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IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 52, NO. 3, MARCH 2004
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Realizable Linear and Decision Feedback Equalizers: Properties and Connections Roberto López-Valcarce, Member, IEEE
Abstract—Recently, there has been renewed interest in the use of infinite impulse response (IIR) linear equalizers (LEs) for digital communication channels as a means for both improving performance and blindly initializing decision feedback structures (DFEs). Theoretical justification for such an approach is usually given assuming unconstrained filters, which are not causal and therefore not implementable in practice. We present an analysis of realizable (i.e., causal, stable, and of finite degree) minimum mean square error (MMSE) equalizers for single-input multiple-output channels, both in the LE and DFE cases, focusing on their structures and filter orders, as well as the connections between them. The DFE resulting from rearranging the MMSE LE within a decision feedback loop is given special attention. It is shown that although this DFE does not in general coincide with the MMSE DFE, it still enjoys certain optimality conditions. The main tools employed are the Wiener theory of minimum variance estimation and Kalman filtering theory, which show interesting properties of the MMSE equalizers not revealed by previous polynomial approaches. Index Terms—Decision feedback equalizers, digital communications, intersymbol interference, linear equalizers, realizable equalizers.
I. INTRODUCTION
O
VER the past few decades, several equalization techniques have been proposed in order to combat frequency selective transmission channels, which introduce intersymbol interference (ISI) in digital communication systems. The optimum receiver in terms of symbol error rate (SER) is the maximum likelihood sequence estimation detector [19], but its associated computational cost grows exponentially with the channel memory. This precludes its use in many applications. Linear equalizers (LEs) followed by a symbol-by-symbol detector are attractive in terms of complexity, although they might introduce excessive noise enhancement if the channel frequency response presents deep nulls [20]. Decision feedback equalizers (DFEs), on the other hand, provide postcursor ISI cancellation with reduced noise enhancement and are widely recognized to offer better steady-state performance than LEs [5]. However, due to the presence of a nonlinear decision device inside the DFE feedback loop, erroneous decisions can result in error bursts that degrade SER performance. This error propagation effect makes transient analysis of the DFE
Manuscript received January 31, 2002; revised April 9, 2003. This work was supported by the Ramón y Cajal program of the Spanish Ministry of Science and Technology. The associate editor coordinating the review of this paper and approving it for publication was Dr. Sergios Theodoridis. The author is with the Department of Signal Theory and Communications, University of Vigo, Vigo, Spain (e-mail: [email protected]). Digital Object Identifier 10.1109/TSP.2003.822360
difficult, and a past correct decisions assumption is often invoked to sidestep this problem. This is reasonable if the DFE is operating in a low SER scenario. In this paper, we study the structure and properties of the minimum mean square error (MMSE) equalizers under the only constraint of realizability, i.e., the filter may have an infinite impulse response (IIR), but it must be causal, stable, and have finite degree. In contrast, usual approaches either assume equalizers with a doubly infinite number of taps [5], [20] or consider finite impulse response (FIR) filters of prespecified degree (and thus potentially suboptimal) [1]. Realizable LEs will be discussed in Section III, whereas Section IV deals with the DFE case. A few authors have previously considered the problem of IIR equalization. Using Wiener and Kalman filtering techniques, Mulgrew and Cowan [18] and McLaughlin [17] obtained the realizable MMSE LE and MMSE DFE, respectively, for the single-input single-output (SISO) FIR channel case. The advantage of this approach is that it automatically reveals the optimal degree for the equalizers. Increasing the equalization lag is seen to decrease the MMSE at the price of a higher filter degree so that a direct relation between complexity, performance, and equalization delay is obtained. Ahlén and Sternad [2], [24] used a polynomial approach for the SISO problem, allowing for colored symbols and/or noise, as well as for IIR channels. Tidestav et al. [25] applied these polynomial methods to obtain the realizable MMSE DFE in a multiple-input multiple-output (MIMO) configuration. We assume a single-input multiple-output (SIMO) FIR channel model with white transmitted symbols; the channel noise is taken as temporally white but possibly spatially colored. Particularities of the single output channel case are also highlighted. One of our contributions is the extension of the Wiener/Kalman techniques of [17], [18] to this SIMO setting. We adopt this approach since we believe it provides insights not obtained with the polynomial methods of [2], [24], [25]. For example, the filter structures obtained from the Kalman state-space equations are appealing if the sparsity of the channel present in several applications is to be exploited. The value of the equalization delay is seen to impact the constituent blocks of the realizable MMSE LE and MMSE DFE differently. It is also shown (Section IV-A) that, contrary to previous belief, the error sequence associated with the realizable MMSE DFE is white. New expressions for the asymptotic performances of the equalizers are obtained as well (Section V). IIR LEs have recently received renewed attention [4], [7], [9], [12], [13] because, in addition to their potential to outperform
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conventional FIR structures with the same number of parameters, they provide a means for unsupervised initialization of a DFE. This approach relies on the fact that the denominator of the transfer function of the doubly infinite MMSE LE coincides with the feedback filter of the doubly infinite MMSE DFE. However, as shown in [7], this is no longer true once finite degree structures are considered. With this in mind, we compare the structures and properties of the realizable linear and DFEs in Section V. It will be shown that when the components of the MMSE LE are rearranged into DFE mode, the resulting feedback filter is still optimized in terms of the feedforward filter, provided that the equalization lag is no less than the channel length. This result provides additional support for this DFE initialization strategy. The notation adopted in the paper is as follows. • Scalars are denoted in lowercase. • Vectors are denoted in lowercase bold. • represents a vector of all zeros, except for a 1 in the th position, counting from zero. • Matrices are denoted in uppercase bold. , , and denote respectively conjugation, trans• position, and conjugate transposition. • denotes the “paraconjugate” of the transfer function , i.e., . • The causal part of is extracted by . • The expression stands for . With this notation, there is no need to distinguish between the complex variable and the unit advance operator, which are both denoted by . • With two vector-valued random variables and , cov . The paper is organized as follows. Section II provides a brief review of the minimum variance estimation problem, and it also presents the channel model. MMSE linear equalizers are discussed in Section III, from both the Wiener and Kalman theory viewpoints. This is also done in Section IV for the MMSE DFE. The connections between these structures are analyzed in Section V. We close with a numerical example in Section VI. II. WIENER FILTERING AND CHANNEL MODEL In this section, we briefly review the structure and design of the causal Wiener filter for the general estimation problem [23], and then, we present the channel model that will be used throughout the paper. A. Causal Wiener Filter Consider the general linear estimation problem in which a is sought so that the causal linear filter mean square value of the norm of the error is minimized. Here, is a sequence of observations, and is a reference signal. The processes , are assumed wide-sense stationary with zero mean. In addition, is assumed to be a full rank process [3, Sec. 9.4], meaning that , to be defined shortly, is not its power spectral density singular for all .
is opThe orthogonality principle states that the filter is orthogonal to the observatimized if and only if the error tions, i.e., cov
for all
Hence, by introducing the following correlation sequences and their -transforms: cov cov one finds that the optimum filter’s coefficients satisfy for all Note that if
were white with for some , then the Wiener filter coefficients would be , , that is, readily obtained from . is not white, the Wiener filter includes a When whitening prefilter. The power spectral density (psd) of can be factored as the full rank process (1) , and the transfer function is causal (its where series expansion involves only nonpositive powers of ), monic for ), and minimum phase (all poles lie in ( , and has constant rank in ). Now, let and
Then, the psd of the process is so that the observations have been whitened is a stable transfer function since as desired. Note that is minimum phase. The Wiener filter is therefore given by , where and cov (2) and the filter as We will refer to the white process the innovations process and the innovations postfilter, respectively. , the causal Wiener filter Now, since can be written as
(3) Comparing this with the doubly infinite (and therefore not realizable) Wiener filter (4)
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Fig. 1. Relation between the causal and the doubly infinite Wiener filters.
both devices are seen to include a prewhitening filter addition, if we write
. In
then we see that the innovations postfilters are related via . However, in general , i.e., the causal Wiener filter is not obtained by direct truncation of the doubly infinite Wiener filter. This is illustrated in Fig. 1. is causal and stable, We must point out that although (3) does not reveal whether it also has finite McMillan degree (which is the minimum number of delay units required for its implementation). This depends on the particular context. In our case, this context is the channel equalization problem, for which the developments in Section III-B (LE) and Section IV-A (DFE) will show that in both cases, the resulting causal Wiener filter does have finite degree, and therefore, it is realizable. Finally, if we write the innovations postfilter as , the MMSE attained by the causal Wiener filter is found to be cov
cov
Fig. 2. Channel model.
where is the additive noise, which is assumed to be zero mean, independent of , and temporally white with for and otherwise. We allow cov for spatial noise correlation in the channel model since this is the case when the multichannel configuration arises from oversampling the received continuous-time signal [6]. Symbols, channel coefficients, and noise are complex-valued in general. It is also of interest to consider a state-space representation of the channel. Since, in the equalization problem, one is interested , let us in estimating a delayed version of the channel input . With this, the channel state vector is define
(5)
(7) Then, the state space equations of the channel (6) are
B. Channel Model We consider a single-input multiple-output (SIMO) baseband digital communications model, as represented in Fig. 2. The scalar symbol sequence is assumed to be zero mean white, without loss of generality. This scaled to unit variance sequence is transmitted over a linear time invariant channel with FIR of order . Thus, the channel transfer function is a vector given by
It is assumed without loss of generality that , as any can be eliminated by including zero leading coefficients in . This a corresponding delay in the symbol sequence multichannel model accommodates communications systems where antenna arrays are used to improve the detection, as well as oversampled received signals (i.e., fractionally spaced receivers) and/or systems with frequency diversity. The received 1 vector signal is the (6)
(8) shift matrix with ones on the first where is a matrix subdiagonal and zeros elsewhere, and is a comprised of the channel coefficients: (9) for . The state space description It is assumed that will be useful when discussing the structure of the steady-state Kalman equalizer. III. MMSE LINEAR EQUALIZERS In the linear equalization context, the goal is to design a linear under the constraints of causality and stability in filter order to minimize the mean square value of the error , where is a suitable delay between the channel input and the estimate being made. Thus, in the nota, which tion of Section II-A, the reference signal is is a scalar.
