IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 57, NO. 4, APRIL 2009
1483
Blind, Adaptive Channel Shortening Equalizer Algorithm Which Can Provide Shortened Channel State Information (BACS-SI) Cenk Toker, Member, IEEE, and Gökhan Altın
Abstract—Channel shortening equalization plays an important role in multicarrier modulation (MCM) systems. In this paper, we propose a blind channel shortening equalizer structure named blind, adaptive channel shortening equalizer which can provide the shortened channel state information (BACS-SI). The algorithm depends on the minimization of a cost function defined as the sum-squared difference of the autocorrelations of the shortened channel impulse response (CIR) and a target impulse response. The surface is proven to be multimodal; however, minima are shown to be related to each other in a certain way. A two-phase approach is proposed. In the first phase, the cost function is minimized by a stochastic gradient descent algorithm in order to find an arbitrary minimum. In the second phase using the relation between minima, genetic algorithms are employed to find the best minimum according to a fitness function. The algorithm can both successfully shorten the channel and also explicitly provides shortened CIR which is a necessary information for the proper operation of a MCM receiver, in contrast to many other algorithms proposed in the literature which cannot directly provide this information. Index Terms—Adaptive equalization, blind equalization, channel shortening, DMT, multicarrier modulation, OFDM.
I. INTRODUCTION DVANCED wireline and wireless communication systems such as IEEE 802.11a/g (Wi-Fi), IEEE 802.16 (WiMAX), ADSL, ADSL2/+, etc., have adopted multicarrier modulation (MCM) as the signaling technique either in the form of orthogonal-frequency-division multiplexing (OFDM) for wireless systems, or in the form of discrete multitone (DMT) for wireline systems, due to its ability to combat the dispersive effect of the communication channel. For the proper operation of MCM, a cyclic prefix that is at least as long as the channel delay spread has to be appended to the transmitted frame [1]. However, if the length of the channel is comparable to the dimension of the FFT employed in MCM, the throughput efficiency of the systems deteriorates significantly with this additional load which does not convey user data. Moreover, if
A
Manuscript received March 12, 2008; revised December 10, 2008. First published January 13, 2009; current version published March 11, 2009. The associate editor coordinating the review of this paper and approving it for publication was Dr. Mounir Ghogho. This work was supported by TUBITAK (The Scientific and Technological Research Council of Turkey) under project 107E056. This work was presented in part at the EURASIP European Signal Processing Conference (EUSIPCO), Poznan, Poland, September 2007. The authors are with the Department of Electrical and Electronics Engineering, Hacettepe University, 06800 Ankara, Turkey (e-mail:
[email protected];
[email protected]). Digital Object Identifier 10.1109/TSP.2009.2012897
the cyclic prefix is shorter than the channel length, intercarrier and intersymbol interferences are observed which significantly reduce the system performance. Hence, channel shortening becomes a crucial component of an MCM system to overcome this problem. A channel shortening equalizer [widely known as the time-domain equalizer (TEQ) for DMT] is a receiver front-end filter which effectively decreases the length of the communication channel experienced by the receiver, to a length less than or equal to the cyclic prefix length which is fixed according to the system standards, e.g., ADSL [2]. Another crucial component of an MCM receiver is the frequency-domain-equalizer (FEQ) which eliminates the degradation due to the scaling and phase rotation caused by the shortened channel even if its length is less than the cyclic prefix. Channel shortening filter design is a widely investigated topic in the literature. The minimum mean-square error (MMSE) method, initially proposed for the maximum-likelihood sequence estimation problem [3] during the 1970s, was later adapted to MCM [4] following the advances in OFDM during the 1990s. Many other shortening techniques have been developed since then, including the maximum shortening signal-to-noise ratio (MSSNR) [5]. An extensive survey of these techniques has been composed by Martin et al. in [6]. Most of the channel shortening equalizer algorithms including the MMSE and MSSNR require channel state information (CSI) at the receiver, either assuming that it is perfectly known or obtained by training, which may be neither possible nor desirable due to several reasons. In this case, channel equalization can still be performed by blind equalization techniques. Although blind channel equalization is a well-studied topic in the literature [7], e.g., the constant modulus algorithm (CMA) [8], finite alphabet (FA), etc., blind channel shortening is a rather unexplored field [9]. Multicarrier equalization by restoration of redundancy (MERRY) [10], sum-squared autocorrelation minimization (SAM) [11], single lag autocorrelation minimization (SLAM) [12], [13], Miyajima and Ding [14], and Darsena and Verde’s [15] algorithms are among the blind channel shortening equalizer proposals in the literature. However, these algorithms have a major shortcoming. Although they develop the TEQ and successfully shorten the channel, they do not provide the shortened channel impulse response (CIR) after shortening, which is necessary for the proper operation of the FEQ. In these techniques, either the shortened CIR is assumed to be provided through channel estimation relying on time or frequency-domain training [11] which is in contrast to the philosophy of blind signal processing, or
1053-587X/$25.00 © 2009 IEEE
1484
