IEEE SIGNAL PROCESSING LETTERS, VOL. 9, NO. 7, JULY 2002
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Optimal and Suboptimal Decision Rules for Parallel Code Acquisition in Chip-Asynchronous DS/SS Systems Seokho Yoon, Member, IEEE, Iickho Song, Senior Member, IEEE, and Cheol Hoon Park, Senior Member, IEEE
Abstract—We investigate optimal and suboptimal decision rules for parallel code acquisition in chip-asynchronous direct-sequence spread-spectrum (DS/SS) systems. The conventional decision rule for parallel acquisition is to choose the largest correlator output of a receiver. However, such a scheme is optimum only for chip-synchronous models. In this letter, an optimal decision rule is derived based on the maximum-likelihood criterion for chipasynchronous models. A simpler suboptimal decision rule is also discussed. The performance of the optimum and suboptimum decision rules is compared to that of the conventional decision rule. Numerical results show, that for chip-asynchronous models, both the optimal and suboptimal decision rules outperform the conventional decision rule.
In this letter, our aim is to obtain optimal and suboptimal decision rules of the correlator outputs of the parallel-acquisition receiver for chip-asynchronous DS/SS systems. Using the ML criterion, an optimal decision rule is derived. Since the optimal decision rule is difficult to implement, a simpler suboptimal decision rule is derived based on the criterion of local detection power [3].
Index Terms—Chip-asynchronous, code acquisition, directsequence spread-spectrum (DS/SS) systems.
A. Distribution of Noncoherent Correlator Outputs
II. OPTIMAL AND SUBOPTIMAL DECISION RULES
In a DS/SS system, the received signal can be expressed as I. INTRODUCTION
I
N A direct-sequence spread-spectrum (DS/SS) system, it is essential to synchronize a locally generated pseudonoise (PN) code with the received one. The synchronization process is usually achieved in two steps: acquisition (coarse alignment) and tracking (fine alignment), of which the former is dealt with in this letter. Specifically, we are concerned with parallel code acquisition (which outperforms the serial and hybrid schemes). The conventional decision rule used for parallel acquisition is to choose the largest among all correlator outputs of the parallelacquisition receiver [1]. The decision rule satisfies the maximum-likelihood (ML) criterion for chip-synchronous models [2] in which the time delay of the received PN code, normalized to chip duration , is assumed to be an integer (i.e., the chip boundaries of the received PN code are assumed to be perfectly aligned with those of the locally generated PN code). For chip-asynchronous models, in which a mismatch between the chip boundaries of the two PN codes exists, however, the conventional decision rule is not optimum. Clearly, chip-asynchronous models are more general and realistic than chip-synchronous models. Manuscript received July 11, 2001; revised February 14, 2002. This work was supported by the Postdoctoral Fellowship Program of Korea Science and Engineering Foundation (KOSEF). The associate editor coordinating the review of this manuscript and approving it for publication was Dr. Geert Leus. S. Yoon is with the Division of Engineering and Applied Sciences, Harvard University, Cambridge, MA 02138 USA (e-mail:
[email protected]). I. Song and C. H. Park are with the Department of Electrical Engineering and Computer Science, Korea Advanced Institute of Science and Technology (KAIST), Yuseong-gu, Daejeon 305-701, Korea. Publisher Item Identifier 10.1109/LSP.2002.800678.
(1)
is the data sequence waveform; , where is the th is the PN chip of a PN code sequence of period , and ; code waveform defined as a unit rectangular pulse over is the time delay normalized to ; is the carrier angular frerepresents the additive quency; is the carrier phase; and white Gaussian noise (AWGN) with two-sided power spectral . In this letter, it is assumed that there is a preamble density for acquisition, so that no data modulation is present during ac(since acquisition is more rapid in the quisition, i.e., absence of data modulation on the transmitted signal, DS/SS transmissions often include a preliminary training period (preamble) [1]). In addition, we assume the parallel-acquisition receiver is a bank of noncoherent correlators: each correlator is composed of in-phase and quadrature-phase ( – ) branches. The noncoherent correlator assumption is reasonable, since initial code acquisition should typically be achieved before carrier phase synchronization. , where is the integer part of and Let (for the chip-synchronous model, is set to 0). Since has period , we assume without loss of generality that . In each of the noncoherent – correlators, the reis first down-converted to the and compoceived signal nents. Then, the and signals are correlated with the locally and of the generated PN code. Thus, the outputs
In (1),
is the signal power;
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and for
IEEE SIGNAL PROCESSING LETTERS, VOL. 9, NO. 7, JULY 2002
branches, respectively, of the th correlator are given by, and ,
B. Optimal Decision Rule In this section, we derive the optimal decision rule of the correlator outputs using the ML criterion: we are to esusing the ML criterion. Let the set timate or of outputs be denoted as . For notato denote tional convenience, we use the notation for [i.e., ]. Thus, the pdf can be expressed as of given
if
(6)
if otherwise (2)
where it should be noted that uniformly distributed in such that
. Since is an integer , the ML criterion is to choose
, , and are zerowhere mean independent and identically distributed (i.i.d.) Gaussian , and is random variables with variance the autocorrelation function of the PN code (7)
(3)
using (5). Practical implementation of the decision rule (7) is difficult unfortunately, because the values of and must be is required. Thus, we have known and the computation of to search further in order to obtain a suboptimal decision rule, which allows us a practical implementation.
