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K -Means Clustering-Based Data Detection and Symbol-Timing Recovery for Burst-Mode Optical Receiver
Tong Zhao, Student Member, IEEE, Arye Nehorai, Fellow, IEEE, and Boaz Porat, Fellow, IEEE
Abstract—Burst-mode receivers are key components of optical transmission systems, including passive optical networks, and have received much attention in recent years. We present new, efficient methods for burst optical signal detection in burst-mode data transmission using a modified -means clustering technique. We also develop a data-aided feedforward symbol-timing recovery method based on a polynomial interpolation and maximum-likelihood estimation theory. A performance criterion considering the error caused by the interpolation approximation is derived for this method. The proposed detection and timing recovery approaches can be implemented effectively and rapidly; therefore, they are very suitable for burst-mode receivers. We also provide some numerical examples to demonstrate the performance of the proposed methods.
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Index Terms—Burst-mode receiver, -means clustering, polynomial interpolation, signal detection, symbol-timing recovery.
I. INTRODUCTION
A
S HIGH-SPEED optical communications rapidly develop, burst-mode data-transmission systems in passive optical networks (PONs) are being investigated extensively. A PON is a point-to-multipoint optical network. Burst-mode transmission is typically used in the upstream in order to facilitate time-division multiple access (TDMA) of multiple optical network units (ONUs) to a single optical line terminal (OLT) [1], [2]. A significant feature of burst-mode data transmissions is that due to unequal distances between the central office and ONUs, optical signal attenuation in PONs is not the same for all ONUs. Therefore, conventional receivers are not suitable for burst-mode data transmissions, because they cannot handle different arriving frames with large differences in optical power levels. Recently, burst-mode receivers have received much attention [3]–[8]. Except for being employed in PONs, burst-mode receivers
Paper approved by C. Tepedelenlioglu, the Editor for Transmission Systems of the IEEE Communications Society. Manuscript received March 25, 2005; revised October 24, 2005 and January 23, 2006. This work was supported in part by the Air Force Office of Scientific Research under Grant F49620-02-1-0339, and in part by the National Science Foundation under Grants CCR-0105334 and CCR-0330342. This paper was presented in part at the IEEE International Conference on Acoustics, Speech, and Signal Processing, Montreal, QC, Canada, May 2004. T. Zhao and A. Nehorai are with the Department of Electrical and Systems Engineering, Washington University, St. Louis, MO 63130 USA (e-mail:
[email protected]). B. Porat is with the Department of Electrical Engineering, Technion-Israel Institute of Technology, Haifa 32000, Israel. Digital Object Identifier 10.1109/TCOMM.2006.878840
are also suitable for many other high-speed optical multiaccess network applications, for example, optical bus networks, wavelength-division multiple-access (WDMA) optical star network, multichannel parallel optical data link, supervisory system for undersea/long-hauled erbium-doped fiber amplifier transmission systems, etc. In this paper, we address two key issues on burst-mode receivers, namely, the burst optical signal detection and the synchronization. First, we present a new, efficient burst-mode signal-detection scheme. Conventionally, to realize burst-mode reception, a preamble is added to the beginning of each burst, consisting of a guard field, an amplitude recovery field, and a clock recovery field. An adaptive threshold control (AGC) circuit is used to determine the detection threshold from the preamble [4], [6]. Due to the short length of the preamble, the established threshold is susceptible to corruption by the reception noise, resulting a detection performance degradation, as compared with continuousmode systems which average a large number of bits to create the threshold [5]. In our proposed new scheme, a burst optical signal detection is realized by applying a two-step data-clustering method based on a -means algorithm. As is known, the -means clustering technique is a simple and efficient method to realize a high processing speed that has recently been applied successfully to digital communications [9]–[11]. Hence, by using a modified -means data-clustering algorithm, we develop a simple and rapidly processed detection method. In our algorithm, since we use all the bits in a burst to establish the threshold, and we also consider the effects of the unbalanced ratio of bits zero and one, we improve the detection performance substantially, compared with the ordinary burst-mode receiver. Another advantage is that the preamble field for the amplitude recovery is not necessary in our scheme, which increases the burst-mode transmission efficiency. We also analyze the performance penalty of our algorithm when there exists intersymbol interference (ISI) in the channel. The task of timing synchronization is another important issue in burst-mode receivers. In burst-mode data transmission, synchronization must be performed very quickly, normally within a limited period of time at the start of each data burst (frame preamble). However, some synchronization methods use feedback schemes, which lead to long acquisition times. Some feedforward timing-recovery techniques are based on restrictive assumptions concerning the statistical distribution of the received signals, and will be effective only when a lot of data are collected [12], [13]. Recently, polynomial interpolation has been introduced as a simple and efficient technique to implement the
