Detection of Aperiodically Embedded Synchronization ... - IEEE Xplore

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IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 54, NO. 11, NOVEMBER 2006

Detection of Aperiodically Embedded Synchronization Patterns on Rayleigh Fading Channels Arkady Kopansky and Maja Bystrom

Abstract—Correct reception of synchronization patterns aperiodically embedded in random data is vital for robust data transmission. In this letter, we derive general metrics and high-signalto-noise ratio approximations to the metrics for random-length frames transmitted over Rayleigh fading channels. Simulation results are presented for commonly used modulation schemes with coherent and noncoherent demodulation. Index Terms—Fading, signal detection, synchronization.

I. INTRODUCTION

I

N ORDER TO aid in the reception or decoding of transmitted data, synchronization patterns may be embedded in the data prior to transmission. The field of fixed-length frame synchronization, in which patterns are periodically embedded in the random data, has been thoroughly explored [1]–[5]. Transmission of data in equal length frames over the flat-fading channel was considered by Robertson [6] and by Kopansky and Bystrom [7] for the cases of -ary coherent and noncoherent detection, respectively. Of interest is the case of aperiodically embedded synchronization patterns which may arise when framed data is transmitted at unequal intervals, or when patterns are randomly embedded in compressed data. Matzner et al. [8] considered this problem for binary phase-shift keying (BPSK) transmission over an additive white Gaussian noise (AWGN) channel, and derived a detection metric for this case. Their experimental results, however, indicated that the metric for detection of aperiodically embedded patterns has performance similar to that of the periodic metric. The results in [9] for BPSK transmission over an AWGN channel suggest that, while for some patterns at signal-to-noise ratios (SNRs) above 3 dB, performance of the periodic metric approaches that of the aperiodic metric of [9], the aperiodic metric performs substantially better than the periodic metric for other patterns and with higher-order modulation schemes. To summarize the differences between [8] and this letter, a different, more easily implementable metric is derived, applicable to higher order modulation schemes and allowing for the same or different patterns to denote beginning of frames. Paper approved by L. Vandendorpe, the Editor for Transmission Systems of the IEEE Communications Society. Manuscript received March 15, 2001; revised June 15, 2004. This work was supported by the National Science Foundation under Award CCR-9875582. This paper was presented in part at the Conference on Information Sciences and Systems, Baltimore, MD, March 2001. A. Kopansky was with the Electrical and Computer Engineering Department, Drexel University, Philadelphia, PA 19104 USA. He is now with Sarnoff Corporation, Princeton, NJ 08540 USA (e-mail: [email protected]). M. Bystrom was with the Electrical and Computer Engineering Department, Drexel University, Philadelphia, PA 19104 USA. She is now with Boston University, Boston, MA 02215 USA (e-mail: [email protected]). Digital Object Identifier 10.1109/TCOMM.2006.884822

Furthermore, in this letter, we extend the results of [9] and [10] to the Rayleigh fading channel and noncoherent demodulation. It is assumed that the distribution of synchronization codes within the random data is known, and that carrier offset and symbol timing have been acquired. In the following section, the aperiodic detection rule for coherent data transmission in flat fading is first derived as an extension of [10] for the case of an arbitrary number of synchronization patterns with constant or varying lengths. Then, the corresponding metric for noncoherent data transmission is derived. Approximations -ary detection rules for to the coherent and noncoherent high channel SNRs are given in Section III. In Section IV, performance simulations on a Rayleigh fading channel for -ary transmission are given.

II. DETECTION METRICS The optimum detection rules for both coherent and non-ary transmission on Rayleigh fading channels coherent are derived in this section for arbitrary numbers and lengths of synchronization patterns. Consider a data stream in which synchronization codes are aperiodically embedded with a known distribution. Suppose that random data immediately followed by a synchronization pattern is defined as a frame of unknown length. The lengths of both the random data portion of the frame and the synchronization pattern may vary by frame. A total of frame lengths is permitted. There are possible synchronization patterns with corre. Then a frame of length sponding lengths includes synchronization pattern occurring .1 The frame consists of symbols of with probability random data , followed by synchronization symbols . Since the frame data is random, the synchronization pattern may denote a set of obreappear in the data. Let served received symbols, where is the size of the largest frame. These symbols correspond to transmitted data , is random data of length . At least one where pattern will be present in the observed data. To minimize probamust be maximized. bility of error, is fixed for the symbols under consideration, we Since instead may maximize . The joint probability of trans. mitting and observing is Summing over all random data and , the probability may be obtained. Finally, eliminating factors independent of will result in the desired metric , 1Note that to account for the case of a synchronization pattern occurring at different frame lengths s = s , n = n for some i = j; 1 i; j K .