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A. Wiener Filtering Approach
Low SNR Case: It is instructive to consider the high noise , with and scenario. Let us write as a matrix with unit 2-norm. Then
The psd of the received signal takes the form
Observe that the series expansion of contains only to . Hence, the minimum powers of ranging from appearing in (1) is an FIR transfer phase spectral factor function of order , and is said to be a moving average transfer function of process of order , or MA( ). The the whitening prefilter can be written as adj
(10)
where “adj” denotes the adjugate matrix. Let . Then, is monic and minimum phase. From (1) (11) Note that
which tends to 1 as the noise power, and, hence, tends to infinity. Thus, in that case, i.e., the effect of noise is to pull the poles of the whitening prefilter toward the origin. Noiseless Case: In the opposite direction, let us assume that so that . If , i.e., if there is a single channel, the received signal is a scalar process and, therefore, full rank [3]. Hence, the spectral factorization of its psd is still well defined, and the preceding development of is the MMSE LE holds. On the other hand, always singular for . In that case, it is well known that as for all , i.e., the subchannels do not share long as of order a common root, an FIR equalizer exists such that , thus yielding zero MSE [26]. , of the subchannels is If the greatest common divisor, say , where now, nontrivial, one can write admits an FIR equalizer so that one recovers the single-channel case. B. Kalman Filtering Approach
(12) 1 where the third line follows from the fact that for arbitrary . (This can vectors , , one has be easily shown by induction on .) Therefore, from (11) and is seen to be the monic minimum-phase spectral (12), . This also shows that is a factor of polynomial of order . Observe that adj is a FIR transfer function, which, . In the single-channel case ( in general, has order ), this adjugate matrix reduces to a scalar constant so that the . whitening filter is just an all-pole filter Concerning the innovations postfilter , observe that with the innovations process cov
cov
(13)
which is zero if since is white, and the transfer function from to is causal. Therefore, the inis FIR with order . (Incidentally, this novations postfilter holds true even if the noise were colored and/or the channel were IIR.) In view of this, the MMSE LE for the delay is seen to be FIR filter of the series interconnection of three blocks: a ,a1 FIR filter of order , and a scalar all-pole order filter of order : adj where is the equalizer output. Hence, the order of the overall adj could be as high as . HowFIR part ever, examination of the Kalman structure in the next section . will reveal that this order is in fact just
The Kalman filter provides a means to obtain the minimum variance linear estimate of the channel state vector from (7), given the observations [3]. Although the Kalman filter is in general time-varying, if the channel is time invariant and the noise and symbol sequences are stationary, it will asymptotically reach time invariance. This steady-state Kalman filter coincides with the realizable Wiener filter, providing valuable information about its structure. Given the state-space channel description (8), the steady-state Kalman filter operation can be described by (14) is the innovations sequence as before, whereas the Here, asymptotic Kalman gain is a matrix. The filter , is extracted from the vector esoutput, i.e., the estimate of . The structure of the Kalman equalizer timate via is shown in Fig. 3. We should note that the asymptotic Kalman gain can be obtained via the solution of a discrete-time Riccati equation [3]; here, our focus is just on the structure of the equalizer, and therefore, we will not worry about the exact value of . Observe that this implementation of the MMSE LE requires delay elements, and hence, this is its McMillan degree. In addition, observe that the Kalman structure effectively incorporates a whitening filter since the innovations process is directly available, and the channel coefficients (with the exception of ) appear directly in the equalizer parameterization. It is interesting as well to note that not only does this structure implement the MMSE LE for the delay but all those for the delays as well since the corresponding estimates are present in the filter’s state vector . If we partition row-wise as (15)
LÓPEZ-VALCARCE: REALIZABLE LINEAR AND DECISION FEEDBACK EQUALIZERS
Fig. 3.
Fig. 4.
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Structure of the Kalman equalizer (in steady state).
Flow diagram of the Kalman MMSE LE, assuming > L.
then it is seen from (14) that and for . This shows that the Kalman implementation of the MMSE LE is modular: The MMSE LE for the delay is obtained directly from that for the delay by adding a delay element and a multiplier . The flowgraph of the Kalman structure, which is depicted in Fig. 4, illustrates this property. polynomials By introducing the 1
It is instructive to write the transfer function of the MMSE LE for the delay as
where represents the recursive (all-pole) part, and denotes the transversal (all-zeros) component of the polynomials, one has equalizer. Hence, in terms of the
(16) adj
(18)
one can write the transfer function from the innovations vector to the equalizer state vector as (19)
.. .
Achieved MSE: From (5), the MSE attained by the delay MMSE LE is
.. .
(20)
In addition, observe that since
the whitening prefilter and innovations postfilter transfer functions are, respectively, given by
(17)
cov ). Therefore, (recall that with equality holding iff , in which case, . This means that, as it could be expected, the MMSE is a nonincreasing function of the equalization delay . An alternative expression for the MMSE is as follows. Let the overall channel-equalizer transfer function be (21)
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Then, for is given by
, the th autocorrelation coefficient of the error cov cov
cov
(22) The first line is a consequence of the orthogonality principle, whereas the second follows from independence of symbols and , we obtain the MMSE noise. For (23) Observe that is the cursor of the overall system . Comparing (20) and (23), we see that . Thus, is real, and . The fact that the cursor is not equal to 1 shows that the decision rule operating on the equalis izer output will be biased, that is, the equalizer output a conditionally biased estimate of since . This bias can be removed by scaling the . Although this worsens equalizer transfer function by the MSE [it is easily shown that the resulting MSE is ], it improves the symbol error rate [8]. Interestingly, (23) shows a direct relation between the MSE and the bias. Comparison With FIR Design: The standard approach to LE design is to assume an FIR structure for the filter. In that case, the MSE becomes a complicated function of the delay. For the IIR structure, however, as we have seen, there exists a direct relationship between MSE, filter order, and delay. In addition, for the same filter order, the IIR LE always yields a smaller MSE than the lag-optimized FIR approach [18]. Single-Channel Case: In the single-channel case, (18) reduces to (24) , then Thus, if we write for . This shows that in the case, the numerators of the MMSE LEs transfer functions associated with different delays are related to each other by simple shift operations, which is a property recognized and exploited in [16] in the context of blind equalization: (25) However, this property is not satisfied in general in the multi). channel case ( An additional property is found in the relation between and for the single-channel case. If we write , then (24) together with shows that, for
..
.
.. .
.. .
.. .
(26)
Fig. 5.
DFE.
This relation means that, if , then the transfer funcreduces to a polynomial of degree tion . As it will be discussed in Section V, this translates into the cancellation of all postcursor ISI once the constituent blocks of the MMSE LE are placed in a decision feedback loop. Equation (26) also suggests a means to achieve channel identification via and are available. blind equalization once IV. MMSE DFES In a DFE, equalization is achieved by means of a causal, -input 1-output feedforward filter and a strictly causal, . Thus, the equalizer output is a SISO feedback filter linear combination of the current and past samples of the re, and past hard decisions , , ceived signal produced by a quantization device (slicer). Again, represents the equalization delay. This is shown in Fig. 5. is the symbol in the transmitted conThe slicer output stellation that is closest to . Note that, due to the inclusion of the slicer in the feedback loop, the DFE is a nonlinear device. This makes the analysis extremely difficult, and therefore, we will adopt the common assumption of correct past decisions, for all . This is reasonable if the receiver is i.e., operating at a low error rate, as it is usually the case. Under this , assumption, the DFE output becomes which is a linear function of the received and transmitted signals. Hence, in that case, the DFE can be regarded as linear. When the DFE operates in a high error rate environment, the correct past decisions assumption ceases to be valid, and error ) are propagation happens as symbol errors (i.e., fed back by the filter . Different strategies must be used for filter design in that case [11], [22]. A. Wiener Filtering Approach MMSE DFE design can be carried out using the realizable Wiener filter theory of Section II-A. The desired signal is as in the linear equalizer case, but now, the observed se, where is the augmented 1 vector quence is (27) As usual, we first compute the whitening prefilter and then the is found to innovations postfilter. The psd of the process be (28)
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Using the fact that
,
one has
which is a constant independent of . Let be the spectral factorization of . Note contains only powers of that the series expansion of ranging from to , again with , and therefore, is FIR of order . In addition
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. Now, (31) is just the negative of the ( )th coeffifor , but this is cient of the impulse response of , and therefore, this a polynomial of degree not exceeding coefficient is zero if . Thus for all
cov On the other hand cov cov
cov
cov cov (33)
constant is constant. Thus, the whitening and it follows that adj is FIR. prefilter Realizable MMSE DFE Is FIR: The reasoning around (13) readily applies to the DFE case as well, showing that the innovations postfilter is again an FIR filter of order not exceeding . Therefore, the overall realizable Wiener equalizer (whitening prefilter plus innovations postfilter) given by the is, rather surprisingly, an FIR 1 vector filter. Degree of the Realizable MMSE DFE: The degrees of the , remain to be determined. Note that adj filters has order in principle. Nevertheless, the following holds. Lemma 1: Under the correct past decisions assumption, the feedforward and feedback filters of the realizable MMSE DFE for the delay have degrees and , respectively. A proof of Lemma 1 using polynomial techniques can be found in [25]. Here, we provide an alternative and more intuitive proof using the orthogonality principle. Proof: Consider a DFE with FIR feedforward and feed, having orders , , respectively. Asback filters suming correct past decisions, the output of such an equalizer is
Now, assume that , are optimal in the MSE sense for such a constrained DFE structure. Then, from the orthogonality must be uncorrelated with principle, the error the data:
Note that
is a linear combination of the symbols , which from (32) are uncorrelated for . Thus, the first term in (33) is zero for . with The second term equals the th coefficient of the impulse response of (times ), and therefore, it vanishes for since has order not exceeding . Thus
It suffices to show that (29) and (30) actually hold for all and all , and therefore, , implement the realizable MMSE DFE for the delay . First, note that
This, together with whiteness of bols and noise, yields cov
cov cov
and independence of symcov (31)
for all
cov
(34)
as was to be shown. We will denote by , the feedforward and feedback filters of the MMSE DFE in order to make explicit their dependence with the delay . Cioffi et al. [1], [27] observed that for a DFE design in which both feedforward and feedback filters are constrained to be FIR, the optimum equalization lag equals the feedforward filter order, provided that the feedback filter order is at least as large as the channel length. Lemma 1 confirms this observation but also states that for a given delay , there is no need to increase the filters’ orders beyond and . Whiteness of Prediction Error: The orthogonality principle also reveals the following property of the realizable MMSE DFE. Lemma 2: Under the correct past decisions assumption, the associated with the realizable MMSE DFE error sequence of delay is white. Proof: Let be the realizable MMSE DFE output. Then, , so that cov
(29) (30)
cov cov
(32)
cov
cov
(35)
is the realizable MMSE DFE error sequence, is Since as well as with uncorrelated with [cf. (32) and (34)]. Noting that is a linear combination and , it follows that the of , as was two terms in the right-hand side of (35) vanish for to be shown. Al-Dhahir and Cioffi [1] studied the properties of an MMSE DFE for the delay having a degree feedforward filter and a degree feedback filter, allowing for colored symbols and noise. They concluded [1, Sec.V-B] that the associated error sequence is not white in general. However, when symbols and noise are temporally white, Lemma 1 shows that such MMSE DFE is the realizable MMSE DFE, and then, Lemma 2 shows that in that case, the error sequence does become white.