differential modulation is employed to eliminate the effect of the phase rotation caused by the shortened channel [16], which has a penalty of 3-dB loss in performance [17]. In this paper, we propose a blind adaptive channel shortening equalizer design which can explicitly provide the shortened channel state information (BACS-SI), which basically combines the advantages of the MMSE channel shortener and the SAM algorithm, while eliminating their disadvantages, i.e., while MMSE channel shortening equalizer can provide the shortened CIR in the form of a target impulse response (TIR), it strictly requires CSI at the receiver, i.e., MMSE is not a blind method; whereas, SAM can perform blind channel shortening, however it cannot provide the shortened CIR. The proposed BACS-SI algorithm can both shorten the channel in a blind manner, and also explicitly provides the shortened CIR. The proposed BACS-SI method aims at minimizing the cost function which is defined as the sum squared difference between the autocorrelations of the shortened CIR and a TIR. It is proven that the cost surface is multimodal, and a minimum of this surface is related to others in a certain fashion, i.e., if any one of the minima is known, direct access to a group of others is possible. As a result of the blind nature of the problem, the best minimum (in the sense of e.g., maximum bit rate) cannot be directly found due to the lack of prior channel knowledge. Hence, a two-phase algorithm is proposed. In the first phase, a stochastic gradient descent algorithm is developed to find an arbitrary minimum of the surface. An adaptive step size technique is also derived which significantly increases the system performance. As the second phase of the algorithm, a global search technique based on genetic algorithms is proposed. We define a fitness function depending on a pilot tone used in ADSL (adapting the proposal to other systems is straightforward). Starting from the minimum found in the first phase, possible solutions are searched by genetic algorithms to find the one which minimizes the fitness function. This solution both provides the equalizer coefficients, and the shortened CIR in the form of the TIR. The paper is organized as follows. Since the proposal is strongly connected to the MMSE channel shortening, the cost function of the MMSE criterion is reviewed in Section II. In Section III, we present the system model and derive the proposed cost function. A stochastic gradient descent algorithm is developed, together with the adaptive step size, in Section IV to minimize this cost function. The global search algorithm which depends on genetic algorithms is examined in Section V. Section VI provides simulations and their results, and conclusions are drawn in Section VII. The notation used throughout the paper is as follows: a scalar, a vector and a matrix will respectively be denoted by a lowercase italic , a lower-case boldface and an upper-case boldface letter. The convolution operator is represented , the complex conjugate is represented by , and by is used for matrix transpose. Since ADSL is considered as an example, all time-domain signals and coefficients are assumed to be real valued apart from the frequency-domain signals; however, adapting the algorithm to the complex case is straightforward.
IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 57, NO. 4, APRIL 2009
Fig. 1. Block diagram of MMSE channel shortening equalizer. The mean-square error E je (n)j is minimized so that the convolution of the original communication channel h and the equalizer w is as close as possible to the TIR b. The delay term 1 is also an optimization parameter. Note that the lower branch does not physically exist, and it represents the impulse response of the ideal shortened channel.
II. MMSE CHANNEL SHORTENING CRITERION The BACS-SI algorithm has strong connections with the MMSE channel shortening equalizer design. Therefore, it is informative to first introduce the optimization criterion for the MMSE algorithm. Interested readers can find further information on MMSE shortening in, e.g., [4], [18], and [19]. The block diagram of the MMSE algorithm is demonstrated in Fig. 1. The primary purpose of the algorithm is to minimize the mean-square error (MSE) which is defined as the variance of the error defined as the difference between the outputs of the equalizer (upper branch) and a TIR (lower branch). On the upper branch, the actual communication channel, , of length , is represented by the vector . Zero-mean, unit variance signal is transmitted through this channel, and zero-mean addiwith variance is added to tive white Gaussian noise obtain the received signal ,
where and Toeplitz convolution matrix
the
.. .
..
.
..
.
..
.
..
.
..
.
, , is
.. .
Samples of the transmitted signal and noise are assumed to be mutually statistically independent. The received signal is processed by the equalizer of length , , generating the output signal of the upper branch
(1) Here, due to the convolution of the channel and the equalizer, the length of the equalized channel, is . However, since
TOKER AND ALTIN: BACS-SI
1485
the intention of the equalizer is to shorten the channel, only consecutive coefficients are expected to be non-zero. The constant length of the TIR on the lower branch is assumed taps, i.e., . The output of to be this ‘virtual’ filter can be written as
Similar to the above analysis, the autocorrelation of the seis quence
(2)
(8)
where the vector represents the augmented TIR of length taps. Here, is the all-zeros vector, the represents the system delay parameter . [4] and Using the definitions in (1) and (2) the MSE is defined as
(7)
Moreover, under the i.i.d. input assumption, the autocorrelation of the signal at the output of the lower branch is equal to the autocorrelation of the (augmented) TIR, i.e.,
(9) (3) which is minimized over , , and under some constraint in and . Several order to avoid the trivial solution of constraints have been proposed in the literature [6]; however, is of the unit norm constraint on the equalizer taps particular importance for our case.