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similar to [10], for detection of aperiodically embedded synchronization patterns. This metric may be rewritten as

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It is assumed that perfect channel state information (CSI) is denote the available at the receiver. Let set of possible symbols. Then, a -dimensional vector can also represent each symbol , and (4)

(1) where (2) is the probability of receiving symbol , given that was transmitted, and spans the set of input symbol symbols. The last term in (1) is a correction term, taking into account the synchronization pattern probabilities. Note that this is similar to the metric of [11] for decoding variable-length codes. To determine the metric for reception in Rayleigh fading, we first consider the coherent detection case. The transmitted -symbol synchronization pattern will be denoted by , where , , and will be represented by -dimensional vectors in the signal space, . Let denote the norm, and the fading with amplitude for the th symbol. Then

(3)

Making use of (2)–(4) in (1), the metric shown in (5) at the bottom of the page can be obtained, where denotes the inner product. The first term of (5) corresponds to the correlation beand received tween the transmitted synchronization pattern symbols , while the remaining terms are correction terms. A similar derivation holds for the case when noncoherent detection is used. Once again, (1) is used as a starting point. In [7], we get (6), as shown it was shown that with and are the in-phase at the bottom of the page, where and the quadrature components of the demodulator output, re. Making use of spectively. A similar result holds for (2) and (6) in (1), the metric for noncoherent detection is then as shown in (7) at the bottom of the page. Note that for equiprob, or if all synchronization patterns are of equal able symbols length, (5) and (7) can be further simplified. For location of synchronization patterns using the above detection metrics, the following method is employed. A set of symbols starting from a reference point is observed. For a sequence of frames with lengths , drawn from , the synchronization pattern is dea distribution , termined to start at position in the observed data if (8)

(5)

(6)

(7)

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where is the number of possible frame lengths. Then serves as the new reference point for detection of the next synchronization pattern in the stream. To summarize the synchronization pattern-detection algorithm, the frame-length distribution, or equivalently, the probabilities for synchronization pattern-starting positions, , is assumed to be known prior to detection. CSI, that is, and , is assumed to be known at the synchronizer. The received symbols, and synchronizer observes the initial searches for each pattern starting position by examining all positions at which synchronization patterns are permitted to start and computing the relevant metric, either (5) or (7). The at which the maximum of the metric position is achieved, according to (8), is declared the synchronization pattern-starting position and, in order to locate the next synchronization pattern in the symbol stream, the search is repeated symbols following the detected pattern. Given this with the detection method, if a synchronization pattern is incorrectly detected, there is the possibility of error propagation. III. HIGH-SNR APPROXIMATIONS TO THE APERIODIC DETECTION RULES In order to slightly reduce computation time, approximations to the metrics derived above can be developed. As in [6], assume and approximate the second term of (5) by equiprobable

(9)

where . This approximation is valid only for low-noise cases. Thus, the high-SNR rule for coherent detection is shown in (10) at the bottom of the page. In the case of noncoherent detection, following the derivations of [5] and [7] yields (11), shown at the bottom of the page, where . These high-SNR approximations are similar

to those given in [6] and [7] for the periodic detection metric, . except for the addition of the correction terms IV. RESULTS Performance of the aperiodic detection rules and their high-SNR approximations was evaluated through simulations, with 20 000 iterations in each considered case. Transmission over a Rayleigh fading channel using quaternary (Q)PSK for coherent detection, and 8-frequency-shift keying (FSK) for noncoherent detection was examined. The performance of the aperiodic rule is compared with that of the periodic rule [the first two terms of (5) or (7)] and with that of the correlation rule [the first term of (5) or (7)]. Two sets of frame lengths with cardinaliand were employed and the distributions ties of the lengths were discretized Gaussian. The frame length is in symbols for all cases and the average symbol energy is . For the case of 100 possible frame normalized, i.e., lengths, the length in symbols of the th frame is given by the , set of frame lengths which was chosen to roughly correspond to MPEG-4 interval for resynchronization marker insertion. The distribution of the frame lengths has a mean of 160 and standard deviation of 10. The case of 28 possible frame lengths was also considered in order to compare results presented here with those of [6]. In this case, the set of frame lengths in symbols is , with a mean of 132 and standard deviation of 3. The percentage of correctly detected synchronization patterns as a function of the average , is shown. symbol SNR, A standard Gray-mapped constellation from [12] was used for QPSK. The synchronization patterns and their corresponding vector representations are given in Table I. Patterns 1 and 2 are the 7-symbol and 13-symbol synchronization patterns, with the Barker sequence or the Neuman–Hofman sequence [2] present in the in-phase component of the vector representation, respectively. Patterns 3–5 are consistent with those of [5]. Note that in most of the simulations, the same pattern is used to denote beginning of frames of different sizes, unless otherwise noted. Simulation results for QPSK using the 7-symbol and 13-symbol synchronization patterns, Patterns 1 and 2, are frame lengths at an SNR of 3 dB, shown in Fig. 1. For

(10)

(11)


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TABLE I SYNCHRONIZATION PATTERNS AND CORRESPONDING CONSTELLATION POINTS

Fig. 1. QPSK synchronization performance,

K = 28.