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-MEANS CLUSTERING-BASED DATA DETECTION AND SYMBOL-TIMING RECOVERY FOR BURST-MODE OPTICAL RECEIVER
synchronization in digital modems [14]–[17]. In this paper, we present a data-aided (DA) feedforward symbol-timing recovery method based on interpolations. Our method can be applied to a very low oversampling ratio (even for a noninteger oversampling ratio between 1 and 2), while maintaining an acceptable synchronization performance. This method is effective, can be performed quickly, and therefore, is very suitable for burstmode data transmissions. We also derive a performance criterion for this type of synchronizer, in which the errors caused by the interpolation approximation are considered. This criterion is useful for evaluating the performance of different symboltiming recovery methods and optimally selecting of the system parameters. The received signal-model and signal-detection methods based on the -means clustering techniques are presented in Section II. The interpolation-based symbol-timing recovery method and its performance analysis are given in Section III. Numerical examples are presented in Section IV to demonstrate the performance of the proposed signal-detection and timing-recovery methods. Finally, conclusions and future work are given in Section V. II. BURST OPTICAL SIGNAL DETECTION In this section, we present a new, efficient signal-detection approach in a burst-mode optical signal transmission. We first develop the received signal model in the burst-mode receiver; then we derive a two-step burst signal-detection algorithm based on -means clustering technology; finally, we analyze the performance penalty when there exists ISI in the channel. A. Received Signal Model
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followed by high impedance or transimpendance preamplifiers, and some type of avalanche photodiodes (APD) receiver, that are far from the sensitivity limit and limited by amplifier rather than detector noise. The AWGN assumption is commonly used in the performance analysis for a burst-mode optical receiver, e.g., in [3], [5], and [6]. We will find that even though we use a Gaussian assumption, the proposed method can be applied to other reception noise-distribution assumptions, since our method is based on clustering technology that is a nonparametric detection approach (However, when the noise satisfies the Gaussian approximation, the detection algorithm will obtain the best performance.) We only implement the Gaussian assumption when we analyze the performance penalty in the case of ISI. According to this model, the burst-mode signal-detection problem can be considered as the following binary hypothesis test [18]:
(2) represents that bit 0 is transmitted, and represents where that bit 1 is transmitted. Conventionally, we could solve this detection problem using maximum-likelihood sequence estimation (MLSE) implemented by the Viterbi algorithm [19]. But this method is very computationally complex, and not suitable for high-speed burst-mode data transmission. In the following sections, we will present a novel and efficient approach suitable for the burst signal detection. B. Burst Optical Signal-Detection Algorithm
A typical burst-mode optical signal is in the form of a frame with a number of binary signals. Each frame has a preamble of known bits that should be kept as short as possible to increase transmission efficiency. After the preamble comes the payload. Each binary signal is transmitted in the form of very and correspond to short pulses. Pulses of power level binary 0 and 1, respectively. Usually the received signal is distorted by the channel, which, in this case, consists of the optical fiber, the photodetector, the preamplifier, and a noise filter. Therefore, we cannot rely on the knowledge of power levels and to design the burst-mode receiver. Under this situation, we propose the following received signal mode. We first assume that there is no ISI in the reception, and each pulse is sampled at the optimal sampling instant. Then the received burst signal model is in the form of
(1) is the noise-free where is the number of bits in one frame, signal which is modeled as a deterministic but unknown is the reception noise that is approximated by constant, a zero-mean white Gaussian random variable with unknown . Additive white Gaussian variance , i.e., noise (AWGN) approximation applies to the usual optical receiver based on positive-intrinsic-negative (PIN) photodiodes,
In the conventional burst-mode receiver, the pulse amplitudes and are recovered from the preamble, and the detection threshold is set to be halfway between both bits’ amplitudes, assuming that noise is averaged to zero, i.e.,
(3) This approach to establish the decision threshold has two intrinsic effects that degrade the detection performance of the burst-mode receiver. First, since the length of the preamble is short, the decision threshold is corrupted by the reception noise, which introduces a sensitivity penalty. Next, only under the situation that the number of bits 1 equals 0, the threshold in (3) is optimal in the sense that the bit-error rate (BER) is minimized. However, in burst-mode signal transmission, the ratio of 0’s and 1’s is not constrained [4], [6]. That means an unbalanced ratio of 0’s and 1’s exists. Under this case, the optimal decision threshold is derived as follows. When we apply the Bayesian approach of minimizing the probability of error to the hypothesis test (2), we find that we should decide that is true if