where in the second line we used the fact that attaching zeros to the beginning or to the end of a sequences does not alter the autocorrelation of the augmented sequence [20]. A direct consepreserves the same value quence of this property is that, for all values of the delay Using the definitions in (6) and (9), the cost function of the , can be written as proposed BACS-SI algorithm, (10)
III. PROBLEM FORMULATION From (3), it is observed that the MMSE channel shortening equalizer design requires CSI which is assumed to be known at the receiver, or can be obtained by training. However, in many cases channel estimation may not be possible or can be undesirable due to several reasons. In such cases, blind equalization offers a solution to the equalization problem. Noting that the length of the autocorrelation of a sequence is determined by the length of that sequence, we propose replacing the signals at the outputs of the upper and lower branches in Fig. 1 by their autocorrelations. This approach is similar to the SAM algorithm [11]; however, SAM can be considered as a special case of our proposal, as will be demonstrated shortly. Using (1), the autocorrelation function of the sequence is written as (4) (5) where is the signal covariance matrix and is the noise covariance matrix. If both and are and assumed i.i.d., it can be shown that both become shifting matrices as explained in [11] with only one diagonal with nonzero entries where the diagonal is determined by the lag . becomes In this case,
(6)
(11) (12) where in the last line we assumed that for the operating condiin (6) can be neglitions of ADSL, the noise term gible due to the reasons discussed in [11]. Moreover, in order to avoid the trivial solution, following [11], we use the unit norm . constraint on the equalizer taps Equation (9) suggests that the one-sided length of the au, is at most , and tocorrelation sequence of the TIR, for . Consequently, from (12) it can be consuppresses the lags of cluded that the cost function for , or equivalently the effective length of the shortened CIR is constrained to be less than taps. Moreover, tries to fit and for lags as seen from (12), . Hence, as the cost function decreases with the progress of an iterative algorithm, the autocorrelation sequences of the shortened channel and target impulse responses get closer, and ultimately they are expected to be the same. Therefore, at the end of the iterations, both the channel is shortened and also the autocorrelation sequence of the shortened CIR appears as the autocorrelation sequence of the TIR. This sequence will later be used to find the shortened CIR. A common issue related to blind signal processing algorithms is multimodality [7], [8]. That is, the cost function may have multiple minima and generally there is no direct way of accessing the global minimum. Hence, a descent algorithm initiated from an arbitrary initial point can easily converge to a local minimum with inferior performance as compared to the global
1486
IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 57, NO. 4, APRIL 2009
Fig. 2. Cost surface of the proposed blind channel shortening equalization al: : and a fixed TIR, b : . gorithm for the channel h The equalizer has three taps w w w w with w w , and the w w =w and the horizontal axis is vertical axis is . arctan w =w
(
= [1 0 3 0 2] =[ = arctan + )
]
= [1 0 5] =1
=
minimum. The cost surface of BACS-SI defined in (11) is also multimodal as can be observed from Fig. 2. Here, the channel is , and for demonstration purposes the TIR is kept fixed at . The equalizer has three taps where . The cost surface is demonstrated in spherical coordinates with unit radius, the horizontal axis is and the vertical . axis is In the following, it will be proven that the minima of are related to each other in a group-wise manner, and the value of the cost function at the minima of a particular group are equal. Proposition 1: Either taking the conjugate-reciprocal of any , and combination of the zeros of a sequence [e.g., ] w.r.t. the unit circle in the -plane or negating the algebraic sign of that sequence, does not alter its autocorrelation , and in (10)–(12)]. function [e.g., Proof: The proof can be summarized as follows. The power spectral density (PSD) of a sequence is the Fourier transform of the autocorrelation function of that sequence. Since the PSD provides only the squared-magnitude information but not the phase information, a zero or its conjugate-reciprocal w.r.t. the unit circle in the -plane has the same magnitude response on the unit circle, also including the negative of the sequence. different combinations of the zeros of a Hence, there are sequence which result in the same PSD (and autocorrelation) is the number of zeros of that sequence. where Another interpretation of the property in Proposition 1 can be found in [21]. Corollary 1: A direct consequence of Proposition 1 is the cost function in (11) is invariant to flipping the zeros of both the equalizer and the TIR coefficients w.r.t. the unit circle, and also negating their values. In other words, there are combinations for and combinations for with the same autocorrelation functions and respectively.
is known, the Hence, if the position of a minimum of minima, of equal BACS-SI positions of the other cost, can be found from this solution by simple zero-flipping and negation operations for and . These properties will later be pair among all possible exploited to find the optimum pairs. Remark 1: Upon a detailed examination of (11), it can be observed that the cost surface is very complicated and it may not be possible even to find the number of all minima. Therefore, the property stated in the above corollary may possibly apply to a subset of all minima. In other words, there may be many members and separate groups of minima consisting of by zero-flipping and negation we can only move within a group. However, also note that, this issue not necessarily have to limit the capabilities of the proposed algorithm since for typical filter can be quite large (e.g., for a TEQ of 16 taps and lengths ), hence the number of accesa TIR of 33 taps we have sible minima can still be large enough to scan the search space with a very fine resolution. Note that grouping is a common property of the autocorrelation based blind channel shortening algorithms [11], [12], and [13], besides BACS-SI. Remark 2: Consider the example in [21], i.e., let the zeros , of the channel be located at and . This channel is almost a single spike, hence its shortener found by an autocorrelation based shortener (SAM [11], SLAM [12] and [13], and BACS-SI) will also be a single spike. Flipping any real zero, or any complex conjugate zero pair of the channel (for a real valued channel) will result in a non-single spike channel, however this new channel will not make any difference in the cost function. Consequently, the corresponding shortener may still be found as a single spike, which obviously do not have to be a good one. At this point, SAM or SLAM has no control over the shortening equalizer coefficients to overcome this problem. On the other hand, although it cannot alter the channel zeros, BACS-SI has the ability to improve the shortening performance of the equalizer by flipping any combination of the zeros of the equalizer, as will be discussed in Section V. Remark 3: Using the method explained in Section V, BACS-SI can access the shortened CIR through the autocorrelation of the TIR which is found as one of the minima of the BACS-SI surface. As a result, determining the appropriate matched filter problem mentioned in [21] can easily be handled with BACS-SI. Remark 4: Comparing the BACS-SI cost function (11) and the SAM cost function [11, eq. (10)]
it is clear that SAM is a special case of BACS-SI. Actually, the main difference is, SAM treats the lags as a “don’t care” region and tries to suppress the coefficients of the autocorrelation in the range This truly shortens the channel, however, due to the “don’t care” region, SAM has neither control of, nor the ability to observe the shortened CSI. As a consequence, if the SAM algorithm is to be employed, some form of channel estimation has to be
TOKER AND ALTIN: BACS-SI
1487
performed after blind channel shortening in order to obtain this information. The same comments are also valid for the SLAM only algorithm, [12] and [13], which aims at suppressing at . On the other hand, since BACS-SI directly incorporates the ‘don’t care’ region into the problem in the form of a TIR, it has direct access to the shortened CSI, without any additional effort. Therefore, BACS-SI does not suffer from the disadvantage of SAM and SLAM mentioned above.
