þ qn;1 qn;1 qn;0 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi!, pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi! 2 rð1 þ rÞen;1 zn 2 rð1 þ rÞen;0 zn I0 .
I 0 qn;1 qn;0 ð10Þ When there is no fading (r ! 1), (10) reduces to Ln ðzn Þ ¼ e
pffiffiffiffiffiffiffiffiffiffiffi en;1 zn Þ pffiffiffiffiffiffiffiffiffiffiffi . I 0 ð2 en;0 zn Þ
ðen;0 en;1 Þ I 0 ð2
(11)
It is noteworthy that, due to the computational complexities of the Bessel and exponential functions, (10) and (11) are quite difficult to implement for real-time processing [7]. Specifically, for the Bessel function, a great quantity of complicated computations is required. In addition, since there is no closed form expression for the Bessel function, one must resort to numerical integration techniques to provide quantitative answers for a given argument [8].
3. Proposed schemes The modified Bessel function I 0 ðxÞ of order zero can be approximated as [6] 2k 1 X 1 1 x I 0 ðxÞ ¼ 2 2 k¼0 ðk!Þ 8 1 2 > > 0pjxj51; > > p ffiffiffiffiffiffiffiffi ; jxjb1: > : 2px With the two approximated expressions in (12), we propose simplifications of (10) and (11) for Ln ðzn Þ in fading and AWGN channels. 3.1. AWGN (no fading) channel Let us first focus on the ratio containing I 0 ð Þ in (11): pffiffiffiffiffiffiffiffiffiffiffi I 0 ð2 en;1 zn Þ (13) pffiffiffiffiffiffiffiffiffiffiffi . I 0 ð2 en;0 zn Þ Approximation 1 for AWGN. Using I 0 ðxÞ pffiffiffiffiffiffiffiffi ex = 2px; (13) can be written as rffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffi pffiffiffi pffiffiffiffiffi pffiffiffiffiffi I 0 ð2 en;1 zn Þ en;0 2 z ð e e Þ . (14) pffiffiffiffiffiffiffiffiffiffiffi e n n;1 n;0 4 I 0 ð2 en;0 zn Þ en;1 Then, the likelihood ratio (11) becomes rffiffiffiffiffiffiffi pffiffiffi pffiffiffiffiffi pffiffiffiffiffi en;0 Ln ðzn Þ efen;0 en;1 þ2 zn ð en;1 en;0 Þg 4 en;1 e pffiffiffiffiffi ¼ zn ,
ð15Þ
e
where ‘¼’ means ‘equals equivalently to’: that is, the likelihood ratios using the quantities before e and after ¼ result in the equivalent detector. Approximation 2 for AWGN. We apply I 0 ðxÞ pffiffiffiffiffiffiffiffi ex = 2px and I 0 ðxÞ 1 þ 14 x2 to the numerator and denominator of (13), respectively, and then we can rewrite Ln ðzn Þ in (11) as pffiffiffiffiffiffiffiffi eðen;0 en;1 þ2 en;1 zn Þ ffi Ln ðzn Þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffi 4p en;1 zn ð1 þ en;0 zn Þ 1 e pffiffiffiffiffiffiffiffiffiffiffi ¼ 2 en;1 zn ln zn lnð1 þ en;0 zn Þ. ð16Þ 4
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Approximation 3 for AWGN. Applying I 0 ðxÞ 1 þ 14x2 to (13), it is easy to see that (11) can be expressed as Ln ðzn Þ eðen;0 en;1 Þ
e
¼
1 þ en;1 zn 1 þ en;0 zn
1 þ en;1 zn . 1 þ en;0 zn
ð17Þ
Note that it is meaningless pffiffiffiffiffiffiffiffi to apply I 0 ðxÞ 1 þ 14x2 and I 0 ðxÞ ex = 2px to the numerator and denominator of (13), respectively, since en;1 4en;0 : 3.2. Fading channel We now focus on the ratio of the zeroth-order modified Bessel functions pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi!, pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi! 2 rð1 þ rÞen;1 zn 2 rð1 þ rÞen;0 zn I0 I0 qn;1 qn;0 (18) in the maximum likelihood ratio (10) for fading channels. Approximation 1 for fading. With I 0 ðxÞ pffiffiffiffiffiffiffiffi ex = 2px and some manipulations as we have done for the AWGN channel case, the likelihood ratio (10) can be written as rffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi qn;1 en;0 qn;0 en;1 pffiffiffiffiffi r Ln ðzn Þ zn 2 zn . qn;1 qn;0 1þr (19) Approximation 2 p forffiffiffiffiffiffiffiffi fading. Using the approximations I 0 ðxÞ ex = 2px and I 0 ðxÞ 1 þ 14 x2 in the numerator and denominator of (18), respectively, the likelihood ratio (10) becomes
1 1 Ln ðzn Þ ð1 þ rÞ zn qn;0 qn;1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 rð1 þ rÞen;1 pffiffiffiffiffi þ zn qn;1 1 ln zn ln½q2n;0 þ rð1 þ rÞen;0 zn . ð20Þ 4 Approximation 3 for fading. With thepapproximaffiffiffiffiffiffiffiffi tions I 0 ðxÞ 1 þ 14 x2 and I 0 ðxÞ ex = 2px in the numerator and denominator of (18), respectively,