(4)
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where is the likelihood function when the hypothesis is true, for is the prior probability of the corresponding hypothesis [18]. Under the Gaussian noise assumption, the likelihood ratio in (4) is given by
the optimal one, as we derived in (6). Since the parameters in (6) are unknown at the receiver, we will estimate them using the results of the initial partition. Then the final detection threshold is calculated as
(8) where the unknown parameters are estimated by applying the initial partition results into the following formulas: (5) After the simplification of the above equations, the Bayes test in itself with another threshold (4) is equivalent to comparing , which is (6) for . where In order to overcome the problems of the conventional burstmode receiver, we propose a new detection method in which we use all the received data to average out the noise, and we also take into account the unbalanced ratio of 0’s and 1’s in the received burst. The basic idea is that by observing that there are and in the distribution of the two clusters centered at received signals, we implement a data-clustering algorithm to divide the received data into two clusters corresponding to the binary 0 and 1, respectively. For this type of clustering technology, we use the -means algorithm since it is simple, efficient, and therefore suitable for realizing high-speed processing while maintaining a good performance. However, the conventional -means is an iterative method. It may take several iterations to converge, which is not suitable for high-speed processing in burst-mode receivers. And also according to the criterion used by the -means algorithm, only under the assumption that the prior probabilities of each cluster are the same, the algorithm may obtain the optimal partition result [9]. This is not suitable for the case of the unbalanced ratio of 0’s and 1’s in burst-mode signal. Hence, we modify the conventional -means and propose a new and efficient two-step burst-mode signal-detection algorithm. It includes an initial partition step and a partition adjustment step. Initial Partition: In this step, we partition the received data into two clusters by comparing the received data with an initial threshold. The initial threshold is set to be the average value of denote the received the received signal. Let data in a burst to be clustered, then the initial threshold, denoted as , is
(7) We denote the partition results as and that are the subsets of and correspond to the binary 0 and 1, respectively. We also and to be and . denote the number of data in Partition Adjustment: In this step, we adjust the initial partition result by replacing the initial threshold to be
(9a) (9b)
(9c) We can find that the proposed detection algorithm is similar, with a -means clustering with only two iterations. However, as we investigated using the simulations, the detection performance of the new algorithm is close to the analytical result, in which we use the optimal threshold in (6) to make the detection and assume the parameters are all known. Of course, if we process more iterations in our algorithm, the performance will be improved; but it will decrease the processing speed. This means that there exists a tradeoff between the detection performance and processing speed. Several discussion comments about the proposed burst signal-detection method are as follows. 1) The computation cost of the proposed algorithm is very low; hence, it can be implemented to realize a very fast processing speed, which is very important for the burst-mode receiver. First, our algorithm is based on the -means clustering technique, which itself is very simple. Also, in our detection problem, the received data has only one dimension and we need to partition it into only two clusters, which is the simplest case of a clustering problem. Next, we modify the conventional -means to a noniterative algorithm, which avoids the possible long convergence time of the algorithm. 2) In the initial partition step, we set the initial detection threshold to be the average of the received data, since we consider that if the ratio of 0’s and 1’s is equivalent or close to 1, this initial threshold is very close to the optimal threshold. In the proposed algorithm, a preamble for the detection is not needed. However, if such a preamble exists in the burst frame, there is an alterative method to determine the initial threshold. That is, we estimate the and from the preamble, and the pulse amplitudes and , initial threshold is set to be halfway between which is the approach to establish the detection threshold in conventional burst-mode receivers. We will study the performance of these two approaches to establish the initial threshold using numerical examples. 3) Compared with the conventional burst-mode signal-detection methods, the proposed approach improves the detection performance and increases the transmission efficiency,
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-MEANS CLUSTERING-BASED DATA DETECTION AND SYMBOL-TIMING RECOVERY FOR BURST-MODE OPTICAL RECEIVER
while still maintaining a fast processing speed. The cost is that at the beginning of the transmission, there exists a delay equivalent to the time required to buffer and process the first arrived burst. Since in our algorithm, the procedures of buffering and detection can be done in parallel, if the bursts are received continuously, the delay will be very small or will not appear after the first burst.
