IEEE SIGNAL PROCESSING LETTERS, VOL. 12, NO. 8, AUGUST 2005
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Low-Complexity Search for Optimal Delay in Linear FIR MMSE Equalization Leszek Szczecinski
Abstract—In this letter, an efficient algorithm to determine the optimal delay in linear finite impulse response equalizers based on the minimum mean-square-error criterion is proposed. The algorithm uses the Levinson–Durbin (L–D) recursion as a starting point to find the values of the mean-square error for equalizers with all nontrivial delays. Despite the exhaustive search approach, the complexity of the proposed algorithm is only doubled when compared to the calculation of the equalizer with one prescribed delay. Such increase in complexity may be fully justified in practice because it yields globally optimal equalizer’s design. Index Terms—Equalization, equalization delay, fast algorithm, finite impulse response (FIR) equalization, Levinson–Durbin (L–D) algorithm, minimum mean-square-error (MMSE) equalization, optimal delay, recursive algorithm.
I. INTRODUCTION
C
HANNEL equalization is one of the fundamental problems in digital communications and consists in removing/reducing the effect of the intersymbol interference (ISI) introduced by frequency selective channels [1, Ch. 10]. Equalizers minimizing the bit-error rate (BER) are nonlinear devices [1, Ch. 10] whose complexity may be prohibitive in practical applications. Therefore, the linear equalizers became very popular due to their low implementation complexity [2], [3, Ch. 8] and very simple adaptation of the coefficients. By far, the most popular and practical adaptation techniques are those using the minimum mean-square-error (MMSE) criterion [1, Ch. 10], [2], [3, Ch. 8.3], [4]. In general, ideal linear equalizers are noncausal infinite impulse response (IIR) filters [3, Ch. 8.3], [2]; however, finite impulse response (FIR) implementation is preferred to avoid instability and to simplify the design [2], [4]. Then, the equalizer’s performance is determined by two parameters : its order (determining the number of taps) and the so-called equalization delay (introduced to mimic noncausal behavior of the IIR equalizer). The order is limited by complexity considerations, i.e., the number of operations per symbol that can be executed at the receiver. Having the order fixed, the equalizer’s performance may be strongly deteriorated by an inappropriately chosen value of the delay [5], [6]. However, no expression for the optimal delay is available, so it is usually chosen using some
heuristics, e.g., the knowledge of the channel to be equalized [4] or as equal to the length of the channel [3, Ch. 8]. To determine the optimal delay, one might also find the equalizers for all nontrivial delays1 and choose the one producing at its output the smallest mean-square error (MSE) [5]. Although this “brute force” approach guarantees the global optimality (in the MMSE sense) of the design, it has the complexity that is prohibitive in practical applications, especially for very long impulse responses of the channel. The contribution of this letter is the following: To find the optimal delay, we calculate the MSE produced by the equalizers with all nontrivial delays (where of order is the maximal nontrivial delay) without explicitly calculating the equalizers, which drastically lowers the complexity ) when compared to the “brute force” ap(by the factor of proach (calculating the equalizers for all delays). We conclude also that our algorithm is twice as complex as calculating the equalizer for one predetermined (i.e., suboptimal, in general) delay. Implementation issues and system-level analysis are out of the scope of this letter, but it is reasonable to say that doubling the complexity (of the equalizer’s design and not equalization itself) is a low price to pay for the globally optimal design. Moreover, the algorithm we propose actually allows us to find the optimal delays not only for the equalizer of order but also for all orders smaller than . Thus, among all FIR equalizers, we may choose the one with the smallest order satisfying the prescribed constraints imposed on the value of MSE. This results in globally optimal design with respect to the parameters and . The following organization of this letter is adopted. Section II introduces the signals’ model and notation and shows the principle of the MMSE equalization. Section III develops the efficient algorithm for finding the optimal delay, starting from the well-known Levinson–Durbin (L–D) algorithm; a numerical example is also shown. The complexity of the new algorithm is evaluated in Section IV. We conclude the letter in Section V. II. SIGNAL MODEL AND MMSE EQUALIZATION The output
of the channel is modeled as (1)
Manuscript received January 19, 2005; revised February 23, 2005. This work was supported by research funds of the government of Quebec under Grant FCAR 2003-NC-81788 and of Canada under Grant NSERC 249704-02. The associate editor coordinating the review of this manuscript and approving it for publication was Dr. Markus Pueschel. The author is with the Institut National de la Recherche Scientifique-EMT, Montreal, QC H5A 1K6, Canada (e-mail:
[email protected]). Digital Object Identifier 10.1109/LSP.2005.849541
is a causal FIR of the where is a discrete time, and nonzero elements . Both the channel with and the observation noise are modeled input signal 1For some delays, called herein trivial, the optimal equalizer has all coefficients equal to zero and, thus, performs no useful filtering.