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Achieved MMSE: The MSE of the MMSE DFE can be written in terms of the overall channel-equalizer response , which is now defined as
vector is defined as has size
, and the channel matrix and is given by (38)
The steady-state Kalman filter has a form similar to (14): so that
. Then cov
cov cov
(39) The asymptotic Kalman gain matrix, which we partition as
cov cov
(40)
similarly to (23). Hence, the MMSE is directly proportional to the bias, as in the LE case. In addition, it is true that . To see this, observe that the DFE defined by the , (which have orders and respecfilters for the delay . Thus, tively) result in an MSE of the performance of the MMSE DFE for the delay cannot be worse than this value. In addition, note that . This is because the realizable MMSE LE produces the MMSE , given , whereas the MMSE estimate of given DFE yields the MMSE estimate of as well as . Cancellation of Postcursor ISI: Note that cov , which, from (32), equals zero for , has degree . i.e., the overall channel-equalizer response This is to say that the feedback filter has canceled all postcursor ISI, which is a well-known property of DFEs. , then in (28) beNoiseless Case: When is not of full comes singular for all so that the process is not well derank, and the spectral factorization of fined. However, in that case, it is possible to achieve zero MSE and having orders and , respectively. This with , can be chosen to satisfy is because (36) eliminates all precursor ISI, whereas takes care i.e., , of all postcursor ISI as before. In the single-channel case , under the assumption of (36) uniquely determines a nonzero leading channel tap. If , there may be multiple filters satisfying (36). B. Kalman Filtering Approach Kalman filtering techniques can also be applied to the DFE problem [17]. Let us consider first the case of an equalization . The statelag at least as large as the channel length, i.e., space description of the channel model generating the vector in (27) has now dimension : (37) where now, 1 vector,
is a
is now a
is a shift matrix, the noise
with 1 vectors and scalars. is the minimum variance linear estimate of given , . Note that the is . Therefore, from (27) and (39), the last last row of equals element of the innovations vector , which is the previous sample of the prediction error. This also shows the whiteness of the error seis white. quence, since the innovations process contains the last element In addition, note from (27) that . Therefore, this element is estimated with zero variance, of i.e., . This is achieved with and . The flowgraph of the Kalman DFE with is shown in Fig. 6, where we have partitioned the innovations . vector as The channel model (37) remains valid if the equalization ), in which case, delay is shorter than the channel length ( and have sizes and , respectively, with
and . The Kalman equalizer partitioned as in is still given by (39), with the Kalman gain (40) but now having size . For , the last elements of the channel state vector are estimated , which is with zero variance since these are delays of present in the observed vector . This is achieved with for and , for . The corresponding flowgraph is depicted in Fig. 7. The Kalman structure is recursive in nature, and thus, the transfer functions of the feedforward and feedback filters would seem to be IIR. Since we know from the Wiener approach that these filters are in fact FIR, this means that the value of the results in all the filter poles being asymptotic Kalman gain placed at the origin. Observe that, as in the Kalman implementation of the MMSE LE, the channel coefficients appear directly in the structures of Figs. 6 and 7. This could be useful when designing DFE structures that exploit the sparsity of the channel in certain applications, such as high-definition television (HDTV) terrestrial broadcast systems [10]. Lack of Modularity: One could think that the state variables of the Kalman DFE structure with delay provide the outputs of the DFE filters with delays as well, but this is not
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Fig. 6.
Flow diagram of the Kalman MMSE DFE, assuming
Fig. 7.
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L.
Flow diagram of the Kalman MMSE DFE, for < L.
true. To see this, note that the th element of in the delay DFE filter is the minimum variance linear estimate of , and . On the other hand, given the output of the delay MMSE DFE is the minimum variance , given and linear estimate of . For , these two estimates are different in general. V. MMSE LE VERSUS THE MMSE DFE Recent works [7], [9], [12], [13] have drawn attention to a striking connection between the MMSE LE and MMSE DFE filters in the unconstrained case. Namely, the transfer function of the (doubly infinite) MMSE LE is, for our channel model and equalization lag (41) is the monic minimum-phase spectral where, as before, factor of . Observe that is not causal. On the other hand, the feedforward and feedback filters of the MMSE DFE obtained when the only constraint is the strict causality of the feedback section are, respectively, [15] (42) This relation between the unconstrained MMSE LE and MMSE DFE is depicted in Fig. 8. Basically, both equalizers share the same building blocks, the only difference being the inclusion of
Fig. 8. Illustrating the relation between the unconstrained MMSE LE and MMSE DFE.
the slicer in the feedback loop for the MMSE DFE. Based on this observation, Labat et al. [12] proposed a means for blind (i.e., unsupervised) initialization of the DFE: First, obtain the MMSE LE via some blind equalization scheme for linear equalizers, and then, throw the switch in Fig. 8 to DFE mode. The main drawback of this approach, as noted in [7], is that it rests on the assumption that the feedforward filter is unconis not causal, and therefore, pracstrained. The resulting tical implementation requires a sufficiently large delay and discarding the remaining noncausal part. If the length of the feedforward transversal filter used is not large enough, then the parameters of the MMSE LE for such constrained structure need not be close to those of the corresponding MMSE DFE. In view of this, it is interesting to compare the structures of the realizable MMSE LE and MMSE DFE developed in the
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Fig. 9. Realizable MMSE LE (a) and MMSE DFE (b) for the delay .
previous sections, as well as their properties. Hopefully, this will aid the designer in the task of developing adequate algorithms and structures for linear and/or decision feedback equalization. A. Same Structure for LE and DFE The realizable MMSE LE and MMSE DFE for the delay consist of a multiple-input single-output transversal section and respectively), both of order , followed by an ( SISO recursive/feedback section [ and , respectively], both of order . See Fig. 9. B. Effect of Delay Observe that the recursive section of the realizable MMSE , which is independent of the delay. It coincides with LE is the recursive section of the unconstrained MMSE LE, as well as with the feedback filter of the unconstrained MMSE DFE. is the spectral factor of , it can Being that be determined from the second-order statistics of the received . signal In contrast, the feedback filter of the realizable MMSE DFE varies with the choice of the delay . This is due to the fact that the goal of is to cancel the postcursor ISI of the overall channel-feedforward filter setting, which remains a function of . Another difference between the realizable MMSE LE and DFE regarding the effect of the delay is as follows. If , then the block of the MMSE LE has degree in general. However, the feedforward filter of the DFE has degree , and therefore, its computational complexity is smaller.
E. Single-Channel Case , if the order of the MMSE LE sections When and in Fig. 9(a) is reversed, then the output of the all-pole section [which is the input to the transversal sec] has been whitened. Having operate with an tion uncorrelated tap-input vector may be beneficial in adaptive implementations [14], [16]. As noted in Section III, in the single-channel case, the inherits the features of the Kalman transversal part implementation, namely, the shift property (25) and modularity (the MMSE LE of delay incorporates in its structure all those for the delays as well). These, however, do not hold for the realizable MMSE DFE: the feedforward filters and are not related by simple shift operations, although for large values of , a relation similar to (25) seems to yield a good approximation. F. Placing the MMSE LE in a Decision Feedback Loop From the preceding observations, it is clear that if the constituent blocks from the MMSE LE with delay are placed within the decision feedback loop (i.e., if one takes and ), the resulting system will be different, in general, from the MMSE DFE. Hence, this DFE solution (which will be referred to as MMSE LE in decision feedback loop, or MMSE LE-DFL) is suboptimal; however, it enjoys the following property. Lemma 3: Under the correct past decisions assumption, the MMSE LE-DFL setting for the delay cancels all postcursor . ISI, provided that Proof: Let the delay- MMSE LE output and error signals be, respectively, given by
C. Minimum-Phase Property The poles of the realizable MMSE LE lie all inside the unit is the minimum-phase spectral factor of circle since . Hence, the MMSE LE is stable. On the other hand, may have roots outside , as observed in [1] and [7]. (This does not necessarily mean that the MMSE DFE is an unstable system, due to the presence of the slicer in the feedback loop.) D. Whiteness of Prediction Error As shown in Section IV, the realizable MMSE DFE produces a white error sequence. This is not the case, in general, for that of the realizable MMSE LE, which is colored according to (22).
(43) Assuming that past decisions are correct, let (44) be the output and error signals of a DFE with feedforward , , respectively (i.e., the MMSE and feedback filters LE-DFL setting). Note that (45)
LÓPEZ-VALCARCE: REALIZABLE LINEAR AND DECISION FEEDBACK EQUALIZERS
which is reminiscent of the relation between the so-called equation error and output error in adaptive IIR filtering [21]. Now, to is the transfer function from
Postcursor ISI is canceled if for . , it suffices to show Since cov , . Let that cov , and note that cov for all , since is the MMSE LE error sequence. Then, from (45) cov
cov
cov
where
is the whitening prefilter, and is the innovations postfilter, which can be obtained from the asymptotic Kalman gain [see (15) and to the innovations (16)]. Define the transfer function from as vector
(46) Now, let be the error sequence of the MMSE LE, as in for (43). By the orthogonality principle, cov all , so that
In addition, observe that cov
Proof: In view of Lemma 3, it suffices to show that the of the MMSE LE-DFL setting, which is error sequence for all . We defined in (44), satisfies cov begin by recalling that the transfer function of the MMSE LE can be written as
cov for
cov
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cov cov
cov
cov
cov for all
(47) , the second term in the right-hand side of (47) is zero. For Thus, from (46) and (47)
as well. Therefore, for all cov
cov
cov
for
(48) Note that is a polynomial of order not exceeding , i.e., for . Hence, evaluating (48) for yields . Since by assumption (cf. . Now, (48) with Section II-B), we must have gives , so that . Repeating this process, one finds , as was to be shown. This postcursor ISI cancellation property means that is the optimum feedback filter in a DFE configuration in which . This property need not the feedforward filter is fixed to hold, however, if . In general, is not the opsince the orthogonality conditimum feedforward filter , are not satisfied. However, tions cov , and therefore, one (46) shows that these do hold for all could expect to lie reasonably close to . G. Asymptotic Behavior As noted in [7], the discrepancies between the components of the MMSE LE and the MMSE DFE tend to diminish as the length of the filters is increased. This is reflected in the following result. , the constituent blocks of Lemma 4: As the delay the MMSE DFE tend to those of the MMSE LE, i.e., and .
is a white process with since innovations postfilter satisfies
cov
Hence, as , one has is a filter matched to the effective channel
(49) . Hence, the
, which . Note that .
Therefore, as
(50) which is seen to be equal to [the filter appearing in the expression of the doubly infinite MMSE LE (41)]. Hence, not surprisingly, as the equalization lag becomes infinite, the realizable MMSE LE approaches the unconstrained MMSE LE. Now write cov cov cov
cov cov (51)
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and recall that, for a stationary process and filters , , one has
with psd
cov which is Parseval’s relation. The path of integration is the counterclockwise unit circle. Using this, (50) shows that the second term in the right-hand side of (51) becomes, for large
where is the covariance matrix of the innovations vector for the LE [cf. (11)]. Thus, the larger the variations of the function , the larger the asymptotic performance improvement of the DFE over the LE. On the other hand, this improvement could be expected to decrease as the SNR is reare pulled duced. This is because in that case, the roots of . toward the origin, resulting in a flatter Finally, closed-form expressions for the asymptotic values of and can also be obtained. , one has Lemma 6: As
cov cov (55)
cov and therefore, from (51), cov for all when , as was to be shown. Thus, for large equalization lags, the structural blocks of the MMSE LE and the MMSE DFE coincide. This fact can be used to determine the MSE improvement of the DFE over the LE. Lemma 5: The asymptotic MSE advantage of the realizable MMSE DFE over the realizable MMSE LE is given by (52) be the error sequence of the MMSE Proof: Again, let be that of the MMSE LE setting. LE-DFL setting, and let These are related by (45), or equivalently, by
Proof: Recall from (23) that is the th tap of the combined response , becomes In view of (50), as
, where given in (21).
where the third line follows from the matrix inversion lemma,1 whereas the fourth follows from (12). Therefore, the MMSE becomes asymptotically (56)
(53) , becomes the error seIn view of Lemma 4, as quence associated with the MMSE DFE, and therefore, it is white (cf. Section IV). From this observation and (53)
yielding the first part of (55) in view of (54). Then, the second part follows from Lemma 5. The asymptotic values (55) can be recast in a more familiar form. Noting from (11) the spectral factorization (57)
cov cov which is (52). Lemma 5 is in agreement with the linear prediction interpretation of the unconstrained MMSE DFE from [5]. The LE is still colored due to the limitations of the LE. Then, error is seen as a linear prediction filter that removes this correlation to whiten the DFE error , therefore reducing the , error variance. Equality holds in (52) if and only if whereas in view of (11) and (12), the integrand in (52) can be written as
(54)
it is seen from (57) that the asymptotic MMSE of the DFE can be rewritten as (58) The expressions (56) and (58) of the asymptotic MMSEs in the form of arithmetic and geometric means agree with known results for the single-channel case [15], for which the integrand reduces to a scaled version of the inverse of the received signal psd (for , it is a scaled version of the inverse of the determinant of the received signal psd).
