Channel equalization using simplified least mean-fourth algorithm
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Channel equalization using simplified least mean-fourth algorithm Musa U. Otaru a , Azzedine Zerguine a,∗ , Lahouari Cheded b a b
MTN Nigeria Communications Ltd, 1612 Adeola Hopewell Street, Victoria Island, Lagos, Nigeria Systems Engineering Department, King Fahd University of Petroleum and Minerals, Dhahran 31261, Saudi Arabia
a r t i c l e
i n f o
a b s t r a c t
Article history: Available online xxxx
The steady flow of new research results and developments in the field of adaptive equalization that was witnessed for at least the last four decades is clearly evidenced by the many footprints of success it left behind and shows no sign of ending. The thrust of research and implementation in this field is mainly powered by the use of the well-known mean-square cost function upon which relies the ubiquitous least-mean square (LMS) algorithm. However, such an algorithm is well-known to lead to sub-optimal solutions in the real world that is largely dominated by non-Gaussian interference signals. The use of a non-mean-square cost function would successfully tackle these types of interference signals but would invariably involve a higher computational cost. To address these important practical issues, this paper proposes a new adaptive equalization technique that combines both the least-mean-fourth (LMF) algorithm, which is governed by a nonmean-square cost function, with a power-of-two quantizer (PTQ) in the coefficient update process, which greatly reduces the computational cost involved and which therefore makes the proposed technique applicable to time-varying environments. This paper not only elaborates on the basic idea behind the proposed technique but also defines the necessary assumptions and provides a thorough statistical performance analysis (including a study of the convergence behavior) of the combined algorithm LMF-PTQ that is at the core of the proposed technique. An extensive simulation work was carried out and showed that the theoretical predictions are very well substantiated. © 2010 Elsevier Inc. All rights reserved.
Keywords: Adaptive equalization LMF algorithm Power-of-two quantizer
1. Introduction Since its introduction in digital communication by Lucky [1], adaptive equalization [2] has known much progress and still remains a very popular and active field of research [3–5]. The wide applicability of this technique, as evidenced by its many footprints of success, has allowed the pace of research and developmental activity in this field to grow unabatedly. The implementation of adaptive equalization has almost exclusively relied on the use of the well-known and ubiquitous leastmean square (LMS) algorithm [6,7]. The optimality of the solution provided by all of the LMS-based equalization techniques hinges upon the assumption that the interference signals involved are Gaussian. Such an assumption is however rarely justified as the real world is largely dominated by non-Gaussian interference signals. Thus, the real-world performance of the LMS-based equalization techniques remains in effect sub-optimal. Moreover, although LMS-based equalization techniques proved successful for time-invariant channels, they did not enjoy the same level of success with time-varying channels which are both time- and frequency-dispersive and where long transmission bursts may get unacceptably corrupted if a fast tracking operation is not in place. Thus, for time-varying channels, there is a need for a mechanism to reduce the computational load of the LMS algorithm to allow it to successfully track fast variations in these channels. Several attempts
*
Corresponding author. E-mail addresses: [email protected] (M.U. Otaru), [email protected] (A. Zerguine), [email protected] (L. Cheded).