Thus
C. Suboptimal Decision Rule for , where represents the Gaussian distribution with mean and variance . Finally, the outputs of the and branches are squared and summed, and the final th , correlator output is thus given by, for (4) and are uncorrelated Gaussian Since random variables, they are also statistically independent. As a is a noncentral chi-square distributed random consequence, , and is a variable with two degrees of freedom for central chi-square distributed random variable with two degrees . Thus, the probability density of freedom for of , normalized by , is given by, for function (pdf) ,
In this section, to derive a suboptimal decision rule which is practical and relatively easy to implement, we use the criterion of local detection power leading to locally optimum (LO) detectors in signal detection theory. The motivation of using this criterion is as follows. First, since an LO detector has the maximum slope of its power function when the signal-to-noise ratio (SNR) approaches zero [3], it is expected to have quite good performance when the SNR is low. Second, an LO detector can always be obtained and is usually much easier to implement than other detectors including uniformly most powerful and optimum detectors [4], [5]. The LO decision rule is to choose such that (8) where is the order of the first nonzero derivative of (with respect to ) at . We compute the first derivative of at . Using (6), we have
if (5) if otherwise where Bessel function of order 0.
is a measure of the SNR and where is the first kind modified
(9)
YOON et al.: PARALLEL CODE ACQUISITION IN CHIP-ASYNCHRONOUS DS/SS SYSTEMS
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where
Here, noting that , and we have . Then (9) reduces to 0. Thus, we are required to at : we get compute the second derivative of Fig. 1. False lock probabilities of the optimal, suboptimal, and conventional decision rules when = 0 and 0:5.
(10) where
loss of generality, we assume : thus, all of the signal comand ponents are contained in the first two correlator outputs . Then, the false lock probability of a decision rule is , given by, for (13)
By noting that , we have . Using these results and the fact that of , we are to choose such that
, and is independent (11)
from (8) and (10). The decision rule (11) changes as the value of changes. In a practical system, the value of is not usually known. Therefore, to obtain a decision rule which does not depend on and yet performs reasonably well for all possible , we have to consider some options. First, we can regard (11) as a conditional decision rule given . Then, by averaging (11) over with the assumption , we would obtain that is distributed uniformly over . Second, we may use the average value of or pick a value of arbitrarily. Since can be assumed as mentioned above, the to be distributed uniformly over (we would like to mention in passing average value of is that this value of is also the “worst-case” value). Thus, the is reasonable in view of system design. If we value or in (11), the decision rule is again use . From these discussions, it is clear that one of the reasonable choices as the suboptimal decision rule is to choose such that (12)
III. NUMERICAL RESULTS In this section, the optimal, suboptimal, and conventional decision rules are compared on the basis of the false lock proba. Without bility ( ), the probability that is neither nor
, , and represent the false lock probabiliwhere ties for the optimal, suboptimal, and conventional decision rules, respectively, and , , and . The performance of the three decision rules is simulated with a PN chips generated from an -sequence with the code of . primitive polynomial Fig. 1 shows the false lock probabilities of the three deciand . In this figure, is almost sion rules when when , while is very close to the same as when . When , the acquisition model reduces to that for the chip-synchronous case. Thus, as we mentioned in the Introduction, the conventional decision rule is optimum. In fact, since the chip-synchronous model is a special case of the chip-asynchronous model, the optimum decision rule (7) becomes the conventional decision rule (which is optimum for the : specifically, becomes chip-synchronous case) when when . Due to the monotonicity of the modi, simplifies to , which is the confied Bessel function , all of the signal ventional decision statistic . When components are contained in . Thus, the addition of any other as in is simply to increase the noise correlator outputs to variance of the statistic, and consequently the performance bewhich uses only . comes slightly worse than that of On the other hand, the optimal and suboptimal decision rules dB over the convenyield an improvement of roughly when . When tional decision rule in most values of , the signal components are equally divided into and . In the optimal and suboptimal decision rules, due to and being used jointly for detection, the average energy for detection is (ideally) twice that of the conventional decision rule in and are individually used. Hence, under the same which
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Fig. 2. The averaged false lock probabilities of the optimal, suboptimal, and conventional decision rules.
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for averaging over , samples of generated uniformly over were used. In this figure, we can see that the performance of the suboptimal decision rule is very close to that of the optimal decision rule. It is also observed that the optimal and suboptimal decision rules can achieve about 1 dB gain over the conventional decision rule. This can be explained as follows. In the conventional decision rule, the two correlator outputs which contain the signal components are individually used for detection: in the optimal and suboptimal decision rules, on the other hand, the two correlator outputs are used jointly for detection. Thus, the signal energy corresponding to each output is efficiently combined and then used for detection. That is, in the optimal and suboptimal decision rules, we can obtain more accurate information on the signal from the two reliable outputs during the detection process. Therefore, the performance of the optimal and suboptimal decision rules is better than that of the conventional decision rule. REFERENCES
condition, the optimal and suboptimal decision rules might lead to up to 3 dB of SNR/chip increase compared to the conventional decision rule. However, in optimal and suboptimal decision rules, the noise variances also increase because the noise components in the two correlator outputs are also combined. Thus, the maximum 3 dB gain is not fully achieved. The actual gain is lower than 3 dB, as shown in Fig. 1. Fig. 2 shows the averaged false lock probabilities ( denotes expectation over ) of the three decision rules. Here,
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