A + vv
1(
)
=
A 0 (A vv A
)=(1 +
v A v)
:
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Fig. 10.
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Zeros of the two subchannels in the example. Fig. 11. MSE of the realizable equalizers as a function of delay. Dotted lines denote asymptotic values. SNR = 14 dB.
VI. NUMERICAL EXAMPLE Consider the following single-input two-output channel of order , which represents a simple model of a two-ray propagation channel:
(59) The zeros of the two subchannels are plotted in Fig. 10. Observe the nearly common subchannel roots, which are also close to the unit circle. This makes this channel rather severe. The noise covariance matrix is assumed to be
This covariance matrix would arise, for example, if a square root raised cosine filter with 100% excess bandwidth is used at the receiver front end and the received signal is sampled at twice the baud rate [6]. With , this results in SNR dB in the received signal. Fig. 11 depicts the MSE achieved by the realizable MMSE LE and MMSE DFE as a function of the delay . The MSE obtained with the MMSE LE-DFL approach is also shown, assuming as before that past decisions are correct. Observe the significant drops in the curves for , 2 (for which the first ray of the channel is captured) and , 8 (which capture the energy in the second ray). The asymptotic values of and are found to be and dB, respectively. Thus, the asymptotic improvement of the DFE over the LE is 2.3 dB. The nonincreasing property of the MSE with can be observed. This property provides the designer with a direct tradeoff between complexity and performance: The equalization delay equals the filter order, which should be taken as large as possible within complexity constraints in order to reduce the MSE.
In addition, if the MMSE LE-DFL strategy is to be used in order should to initialize the DFE, Lemma 3 suggests that be adopted in order to completely remove postcursor ISI. Note in Fig. 11 that the excess MSE of the MMSE LE-DFL approach reduces considerably for delays larger than the channel length. Except for , the (suboptimal) MMSE LE-DFL approach always yields, in this case, a lower MSE than that of the MMSE LE for the same delay. In order to illustrate the relation between the LE and DFE sec(the “short feedforward tions, we will consider two cases: filter” case) and (the “long feedforward filter” case). The taps of the feedforward and the feedback/recursive portions of the MMSE equalizers for the delay are shown in Fig. 12. As expected, the sections of the MMSE LE do not coincide with those of the MMSE DFE, although the coefficients of and are reasonably close. Observe that the feedforward secof the MMSE DFE has order equal to (i.e., its tion taps with are zero) as expected, whereas this is not of the MMSE LE, which true for the feedforward section . in general has order The overall channel-equalizer impulse response for the MMSE DFE, as well as that for the MMSE LE-DFL approach , are shown in Fig. 13. Observe that some residual postcursor ISI remains in (since ) but not in . In this setting, dB, whereas the MSE obtained in the MMSE LE-DFL setting is dB, i.e., the excess MSE of this suboptimal DFE is of 1.71 dB. Now, we examine the equalizers that result for , with the same SNR dB. Fig. 14 shows the coefficients of the feedforward and the feedback/recursive portions of the MMSE LE and MMSE DFE. Although these sections are not yet quite the same, they do lie reasonably close to each other, as expected. The overall channel-equalizer impulse responses in the decision feedback loop and are shown in
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Fig. 12.
Fig. 13.
Comparison of the feedforward and feedback/recursive sections of the LE (circles) and DFE (squares). Delay = 4, SNR = 14 dB.
Overall channel-equalizer impulse responses of the MMSE LE in decision feedback loop and MMSE DFE. Delay = 4, SNR = 14 dB.
Fig. 15. Note that now, the entire postcursor ISI has been cancelled in both systems, as predicted by Lemma 3. The excess MSE of the MMSE LE-DFL approach obtained with has dropped to 0.5 dB.
The ultimate performance measure of any equalizer structure is the symbol error rate (SER). Fig. 16 shows the SER as a function of SNR for the channel (59) and the same noise crosscorrelation coefficient of 0.5 between the two subchannels, for the
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Fig. 14.
Fig. 15.
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Comparison of the feedforward and feedback/recursive sections of the LE (circles) and DFE (squares). Delay = 10, SNR = 14 dB.
Overall channel-equalizer impulse responses of the MMSE LE in decision feedback loop and MMSE DFE. Delay = 10, SNR = 14 dB.
MMSE LE, MMSE DFE, and MMSE LE-DFL designs with and . A real 4-PAM constellation was used. It must be emphasized that the curves for the DFE and LE-DFL approaches were obtained with hard decisions being fed back, so that the filters were subject to error propagation. Even in that
situation, the LE-DFL strategy provides a sizable SER improvement over the MMSE LE, and for , its performance is very close to that of the MMSE DFE, as expected. These observations lend additional support for the LE-DFL initialization scheme of [12] and [13].
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Fig. 16.
Symbol error rate of the realizable equalizers as a function of SNR.
VII. CONCLUSIONS
ACKNOWLEDGMENT
We have presented a systematic analysis of the structure and properties of MMSE linear and decision feedback equalizers under realizability constraints, meaning that the filters should be causal and stable and of finite degree. Our focus has been on finite impulse response, single-input multiple-output channels with uncorrelated symbols and noise. The use of Wiener as well as Kalman filtering ideas reveals interesting properties of the equalizers concerning their structure, order, and error characteristics. In addition, the properties of the MMSE linear equalizer when placed in a decision feedback loop have been explored. The orthogonality principle provides an intuitively more appealing tool for the analysis than previous polynomial approaches. We have shown that the McMillan degree of both MMSE LE and MMSE DFE is no less than the channel length. In many practical situations, this requirement may be too demanding. In that case, one is faced with a constrained design, which is usually carried out by assuming a transversal structure for the equalizer. Clearly, it is of interest to explore the structure and properties of IIR configurations in such undermodeled scenarios in order to determine whether or not they may provide advantages over traditional FIR filters in terms of performance and/or complexity reduction. Our analysis of the realizable MMSE DFE hinges on the standard assumption of correct decisions being fed back. In severe channel conditions, this assumption will be violated, error propagation will take place, and designs different from the MMSE approach could provide smaller SER [11], [22]. An interesting open question is whether it is possible to obtain initialization strategies similar to the MMSE LE-DFL approach for these alternative designs.
The author would like to thank Prof. P. A. Regalia for helpful comments on the subject of this paper. REFERENCES [1] N. Al-Dhahir and J. M. Cioffi, “MMSE decision-feedback equalizers: Finite-length results,” IEEE Trans. Inform. Theory, vol. 41, pp. 961–975, July 1995. [2] A. Ahlén and M. Sternad, “Optimal deconvolution based on polynomial methods,” IEEE Trans. Acoust., Speech, Signal Processing, vol. 37, pp. 217–226, Feb. 1989. [3] B. D. O. Anderson and J. B. Moore, Optimal Filtering. Englewood Cliffs, NJ: Prentice-Hall, 1979. [4] A. de Baynast and I. Fijalkow, “Prefiltered blind equalization: How to fully benefit from spatio-temporal diversity?,” in IEEE Int. Conf. Acoust., Speech, Signal Process., vol. 4, Salt Lake City, UT, May 2001, pp. 2069–2072. [5] C. Belfiore and J. Park, Jr., “Decision feedback equalization,” Proc. IEEE, vol. 67, pp. 1143–1156, Aug. 1979. [6] D. K. Borah, R. A. Kennedy, Z. Ding, and I. Fijalkow, “Sampling and prefiltering effects on blind equalizer design,” IEEE Trans. Signal Processing, vol. 49, pp. 209–218, Jan. 2001. [7] R. A. Casas, T. J. Endres, A. Touzni, C. J. Richard Jr., and J. R. Treichler, “Current approaches to blind decision feedback equalization,” in Signal Processing Advances in Wireless & Mobile Communications, G. B. Giannakis, Y. Hua, P. Stoica, and L. Tong, Eds. Englewood Cliffs, NJ: Prentice-Hall, 2001, ch. 11. [8] J. M. Cioffi, G. P. Dudevoir, M. V. Eyuboglu, and G. D. Forney Jr., “MMSE decision-feedback equalizers and coding—Parts I and II,” IEEE Trans. Commun., vol. 43, pp. 2582–2604, Oct. 1995. [9] T. J. Endres et al., “Carrier independent blind initialization of a DFE,” in Proc. 2nd IEEE Workshop Signal Process. Adv. Wireless Commun., May 1999, pp. 239–242. [10] I. J. Fevrier, S. B. Gelfand, and M. P. Fitz, “Reduced complexity decision feedback equalization for multipath channels with large delay spreads,” IEEE Trans. Commun., vol. 47, pp. 927–937, June 1999. [11] M. Ghosh, “Analysis of the MMSE DFE with error propagation,” in Proc. IEEE GLOBECOM Commun. Theory Mini-Conf., vol. 4, Nov. 1997, pp. 85–89. [12] J. Labat, O. Macchi, and C. Laot, “Adaptive decision feedback equalization: Can you skip the training period?,” IEEE Trans. Commun., vol. 46, pp. 921–930, July 1998.
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[13] J. Labat and C. Laot, “Blind adaptive multiple-input decision feedback equalizer with a self-optimized configuration,” IEEE Trans. Commun., vol. 49, pp. 646–654, Apr. 2001. [14] J. P. LeBlanc and I. Fijalkow, “Blind adapted, pre-whitened constant modulus algorithm,” in Proc. IEEE Int. Conf. Commun., vol. 8, Helsinki, Finland, June 2001, pp. 2438–2442. [15] E. A. Lee and D. G. Messerschmitt, Digital Communications. Boston, MA: Kluwer, 1994. [16] R. López-Valcarce and F. Pérez-González, “Efficient reinitialization of the prewhitened constant modulus algorithm,” IEEE Commun. Lett., vol. 5, pp. 488–490, Dec. 2001. [17] S. McLaughlin, “Adaptive equalization via Kalman filtering techniques,” Proc. Inst. Elect. F Radar Signal Processing, vol. 138, pp. 388–396, Aug. 1991. [18] B. Mulgrew and C. F. N. Cowan, “An adaptive Kalman equalizer: Structure and performance,” IEEE Trans. Acoust., Speech, Signal Processing, vol. ASSP-35, pp. 1727–1735, Dec. 1987. [19] J. G. Proakis, Digital Communications, 4th ed. New York: McGrawHill, 2001. [20] S. U. H. Qureshi, “Adaptive equalization,” Proc. IEEE, vol. 73, pp. 1349–1387, Sept. 1985. [21] P. A. Regalia, Adaptive IIR Filtering in Signal Processing and Control. New York: Marcel Dekker, 1995. [22] M. Reuter, J. C. Allen, J. R. Zeidler, and R. C. North, “Mitigating error propagation effects in a decision feedback equalizer,” IEEE Trans. Commun., vol. 49, pp. 2028–2041, Nov. 2001. [23] M. D. Srinath, P. K. Rajasekaran, and R. Viswanathan, An Introduction to Statistical Signal Processing with Applications. Englewood Cliffs, NJ: Prentice-Hall, 1996.