1051-2004/$ – see front matter doi:10.1016/j.dsp.2010.11.005
©
2010 Elsevier Inc. All rights reserved.
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were made in this direction by reducing the structural complexity of the LMS algorithm [8–15]. Most notable amongst these various contributions are those that replace the standard analog-to-digital converter (ADC) with a power-of-two quantizer (PTQ). In [9], the performance of adaptive filters based on a class of correlation multipliers using a power-of-two quantizer was studied. In [10], Xue and Liu showed that an improvement in the overall performance of the LMS-based algorithm can be achieved by using a power-of-two quantizer. The key advantage to using a power-of-two quantizer is that the binary code representing the signal samples consists of only one non-zero bit, thus reducing all required multiplications to mere logical shifts. This in turn leads to an attractive increase in the processing speed of the LMS algorithm, thus rendering it capable of successfully tracking at least reasonably fast-changing channels. Although this structural simplicity enhances the computational speed of the LMS algorithm, it does not however improve its sub-optimal performance in the face of nonGaussian interference signals which can lead to an unacceptable degradation in performance levels. Several attempts were made at improving the equalizer’s performance in real-world situations by exploiting an adaptive mechanism that rely on a non-mean square error criterion. This situation was later remedied with the proposed use, in [16], of the least-mean fourth (LMF) algorithm instead of the LMS one. This proved to yield an improved performance in non-Gaussian environments. It must be pointed out here that later on, a number of equalization techniques based on higher (than 2) order statistics (HOS) were proposed to handle channels that were non-Gaussian, nonlinear and non-stationary. Although these techniques did produce some notable contributions, they nevertheless require a lot more data and suffer from a higher statistical variance of the sampled estimators used than do their second-order (SOS)-based counterparts [17]. As such, they were therefore not found suitable for use with fast-varying channels as in the case for example in wireless communications. However, while the proposed LMF algorithm of [16] led to an improved performance in non-Gaussian environments, it did so at the cost of an increase in the complexity and hence in the computational speed of the equalization technique based on it. It thus follows that any attempt at reducing the structure of the traditional LMF would yield a much-desired improvement in its computational load, and hence in its processing time, as well as in the cost of its implementation. The main contribution of this paper is therefore to introduce and carry out a thorough performance analysis of a new LMF-PTQ-based equalization algorithm which incorporates a power-of-two quantizer and which is capable of providing an improved performance for both time-invariant and time-varying channels, thus extending the improved performance of the traditional LMF algorithm to the important realm of fast time-varying channels. Due to the inherent complexity of the LMF algorithm and in order to make the performance analysis mathematically tractable, we have adopted the approach of simplification-through-linearization wherein a linearized description of the PTQ is first obtained and then used to carry out our convergence and steady-state mean-square analyses of the proposed LMF-PTQ algorithm. It is worth pointing out at this juncture that another contribution, in the context of adaptive channel equalization, using a combined scheme based on switching between the LMS and LMF algorithms was proposed in [18]. However, our approach is radically different from that in [18] in that ours is solely based on the LMF. The equalization performance of our new algorithm is tested by simulation on both time-invariant channels, using a standard transversal (FIR type) equalizer, as well as on time-varying ones using a decision feedback equalizer (DFE). The paper is organized as follows: following the introduction is Section 2 which introduces the mean-square analysis of the LMF algorithm in adaptive equalization. Section 3 introduces the new LMF-PTQ and both its defining and weight update equations. Then Sections 4 and 5 give respectively the convergence analysis and the steady-state mean-square error analysis, both based on a linearized version of this proposed algorithm. Section 6 presents and discusses several simulation results on the performance and the existence of an optimal step size of the proposed equalizer with different channels, including a time-varying one. The results substantiate very well our theoretical predictions. Finally some conclusions are given in Section 7. 2. Mean-square analysis for LMF algorithm in adaptive equalization Consider the model of a linear channel followed by an N-tap equalizer, whose input is expressed by:
x(n) =
N −1
h(i )a(n − i ) + ν (n),
(1)
i =0
where h(i ), i = 0, 1, . . . , N − 1, is the channel impulse response, a(n) is the binary input sequence drawn from {±1}, ν (n) is a non-Gaussian additive noise with power spectral density σν2 , and N represents the length of the equalizer. The estimated output, y (n), is defined as:
y (n) = w T (n)x(n),
(2)
where
T
w(n) = w (0), w (1), . . . , w ( N − 1)
is the ( N × 1) current value of the adaptive weights, superscript T denotes transpose operation, and
T
x(n) = x(n), x(n − 1), . . . , x(n − N + 1)
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represents the ( N × 1) vector of the last n samples of the equalizer input. The system output error sample at the nth moment, e (n) can be found as:
e (n) = a(n − n0 ) − y (n)
= d(n) − xT (n)w(n),
(3)
where d(n) is the desired signal. The adaptation is done by an adaptive filter whose weight vector, w(n), is updated by LMF algorithm [16] according to
w(n + 1) = w(n) + 2μe 3 (n)x(n),
(4)
where μ is the step-size constant which controls stability and rate of convergence. Let v(n) denotes the weight error vector which is defined as:
v(n) = w(n) − wopt ,
(5)
where wopt is the optimal weight vector which minimizes the mean square error and is given by
wopt = R−1 p,
(6)
where R = E [x(n)x (n)] is the autocorrelation matrix of the tap input vector, p = E [x(n)d(n)] is the cross-correlation vector between the tap input vector and the desired response, and E [·] denoting statistical expectation. It is assumed that the weight error vector v(n) and the equalizer input x(n) are statistically independent. The optimal estimation error e min is given by T
e min = d(n) − x T (n)wopt .