99.79% of the synchronization patterns can be detected using the 13-symbol synchronization pattern. It should be noted, as pointed out in [3] and [4], that since the synchronization pattern is allowed to reappear in the random data, 100% synchronization is not possible, even on high-quality channels. When applying the periodic rule, achieved performance is consistent with that reported in [6]. Note that at lower SNRs, the performance gains provided over the periodic rule by employing the aperiodic rule are substantial (in this case, as high as 3 dB). Therefore, employing knowledge of the pattern distribution improved detection significantly. This advantage is especially apparent for the case when multiple patterns are embedded in the bitstream, for example, when both 7- or 13-symbol patterns are permitted. At SNRs above 3 dB, there is little performance penalty, as the figure demonstrates. This can be attributed to the fact that the metric (5) takes into account unequal pattern lengths. Employing the high-SNR rule, (10) carries essentially no performance penalty at SNRs above approximately 3 dB, and small performance loss at all channel SNRs considered. Simulation results for QPSK transmission for the case of a data packet embedded in noise are shown in Fig. 2. The 13-symbol Pattern 2 is used to denote the beginning of the packet, and the packet length is 124 symbols. Note that packet length in this case is not important, since only the valid synchronization-pattern starting positions must be considered for metric evaluation. It is assumed that probability of a packet

Fig. 2. QPSK synchronization performance for packets in noise,

Fig. 3. 8-FSK synchronization performance,

K = 28.

K = 100.

, is exponentially starting at position . As the figure shows, distributed with parameter employing the rule of (5) results in performance gain of approximately 2 dB, as compared with the correlation rule at lower SNRs. Finally, results for 8-FSK transmission are presented in Fig. 3. Here, Patterns 3–5 are employed. As expected, using

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the long 16-symbol Pattern 5 provides the best performance, with detection of 98.3% of embedded synchronization patterns at SNR of 3 dB and approximately 100% at SNR of 11 dB. There is approximately 1 dB of performance loss associated with employing the 12-symbol synchronization pattern instead of the 16-symbol synchronization pattern. Furthermore, using the short 8-symbol synchronization pattern results in additional performance loss of approximately 1 dB. V. CONCLUSIONS This letter presented development of the optimum rules for detection of synchronization patterns aperiodically embedded in random data transmitted over the Rayleigh fading channel. Simulated results for both coherent detection of data transmitted over the Rayleigh fading channel with PSK, as well as noncoherent detection with FSK signaling were shown. Discretized Gaussian distributions were selected for 100 or 28 different frame lengths to simulate compressed video synchronization. A simulation for the case of a data packet embedded in noise with an exponential synchronization-pattern starting position distribution was also performed. Results indicate that in many cases, synchronization patterns can be reliably detected, even at channel SNRs of approximately 5 dB, and the use of knowledge of the pattern distribution improves detection significantly. Reappearance of the synchronization sequence in the random frame data is the only factor limiting performance at high SNRs. For channel SNRs of 3 dB and above, in order to reduce computational complexity, a high-SNR approximation

to the exact rule may be used with essentially no performance penalty for coherent detection, and a small performance loss for noncoherent detection. REFERENCES [1] R. H. Barker, “Group synchronization of binary digital systems,” in Communication Theory, W. Jackson, Ed.. London, U.K.: Butterworth, 1953, pp. 273–287. [2] J. L. Massey, “Optimum frame synchronization,” IEEE Trans. Commun., vol. IT-20, no. 4, pp. 115–119, Apr. 1972. [3] P. T. Nielsen, “Some optimum and suboptimum frame synchronizers for binary data in Gaussian noise,” IEEE Trans. Commun., vol. IT-21, no. 6, pp. 770–772, Jun. 1973. [4] G. L. Lui and H. H. Tan, “Frame synchronization for direct detection optical communications systems,” IEEE Trans. Commun., vol. COM-34, no. 3, pp. 227–237, Mar. 1986. [5] ——, “Frame synchronization for Gaussian channels,” IEEE Trans. Commun., vol. COM-35, no. 8, pp. 818–829, Aug. 1987. [6] P. Robertson, “Maximum likelihood frame synchronization for flat fading channels,” in Proc. Int. Conf. Commun., Jun. 1992, pp. 1426–1430. [7] A. Kopansky and M. Bystrom, “Frame synchronization for noncoherent demodulation on flat fading channels,” in Proc. Int. Conf. Commun., 2000, pp. 312–316. [8] R. Matzner, P. Eck, and X. Changsong, “On MPEG-2 decoding of noisy input data,” in Proc. IEEE Int. Conf. Image Process., Lausanne, Switzerland, 1996, pp. 751–754. [9] A. Kopansky and M. Bystrom, “Detection of aperiodically embedded synchronization patterns,” IEEE Trans. Wireless Commun., vol. 3, no. 5, pp. 1386–1392, Sep. 2004. [10] ——, “Detection of aperiodically embedded synchronization patterns in Rayleigh fading,” in Proc. Conf. Inf. Sci. Syst., Baltimore, MD, Mar. 2001, pp. 531–536. [11] J. L. Massey, “Variable-length codes and the Fano metric,” IEEE Trans. Inf. Theory, vol. IT-18, no. 1, pp. 196–198, Jan. 1972. [12] J. G. Proakis, Digital Communications. New York: McGraw-Hill, 1995.