1567
we can rewrite (10) as
1 1 zn Ln ðzn Þ ð1 þ rÞ qn;0 qn;1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 rð1 þ rÞen;0 pffiffiffiffiffi zn qn;0 1 þ ln zn þ ln½q2n;1 þ rð1 þ rÞen;1 zn . ð21Þ 4 Approximation 4 for fading. Using I 0 ðxÞ 1 þ 14x2 ; we obtain the likelihood ratio Ln ðyn Þ in (10) as
1 1 Ln ðzn Þ ð1 þ rÞ zn qn;0 qn;1 " # q2n;1 þ rð1 þ rÞen;1 zn þ ln 2 . ð22Þ qn;0 þ rð1 þ rÞen;0 zn Unlike in the case of AWGN channel, we have also considered the papproximations I 0 ðxÞ 1 þ ffiffiffiffiffiffiffiffi 1 2 x x and I ðxÞ e = 2px to the numerator and 0 4 denominator of (18), respectively, in the fading channel. This is due to the fact that we pffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffi could have both e z =q 4 en;0 zn =qn;0 and pffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffi n;1 n n;1 en;1 zn =qn;1 o en;0 zn =qn;0 depending on the specific values of the four parameters en;0 ; en;1 ; qn;0 ; and qn;1 :
4. Performance evaluation We have so far obtained simplifications to the maximum likelihood ratios using the two approximations in (12). We now investigate the performance of the proposed and original acquisition schemes with the TSPRT. 4.1. Decision processor with TSPRT In the SPRT scheme, we compare the likelihood ratio Ln ðzn Þ with two thresholds A and B, where A4B40; for n ¼ 1; 2; 3; . . . ; until one of the thresholds is reached. When the detected value corresponds to neither H0 nor H1 ; the SPRT may not end for a long period of time. To avoid such an excessive delay associated with the SPRT, the TSPRT [2] puts an upper bound n^ on the
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test length:
ASN function
350
No approximation Approximation 1 for AWGN Approximation 2 for AWGN Approximation 3 for AWGN
say H1 ; 300
say H0 ; continue; H1 ; H0 :
ð23Þ
250
ASN
8 XA^ > > < ^ Ln ðzn Þ pB^ TSPRT : if non; > > : otherwise ( X^t say ^ Ln^ ðzn^ Þ if n ¼ n; o^t say
200
SNR=-10dB
Therefore, the test is truncated at n ¼ n^ if it has not been previously terminated.
=βdTSPRT=0.005 αTSPRT d
150
p0=p1=0.5
4.2. Simulation results and discussions
100
(a)
(1) PN code sequence of 1023 chips generated from the primitive polynomial 1 þ x2 þ x5 þ x6 þ x10 ; (2) chip SNR ¼ 10 dB; (3) advancing step size D ¼ 1=2; (4) residual code phase offset g ¼ 1=2: Fig. 2 shows the performance of the original and proposed schemes for AWGN, obtained from (11), (15)–(17). In this figure, the TSPRT is designed with ad ¼ bd ¼ 0:005 and p0 ¼ p1 ¼ 0:5; where p0 and p1 are the design constants of the TSPRT [2]. If p0 ¼ p1 ¼ 0; the TSPRT becomes the FSS test and if p0 ¼ p1 ¼ 1; the TSPRT becomes the SPRT. For values of p0 and p1 in between, the TSPRT can
1
1.5
2
|j+γ| Power function
1
No approximation Approximation 1 for AWGN Approximation 2 for AWGN Approximation 3 for AWGN
0.9 0.8
Power function
Let us employ the average sample number (ASN) and power function as the performance measures. The ASN is the average number of chips needed for a test to terminate. The power function is the probability of accepting H1 as a function of the absolute phase difference jj þ gj: If jj þ gj is consistent with the hypothesis H1 ; then the power function value is the detection probability. Similarly, if jj þ gj is consistent with the hypothesis H0 ; the power function value is the false alarm probability. Briefly, the ASN represents the price we have to pay, in terms of the number of observations required for the sequential test, and the power function describes how well the sequential test procedure achieves its objective of making correct decisions. In evaluating the performance, we use the following parameters:
0.5
0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0.5
(b)
1
1.5
2
|j+γ|
Fig. 2. ASN and power functions of the TSPRT in AWGN channel. (a) ASN function and (b) Power function.