The gradient of
w.r.t.
is
(16) IV. ADAPTIVE ALGORITHM As indicated in Corollary 1, it is adequate to find the location surface in order to obtain all of one minimum of the other minima in a large group. For this purpose, we will employ a stochastic gradient descent algorithm where the coefficients of and the TIR constitute the variables. Also, the equalizer since the purpose of BACS-SI is blind equalization, we assume that no access is possible to neither the channel coefficients, , nor the input symbols, . However, it is assumed that the statistical properties of the input is known (zero-mean, unit and the variance), and direct access to the channel output, equalizer and TIR coefficients is available. Although two separate update equations could be written for the variables and , the convergence rate of the iterative algorithm would be determined by the slowest one. Therefore, in for the the sequel, we define a composite variable clarity of notation, and employ a single step size. The composite update equation for the stochastic gradient descent algorithm can be written as (13) where
and are given The derivation of the vectors in the appendix and their structures are and . Then, using (16) and (15), the update rule for can be written as
(17) where the first vector component of the right-most vector corresponds to the update of the equalizer coefficients, and similarly the second vector component corresponds to the update of the TIR. In (17), three expectation terms appear. Since direct access to the CSI is not possible, these expectations have to be estimated empirically over signal samples. As stated in [22], there are several unbiased estimates for this calculation. In the sequel, we will investigate the moving average (MA) and autoregressive (AR) estimates. A. Moving Average Estimates
is the step size, and , , and are the gradient operators , (11), w.r.t. , and , respectively. The gradient of w.r.t. can be calculated as
If we follow the methodology in [22] and calculate the expectations in a block-by-block manner, where each block contains samples, for the th block will be (18) Similarly, we have
(14) Recalling that the output of the equalizer is its autocorrelation is in (14) becomes
and , the gradient
B. Autoregressive Estimates An alternative method to calculate the expectations in (17) is to use AR estimates. For this purpose, we define the following update equations for each lag :
.. . (15)
1488
IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 57, NO. 4, APRIL 2009
and similarly the gradient
can be calculated as
.. .
(22)
where can be considered as a forgetting factor. Decreasing the value of will increase the contribution of the past estimates to the AR estimates. By using these definitions, the auto/cross-correlation functions are obtained as
C. Adaptive Step Size Although, in general, a constant step size is utilized for gradient descent algorithms in the literature, it is of great benefit to use an adaptive step size instead. Two major advantages are significant improvements in convergence speed and final residual error [7]. In most cases, finding the optimum step size by exact line search [23] may not be analytically possible or straightforward, hence here we opt for searching the most suitable step size by utilizing a secondary adaptation [24], [25]. In order to automatically adapt the step size in (17), an additional update equation is required (19) where
is a secondary constant step size. The gradient of w.r.t. can be implemented using the chain rule
(20) and where the gradients have already been calculated in (15) and (16), respectively. Then, considering the update equations for the TEQ and TIR taps
the gradient
in (20) clearly is (21)
Substituting (21) and (22) into (19) and (20), we obtain the update rule for the step size, . Remark 5: Although convergence of the BACS-SI algorithm is highly sensitive to the selection of the step size , it is less sensitive to the secondary step size , as long as has a proper value to guarantee convergence. Remark 6: From (20), it can be observed that the additional computational complexity introduced by the adaptation of the step size is very low. Actually the gradients and have already been calculated by the stochastic gradient descent algorithm. For the adaptation of the step size , only the previous values of these gradients have to be stored and two inner product operations together with an addition has to be performed. However, as it will be demonstrated in the simulations, the step size adaptation introduces significant performance improvement. V. OPTIMIZATION BY GENETIC ALGORITHMS Since the cost surface of BACS-SI is multimodal, the minimum which the iterative algorithm proposed in Section IV converges to strictly depends on the initial point. Similar to the SAM algorithm [11], some of the minima will correspond to “good” TEQs and others to “bad” TEQs in terms of a certain criterion (e.g., bit rate). Hence, for an arbitrary initial point, the gradient descent algorithm of the previous section may not necessarily converge to a good solution. Therefore, the best min, imum, i.e., the optimum equalizer and TIR pair which satisfies a certain criterion such as minimum bit error rate, maximum bit rate, etc., has to be found, again in a blind manner. As proven in Proposition 1, the minima of the cost function in a group are related to each other in a certain way. Although the minimum found after running the iterative algorithm may not be a good one, it can be used to find the best minimum among the group of minima. As explained in Corollary 1, this can be achieved by simple zero flipping and negation operations. Hence, the search space of the second phase of the proposed algorithm is composed of the positions of the zeros of and w.r.t. the unit circle (inside or outside). This clearly reveals the discrete nature of the problem. It is well known that genetic algorithms [26] are efficient optimization techniques based on the principles of genetics and natural selection which can perform global search, especially when the search space is discrete. Hence, as the second phase of the optimization, we employ genetic algorithms. The parameters to be optimized are represented by “genes,” and the combination of genes for a particular solution is called a “chromosome.” In our case, a binary valued gene will represent the position of a particular zero w.r.t. the unit circle (i.e., 0: inside, 1: outside), whereas a chromosome will be a combination of the positions of all zeros of both the equalizer and the TIR. Therefore, regarding the discussion in Corollary 1, it can combinations for the and be concluded that there are
TOKER AND ALTIN: BACS-SI
1489
Fig. 3. A chromosome is composed of three parts; positions of the zeros of the equalizer ( ) w.r.t. the unit circle, positions of the zeros of the TIR ( ), and the relative sign of the TIR coefficients w.r.t. to the equalizer. Position of a zero is represented by a gene which is binary valued (0: inside, 1: outside the unit circle (conjugate-reciprocal)).