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[24] M. Sternad and A. Ahlén, “The structure and design of realizable decision feedback equalizers for IIR channels with colored noise,” IEEE Trans. Inform. Theory, vol. 36, pp. 848–858, July 1990. [25] C. Tidestav, A. Ahlén, and M. Sternad, “Realizable MIMO decision feedback equalizers: Structure and design,” IEEE Trans. Signal Processing, vol. 49, pp. 121–133, Jan. 2001. [26] L. Tong, G. Xu, B. Hassibi, and T. Kailath, “Blind identification and equalization based on second order statistics: A frequency domain approach,” IEEE Trans. Inform. Theory, vol. 41, pp. 329–334, Jan. 1995. [27] P. A. Voois, I. Lee, and J. M. Cioffi, “The effect of decision delay in finite-length decision feedback equalization,” IEEE Trans. Inform. Theory, vol. 42, pp. 618–621, Mar. 1996.
Roberto López-Valcarce (M’01) was born in Spain in 1971. He received the telecommunications engineer degree from Universidad de Vigo, Vigo, Spain in 1995, and the M.S. and Ph.D. degrees in electrical engineering from the University of Iowa, Iowa City, in 1998 and 2000 respectively. From 1995 to 1996 he was a systems engineer with Intelsis. He was the recipient of a Fundación Pedro Barrié de la Maza fellowship for continuation of studies. He is currently a research associate (Ramón y Cajal fellow) with the Signal Theory and Communications Department at Universidad de Vigo. His research interests are in adaptive signal processing, communications, and traffic monitoring systems.
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Realizable Linear and Decision Feedback Equalizers: Properties and Connections Roberto López-Valcarce, Member, IEEE
Abstract—Recently, there has been renewed interest in the use of infinite impulse response (IIR) linear equalizers (LEs) for digital communication channels as a means for both improving performance and blindly initializing decision feedback structures (DFEs). Theoretical justification for such an approach is usually given assuming unconstrained filters, which are not causal and therefore not implementable in practice. We present an analysis of realizable (i.e., causal, stable, and of finite degree) minimum mean square error (MMSE) equalizers for single-input multiple-output channels, both in the LE and DFE cases, focusing on their structures and filter orders, as well as the connections between them. The DFE resulting from rearranging the MMSE LE within a decision feedback loop is given special attention. It is shown that although this DFE does not in general coincide with the MMSE DFE, it still enjoys certain optimality conditions. The main tools employed are the Wiener theory of minimum variance estimation and Kalman filtering theory, which show interesting properties of the MMSE equalizers not revealed by previous polynomial approaches. Index Terms—Decision feedback equalizers, digital communications, intersymbol interference, linear equalizers, realizable equalizers.
I. INTRODUCTION
O
VER the past few decades, several equalization techniques have been proposed in order to combat frequency selective transmission channels, which introduce intersymbol interference (ISI) in digital communication systems. The optimum receiver in terms of symbol error rate (SER) is the maximum likelihood sequence estimation detector [19], but its associated computational cost grows exponentially with the channel memory. This precludes its use in many applications. Linear equalizers (LEs) followed by a symbol-by-symbol detector are attractive in terms of complexity, although they might introduce excessive noise enhancement if the channel frequency response presents deep nulls [20]. Decision feedback equalizers (DFEs), on the other hand, provide postcursor ISI cancellation with reduced noise enhancement and are widely recognized to offer better steady-state performance than LEs [5]. However, due to the presence of a nonlinear decision device inside the DFE feedback loop, erroneous decisions can result in error bursts that degrade SER performance. This error propagation effect makes transient analysis of the DFE
Manuscript received January 31, 2002; revised April 9, 2003. This work was supported by the Ramón y Cajal program of the Spanish Ministry of Science and Technology. The associate editor coordinating the review of this paper and approving it for publication was Dr. Sergios Theodoridis. The author is with the Department of Signal Theory and Communications, University of Vigo, Vigo, Spain (e-mail: [email protected]). Digital Object Identifier 10.1109/TSP.2003.822360
difficult, and a past correct decisions assumption is often invoked to sidestep this problem. This is reasonable if the DFE is operating in a low SER scenario. In this paper, we study the structure and properties of the minimum mean square error (MMSE) equalizers under the only constraint of realizability, i.e., the filter may have an infinite impulse response (IIR), but it must be causal, stable, and have finite degree. In contrast, usual approaches either assume equalizers with a doubly infinite number of taps [5], [20] or consider finite impulse response (FIR) filters of prespecified degree (and thus potentially suboptimal) [1]. Realizable LEs will be discussed in Section III, whereas Section IV deals with the DFE case. A few authors have previously considered the problem of IIR equalization. Using Wiener and Kalman filtering techniques, Mulgrew and Cowan [18] and McLaughlin [17] obtained the realizable MMSE LE and MMSE DFE, respectively, for the single-input single-output (SISO) FIR channel case. The advantage of this approach is that it automatically reveals the optimal degree for the equalizers. Increasing the equalization lag is seen to decrease the MMSE at the price of a higher filter degree so that a direct relation between complexity, performance, and equalization delay is obtained. Ahlén and Sternad [2], [24] used a polynomial approach for the SISO problem, allowing for colored symbols and/or noise, as well as for IIR channels. Tidestav et al. [25] applied these polynomial methods to obtain the realizable MMSE DFE in a multiple-input multiple-output (MIMO) configuration. We assume a single-input multiple-output (SIMO) FIR channel model with white transmitted symbols; the channel noise is taken as temporally white but possibly spatially colored. Particularities of the single output channel case are also highlighted. One of our contributions is the extension of the Wiener/Kalman techniques of [17], [18] to this SIMO setting. We adopt this approach since we believe it provides insights not obtained with the polynomial methods of [2], [24], [25]. For example, the filter structures obtained from the Kalman state-space equations are appealing if the sparsity of the channel present in several applications is to be exploited. The value of the equalization delay is seen to impact the constituent blocks of the realizable MMSE LE and MMSE DFE differently. It is also shown (Section IV-A) that, contrary to previous belief, the error sequence associated with the realizable MMSE DFE is white. New expressions for the asymptotic performances of the equalizers are obtained as well (Section V). IIR LEs have recently received renewed attention [4], [7], [9], [12], [13] because, in addition to their potential to outperform
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conventional FIR structures with the same number of parameters, they provide a means for unsupervised initialization of a DFE. This approach relies on the fact that the denominator of the transfer function of the doubly infinite MMSE LE coincides with the feedback filter of the doubly infinite MMSE DFE. However, as shown in [7], this is no longer true once finite degree structures are considered. With this in mind, we compare the structures and properties of the realizable linear and DFEs in Section V. It will be shown that when the components of the MMSE LE are rearranged into DFE mode, the resulting feedback filter is still optimized in terms of the feedforward filter, provided that the equalization lag is no less than the channel length. This result provides additional support for this DFE initialization strategy. The notation adopted in the paper is as follows. • Scalars are denoted in lowercase. • Vectors are denoted in lowercase bold. • represents a vector of all zeros, except for a 1 in the th position, counting from zero. • Matrices are denoted in uppercase bold. , , and denote respectively conjugation, trans• position, and conjugate transposition. • denotes the “paraconjugate” of the transfer function , i.e., . • The causal part of is extracted by . • The expression stands for . With this notation, there is no need to distinguish between the complex variable and the unit advance operator, which are both denoted by . • With two vector-valued random variables and , cov . The paper is organized as follows. Section II provides a brief review of the minimum variance estimation problem, and it also presents the channel model. MMSE linear equalizers are discussed in Section III, from both the Wiener and Kalman theory viewpoints. This is also done in Section IV for the MMSE DFE. The connections between these structures are analyzed in Section V. We close with a numerical example in Section VI. II. WIENER FILTERING AND CHANNEL MODEL In this section, we briefly review the structure and design of the causal Wiener filter for the general estimation problem [23], and then, we present the channel model that will be used throughout the paper. A. Causal Wiener Filter Consider the general linear estimation problem in which a is sought so that the causal linear filter mean square value of the norm of the error is minimized. Here, is a sequence of observations, and is a reference signal. The processes , are assumed wide-sense stationary with zero mean. In addition, is assumed to be a full rank process [3, Sec. 9.4], meaning that , to be defined shortly, is not its power spectral density singular for all .
is opThe orthogonality principle states that the filter is orthogonal to the observatimized if and only if the error tions, i.e., cov
for all
Hence, by introducing the following correlation sequences and their -transforms: cov cov one finds that the optimum filter’s coefficients satisfy for all Note that if
were white with for some , then the Wiener filter coefficients would be , , that is, readily obtained from . is not white, the Wiener filter includes a When whitening prefilter. The power spectral density (psd) of can be factored as the full rank process (1) , and the transfer function is causal (its where series expansion involves only nonpositive powers of ), monic for ), and minimum phase (all poles lie in ( , and has constant rank in ). Now, let and
Then, the psd of the process is so that the observations have been whitened is a stable transfer function since as desired. Note that is minimum phase. The Wiener filter is therefore given by , where and cov (2) and the filter as We will refer to the white process the innovations process and the innovations postfilter, respectively. , the causal Wiener filter Now, since can be written as
(3) Comparing this with the doubly infinite (and therefore not realizable) Wiener filter (4)
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Fig. 1. Relation between the causal and the doubly infinite Wiener filters.
both devices are seen to include a prewhitening filter addition, if we write
. In
then we see that the innovations postfilters are related via . However, in general , i.e., the causal Wiener filter is not obtained by direct truncation of the doubly infinite Wiener filter. This is illustrated in Fig. 1. is causal and stable, We must point out that although (3) does not reveal whether it also has finite McMillan degree (which is the minimum number of delay units required for its implementation). This depends on the particular context. In our case, this context is the channel equalization problem, for which the developments in Section III-B (LE) and Section IV-A (DFE) will show that in both cases, the resulting causal Wiener filter does have finite degree, and therefore, it is realizable. Finally, if we write the innovations postfilter as , the MMSE attained by the causal Wiener filter is found to be cov
cov
Fig. 2. Channel model.
where is the additive noise, which is assumed to be zero mean, independent of , and temporally white with for and otherwise. We allow cov for spatial noise correlation in the channel model since this is the case when the multichannel configuration arises from oversampling the received continuous-time signal [6]. Symbols, channel coefficients, and noise are complex-valued in general. It is also of interest to consider a state-space representation of the channel. Since, in the equalization problem, one is interested , let us in estimating a delayed version of the channel input . With this, the channel state vector is define
(5)
(7) Then, the state space equations of the channel (6) are
B. Channel Model We consider a single-input multiple-output (SIMO) baseband digital communications model, as represented in Fig. 2. The scalar symbol sequence is assumed to be zero mean white, without loss of generality. This scaled to unit variance sequence is transmitted over a linear time invariant channel with FIR of order . Thus, the channel transfer function is a vector given by
It is assumed without loss of generality that , as any can be eliminated by including zero leading coefficients in . This a corresponding delay in the symbol sequence multichannel model accommodates communications systems where antenna arrays are used to improve the detection, as well as oversampled received signals (i.e., fractionally spaced receivers) and/or systems with frequency diversity. The received 1 vector signal is the (6)
(8) shift matrix with ones on the first where is a matrix subdiagonal and zeros elsewhere, and is a comprised of the channel coefficients: (9) for . The state space description It is assumed that will be useful when discussing the structure of the steady-state Kalman equalizer. III. MMSE LINEAR EQUALIZERS In the linear equalization context, the goal is to design a linear under the constraints of causality and stability in filter order to minimize the mean square value of the error , where is a suitable delay between the channel input and the estimate being made. Thus, in the nota, which tion of Section II-A, the reference signal is is a scalar.