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TABLE I VALUES OF y WHEN IGNORING THE ADDITIVE NOISE ACCORDING TO THE b1x b1x SYMMETRIC ISI MODEL y x
=
+
+
C. Performance Penalty Analysis in the Presence of ISI In addition to the additive noise, the burst-mode data transmission in optical channels is often impaired by channel ISI. A typical optical channel tends to generate fairly symmetric ISI, for example, chromatic dispersion is totally symmetric; moreover, electrical filtering at the receiver is usually very tight and also creates symmetric ISI. We assume the ISI is mainly due to the effect of the nearest symbols, and the channel characteristics do not change during each frame (since the time of each burst is very short, this assumption is reasonable). Based on the above discussion and assumptions, we propose the following ISI channel model for the burst-mode data transmission: (10) where is the ISI parameter, representing the effects of the nearest symbols at the current sampling instant, and is assumed and are the same as to be deterministic and unknown; defined in (1). In the presence of ISI as in (10), our proposed burst signaldetection algorithm in Section II-B still works. However, there exists a performance degradation, compared with the case when there is no ISI in the channel. In this section, we will analyze this performance penalty. First, we derive the detection performance in terms of prob(which is also called the BER) of the proability of error posed burst signal-detection method when there is no ISI in the channel. According to the hypothesis test in (2) and our detecis determined as tion rule,
, the received signal that when we ignore the additive noise will take one of the six different values according to all pos, and . The relations besible values for the signal and the combinations of , tween different values of and their probabilities are summarized in Table I. Here and are independent. we apply the condition that , According to these relations, we can conclude that the detection problem in the presence of ISI includes six hypotheses, as follows:
(13) correspond to the case in which the hypotheses , that the bit 0 is received; the hypotheses , correspond to the case that the bit 1 is received. It can be verified that when we use the proposed burst signaldetection method in Section II-B to make the decision in the case of symmetric ISI model (10), the detection threshold is
(14) Here, for the simplicity of the derivation, we assume the equal prior . Therefore, following a similar derivation process to (11), the probability of error in the presence of ISI is (11) is the conditional probability that indicates the where is true, probability of determining bit is received when , is the optimal detection threshold, as derived in (6), is defined as . , the probability of Under the situation of equal prior error is simplified as (12) is a simple version of a signal-to-noise where ratio (SNR). In the presence of ISI, the detection performance analysis is more complex. From the symmetric ISI model (10), we observe
(15) where . We note that when , i.e., there is no ISI in the channel, the results in (12) and (15) are the same. as a function In Fig. 1, we draw the probabilities of error of under both cases, where there is no ISI, and there exists ISI, in the channel. We find that the ISI causes a performance penalty, the amount of which varies according to different values
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Fig. 1. BER for the case when there is no ISI, and the cases when there exists ISI under different ISI parameters b.
sampling clock, passes through an interpolator to form a polynomial approximation of the matched filtered signal, and then an ML estimation of the unknown synchronization parameters is applied to this approximated signal. By using this interpolation approximation, we obtain an analytical form of the likelihood function. Numerical methods can be used to calculate the estimation of the synchronization parameters; therefore, a tracking loop, which would lead to a long acquisition time [12], [19], is not needed. As shown in Fig. 2, after passing through the matched filter, the received baseband signal is sampled using the oversampling , where and are the symbol and samratio of pling intervals, respectively. In the digital part of the receiver, and are estimated. For consynchronization parameters venience, we assume that the sampling rate at the output of the interpolator is equal to the symbol rate. We also assume the overto be rational, but not necessary to be sampling ratio an integer. 1) Polynomial Interpolation Approximation: Denote the im. Then, pulse response of the analog interpolating filter by the continuous time output of the filter is
(16)
Fig. 2. Receiver with nonsynchronized sampling and interpolator.
of . For example, if the probability of error is fixed to and , there is a 1.10 dB performance penalty. An intuitive explanation of this performance penalty is that when there exists ISI in the channel, if we apply the proposed detection method to partition the received data into two clusters and use Gaussian density to approximate them, then the variance of each Gaussian density increases due to the influence of ISI. Therefore, the SNR in (12) will decrease, resulting a degradation of the detection performance. III. SYMBOL-TIMING RECOVERY In digital communication systems, ML estimation theory provides a general framework for developing optimal [with respect to the Cramér–Rao bound (CRB)] synchronization schemes. However, conventional ML-based symbol-timing recovery is usually accomplished using a feedback method; that is, a tracking loop is needed to estimate the synchronization parameters. This leads to long acquisition times, which are not suitable for burst-mode data transmission. In this section, we present a DA feedforward symbol-timing recovery method, based on a polynomial interpolation and ML estimation theory. We also derive a performance criterion for this type of synchronizer. The proposed method is implemented effectively and rapidly, and thus, is suitable for the burst-mode data transmission.
at the time instants , where is We resample synchronized with the signal symbols, and is the time delay between this resampling and the original free-running sampling which is modeled as an unknown constant. The new samples are represented by (17) A more useful format is obtained by rearranging the indexing in (17). Define a filter index
(18) means the largest integer not exceeding . Also where define a basepoint index
(19) and a fractional interval
(20) . When the oversampling ratio is where an integer, will be a constant; otherwise, will cyclicly repeat a finite set of values when the timing loop is in equilibrium. Using the above definitions, (17) can be rewritten as
A. Interpolation-Based Feedforward Symbol-Timing Recovery The proposed method for symbol-timing recovery is represented in Fig. 2. The data, after being sampled by a free-running
(21)
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If the interpolation filter has a finite impulse response (FIR), then and are fixed finite numbers, and the digital filter actaps. tually used for computing the interpolants has We design our interpolator to be efficiently implemented by applying the approach devised by Farrow [20]. According to the relationships in (18)–(20), the classical Lagrange polynomial can be rewritten as a polynomial with interpolating filter respect to
(22) are interpolator coefficients that are fixed numHere, bers, independent of , determined solely by the filter’s im. Substitute (22) into (21) and rearrange pulse response terms to show that the interpolants can be computed from
Fig. 3. DA symbol-timing synchronization scheme.