be viewed as a mixture of the SPRT and FSS test. The hypotheses H1 and H0 are represented by jj þ gj ¼ 0:5 and jj þ gj ¼ 2:0; respectively, in the figure. From this figure, we can see that Approximation 2 (or the approximation (16)) is better than the other approximations and has some performance improvements over the original scheme at the cost of false alarm probability. Compared with the original scheme, the approximation (16) has no larger ASN at all phase difference. Approximation (16) also exhibits a similarly decreasing characteristic of power function compared with the original scheme. Note that the approximation (16) has a
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The ratio of the zeroth– order modified Bessel functions
The ratio of the zeroth – order modified Bessel functions
1070 1060 1050
O
SNR=0dB
1040 1030
No Approximation (SNR=0dB) Approximation 1 for AWGN (SNR=0dB) Approximation 2 for AWGN (SNR=0dB) Approximation 3 for AWGN (SNR=0dB) No Approximation (SNR=20dB) Approximation 1 for AWGN (SNR=20dB) Approximation 2 for AWGN (SNR=20dB) Approximation 3 for AWGN (SNR=20dB)
O
1020 1010
O
O
100 SNR=20dB 10-10
0
20
40
60
: SNR=0dB : SNR=20dB
80
100
zn pffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffi Fig. 3. The ratio I 0 ð2 en;1 zn Þ=I 0 ð2 en;0 zn Þ with the proposed approximations for the AWGN channel.
small loss in false alarm probability in comparison with the original scheme. To account for these results, let us consider pffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffi Fig. 3 showing the ratio I 0 ð2 en;1 zn Þ=I 0 ð2 en;0 zn Þ for the AWGN channel. In this figure, defining the signal-to-noise ratio (SNR) per chip as A20 T c =ð2N 0 Þ; we have assumed the approximations en;1 ¼ 1n ðSNRÞS 2n;1 nðSNRÞð1 jgjDÞ2 and en;0 ¼ 2 1 n ðSNRÞS n;0 SNR which have been proposed and proved very effective in [2]. In this figure, n ¼ 100; g ¼ 12 (the worst case value of the residual code phase offset), and D ¼ 12 are used. Since it has been found empirically that the general tendency does not depend on the value of n, we have chosen n ¼ 100: We can see that all the proposed approximations exhibit a similarly increasing characteristic in comparison with the original scheme at the lower value 20 dB of SNR. At the higher value of SNR, the original scheme, Approximation 1, and Approximation 2 show an increasing characteristic, while Approximation 3 does not follow the original scheme, being stationary at a relatively small value.
Note that the ratios in Approximation 2 grow up faster than the others regardless of the SNR. This result means that Approximation 2 generates a larger value of Ln ðzn Þ than the others for a given value of zn : Thus, Ln ðzn Þ of Approximation 2 is more likely to be larger than a certain threshold, and would consequently accept H1 more frequently. Another consequence is that using Approximation 2 may produce some adverse effects when Ln ðzn Þ is desired to be smaller than a certain threshold (that is, when it is desired to accept H0 ). In short, we find that approximation (16) yields a higher value than the original ratio (11), and thus is more likely to be larger than the upper threshold. Consequently, the ASN decreases and the power function value increases when the absolute phase difference is within a half of the chip duration ðjj þ gjo1:25Þ: On the other hand, when the absolute phase difference is larger than a half of the chip duration ðjj þ gj41:25Þ; the ASN still decreases and the power function tends to be higher (than that for the original scheme) since the event fLn ðzn Þ4upper thresholdg occurs more
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ASN function 700 SNR=10dB, r=10 TSPRT
αd
600
TSPRT
=βd
=0.01
p0=p1=0.5
ASN
500
400
300 No approximation Approximation 1 for Fading Approximation 2 for Fading Approximation 3 for Fading Approximation 4 for Fading
200
100 0.5
1
1.5
2
|j+γ|
(a)
Power function 1 No approximation Approximation 1 for Fading Approximation 2 for Fading Approximation 3 for Fading Approximation 4 for Fading
0.9 0.8
Power function
0.7 0.6 0.5 0.4
ability when jj þ gjo1:25: Similarly, approximation (19) (Approximation 1) has the best performance among the approximations and improved ASN characteristic over the original scheme at the expense of false alarm probability when jj þ gj41:25 in fading channels. These observations can be similarly explained with a rationale similar to that described for AWGN channels. Fig. 5 shows the ratio (18) with the proposed approximations for the fading channel when r ¼ 10: When the SNR is low, it is observed that the ratios of I 0 ð Þ in the original scheme, Approximations 1 and 2 have a similarly increasing characteristic, but those of Approximations 3 and 4 do not have such a characteristic. At the higher value of SNR, the ratios in the original scheme, Approximations 1 and 2 still show a similar pattern, although the ratios in Approximation 2 become somewhat larger than the others. On the other hand, the ratios of Approximations 3 and 4 again do not follow that of the original scheme. We can thus anticipate that Approximations 1 and 2 will have a better performance than Approximations 3 and 4, as is confirmed in Fig. 4.