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A. Wiener Filtering Approach
Low SNR Case: It is instructive to consider the high noise , with and scenario. Let us write as a matrix with unit 2-norm. Then
The psd of the received signal takes the form
Observe that the series expansion of contains only to . Hence, the minimum powers of ranging from appearing in (1) is an FIR transfer phase spectral factor function of order , and is said to be a moving average transfer function of process of order , or MA( ). The the whitening prefilter can be written as adj
(10)
where “adj” denotes the adjugate matrix. Let . Then, is monic and minimum phase. From (1) (11) Note that
which tends to 1 as the noise power, and, hence, tends to infinity. Thus, in that case, i.e., the effect of noise is to pull the poles of the whitening prefilter toward the origin. Noiseless Case: In the opposite direction, let us assume that so that . If , i.e., if there is a single channel, the received signal is a scalar process and, therefore, full rank [3]. Hence, the spectral factorization of its psd is still well defined, and the preceding development of is the MMSE LE holds. On the other hand, always singular for . In that case, it is well known that as for all , i.e., the subchannels do not share long as of order a common root, an FIR equalizer exists such that , thus yielding zero MSE [26]. , of the subchannels is If the greatest common divisor, say , where now, nontrivial, one can write admits an FIR equalizer so that one recovers the single-channel case. B. Kalman Filtering Approach
(12) 1 where the third line follows from the fact that for arbitrary . (This can vectors , , one has be easily shown by induction on .) Therefore, from (11) and is seen to be the monic minimum-phase spectral (12), . This also shows that is a factor of polynomial of order . Observe that adj is a FIR transfer function, which, . In the single-channel case ( in general, has order ), this adjugate matrix reduces to a scalar constant so that the . whitening filter is just an all-pole filter Concerning the innovations postfilter , observe that with the innovations process cov
cov
(13)
which is zero if since is white, and the transfer function from to is causal. Therefore, the inis FIR with order . (Incidentally, this novations postfilter holds true even if the noise were colored and/or the channel were IIR.) In view of this, the MMSE LE for the delay is seen to be FIR filter of the series interconnection of three blocks: a ,a1 FIR filter of order , and a scalar all-pole order filter of order : adj where is the equalizer output. Hence, the order of the overall adj could be as high as . HowFIR part ever, examination of the Kalman structure in the next section . will reveal that this order is in fact just
The Kalman filter provides a means to obtain the minimum variance linear estimate of the channel state vector from (7), given the observations [3]. Although the Kalman filter is in general time-varying, if the channel is time invariant and the noise and symbol sequences are stationary, it will asymptotically reach time invariance. This steady-state Kalman filter coincides with the realizable Wiener filter, providing valuable information about its structure. Given the state-space channel description (8), the steady-state Kalman filter operation can be described by (14) is the innovations sequence as before, whereas the Here, asymptotic Kalman gain is a matrix. The filter , is extracted from the vector esoutput, i.e., the estimate of . The structure of the Kalman equalizer timate via is shown in Fig. 3. We should note that the asymptotic Kalman gain can be obtained via the solution of a discrete-time Riccati equation [3]; here, our focus is just on the structure of the equalizer, and therefore, we will not worry about the exact value of . Observe that this implementation of the MMSE LE requires delay elements, and hence, this is its McMillan degree. In addition, observe that the Kalman structure effectively incorporates a whitening filter since the innovations process is directly available, and the channel coefficients (with the exception of ) appear directly in the equalizer parameterization. It is interesting as well to note that not only does this structure implement the MMSE LE for the delay but all those for the delays as well since the corresponding estimates are present in the filter’s state vector . If we partition row-wise as (15)
LÓPEZ-VALCARCE: REALIZABLE LINEAR AND DECISION FEEDBACK EQUALIZERS
Fig. 3.
Fig. 4.
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Structure of the Kalman equalizer (in steady state).
Flow diagram of the Kalman MMSE LE, assuming > L.
then it is seen from (14) that and for . This shows that the Kalman implementation of the MMSE LE is modular: The MMSE LE for the delay is obtained directly from that for the delay by adding a delay element and a multiplier . The flowgraph of the Kalman structure, which is depicted in Fig. 4, illustrates this property. polynomials By introducing the 1
It is instructive to write the transfer function of the MMSE LE for the delay as
where represents the recursive (all-pole) part, and denotes the transversal (all-zeros) component of the polynomials, one has equalizer. Hence, in terms of the
(16) adj
(18)
one can write the transfer function from the innovations vector to the equalizer state vector as (19)
.. .
Achieved MSE: From (5), the MSE attained by the delay MMSE LE is
.. .
(20)
In addition, observe that since
the whitening prefilter and innovations postfilter transfer functions are, respectively, given by
(17)
cov ). Therefore, (recall that with equality holding iff , in which case, . This means that, as it could be expected, the MMSE is a nonincreasing function of the equalization delay . An alternative expression for the MMSE is as follows. Let the overall channel-equalizer transfer function be (21)
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Then, for is given by
, the th autocorrelation coefficient of the error cov cov
cov
(22) The first line is a consequence of the orthogonality principle, whereas the second follows from independence of symbols and , we obtain the MMSE noise. For (23) Observe that is the cursor of the overall system . Comparing (20) and (23), we see that . Thus, is real, and . The fact that the cursor is not equal to 1 shows that the decision rule operating on the equalis izer output will be biased, that is, the equalizer output a conditionally biased estimate of since . This bias can be removed by scaling the . Although this worsens equalizer transfer function by the MSE [it is easily shown that the resulting MSE is ], it improves the symbol error rate [8]. Interestingly, (23) shows a direct relation between the MSE and the bias. Comparison With FIR Design: The standard approach to LE design is to assume an FIR structure for the filter. In that case, the MSE becomes a complicated function of the delay. For the IIR structure, however, as we have seen, there exists a direct relationship between MSE, filter order, and delay. In addition, for the same filter order, the IIR LE always yields a smaller MSE than the lag-optimized FIR approach [18]. Single-Channel Case: In the single-channel case, (18) reduces to (24) , then Thus, if we write for . This shows that in the case, the numerators of the MMSE LEs transfer functions associated with different delays are related to each other by simple shift operations, which is a property recognized and exploited in [16] in the context of blind equalization: (25) However, this property is not satisfied in general in the multi). channel case ( An additional property is found in the relation between and for the single-channel case. If we write , then (24) together with shows that, for
..
.
.. .
.. .
.. .
(26)
Fig. 5.
DFE.
This relation means that, if , then the transfer funcreduces to a polynomial of degree tion . As it will be discussed in Section V, this translates into the cancellation of all postcursor ISI once the constituent blocks of the MMSE LE are placed in a decision feedback loop. Equation (26) also suggests a means to achieve channel identification via and are available. blind equalization once IV. MMSE DFES In a DFE, equalization is achieved by means of a causal, -input 1-output feedforward filter and a strictly causal, . Thus, the equalizer output is a SISO feedback filter linear combination of the current and past samples of the re, and past hard decisions , , ceived signal produced by a quantization device (slicer). Again, represents the equalization delay. This is shown in Fig. 5. is the symbol in the transmitted conThe slicer output stellation that is closest to . Note that, due to the inclusion of the slicer in the feedback loop, the DFE is a nonlinear device. This makes the analysis extremely difficult, and therefore, we will adopt the common assumption of correct past decisions, for all . This is reasonable if the receiver is i.e., operating at a low error rate, as it is usually the case. Under this , assumption, the DFE output becomes which is a linear function of the received and transmitted signals. Hence, in that case, the DFE can be regarded as linear. When the DFE operates in a high error rate environment, the correct past decisions assumption ceases to be valid, and error ) are propagation happens as symbol errors (i.e., fed back by the filter . Different strategies must be used for filter design in that case [11], [22]. A. Wiener Filtering Approach MMSE DFE design can be carried out using the realizable Wiener filter theory of Section II-A. The desired signal is as in the linear equalizer case, but now, the observed se, where is the augmented 1 vector quence is (27) As usual, we first compute the whitening prefilter and then the is found to innovations postfilter. The psd of the process be (28)
LÓPEZ-VALCARCE: REALIZABLE LINEAR AND DECISION FEEDBACK EQUALIZERS
Using the fact that
,
one has
which is a constant independent of . Let be the spectral factorization of . Note contains only powers of that the series expansion of ranging from to , again with , and therefore, is FIR of order . In addition
763
. Now, (31) is just the negative of the ( )th coeffifor , but this is cient of the impulse response of , and therefore, this a polynomial of degree not exceeding coefficient is zero if . Thus for all
cov On the other hand cov cov
cov
cov cov (33)
constant is constant. Thus, the whitening and it follows that adj is FIR. prefilter Realizable MMSE DFE Is FIR: The reasoning around (13) readily applies to the DFE case as well, showing that the innovations postfilter is again an FIR filter of order not exceeding . Therefore, the overall realizable Wiener equalizer (whitening prefilter plus innovations postfilter) given by the is, rather surprisingly, an FIR 1 vector filter. Degree of the Realizable MMSE DFE: The degrees of the , remain to be determined. Note that adj filters has order in principle. Nevertheless, the following holds. Lemma 1: Under the correct past decisions assumption, the feedforward and feedback filters of the realizable MMSE DFE for the delay have degrees and , respectively. A proof of Lemma 1 using polynomial techniques can be found in [25]. Here, we provide an alternative and more intuitive proof using the orthogonality principle. Proof: Consider a DFE with FIR feedforward and feed, having orders , , respectively. Asback filters suming correct past decisions, the output of such an equalizer is
Now, assume that , are optimal in the MSE sense for such a constrained DFE structure. Then, from the orthogonality must be uncorrelated with principle, the error the data:
Note that
is a linear combination of the symbols , which from (32) are uncorrelated for . Thus, the first term in (33) is zero for . with The second term equals the th coefficient of the impulse response of (times ), and therefore, it vanishes for since has order not exceeding . Thus
It suffices to show that (29) and (30) actually hold for all and all , and therefore, , implement the realizable MMSE DFE for the delay . First, note that
This, together with whiteness of bols and noise, yields cov
cov cov
and independence of symcov (31)
for all
cov
(34)
as was to be shown. We will denote by , the feedforward and feedback filters of the MMSE DFE in order to make explicit their dependence with the delay . Cioffi et al. [1], [27] observed that for a DFE design in which both feedforward and feedback filters are constrained to be FIR, the optimum equalization lag equals the feedforward filter order, provided that the feedback filter order is at least as large as the channel length. Lemma 1 confirms this observation but also states that for a given delay , there is no need to increase the filters’ orders beyond and . Whiteness of Prediction Error: The orthogonality principle also reveals the following property of the realizable MMSE DFE. Lemma 2: Under the correct past decisions assumption, the associated with the realizable MMSE DFE error sequence of delay is white. Proof: Let be the realizable MMSE DFE output. Then, , so that cov
(29) (30)
cov cov
(32)
cov
cov
(35)
is the realizable MMSE DFE error sequence, is Since as well as with uncorrelated with [cf. (32) and (34)]. Noting that is a linear combination and , it follows that the of , as was two terms in the right-hand side of (35) vanish for to be shown. Al-Dhahir and Cioffi [1] studied the properties of an MMSE DFE for the delay having a degree feedforward filter and a degree feedback filter, allowing for colored symbols and noise. They concluded [1, Sec.V-B] that the associated error sequence is not white in general. However, when symbols and noise are temporally white, Lemma 1 shows that such MMSE DFE is the realizable MMSE DFE, and then, Lemma 2 shows that in that case, the error sequence does become white.