w
b
pairs which give the same minimum value for (11), including the negatives. In the genetic algorithm employed here, there are binary valued genes in a chromosome as demonstrated in Fig. 3. The position of the zeros of and w.r.t. the unit circle are and genes respectively. The last demonstrated with represents the relative sign of the TIR to the shortening gene equalizer. Another important issue related to genetic algorithms is the fitness function. Chromosomes are adapted so as to minimize/ maximize the fitness function. Since BACS-SI is a blind algorithm, the fitness function has to be chosen accordingly. In the present work, we adapt the following property of the MMSE channel equalization technique; If the equalizer works appropriately, the impulse response of the upper branch (convolution of the channel and the receiver, i.e., the shortened channel) is expected to perfectly match the impulse response of the lower branch (TIR) in the absence of noise. Hence, we define the fitness function as the mismatch between these two impulse responses. The block diagram of a generic MCM system is depicted in Fig. 4. At the transmitter after taking the IFFT of a data block of symbols , the signal is parallel-to-serial converted and a cyclic prefix (CP) is appended to form an MCM frame. Then, the frame is transmitted to the receiver through the channel. At the receiver, the procedure followed at the transmitter is reversed and channel equalization is applied to obtain estimates of the transmitted symbols. The main focus of this paper is the two equalizer blocks at the receiver, specifically, TEQ and FEQ. As discussed above, TEQ performs the actual channel shortening which results in the shortened CIR whose FFT . If this transfer functransfer function is tion is known at the receiver, it can directly be used in FEQ as to perfectly reconstruct the transmitted symbols by eliminating the phase rotation at each tone caused by the shortened channel, under ideal conditions. However, in the present case, direct access to the shortened channel coefficients, , is not possible due to the blind na, we substitute ture of the problem. Hence, in place of the known TIR in the form of the transfer function FFT , with the assumption that ideally and should be identical. With this substitution, the FEQ output be, where is the comes data symbol transmitted at the th tone, and ideally we expect . In Section III, the similarity between the uncertainty related to the minima of the BACS-SI surface, and the phase uncertainty
Fig. 4. Generic structure for MCM transmission.
in blind equalization (e.g., the CM algorithm [8]) was briefly discussed. A common solution to the problem with CMA is to transmit few symbols which are also known at the receiver. We follow a similar approach and assume that some of the DMT tones are known at the receiver. Actually, the sixty-fourth tone in the ADSL standard [2] is reserved as a pilot signal for timing . (In ITU 992.1 Amendment 1 [27], recovery which is set to other tones are also defined as optional pilot tones which can further be exploited by the proposed method for improved performance.) Specifically, if both TEQ and FEQ perform properly, . In the sixty-fourth tone at the FEQ output should be is known at the receiver (i.e., pilot tones) other words, if , then the fitness function for the proposed gefor some netic algorithm is defined as1 (23) where is the set of pilot tones defined in [2] or [27]. Minimization of this function over all accessible minima of the BACS-SI surface will give the best matching pair. Clearly, for an arbitrary minimum of the BACS-SI surand can be very different. In other words, face, will possess a large value even if the pair corresponds to a minimum of . Therefore, we need to pair for a proper operation. Only find the matching [or equivalently ] a matching will result in minimum fitness value. The flowchart of the genetic algorithm employed in this study is provided in Fig. 5. First an initial population consisting of 18 chromosomes is randomly generated. Thereafter, two chromosomes (parents) with the best fitness values are selected from the mating pool to produce two new offsprings. Mating is carried out separately for the and genes as illustrated in Fig. 6, i.e., first, two crossover points are randomly selected in the range for genes and for genes. Then for both and genes, the lower and the upper halves of the genes with respect to the crossover points are swapped separately between the parents to generate the offsprings as depicted in Fig. 6. Note that mating increases the number of chromosomes in a population by two. Another important operation in genetic algorithms is mutation. Fifteen percent of all genes in the population are randomly 1Note that the effect of noise on the pilot tones for ADSL is assumed to be negligible.
1490
IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 57, NO. 4, APRIL 2009
Fig. 5. Flowchart of the genetic algorithm employed during the second phase of the proposed algorithm.