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5. Conclusion
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Fig. 4. ASN and power functions of the TSPRT in fading channel. (a) ASN function and (b) Power function.
frequently under approximation (16) than under the original ratio (11). Next, we consider the performance of code acquisition schemes under fading environment assuming that the fading pdf f c ðcÞ is known. Fig. 4 shows the ASN and power function of the TSPRT designed for the known fading with r ¼ 10; ad ¼ bd ¼ 0:01; and p0 ¼ p1 ¼ 0:5: We can observe that approximation (20) (Approximation 2) outperforms the other approximations and has a performance improvement in ASN over the original scheme at the cost of false alarm prob-
We have considered noncoherent sequential code acquisition with the TSPRT. As the code acquisition schemes using the maximum likelihood estimate are known to be impractical to implement, we have derived simple approximations to the original schemes for use in channels with and without channel fading. We have compared the performance of the proposed and original schemes. It has been found that Approximation 2 for AWGN channel and Approximations 1 and 2 for fading channel outperform the other approximations and have some performance improvements over the corresponding original schemes at the cost of false alarm probability.
Acknowledgements This research was supported by Korea Science and Engineering Foundation (KOSEF) under
ARTICLE IN PRESS H. Kwon et al. / Signal Processing 85 (2005) 1563–1571
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The ratio of the zeroth– order modified Bessel functions (r =10) The ratio of the zeroth – order modified Bessel functions
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SNR = 0dB SNR = 20dB
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100 No approximation (SNR=0dB) Approximation 1 for Fading (SNR=0dB) Approximation 2 for Fading (SNR=0dB) Approximation 3 for Fading (SNR=0dB) Approximation 4 for Fading (SNR=0dB) No approximation (SNR=20dB) Approximation 1 for Fading (SNR=20dB) Approximation 2 for Fading (SNR=20dB) Approximation 3 for Fading (SNR=20dB) Approximation 4 for Fading (SNR=20dB)
10-2
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D
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zn pffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffi Fig. 5. The ratio I 0 ð2 en;1 zn Þ=I 0 ð2 en;0 zn Þ with the proposed approximations for fading channel (r ¼ 10).
Grant R01-2004-000-10019-0, for which the authors would like to express their thanks. The authors also wish to thank the anonymous reviewers for their helpful and constructive comments and suggestions. References [1] S. Tantaratana, H.V. Poor, Asymptotic efficiencies of truncated sequential tests, IEEE Trans. Inform. Theory 28 (November 1982) 911–923. [2] S. Tantaratana, A.W. Lam, P.J. Vincent, Noncoherent sequential acquisition of PN sequences for DS/SS communications with/without channel fading, IEEE Trans. Comm. 43 (February/March/April 1995) 1738–1745. [3] W.-H. Sheen, H.-C. Wang, Performance analysis of the biased square-law sequential detection with signal present, IEEE Trans. Inform. Theory 43 (July 1997) 1268–1273.
[4] S. Yoon, I. Song, S.Y. Kim, S.R. Park, DS/SS code acquisition with joint detection of multiple correct cells using locally optimum test statistics in Rayleigh fading channels, Signal Process. 82 (November 2001) 609–623. [5] H.G. Kim, I. Song, S. Yoon, S.R. Park, Nonparametric PN code acquisition using the signed-rank statistic for DS/ CDMA systems in frequency-selective Rician fading channels, IEEE Trans. Veh. Technol. 51 (September 2002) 1138–1144. [6] R.N. McDonough, A.D. Whalen, Detection of Signals in Noise, Academic Press, CA, 1995. [7] C.F. du Toit, The numerical computation of Bessel functions of the first and second kind for integer orders and complex arguments, IEEE Trans. Antennas and Propagation 38 (September 1990) 1341–1349. [8] F.B. Gross, New approximations to J 0 and J 1 Bessel functions, IEEE Trans. Antennas and Propagation 43 (August 1995) 904–907.