Hence, the synchronization parameters can be estimated as
(27)
(23)
(see [14] and [15] for where from to , details). Therefore, when we vary from 0 to 1 continuously for each , the output of and vary the interpolation filter forms a polynomial approxi, which is the output mation for the continuous-time signal . of the matched filter, in the interval 2) ML Estimation: From [19], we know that in order to estimate the synchronization parameter, the log-likelihood function (LLF) we want to maximize is
(24) where is a constant, and
and the value of the output at the desired time instant is determined by (23). The scheme for this synchronization method is shown in Fig. 3. B. Performance Characteristics In the proposed symbol-timing recovery method, we implement the interpolation-based ML estimator to obtain the estimation of the unknown synchronization parameter . The quality of a signal-parameter estimate is usually measured in terms of its bias and variance, where, in general, the variance is difficult to compute. However, a well-known result in unbiased parameter estimation is the CRB on the variance. Assume we have a , with a probasequence of observations from which we extract an esbility density distribution timate of the parameter ; the CRB is defined as (28)
is defined as
(25) is the which is the output of the matched filter. Here, impulse response of the matched filter. are the transmitted binary digits. Because we use the DA symbol-timing recovery are assumed to be known in the algorithm. scheme, Because in (23) we already used the interpolation to obtain a polynomial approximation of the output signal of the matched filter, by substituting this approximation into (24), we achieve the polynomial approximation of the LLF as
(26)
However, in the proposed new ML estimator, we use the Lagrange interpolation to form an approximation of the received matched-filtered signal. Hence, the conventional CRB for the estimate of synchronization parameter , as derived in [19], cannot be used as a criterion for the performance analysis. The error caused by the interpolation should be considered in the CRB of this new ML method. The error of the Lagrange interpolation polynomial was studied by Henrici in his book [21], and some further research results can be found in [22]. The main results for the Lagrange interpolation error is concluded in the following theorem (see [21] for the details of the proof). Theorem 3.1: Let the real function be defined on an interval , and be distinct points of . Let represent the th-order Lagrange interpolation polynomial of , and be times continuously differentiable on the , there exists a point located interval . Then to each
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in the smallest interval containing the points such that
(29) . The quantity in (29) can be defined as a continuous function of for . By considering that in the proposed ML method we approximate the output signal of the matched filter using a Lagrange interpolation polynomial, and by applying the above theorem, we can conclude that the proposed method could be considered a conventional ML estimation under a different signal model, such that the output of the matched filter in this new model is the same as the Lagrange interpolation polynomial approximation. Therefore, we can directly replace the matched-filter in (24) with its Lagrange interpolation polynomial output in (29) and obtain the CRB of the proposed new ML estimator as where
Fig. 4. BER versus SNR for a prior probability p = 0:2. (a) Analytical result. (b) Result of the new algorithm where the initial threshold is from the preamble. (c) Result of the new algorithm where the initial threshold is set to be the average value.
(30) where
(31) This lower bound is useful for analyzing the performance of parameter estimates, and can also be implemented as a criterion to optimally design the synchronization system. IV. NUMERICAL EXAMPLES In this section, we present some numerical examples to analyze the performance of the proposed burst-mode optical signaldetection algorithm and the interpolation-based symbol-timing recovery method.
Fig. 5. BER versus SNR for a prior probability p = 0:4. (a) Analytical result. (b) Result of the new algorithm where the initial threshold is from the preamble. (c) Result of the new algorithm where the initial threshold is set to be the average value.
A. Burst-Mode Optical Signal Detection We investigate the performance of the proposed burst-mode optical signal-detection method, and the influence of some important elements, such as the ratio of 0’s and 1’s, the initial threshold-establishing scheme, and the SNR. In our examples, . We also the SNR is defined as SNR compare the detection performance of our new algorithm with the detection methods used in ordinary burst-mode receivers. We simulate the proposed detection method in an AWGN channel. We consider the situations when there is no ISI in the channel (the performance penalty caused by ISI in the channel has been analyzed in Section II-C). In the first example, we compute the BERs of the proposed burst signal-detection algorithm as a function of the SNR, and compare them with the analytical result. The analytical result is obtained using the equation in (11). The threshold is set to
be the optimal one, as derived in (6). We also study the effects of the initial threshold-establishing scheme (setting the initial threshold using the preamble and setting the initial threshold to be the average value), and the ratio of 0’s and 1’s. We set . The length of the burst we test is 1000 bits. The results with various prior probabilities of the bit 0, i.e., and , are shown in Figs. 4 and 5, respectively. The BER we obtained is averaged by testing the bursts repeatedly. From these results, we observe that the proposed burst signal-detection algorithm works well, since its performance is very close to the analytical results, especially when the ratio of 0’s and 1’s approaches 1. We also find that when we establish the initial threshold using the preamble, the detection performance is better than using the average value, especially when the ratio of 0’s and 1’s is far from 1. The reason is that when the
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Fig. 6. BER versus SNR for a prior probability p = 0:3. (a) Analytical result. (b) Result of the new algorithm where the initial threshold is from the preamble. (c) Result of the new algorithm where the initial threshold is set to be the average value. (d) Result of the ordinary burst-mode signal-detection method.