(7)
Combining (3), (5) and (7), it follows that
e (n) = e min − x T (n)v(n).
(8)
Now let
ave (n) = E e2 (n) ,
(9)
and
min = E e2min
(10)
denote the average output mean-squared estimation error after the nth iteration and the minimum (Wiener) mean-squared error, respectively. By employing Eq. (8) in Eq. (9), we obtained the average output mean-squared estimation error as:
2
ave (n) = E emin − xT (n)v(n)
= min + E v(n)T x(n)xT (n)v(n) − 2E e min v(n)T x(n) .
(11)
By the independent assumption, the last expectation of (11) becomes zero. Thus Eq. (11) reduces to:
ave (n) = min + tr RK(n) ,
(12)
where K(n) is the autocorrelation matrix of the coefficient error vector v(n) defined as:
K(n) = E v(n)v T (n) ,
(13)
and tr{·} denotes the trace of {·}. Eq. (12) indicates that, the average mean-square estimation error consists of two components—the minimum (Wiener) mean-square error min and a component depending on the transient behaviour of the weight-error correlation matrix, K(n), known as the excess mean square error (EMSE) [7]. The EMSE represents the MSE above min in the steady state. Finally, the minimum mean-square error, min , is obtained by substituting (7) in (10), so that
min = E d2 (n) − pT wopt
= E d2 (n) − pT R−1 p.
(14)
The next section develops the simplified algorithm—power-of-two quantizer based LMF algorithm. As it has been mentioned earlier, this algorithm will make the implementation of the LMF algorithm very simple as it will based upon binary shifts.
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3. Power-of-two quantizer based LMF algorithm Power-of-two quantizer is one of the several simplifications that have been proposed in the literature as a means of simplifying gradient-based algorithms to reduce their hardware complexity. Here, we use it for one of the two inputs of the multiplier to replace its input by a finite bit words having only a single non-trivial bit, thereby reducing the multiplication to a simple shift [9,10]. Quantized gradient algorithms can use quantization of the estimation error e (n), the regressor x(n), or both of them. However, Sethares and Johnson [15] have shown in their work that the quantization of x(n) may lead to instability of the algorithm. Therefore, in this work we have considered the LMF algorithm with power-of-two quantization of the estimation error e(n) only. A power-of-two quantizer is defined by Duttweiler [9] as:
q(u ) = 2ln |u | sgn(u ),
(15)
where u is the largest integer less than u and sgn(u ) is the sign of u defined as:
sgn(u ) =
1,
u 0,
−1, u < 0.
The quantizer defined by (15) is an infinite bit quantizer. However, in a real application, a finite bit quantizer is often used. The analysis of a finite bit power-of-two quantizer incorporated with LMS algorithm is given by Xue and Liu in [10], where they have indicated that a B-bit power-of-two quantizer converts an input u to a “one-bit” word according to:
q (u ) =
⎧ ⎨ sgn(u ), ⎩
| u | 1,
2ln |u | sgn(u ),
2− B +1 | u | < 1,
0,
| u | < 2− B +1 .