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Achieved MMSE: The MSE of the MMSE DFE can be written in terms of the overall channel-equalizer response , which is now defined as
vector is defined as has size
, and the channel matrix and is given by (38)
The steady-state Kalman filter has a form similar to (14): so that
. Then cov
cov cov
(39) The asymptotic Kalman gain matrix, which we partition as
cov cov
(40)
similarly to (23). Hence, the MMSE is directly proportional to the bias, as in the LE case. In addition, it is true that . To see this, observe that the DFE defined by the , (which have orders and respecfilters for the delay . Thus, tively) result in an MSE of the performance of the MMSE DFE for the delay cannot be worse than this value. In addition, note that . This is because the realizable MMSE LE produces the MMSE , given , whereas the MMSE estimate of given DFE yields the MMSE estimate of as well as . Cancellation of Postcursor ISI: Note that cov , which, from (32), equals zero for , has degree . i.e., the overall channel-equalizer response This is to say that the feedback filter has canceled all postcursor ISI, which is a well-known property of DFEs. , then in (28) beNoiseless Case: When is not of full comes singular for all so that the process is not well derank, and the spectral factorization of fined. However, in that case, it is possible to achieve zero MSE and having orders and , respectively. This with , can be chosen to satisfy is because (36) eliminates all precursor ISI, whereas takes care i.e., , of all postcursor ISI as before. In the single-channel case , under the assumption of (36) uniquely determines a nonzero leading channel tap. If , there may be multiple filters satisfying (36). B. Kalman Filtering Approach Kalman filtering techniques can also be applied to the DFE problem [17]. Let us consider first the case of an equalization . The statelag at least as large as the channel length, i.e., space description of the channel model generating the vector in (27) has now dimension : (37) where now, 1 vector,
is a
is now a
is a shift matrix, the noise
with 1 vectors and scalars. is the minimum variance linear estimate of given , . Note that the is . Therefore, from (27) and (39), the last last row of equals element of the innovations vector , which is the previous sample of the prediction error. This also shows the whiteness of the error seis white. quence, since the innovations process contains the last element In addition, note from (27) that . Therefore, this element is estimated with zero variance, of i.e., . This is achieved with and . The flowgraph of the Kalman DFE with is shown in Fig. 6, where we have partitioned the innovations . vector as The channel model (37) remains valid if the equalization ), in which case, delay is shorter than the channel length ( and have sizes and , respectively, with
and . The Kalman equalizer partitioned as in is still given by (39), with the Kalman gain (40) but now having size . For , the last elements of the channel state vector are estimated , which is with zero variance since these are delays of present in the observed vector . This is achieved with for and , for . The corresponding flowgraph is depicted in Fig. 7. The Kalman structure is recursive in nature, and thus, the transfer functions of the feedforward and feedback filters would seem to be IIR. Since we know from the Wiener approach that these filters are in fact FIR, this means that the value of the results in all the filter poles being asymptotic Kalman gain placed at the origin. Observe that, as in the Kalman implementation of the MMSE LE, the channel coefficients appear directly in the structures of Figs. 6 and 7. This could be useful when designing DFE structures that exploit the sparsity of the channel in certain applications, such as high-definition television (HDTV) terrestrial broadcast systems [10]. Lack of Modularity: One could think that the state variables of the Kalman DFE structure with delay provide the outputs of the DFE filters with delays as well, but this is not
LÓPEZ-VALCARCE: REALIZABLE LINEAR AND DECISION FEEDBACK EQUALIZERS
Fig. 6.
Flow diagram of the Kalman MMSE DFE, assuming
Fig. 7.
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L.
Flow diagram of the Kalman MMSE DFE, for < L.
true. To see this, note that the th element of in the delay DFE filter is the minimum variance linear estimate of , and . On the other hand, given the output of the delay MMSE DFE is the minimum variance , given and linear estimate of . For , these two estimates are different in general. V. MMSE LE VERSUS THE MMSE DFE Recent works [7], [9], [12], [13] have drawn attention to a striking connection between the MMSE LE and MMSE DFE filters in the unconstrained case. Namely, the transfer function of the (doubly infinite) MMSE LE is, for our channel model and equalization lag (41) is the monic minimum-phase spectral where, as before, factor of . Observe that is not causal. On the other hand, the feedforward and feedback filters of the MMSE DFE obtained when the only constraint is the strict causality of the feedback section are, respectively, [15] (42) This relation between the unconstrained MMSE LE and MMSE DFE is depicted in Fig. 8. Basically, both equalizers share the same building blocks, the only difference being the inclusion of
Fig. 8. Illustrating the relation between the unconstrained MMSE LE and MMSE DFE.
the slicer in the feedback loop for the MMSE DFE. Based on this observation, Labat et al. [12] proposed a means for blind (i.e., unsupervised) initialization of the DFE: First, obtain the MMSE LE via some blind equalization scheme for linear equalizers, and then, throw the switch in Fig. 8 to DFE mode. The main drawback of this approach, as noted in [7], is that it rests on the assumption that the feedforward filter is unconis not causal, and therefore, pracstrained. The resulting tical implementation requires a sufficiently large delay and discarding the remaining noncausal part. If the length of the feedforward transversal filter used is not large enough, then the parameters of the MMSE LE for such constrained structure need not be close to those of the corresponding MMSE DFE. In view of this, it is interesting to compare the structures of the realizable MMSE LE and MMSE DFE developed in the
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Fig. 9. Realizable MMSE LE (a) and MMSE DFE (b) for the delay .
previous sections, as well as their properties. Hopefully, this will aid the designer in the task of developing adequate algorithms and structures for linear and/or decision feedback equalization. A. Same Structure for LE and DFE The realizable MMSE LE and MMSE DFE for the delay consist of a multiple-input single-output transversal section and respectively), both of order , followed by an ( SISO recursive/feedback section [ and , respectively], both of order . See Fig. 9. B. Effect of Delay Observe that the recursive section of the realizable MMSE , which is independent of the delay. It coincides with LE is the recursive section of the unconstrained MMSE LE, as well as with the feedback filter of the unconstrained MMSE DFE. is the spectral factor of , it can Being that be determined from the second-order statistics of the received . signal In contrast, the feedback filter of the realizable MMSE DFE varies with the choice of the delay . This is due to the fact that the goal of is to cancel the postcursor ISI of the overall channel-feedforward filter setting, which remains a function of . Another difference between the realizable MMSE LE and DFE regarding the effect of the delay is as follows. If , then the block of the MMSE LE has degree in general. However, the feedforward filter of the DFE has degree , and therefore, its computational complexity is smaller.
E. Single-Channel Case , if the order of the MMSE LE sections When and in Fig. 9(a) is reversed, then the output of the all-pole section [which is the input to the transversal sec] has been whitened. Having operate with an tion uncorrelated tap-input vector may be beneficial in adaptive implementations [14], [16]. As noted in Section III, in the single-channel case, the inherits the features of the Kalman transversal part implementation, namely, the shift property (25) and modularity (the MMSE LE of delay incorporates in its structure all those for the delays as well). These, however, do not hold for the realizable MMSE DFE: the feedforward filters and are not related by simple shift operations, although for large values of , a relation similar to (25) seems to yield a good approximation. F. Placing the MMSE LE in a Decision Feedback Loop From the preceding observations, it is clear that if the constituent blocks from the MMSE LE with delay are placed within the decision feedback loop (i.e., if one takes and ), the resulting system will be different, in general, from the MMSE DFE. Hence, this DFE solution (which will be referred to as MMSE LE in decision feedback loop, or MMSE LE-DFL) is suboptimal; however, it enjoys the following property. Lemma 3: Under the correct past decisions assumption, the MMSE LE-DFL setting for the delay cancels all postcursor . ISI, provided that Proof: Let the delay- MMSE LE output and error signals be, respectively, given by
C. Minimum-Phase Property The poles of the realizable MMSE LE lie all inside the unit is the minimum-phase spectral factor of circle since . Hence, the MMSE LE is stable. On the other hand, may have roots outside , as observed in [1] and [7]. (This does not necessarily mean that the MMSE DFE is an unstable system, due to the presence of the slicer in the feedback loop.) D. Whiteness of Prediction Error As shown in Section IV, the realizable MMSE DFE produces a white error sequence. This is not the case, in general, for that of the realizable MMSE LE, which is colored according to (22).
(43) Assuming that past decisions are correct, let (44) be the output and error signals of a DFE with feedforward , , respectively (i.e., the MMSE and feedback filters LE-DFL setting). Note that (45)
LÓPEZ-VALCARCE: REALIZABLE LINEAR AND DECISION FEEDBACK EQUALIZERS
which is reminiscent of the relation between the so-called equation error and output error in adaptive IIR filtering [21]. Now, to is the transfer function from
Postcursor ISI is canceled if for . , it suffices to show Since cov , . Let that cov , and note that cov for all , since is the MMSE LE error sequence. Then, from (45) cov
cov
cov
where
is the whitening prefilter, and is the innovations postfilter, which can be obtained from the asymptotic Kalman gain [see (15) and to the innovations (16)]. Define the transfer function from as vector
(46) Now, let be the error sequence of the MMSE LE, as in for (43). By the orthogonality principle, cov all , so that
In addition, observe that cov
Proof: In view of Lemma 3, it suffices to show that the of the MMSE LE-DFL setting, which is error sequence for all . We defined in (44), satisfies cov begin by recalling that the transfer function of the MMSE LE can be written as
cov for
cov
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cov cov
cov
cov
cov for all
(47) , the second term in the right-hand side of (47) is zero. For Thus, from (46) and (47)
as well. Therefore, for all cov
cov
cov
for
(48) Note that is a polynomial of order not exceeding , i.e., for . Hence, evaluating (48) for yields . Since by assumption (cf. . Now, (48) with Section II-B), we must have gives , so that . Repeating this process, one finds , as was to be shown. This postcursor ISI cancellation property means that is the optimum feedback filter in a DFE configuration in which . This property need not the feedforward filter is fixed to hold, however, if . In general, is not the opsince the orthogonality conditimum feedforward filter , are not satisfied. However, tions cov , and therefore, one (46) shows that these do hold for all could expect to lie reasonably close to . G. Asymptotic Behavior As noted in [7], the discrepancies between the components of the MMSE LE and the MMSE DFE tend to diminish as the length of the filters is increased. This is reflected in the following result. , the constituent blocks of Lemma 4: As the delay the MMSE DFE tend to those of the MMSE LE, i.e., and .
is a white process with since innovations postfilter satisfies
cov
Hence, as , one has is a filter matched to the effective channel
(49) . Hence, the
, which . Note that .