ratio of 0’s and 1’s is far from 1, the average value which can is very different from the be approximated by optimal threshold as in (6). Therefore, the two-step -means algorithm cannot converge to the optimal partition results. In the second example, we compare the performance of the proposed detection algorithm with the method used in ordinary burst-mode receivers. In ordinary burst-mode receivers, the deand which tection threshold is set to be halfway between are recovered from the preamble. In our comparison, we let and the length of preamble be 4 bits. We simulate the proposed methods with both initial threshold-establishing schemes. The result is in Fig. 6. We find that the performances of the proposed detection methods with both initial threshold-establishing schemes are better than the ordinary burst signal-de, the new algotection methods. For example, at a BER of rithm has a nearly 2 dB performance improvement, compared with the ordinary detection methods. Of course, when we increase the length of the preamble, the detection performance of the ordinary method will be improved. However, it will decrease the transmission efficiency. In our new method, it is not necessary to use a preamble. B. Symbol-Timing Recovery In the proposed symbol-timing recovery method, we use the Lagrange polynomial interpolation. For an -point base, its polynomial interpolation can be point set performed by the Lagrange formulas (32) where
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Fig. 7. Comparison of the waveforms between the original signal (dashed line) and the interpolation approximated signal (solid line) under oversampling ratio = 2.
In the following numerical examples, we use the cubic interpolator; that is, we use the four-point Lagrange polynomial interpolation. The proposed DA symbol-timing recovery method is based on approximating the matched-filtered signal using an interpolator. Hence, in the first example, we compare the waveforms of the original received signal with the interpolation approximated signal. We set the length of the training sequence to be 32, and the two signals are compared under the oversampling . We also set the ratio of 0’s and 1’s to 1 and the ratio SNR to 20 dB. The result is shown in Fig. 7, where represents the symbol interval. We find that the interpolated signal approximates the original signal very well. Comparing the two signals, only some detailed information, which mostly represents the noise, is lost. This means when we use the interpolator, it is as if we pass the received signal through a low-pass filter and remove part of the noise. Therefore, we can expect that when we use the low oversampling ratio, we can still obtain an accurate estimation of the time delay. In the next examples, we investigate the performance of the time-delay estimation and the burst signal detection based on the estimated timing information. In Fig. 8, we compare the root mean squared (RMS) errors of the estimated time delay under different oversampling ratios. We find that, as we discussed above, even using very low oversampling ratio, for in, the estimation accuracy is still very high. For stance example, at SNR dB, the RMS error is below 0.03 symbol period. This estimation accuracy is also related to the used polynomial interpolator, as in (32). A higher order interpolator with more points will provide a higher accuracy. In Fig. 9, we calculate the BERs of the burst signal detection with the estimated time delay and compare them with the analytical results under different oversampling ratios. The analytical result is obtained by sampling the received signal with the true time delay. We observe that even when we use a very low oversampling ratio at , there is not a significant detection performance degradation, compared with the analytical results and the cases when we use the higher oversampling ratio.
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Fig. 8. RMS error of the time-delay estimation versus SNR for different oversampling ratios.
a new detection method based on the -means clustering techniques. The theoretical analysis and the numerical examples show that this method is efficient, can be performed quickly, has very good detection performance, and is thus suitable for the burst-mode optical receiver. We also analyzed the performance penalty of the proposed burst signal-detection method in the presence of ISI. For the symbol-timing recovery, we presented a DA interpolation-based feedforward timing-recovery scheme. By using a feedforward approach instead of a feedback loop, we achieved a fast synchronizer. We investigated the performance under different oversampling ratios and obtained the conclusion that even using a very low oversampling ratio, the detection performance does not degrade dramatically, compared with high oversampling ratios. These characteristics make the proposed methods suitable for burst-mode optical signal transmission, and can also be used in other high-speed transmission systems. We also derived a performance criterion for the proposed method, in which we consider the errors caused by the interpolation approximation. In future work, we will investigate the performance of the detection methods under non-Gaussian noise. We will also study the new detection methods that can be processed more efficiently in the presence of non-Gaussian noise and/or ISI. For the proposed timing-recovery method, the interpolator is a very important device. We are also interested in investigating the effects of different interpolators on the system performance.
REFERENCES
Fig. 9. BER versus SNR for different oversampling ratios.