Fig. 6. Crossover in mating is performed separately for both and respect to two random point.
w and b with
selected and mutated by altering the state of the gene (i.e., and ). After mating and mutation, the chromosomes are sorted according to their fitness values calculated by (23), and the worst two chromosomes are deleted from the population. The above operation is carried out iteratively until a certain stopping criterion is satisfied. In this study, we continue the iterations until a maximum iteration number is exceeded, which is set to 750. Remark 7: Note that, as a common issue with the whole class of genetic algorithms, the computational complexity of the proposed genetic algorithm can be high. Actually, we tested a discrete gradient descent-like algorithm which moves from one minimum of the cost surface (12) to another by decreasing the cost function (23), however it was observed that this algorithm can get stuck in a local minimum. This is caused by the local search nature of the gradient descent algorithm. Relying on this observation, we have opted for genetic algorithms, which is a global search technique. VI. SIMULATIONS For the simulations, we used the same ADSL environment as the one used in [11]. Specifically, the cyclic prefix is 32 samples, has 16 taps, and the channel the FFT size is 512, the TEQ is chosen as the CSA test loop 1 [28], unless otherwise stated. The signal-to-noise ratio is set to 40 dB.
Fig. 7. Progress of the stochastic gradient descent algorithm for the BACS and SAM cost functions versus number of iterations when the step size is fixed and adaptive.
As mentioned before, the proposed algorithm has two phases. In the first phase, a stochastic gradient descent algorithm is uticost surlized to find any one of the minima of the face. The equalizer , and TIR are initialized to all-zeros vectors of dimensions respectively and except the middle taps equal to unity. Fig. 7 illustrates the progress of the stochastic gradient descent algorithm proposed in Section IV In the upper where the AR estimates are utilized with figure, the step size is fixed, whereas in the lower figure adaptive step size is utilized. For the former, we used the step size and for the later we initialized the primary step and used the secondary step size as size to Moreover, we also provide the performance of the SAM algorithm with both fixed and adaptive step sizes. Clearly, adaptation of the step size significantly improves both the convergence rate and the residual error. With adaptive step size, convergence is achieved in at most 200 iterations in contrast to the fixed step size where convergence requires more than 1000 iterations. One interesting observation about Fig. 7 is the difference between the excess error of BACS-SI and SAM cost functions which is caused by the “don’t care” region. In SAM, only the autocorrelation function out of the TIR window is being suppressed, whereas in BACS-SI, the difference between the part of the autocorrelation function inside the window and the TIR is also minimized in addition to the suppression of the out of window components. Hence, BACS-SI cost function has an additional error term as compared to that of SAM. Therefore, it is not fair to directly compare the values of the cost functions. However, Fig. 7 provides a good insight of the convergence time properties of both algorithms. In Fig. 8, we investigate the average distance of the zeros of the BACS-SI TIR to the unit circle, similar to the study in [29] which is for the MMSE channel shortening equalizer. The distances of the zeros of the TIR (i.e., 32 zeros) from the unit circle are averaged over CSA Loops 1–8 for TEQ lengths from 1 to 64.
TOKER AND ALTIN: BACS-SI
1491
Fig. 8. Distance of the zeros of the BACS-SI TIR from the unit circle averaged over CSA Loops 1 to 8. The numbers given on the figure are the number of zeros of the TIR exactly on the unit circle for a particular TEQ length.
The results reveal that for some TEQs, the zeros are exactly located on the unit circle, especially for shorter TEQs. However, apart from some impractical cases (TEQs of length 1 and 2), the number of these zeros is no larger than 8–10 (number of zeros exactly on the unit circle is given on the figure), leaving all other zeros being located away from the unit circle with an . This behavior seems quite difaverage distance of ferent from that of the MMSE equalizer [29] and can be concluded that BACS-SI does not significantly null the tones of a MCM transmission compared to MMSE since the average dis. Further detailed investigation tance for MMSE is around of the cost surface is still an open issue. Fig. 9 demonstrates the bit rate performance of the proposed algorithm compared to that of MFB, MSSNR [5] and the SAM algorithm. The bit rate is calculated based on [30], i.e., SNR
SNR
(24)
(25)
where is the SNR gap and is set to 9.8 dB, is signal power, is noise power, and , , are the gains of signal, noise and ISI terms in the th tone, respectively. Since the cost function definitions of BACS-SI and SAM algorithms are similar, they also share similar characteristics in terms of bit rate performance. However, note that BACS-SI achieve relatively higher performance as compared to SAM. Also note that, in SAM the bit rate value is calculated assuming that the shortened CIR is perfectly known which is not a realistic assumption [21]. On the other hand, BACS-SI can provide an estimate of this information in the TIR without any further effort. Only one of the minima of BACS-SI will have the bit rate demonstrated in Fig. 9, which corresponds to the matching
Fig. 9. Comparison of bit rate performance of the proposed BACS algorithm to that of the SAM, matched filter bound and MSSNR solutions. SNR is 40 dB, channel is CSA Loop 1.