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IEEE SIGNAL PROCESSING LETTERS, VOL. 12, NO. 8, AUGUST 2005
as zero-mean wide-sense stationary random processes with respective autocorrelation functions and , where denotes expectation. The design of the MMSE equalizer consists in finding a causal FIR filter whose output is closest to the delayed version in the MSE sense [2], [7, Ch. 2] of (2) where contains equalizer’s coefficients ( is a priori chosen), is the regression vector, and is the fixed delay; , denote, respectively, transpose and transpose-conjugate operations. To calculate the equalizer, first, the channel’s impulse reand the noise variance are identified [8] (the sponse important issue of channel identification is out of the scope of this letter), and next, the equalizer’s coefficients are calculated obtained from using the model of the regression vector (1) (3) where
, , and the convolution matrix is given, as shown in (4) at the bottom of the page. The maximum equalization delay that should be considered because for , does is , so . Consequently, is nontrivial not affect . if We note that
for
(7) III. FINDING THE OPTIMAL DELAY It is known that inappropriately chosen delay may drastically deteriorate the equalizer’s performance [4]–[6]. However, since no closed-form expression for the globally optimal delay exists, it may be found only through the exhaustive , search. The latter consists in calculating all the receivers using (6) and choosing the one producing the , a significant comsmallest MSE (7). However, to obtain putational effort is necessary, so such a “brute-force” approach is too complex to be practical. As an alternative, only a few equalizers may be computed for delays, which are expected to be the best candidates. Such approach, however, does not guarantee the optimality of the final design, and its complexity grows proportionally to the number of candidates to be considered. In practice, the delay and the length are chosen using some heuristics, e.g., knowing the type of the channel to be equalized [4]. In the quest for the globally optimal design, we pursue the idea of the exhaustive search; however, instead of solving the re, we quired Wiener equations for all the delays find only the corresponding values of the MSE. This dramatically reduces the complexity of the resulting algorithm when compared to the “brute-force” approach. We develop the new algorithm staring from the L–D algorithm defined as follows: (8) for
(5)
if . where for convenience, we use The solution of (2) satisfies the so-called Wiener–Hopf equations [7, Ch. 2] (6) is the autowhere ; is the identity matrix. Since the correlation matrix of matrix is Toeplitz, (6) may be efficiently solved using the L–D algorithm [2].
.. .
[cf. (3)], the MSE Because obtained with the designed equalizer is given by [7, Ch. 2]
..
.
..
.
a)
(9)
b)
(10)
c) d)
(11) (12)
is the prediction order, where are the coefficients of the forward prediction error filter, is are the so-called reflection coefficient, the prediction error, and diag . The operator reads the , . The vector backward, i.e., vector contains the elements of the autocorrelation function of the computed straightforwardly from (1) as signal
..
. (4)
SZCZECINSKI: LOW-COMPLEXITY SEARCH FOR OPTIMAL DELAY IN LINEAR FIR MMSE EQUALIZATION
. The lower triangular matrix
is
given by
.. .
..
.
.. .