Therefore, as
(50) which is seen to be equal to [the filter appearing in the expression of the doubly infinite MMSE LE (41)]. Hence, not surprisingly, as the equalization lag becomes infinite, the realizable MMSE LE approaches the unconstrained MMSE LE. Now write cov cov cov
cov cov (51)
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and recall that, for a stationary process and filters , , one has
with psd
cov which is Parseval’s relation. The path of integration is the counterclockwise unit circle. Using this, (50) shows that the second term in the right-hand side of (51) becomes, for large
where is the covariance matrix of the innovations vector for the LE [cf. (11)]. Thus, the larger the variations of the function , the larger the asymptotic performance improvement of the DFE over the LE. On the other hand, this improvement could be expected to decrease as the SNR is reare pulled duced. This is because in that case, the roots of . toward the origin, resulting in a flatter Finally, closed-form expressions for the asymptotic values of and can also be obtained. , one has Lemma 6: As
cov cov (55)
cov and therefore, from (51), cov for all when , as was to be shown. Thus, for large equalization lags, the structural blocks of the MMSE LE and the MMSE DFE coincide. This fact can be used to determine the MSE improvement of the DFE over the LE. Lemma 5: The asymptotic MSE advantage of the realizable MMSE DFE over the realizable MMSE LE is given by (52) be the error sequence of the MMSE Proof: Again, let be that of the MMSE LE setting. LE-DFL setting, and let These are related by (45), or equivalently, by
Proof: Recall from (23) that is the th tap of the combined response , becomes In view of (50), as
, where given in (21).
where the third line follows from the matrix inversion lemma,1 whereas the fourth follows from (12). Therefore, the MMSE becomes asymptotically (56)
(53) , becomes the error seIn view of Lemma 4, as quence associated with the MMSE DFE, and therefore, it is white (cf. Section IV). From this observation and (53)
yielding the first part of (55) in view of (54). Then, the second part follows from Lemma 5. The asymptotic values (55) can be recast in a more familiar form. Noting from (11) the spectral factorization (57)
cov cov which is (52). Lemma 5 is in agreement with the linear prediction interpretation of the unconstrained MMSE DFE from [5]. The LE is still colored due to the limitations of the LE. Then, error is seen as a linear prediction filter that removes this correlation to whiten the DFE error , therefore reducing the , error variance. Equality holds in (52) if and only if whereas in view of (11) and (12), the integrand in (52) can be written as
(54)
it is seen from (57) that the asymptotic MMSE of the DFE can be rewritten as (58) The expressions (56) and (58) of the asymptotic MMSEs in the form of arithmetic and geometric means agree with known results for the single-channel case [15], for which the integrand reduces to a scaled version of the inverse of the received signal psd (for , it is a scaled version of the inverse of the determinant of the received signal psd).
A + vv
1(
)
=
A 0 (A vv A
)=(1 +
v A v)
:
LÓPEZ-VALCARCE: REALIZABLE LINEAR AND DECISION FEEDBACK EQUALIZERS
Fig. 10.
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Zeros of the two subchannels in the example. Fig. 11. MSE of the realizable equalizers as a function of delay. Dotted lines denote asymptotic values. SNR = 14 dB.
VI. NUMERICAL EXAMPLE Consider the following single-input two-output channel of order , which represents a simple model of a two-ray propagation channel:
(59) The zeros of the two subchannels are plotted in Fig. 10. Observe the nearly common subchannel roots, which are also close to the unit circle. This makes this channel rather severe. The noise covariance matrix is assumed to be
This covariance matrix would arise, for example, if a square root raised cosine filter with 100% excess bandwidth is used at the receiver front end and the received signal is sampled at twice the baud rate [6]. With , this results in SNR dB in the received signal. Fig. 11 depicts the MSE achieved by the realizable MMSE LE and MMSE DFE as a function of the delay . The MSE obtained with the MMSE LE-DFL approach is also shown, assuming as before that past decisions are correct. Observe the significant drops in the curves for , 2 (for which the first ray of the channel is captured) and , 8 (which capture the energy in the second ray). The asymptotic values of and are found to be and dB, respectively. Thus, the asymptotic improvement of the DFE over the LE is 2.3 dB. The nonincreasing property of the MSE with can be observed. This property provides the designer with a direct tradeoff between complexity and performance: The equalization delay equals the filter order, which should be taken as large as possible within complexity constraints in order to reduce the MSE.
In addition, if the MMSE LE-DFL strategy is to be used in order should to initialize the DFE, Lemma 3 suggests that be adopted in order to completely remove postcursor ISI. Note in Fig. 11 that the excess MSE of the MMSE LE-DFL approach reduces considerably for delays larger than the channel length. Except for , the (suboptimal) MMSE LE-DFL approach always yields, in this case, a lower MSE than that of the MMSE LE for the same delay. In order to illustrate the relation between the LE and DFE sec(the “short feedforward tions, we will consider two cases: filter” case) and (the “long feedforward filter” case). The taps of the feedforward and the feedback/recursive portions of the MMSE equalizers for the delay are shown in Fig. 12. As expected, the sections of the MMSE LE do not coincide with those of the MMSE DFE, although the coefficients of and are reasonably close. Observe that the feedforward secof the MMSE DFE has order equal to (i.e., its tion taps with are zero) as expected, whereas this is not of the MMSE LE, which true for the feedforward section . in general has order The overall channel-equalizer impulse response for the MMSE DFE, as well as that for the MMSE LE-DFL approach , are shown in Fig. 13. Observe that some residual postcursor ISI remains in (since ) but not in . In this setting, dB, whereas the MSE obtained in the MMSE LE-DFL setting is dB, i.e., the excess MSE of this suboptimal DFE is of 1.71 dB. Now, we examine the equalizers that result for , with the same SNR dB. Fig. 14 shows the coefficients of the feedforward and the feedback/recursive portions of the MMSE LE and MMSE DFE. Although these sections are not yet quite the same, they do lie reasonably close to each other, as expected. The overall channel-equalizer impulse responses in the decision feedback loop and are shown in
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Fig. 12.
Fig. 13.
Comparison of the feedforward and feedback/recursive sections of the LE (circles) and DFE (squares). Delay = 4, SNR = 14 dB.
Overall channel-equalizer impulse responses of the MMSE LE in decision feedback loop and MMSE DFE. Delay = 4, SNR = 14 dB.
Fig. 15. Note that now, the entire postcursor ISI has been cancelled in both systems, as predicted by Lemma 3. The excess MSE of the MMSE LE-DFL approach obtained with has dropped to 0.5 dB.
The ultimate performance measure of any equalizer structure is the symbol error rate (SER). Fig. 16 shows the SER as a function of SNR for the channel (59) and the same noise crosscorrelation coefficient of 0.5 between the two subchannels, for the
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Fig. 14.
Fig. 15.
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Comparison of the feedforward and feedback/recursive sections of the LE (circles) and DFE (squares). Delay = 10, SNR = 14 dB.
Overall channel-equalizer impulse responses of the MMSE LE in decision feedback loop and MMSE DFE. Delay = 10, SNR = 14 dB.
MMSE LE, MMSE DFE, and MMSE LE-DFL designs with and . A real 4-PAM constellation was used. It must be emphasized that the curves for the DFE and LE-DFL approaches were obtained with hard decisions being fed back, so that the filters were subject to error propagation. Even in that
situation, the LE-DFL strategy provides a sizable SER improvement over the MMSE LE, and for , its performance is very close to that of the MMSE DFE, as expected. These observations lend additional support for the LE-DFL initialization scheme of [12] and [13].
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Fig. 16.
Symbol error rate of the realizable equalizers as a function of SNR.
VII. CONCLUSIONS
ACKNOWLEDGMENT
We have presented a systematic analysis of the structure and properties of MMSE linear and decision feedback equalizers under realizability constraints, meaning that the filters should be causal and stable and of finite degree. Our focus has been on finite impulse response, single-input multiple-output channels with uncorrelated symbols and noise. The use of Wiener as well as Kalman filtering ideas reveals interesting properties of the equalizers concerning their structure, order, and error characteristics. In addition, the properties of the MMSE linear equalizer when placed in a decision feedback loop have been explored. The orthogonality principle provides an intuitively more appealing tool for the analysis than previous polynomial approaches. We have shown that the McMillan degree of both MMSE LE and MMSE DFE is no less than the channel length. In many practical situations, this requirement may be too demanding. In that case, one is faced with a constrained design, which is usually carried out by assuming a transversal structure for the equalizer. Clearly, it is of interest to explore the structure and properties of IIR configurations in such undermodeled scenarios in order to determine whether or not they may provide advantages over traditional FIR filters in terms of performance and/or complexity reduction. Our analysis of the realizable MMSE DFE hinges on the standard assumption of correct decisions being fed back. In severe channel conditions, this assumption will be violated, error propagation will take place, and designs different from the MMSE approach could provide smaller SER [11], [22]. An interesting open question is whether it is possible to obtain initialization strategies similar to the MMSE LE-DFL approach for these alternative designs.
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[13] J. Labat and C. Laot, “Blind adaptive multiple-input decision feedback equalizer with a self-optimized configuration,” IEEE Trans. Commun., vol. 49, pp. 646–654, Apr. 2001. [14] J. P. LeBlanc and I. Fijalkow, “Blind adapted, pre-whitened constant modulus algorithm,” in Proc. IEEE Int. Conf. Commun., vol. 8, Helsinki, Finland, June 2001, pp. 2438–2442. [15] E. A. Lee and D. G. Messerschmitt, Digital Communications. Boston, MA: Kluwer, 1994. [16] R. López-Valcarce and F. Pérez-González, “Efficient reinitialization of the prewhitened constant modulus algorithm,” IEEE Commun. Lett., vol. 5, pp. 488–490, Dec. 2001. [17] S. McLaughlin, “Adaptive equalization via Kalman filtering techniques,” Proc. Inst. Elect. F Radar Signal Processing, vol. 138, pp. 388–396, Aug. 1991. [18] B. Mulgrew and C. F. N. Cowan, “An adaptive Kalman equalizer: Structure and performance,” IEEE Trans. Acoust., Speech, Signal Processing, vol. ASSP-35, pp. 1727–1735, Dec. 1987. [19] J. G. Proakis, Digital Communications, 4th ed. New York: McGrawHill, 2001. [20] S. U. H. Qureshi, “Adaptive equalization,” Proc. IEEE, vol. 73, pp. 1349–1387, Sept. 1985. [21] P. A. Regalia, Adaptive IIR Filtering in Signal Processing and Control. New York: Marcel Dekker, 1995. [22] M. Reuter, J. C. Allen, J. R. Zeidler, and R. C. North, “Mitigating error propagation effects in a decision feedback equalizer,” IEEE Trans. Commun., vol. 49, pp. 2028–2041, Nov. 2001. [23] M. D. Srinath, P. K. Rajasekaran, and R. Viswanathan, An Introduction to Statistical Signal Processing with Applications. Englewood Cliffs, NJ: Prentice-Hall, 1996.
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Roberto López-Valcarce (M’01) was born in Spain in 1971. He received the telecommunications engineer degree from Universidad de Vigo, Vigo, Spain in 1995, and the M.S. and Ph.D. degrees in electrical engineering from the University of Iowa, Iowa City, in 1998 and 2000 respectively. From 1995 to 1996 he was a systems engineer with Intelsis. He was the recipient of a Fundación Pedro Barrié de la Maza fellowship for continuation of studies. He is currently a research associate (Ramón y Cajal fellow) with the Signal Theory and Communications Department at Universidad de Vigo. His research interests are in adaptive signal processing, communications, and traffic monitoring systems.