The oversampling ratio is a very important issue in highspeed burst-mode optical transmission. For example, as proposed for ethernet PON (EPON), the data transmission rate is at 1.25 Gb/s. If the oversampling ratio is high, e.g., four samples per bit, then the traffic to be sent to the digital signal processor/ field-programmable gate array implementing the proposed algorithm would be very large, which may not be feasible with low-cost electronics. However, in the numerical examples, we demonstrate that since in our algorithm a fractional oversampling ratio can be implemented, a very low oversampling ratio (between one and two) can be used without a significant performance degradation. This low oversampling ratio could be reasonable for high-data-rate transmission. Except for the burstmode optical signal transmission, the proposed symbol-timing recovery method is also a potential approach to be used in other high-speed data-transmission systems. V. CONCLUSION We investigated burst optical signal detection and symboltiming recovery in burst-mode optical receivers. We proposed
[1] G. Kramer and G. Pesavento, “Ethernet passive optical network (EPON): Building a next-generation optical access network,” IEEE Commun. Mag., vol. 40, pp. 66–73, Feb. 2002. [2] F. J. Effenberger, H. Ichibangase, and H. Yanashita, “Advances in broadband passive optical networking technologies,” IEEE Commun. Mag., vol. 39, pp. 118–124, Dec. 2001. [3] C. A. Eldering, “Theoretical determination of sensitivity penalty for burst mode fiber optic receivers,” J. Lightw. Technol., vol. 11, no. 12, pp. 2145–2149, Dec. 1993. [4] Y. Ota, R. G. Swartz, V. D. Archer, S. K. Korotky, M. Banu, and A. E. Dunlop, “High-speed, burst-mode, package-capable optical receiver and instantaneous clock recovery for optical bus operation,” J. Lightw. Technol., vol. 12, no. 2, pp. 325–331, Feb. 1994. [5] P. Menéndez-Valdés, “Performance of optical direct receivers using noise corrupted decision threshold,” J. Lightw. Technol., vol. 13, no. 11, pp. 2202–2214, Nov. 1995. [6] C. Su, L. Chen, and K. Cheung, “Theory of burst-mode receiver and its applications in optical multiaccess networks,” J. Lightw. Technol., vol. 15, no. 4, pp. 590–606, Apr. 1997. [7] P. Ossieur, Y. C. Yi, J. Bauwelinck, X. Z. Qiu, J. Vandewege, and E. Gilon, “DC-coupled 1.25 Gbit/s burst-mode receiver with automatic offset compensation,” Electron. Lett., vol. 40, no. 7, pp. 447–448, Apr. 2004. [8] T. Zhao, A. Nehorai, and B. Porat, “New data detection and symbol timing recovery approaches for burst optical signal transmission,” in Proc. IEEE Int. Conf. Acoust., Speech, Signal Process., Montreal, QC, Canada, May 2004, vol. 4, pp. 918–920. [9] A. K. Jain and R. C. Dubes, Algorithms for Clustering Data. Englewood Cliffs, NJ: Prentice-Hall, 1998. [10] J. R. M. Filho, B. Dorizzi, and J. C. M. Mota, “Channel estimation by symmetrical clustering,” IEEE Trans. Signal Process., vol. 50, no. 6, pp. 1459–1469, Jun. 2002. [11] Y. Jeng and C. Yeh, “Cluster-based blind nonlinear-channel estimation,” IEEE Trans. Signal Process., vol. 45, no. 5, pp. 1161–1172, May 1997. [12] H. Meyr and G. Ascheid, Synchronization in Digital Communications. New York: Wiley, 1990, vol. 1.
ZHAO et al.:
-MEANS CLUSTERING-BASED DATA DETECTION AND SYMBOL-TIMING RECOVERY FOR BURST-MODE OPTICAL RECEIVER
[13] H. Meyr, M. Moeneclaey, and S. A. Fechtel, Digital Communication Receivers—Synchronization, Channel Estimation, and Signal Processing. New York: Wiley, 1998. [14] G. M. Gardner, “Interpolation in digital modems—Part I: Fundamentals,” IEEE Trans. Commun., vol. 41, no. 3, pp. 502–508, Mar. 1993. [15] L. Erup, F. M. Gardner, and R. A. Harris, “Interpolation in digital modems—Part II: Implementation and performance,” IEEE Trans. Commun., vol. 41, no. 6, pp. 998–1008, Jun. 1993. [16] R. Hamila, J. Vesma, and M. Renfors, “Polynomial-based maximumlikelihood technique for synchronization in digital receivers,” IEEE Trans. Circuits Syst. II: Analog and Digital Signal Process., vol. 49, no. 8, pp. 567–576, Aug. 2002. [17] G. N. Tavares, L. M. Tavares, and M. S. Piedade, “A new ML-based data-aided feedforward symbol synchronizer for burst-mode transmission,” in Proc. IEEE Int. Symp. Circuits Syst., Geneva, Switzerland, May 2000, vol. 2, pp. 357–360. [18] S. M. Kay, Fundamentals of Statistical Signal Processing—Detection Theory. Englewood Cliffs, NJ: Prentice-Hall, 1998, vol. II. [19] J. G. Proakis, Digital Communications, 4th ed. Boston: McGrawHill, 2001. [20] C. W. Farrow, “A continuously variable digital delay element,” in Proc. IEEE Int. Symp. Circuits, Syst., Espoo, Finland, Jun. 1988, pp. 2641–2645. [21] P. Henrici, Elements of Numerical Analysis. New York: Wiley, 1964. [22] R. Radzyner and P. Bason, “An error bound for Lagrange interpolation of low-pass functions,” IEEE Trans. Inf. Theory, vol. IT-18, no. 9, pp. 669–671, Sep. 1972. Tong Zhao (S’02) received the B.Eng. and M.Sc. degrees in electrical engineering from the University of Science and Technology of China, Hefei, China, in 1997 and 2000, respectively. Currently, he is working toward the D.Sc. degree with the Department of Electrical and Systems Engineering, Washington University, St. Louis, MO. His research interests are statistical signal processing and its applications in radar, communication, biochemical sensor, and wireless sensor networks.