pair. This can be justified as follows. As mentioned in Section V, we do not have direct access to the , hence TIR freshortened channel frequency response is substituted into the FEQ, instead. quency response Unless the matching pair is utilized as the TEQ ], the received signal will be and FEQ [i.e., distorted, which results in a lower bit rate. At the top of Fig. 10, the result of the stochastic gradient descent algorithm can be seen where the shortened channel and target impulse responses are very different, although as a pair, cost function. they minimize the Therefore, as the second phase of the algorithm, we employ the genetic algorithm explained in Section V. After 750 iterations, we obtain the shortened channel and target impulse responses demonstrated in the middle of Fig. 10, which clearly are almost identical. Therefore, while the corresponding equalizer coefficients are used as the TEQ, the TIR coefficients can successfully be used in FEQ for the best performance. At the bottom of Fig. 10, we verify that the autocorrelation of the shortened CIR and the TIR, both before applying to the genetic algorithm and also after the genetic algorithm, are identical. Hence, we can conclude that the solutions both before and after applying the genetic algorithm are minima of the surface yielding the same value. However, the genetic algorithm provides the optimum pair in terms of the defined fitness function which gives a better overall performance. VII. CONCLUSION In this paper, a blind, adaptive channel shortening algorithm which can also provide the shortened channel State Information (BACS-SI) is proposed. The algorithm is composed of two phases. In the first phase, a cost function is defined as the sum-squared difference between the autocorrelations of the shortened channel impulse response and a target impulse response. The
1492
IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 57, NO. 4, APRIL 2009
APPENDIX Let
be the autocorrelation of a sequence Then the derivative of w.r.t. [7] can be written as
where is the -elements up-shifted and ements down-shifted versions of the vector , i.e.,
is the -el-
using the MATLAB array notation. FUTURE RESEARCH
Fig. 10. (Top) Shortened channel and target impulse responses obtained after the stochastic gradient descent algorithm; (middle) optimum shortened channel and target impulse response pair found by the genetic algorithm; and (bottom) Autocorrelations of the shortened channel and target impulse responses before and after applying the genetic algorithm.
cost surface is shown to be multimodal with groups of minima having identical values. It is proven that minima in a group are related to each other in a certain manner. Therefore, it is adequate to find an arbitrary minimum in order to access the others. For typical system parameters, the number of minima in a group can be large enough to scan the whole search space with a fine resolution. A stochastic gradient descent algorithm is derived to find one of these minima. In addition, an adaptive step size technique is proposed which significantly improves the performance of this iterative algorithm. Although the minima optimizes the BACS-SI cost surface, only one of them actually performs the best shortening in the sense of a certain criterion. As the second phase, a genetic algorithm based method is proposed to find this minimum and the corresponding equalizer and TIR taps, considering a fitness function derived from the pilot tone utilized in ADSL (application to other systems is straightforward). It is demonstrated that the genetic algorithm can successfully determine the optimum minimum. In spite of being a blind algorithm, the proposed algorithm can retrieve a significant portion of the channel capacity. Another major advantage of the proposed algorithm is that it provides the shortened channel impulse response which is necessary for the FEQ block at the DMT receiver for proper operation. Other algorithms in the literature like SAM, SLAM, and MERRY, do not provide this needed information directly.
As mentioned by one of the anonymous reviewers, with the direct use of the pilot tones (although few of them are utilized), the proposed two-phase algorithm can be considered as a semiblind one. For a completely blind algorithm, one could develop a decision-directed second phase, which could be a topic for future research. Another important open point about the second phase is finding a faster combinatorial programming algorithm to obtain the best minimum of (11) rather than genetic algorithms. Also, extending the ideas in this paper to the fractionally spaced, SIMO, and MIMO equalization scenarios could yield interesting results. ACKNOWLEDGMENT The authors would like to express their gratitude to the anonymous reviewers, Prof. J. A. Chambers, and Dr. R. K. Martin for their inspiring and valuable comments which helped significantly improve the quality and presentation of the paper. REFERENCES [1] A. Goldsmith, Wireless Communications. Cambridge, U.K.: Cambridge Univ. Press, 2005. [2] Asymmetric Digital Subscriber Line (ADSL) Transceivers, ITU G.992.1, Rev. 07/99, 1999. [3] D. D. Falconer and F. R. Magee, “Adaptive channel memory truncation for maximum likelihood sequence estimation,” Bell Syst. Tech. J., pp. 1541–1562, Nov. 1973. [4] N. Al-Dhahir and J. M. Cioffi, “Optimum finite-length equalization for multicarrier transceivers,” IEEE Trans. Commun., vol. 44, pp. 56–64, Jan. 1996. [5] P. J. W. Melsa, R. C. Younce, and C. E. Rohrs, “Impulse response shortening for discrete multitone transceivers,” IEEE Trans. Commun., vol. 42, pp. 1662–1672, Dec. 1996. [6] R. K. Martin et al., “Unification and evaluation of equalization structures and design algorithms for discrete multitone modulation systems,” IEEE Trans. Signal Process., vol. 53, no. 10, pp. 3880–3894, Oct. 2005. [7] S. Haykin, Adaptive Filter Theory, 2nd ed. Englewood Cliffs, NJ: Prentice-Hall, 1991.