(13) is the all-zeros vector. where , which is The complexity of the L-D algorithm is times smaller than solving (6) directly (as a system of linear equations). For the detailed explanation of the L-D algorithm, we refer the reader to textbooks, e.g., [7, Ch. 3] and [9, Ch. 11]. in (12) might be, of course, We note that the equalizers (to lower the complexity), but the calculated only for notation dependent on eases the development of the subsequent formulas. As we already mentioned, our objective is to choose the for all possible delays . smallest MSE Let us calculate them for each of the filter of order with the delay using (5), (7), (12), and (13)
(14) (15) , and ( is trivial for where the order ). Using (14) directly does not offer us any advantage over the “brute-force” approach, so to keep the complexity low, we deusing (5) and (10) velop a recursion to obtain
(16) where
may be similarly expressed as
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To evaluate the complexity and have a better insight into the operation of the proposed algorithm, we summarize it in Table I, , where we used the vector notation , , and . Finally, we observe that the proposed algorithm calculates . Therefore, optimal delays for all equalizers of orders if a criterion on the minimum value of MSE that should be attained was given a priori, the order of the equalizer might be determined during the execution of the algorithm. Thus, the proposed algorithm actually may be used to find the smallest order satisfying the prescribed constraints (the delay is naturally optimal as well through the use of the described algorithm) (18) If the constraint in (18) is not satisfied, the algorithm terminates at , so in Table I becomes the maximum value of the order that can be supported by the receiver. A. Numerical Example First of all, we note that as a double check, we implemented the proposed algorithm and compared the results it produced with those obtained by means of the brute-force technique, i.e., solving Wiener (6) for all nontrivial delays. The perfect match was observed for various randomly generated impulse responses . Second, we compare our algorithm to the one proposed in [10], where the delay of the linear equalizer is chosen using the criterion of the minimum BER. We use the same channel model as in [10], i.e., the channel impulse response is , and the equalizer’s order is . The variance of the noise is , i.e., signal-to-noise dB. Applying our algorithm, we obratio SNR ; thus, tain the optimal delay is , which agrees with the optimal delay determined by the method proposed in [10] and numerical simulations shown therein. Of course, from one example, we cannot conclude that our algorithm and the one presented in [10] always yield the same results, but using observations made in [11], we can expect the results to be very close—especially for long equalizers. The performance being equal, the main difference resides in the implementation complexity. In [10], it is necessary to calcudelays and then calculate the late the receivers for metrics related to the BER performance. Only calculation of all the equalizers is roughly twice as complex as our algorithm. For long channels, the method proposed in [10] would be impractical. IV. IMPLEMENTATION COMPLEXITY
(17) . and where Using (16) and (17) in (15), using scalar operations only.
is recursively calculated
We assume that the channel impulse response is complex; thus, all the variables in the algorithm are complex exand . To simplify the analysis, we consider the cept for number of multiplications as the appropriate complexity measure (in fact, the numbers of additions and multiplications are similar).
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TABLE I PROPOSED ALGORITHM TO SEARCH FOR THE OPTIMAL DELAY ; “:?” DENOTES ELEMENT-BY-ELEMENT MULTIPLICATION OF VECTORS
IEEE SIGNAL PROCESSING LETTERS, VOL. 12, NO. 8, AUGUST 2005
To have the absolute complexity measure, we consider the . Then, the original L-D recursion [points a)–c) case multiplicaof the algorithm in Table I] requires tions, which combined with the complexity for getting the Wiener filter (the last line in Table I) results in . Since the proposed exthe total complexity tension of the L-D algorithm [points d)–f) of the algorithm in Table I] requires multiplications, we conclude that for , the proposed algorithm doubles the complexity of the original L–D algorithm. is comparable to , so the increase of However, in practice, the complexity is less than double. V. CONCLUSION In this letter, we proposed a computationally efficient algorithm to find a globally optimal value of the equalization delay in the linear FIR equalizers based on MMSE criterion. The proposed algorithm is the extension of the L–D algorithm, and its complexity is roughly doubled when compared to the latter. This increase of the complexity (of the equalizer’s design and not the equalization itself) may be considered small, taking into account the global optimality of the resulting design. The algorithm provides also the delay-optimal value of the MSE for all equalizers with successively growing orders. Therefore, the algorithm may be used to find the shortest equalizer yielding the desired performance defined in the sense of the MMSE. REFERENCES
To analyze meaningfully the additional complexity required by the proposed algorithm, we consider the scenario where instead of using the proposed algorithm, one would apply the L-D different values of delays algorithm directly for (chosen a priori) and determine among them the best (in the sense of the produced output MSE) delay. The proposed algorithm needs to execute points d)–f) defined in Table I, which require multiplications.2 On the other hand, using the L-D algorithm, we would additionally need to calculate equalizers (the last line of the algorithm in Table I), and this requires multiplications. Comparing both expressions, we can conclude that it is beneficial to use times the L–D algorithm only for . By the similar analysis, we find that if the Wiener solutions were to be calculated for all delays (“brute force”), the proposed algorithm would reduce the . complexity by the factor of 2Squaring of the absolute value is counted as 1/2 of a complex multiplication and a real multiplication as 1/4 of a complex multiplication.
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