Arye Nehorai (S’80–M’83–SM’90–F’94) received the B.Sc. and M.Sc. degrees in electrical engineering from the Technion, Israel, and the Ph.D. degree in electrical engineering from Stanford University, Stanford, CA. From 1985 to 1995, he was a faculty member with the Department of Electrical Engineering, Yale University, New Haven, CT. In 1995, as Full Professor, he joined the Department of Electrical Engineering and Computer Science, The University of Illinois at Chicago (UIC). From 2000 to 2001, he was Chair of
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the department’s Electrical and Computer Engineering (ECE) Division, which then became a new department. In 2006, he assumed the Chairman position of the Department of Electrical and Systems Engineering at Washington University, St. Louis, MO, where he is also the inaugural holder of the Eugene and Martha Lohman Professorship of Electrical Engineering. He is also the Principal Investigator of the new multidisciplinary university research initiative (MURI) project entitled Adaptive Waveform Diversity for Full Spectral Dominance. Dr. Nehorai was Editor-in-Chief of the IEEE TRANSACTIONS ON SIGNAL PROCESSING during 2000–2002. In 2003–2005, he was Vice President (Publications) of the IEEE Signal Processing Society, Chair of the Publications Board, member of the Board of Governors, and member of the Executive Committee of this Society. He is the founding editor of the special columns on Leadership Reflections in the IEEE Signal Processing Magazine. He was co-recipient of the IEEE SPS 1989 Senior Award for Best Paper with P. Stoica, as well as coauthor of the 2003 Young Author Best Paper Award and of the 2004 Magazine Paper Award with A. Dogandzic. He was elected Distinguished Lecturer of the IEEE SPS for the term 2004 to 2005. He has been a Fellow of the Royal Statistical Society since 1996. In 2001, he was named University Scholar of the University of Illinois.
Boaz Porat (M’82–SM’87–F’93) was born in Haifa, Israel, in 1945. He received the B.S. and M.S. degrees in electrical engineering from the Technion, Haifa, Israel, in 1967 and 1975, respectively, and the M.S. degree in statistics and Ph.D. degree in electrical engineering from Stanford University, Stanford, CA, in 1982. From 1967 to 1972, he served in the Israeli Defense Force as an Electronics Engineer. From 1972 to 1979, and during 1983, he was with RAFAEL, Haifa. From 1979 to 1982, he was a graduate student and a Research Assistant at the Information Systems Laboratory, Stanford University. Since 1983, he has been with the Department of Electrical Engineering at the Technion. During part of 1991 and part of 1995, he was a Visiting Professor at the University of California at Davis. During part of 1993, he was a Visiting Professor at Ben-Gurion University, Beer-Sheba, Israel. During another part of 1993, he was a Visiting Research Associate at Yale University, New Haven, CT. He also spent various periods with Signal Processing Technology, CA, and served as a consultant to electronics industries in Israel on numerous occasions. He was a Co-Founder and Chief Scientist of Savan Communications during the years 1997–2002. Since 2002, he has been an Emeritus Professor of the Technion. He is author of the books Digital Processing of Random Signals: Theory and Methods (Englewood Cliffs, NJ: Prentice-Hall, 1994) and A Course in Digital Signal Processing (New York: Wiley, 1997). Dr. Porat received the European Association for Signal Processing Award for the Best Paper of the Year in 1985; the Ray and Miriam Klein Award for Excellence in Research in 1986; the Technion’s Distinguished Lecturer Award in 1989, 1990, and 1996; the Jacknow Award for Excellence in Teaching in 1994; and the IEEE Signal Processing Society Technical Achievement Award in 1997. He was an Associate Editor of the IEEE TRANSACTIONS ON INFORMATION THEORY from 1990 to 1992, in the area of estimation.