TOKER AND ALTIN: BACS-SI
[8] C. R. Johnson, Jr. et al., “Blind equalization using the constant modulus criterion: A review,” Proc. IEEE, vol. 86, pp. 1927–1950, Oct. 1998. [9] R. K. Martin and C. R. Johnson, Jr., “Adaptive equalization: Transitioning from single-carrier to multicarrier systems,” IEEE Signal Process. Mag., vol. 22, no. 6, pp. 108–122, Nov. 2005. [10] R. K. Martin et al., “A blind adaptive TEQ for multicarrier systems,” IEEE Signal Process. Lett., vol. 9, no. 11, pp. 341–343, Nov. 2002. [11] J. Balakrishnan, R. K. Martin, and C. R. Johnson , Jr, “Blind, adaptive channel shortening by sum-squared auto-correlation minimization (SAM),” IEEE Trans. Signal Process., vol. 51, no. 12, pp. 3086–3093, Dec. 2003. [12] R. Nawaz and J. A. Chambers, “A novel single lag auto-correlation minimization (SLAM) algorithm for blind adaptive channel shortening,” in Proc. IEEE Int. Conf. Acoustics, Speech, Signal Process., Mar. 2005, vol. 5, pp. 885–888. [13] R. Nawaz and J. A. Chambers, “Blind adaptive channel shortening by single lag autocorrelation minimisation,” Electron. Lett., vol. 40, no. 25, pp. 1609–1611, Dec. 2004. [14] T. Miyajima and Z. Ding, “Second order statistical approaches to channel shortening in multicarrier systems,” IEEE Trans. Signal Process., vol. 52, no. 11, pp. 3253–3264, Nov. 2004. [15] D. Darsena and F. Verde, “A blind direct channel-shortening approach for multicarrier systems,” in Proc. IEEE Workshop Signal Process. Advances Wireless Commun., Jun. 2007, pp. 1–5. [16] R. K. Martin et al., “Fast converging blind adaptive channel-shortening and frequency-domain equalization,” IEEE Trans. Signal Process., vol. 55, no. 1, pp. 102–110, Jan. 2005. [17] J. G. Proakis, Digital Communications, 4th ed. New York: McGrawHill, 2000. [18] J. S. Chow and J. M. Cioffi, “A cost-effective maximum likelihood receiver for multicarrier systems,” in Proc. IEEE Int. Conf. Commun., Jun. 1992, vol. 2, pp. 948–952. [19] N. Al-Dhahir, “FIR channel shortening equalizers for MIMO ISI channels,” IEEE Trans. Commun., vol. 49, pp. 213–218, Feb. 2001. [20] A. V. Oppenheim, R. W. Schafer, and J. R. Buck, Discrete-Time Signal Processing, 2nd ed. Englewood Cliffs, NJ: Prentice-Hall, 1999. [21] J. M. Walsh, R. K. Martin, and C. R. Johnson, “Convergence and performance issues for autocorrelation based adaptive channel shortening,” in Proc. 40th Asilomar Conf. Signals, Systems, Computers, 2006, pp. 238–242. [22] R. K. Martin, “Blind, Adaptive equalization for multicarrier receivers,” Ph.D. dissertation, Cornell Univ., Ithaca, NY, 2004. [23] S. Boyd and L. Vandenberghe, Convex Optimization. Cambridge, U.K.: Cambridge Univ. Press, 2005. [24] V. J. Mathews and Z. Xie, “A stochastic gradient adaptive filter with gradient adaptive step size,” IEEE Trans. Signal Process., vol. 41, no. 6, pp. 2075–2087, Jun. 1993. [25] S. C. Douglas, “Generalized gradient adaptive step sizes for stochastic gradient adaptive filters,” in Proc. IEEE Int. Conf. Acoustics, Speech, Signal Process., May 1995, vol. 2, pp. 1396–1399. [26] R. L. Haupt and S. E. Haupt, Practical Genetic Algorithms, 2nd ed. Hoboken, NJ: Wiley, 2004. [27] Asymmetric Digital Subscriber Line (ADSL) Transceivers, ITU G.992.1 Amendment 1, Rev. 03/03, 2003.
1493
[28] K. Sistanizadeh, “Lost characteristics of the proposed connanical ADSL loops with 100-ohm termination at 70, 90, and 120 f,” in ANSI T1E1. 4 Committee Contribution No. 161. Washington, DC: Amer. Nat. Stand. Inst., 1991. [29] R. K. Martin et al., “Infinite length results and design implications for time-domain equalizers,” IEEE Trans. Signal Process., vol. 52, no. 1, pp. 297–301, Jan. 2004. [30] G. Arslan, B. L. Evans, and S. Kiaei, “Equalization for discrete multitone transceivers to maximize bit rate,” IEEE Trans. Signal Process., vol. 49, no. 12, pp. 3123–3135, Dec. 2001.
Cenk Toker (S’97–M’05) was born in Ankara, Turkey, in 1974. He received the B.Sc. degree (with high hons.) from Hacettepe University, Ankara, Turkey, in 1995 and the M.Sc. degree from the Middle East Technical University, Ankara, Turkey, in 1999, both in electrical and electronics engineering, and the Ph.D. degree in electronics engineering from the Centre for Digital Signal Processing (CDSPR), King’s College London (KCL), U.K., in 2004. Between 1995 and 2002, he held several positions in the industry as a Research and Development Engineer, with such organizations as Motorola, Swindon, U.K.; TUBITAK-BILTEN, Ankara, Turkey; and Aselsan Inc., Ankara, Turkey. In 2005, he continued his studies at CDSPR, KCL as a Postdoctoral Research Associate. In September 2005, he joined the faculty of the Department of Electrical and Electronics Engineering, Hacettepe University, where he is currently an Assistant Professor and the Vice-Chair of the department. In 2001, Dr. Toker received the Motorola Sponsorship for his doctoral studies through the University Partnership for Research Programme. From 2002 to 2004, he was sponsored by the School of Physical Sciences and Engineering of King’s College London. His current research interests include resource allocation problems, integer programming methods, joint transceiver design for multicarrier applications, channel equalization—specifically channel shortening, and space-time signal processing.
Gökhan Altın received the B.Sc. degree in electronics engineering from the Turkish Air Force Academy, Istanbul, Turkey, in 2001. In 2003, he graduated first in his class from the Turkish Air Force Communications, Electronics, and Information Systems School, Izmir, Turkey, as a Communications Officer. He received the M.Sc. degree in electrical and electronics engineering from Hacettepe University, Ankara, Turkey in 2007. He is currently a Project Officer in the Turkish Air Force Command, Ankara, Turkey. His research interests include equalization for multicarrier systems and blind adaptive filters. Mr. Altın is a member of Tau Beta Pi, Ohio Eta Chapter.