(16)
To take care of the cubic error power in the LMF recursion formula of Eq. (4), we substitute u = e 3 in Eq. (16), and considering such substitution as bijective i.e. a one-to-one mapping of the two quantities in either direction, we arrived at a modified power-of-two quantizer as:
3
q e
=
⎧ sgn(e ), ⎪ ⎨ ⎪ ⎩
|e | 1,
23 ln |e| sgn(e ),
2
− B +1 3
|e | < 1,
|e | < 2
0,
− B +1 3
(17)
.
Combination of Eqs. (4) and (17) result in a power-of-two quantizer based LMF algorithm (LMF-PTQ). The updating algorithm (4) is then replaced by the following recursion formula:
w(n + 1) = w(n) + 2μq e 3 (n) sgn x(n) .
(18)
Since the input considered here is bipolar binary, and in order to carry out our analysis along lines similar to those in [9– 12], we replaced the regressor with its signed version as this would not change its values. Fig. 1 illustrates the input–output characteristics of such quantizer with B = 4 bits. 4. Convergence analysis of LMF-PTQ algorithm In the ensuing analysis, we will make use of the following assumptions in the derivation of the convergence behaviour of the LMF-PTQ algorithm. These assumptions are valid in most practical applications and have been successfully used in the literature [19–23]: A1 The input x(n) is real, zero mean, independent and identically distributed Gaussian process with a positive definite covariance matrix. A2 The additive noise ν (n) is a zero mean, stationary non-Gaussian random process and independent of the input signal x(n). A3 The input x(n) is independent of the weight error vector. Assumption A3 is the so-called independence assumption [7], which has been extensively used in the literature as it leads to tractable convergence analysis [23–25]. It is interesting to note that the effect of the independence assumption is even less pronounced for independent binary inputs, as in the case of channel equalization. It was demonstrated [26] that, for small values of step-size, the independence assumption is quite accurate in predicting the steady-state performance of the LMS algorithm. Consequently, for values of error raised to the third power, e 3 (n), it will be more accurate for a small step-size and an absolute error satisfying: |e (n)| < 1. Therefore, it is also applicable to LMF algorithm. By subtracting wopt , from both sides of (18) and using (5) accordingly, we have
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Fig. 1. Input–output characteristics of a 4-bit power-of-two quantizer.
v(n + 1) = v(n) + 2μq e 3 (n) sgn x(n) .
(19)
Squaring both sides of (19) and taking the expected values, noting that sgn[x(n)]2 = N, and defining
2 θ(n) = E v(n) ,
(20)
we obtain:
θ(n + 1) = θ(n) + 4N μ2 E q2 e 3 (n) + 4μ E q e 3 (n) vT (n) sgn x(n) .
(21)
As expected, Eq. (21) resulted in a highly nonlinear equation. To study the behavior of this algorithm according to this recursion, very highly complex mathematical tools must be used. As a remedy to this obstacle, only a linearized version of this recursion will be analyzed. 4.1. Linearized analysis The linearized analysis involves approximation of the quantizer function q[e 3 (n)] to a linear function. This is done by drawing a straight line passing through the center of each step of the quantizer transfer characteristic. Such line is shown dotted in Fig. 1. Although, the approach may give less accurate results, but it will surely render the analysis more tractable. Geometrical analysis of Fig. 1 (see Appendix A) leads to the following approximation:
q e 3 (n) ≈
7 8
e (n).
(22)
4.2. Convergence in the mean On using approximation (22), Eq. (18) becomes
w(n + 1) = w(n) +
7 4
μe(n) sgn x(n) .
(23)
Interestingly, Eq. (23) reveals the equivalence between the linearized LMF-PTQ algorithm and the signed-regressor LMS algorithm [23] with a modified step-size of:
7
μ = μ. 4
(24)
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However, it should be noted here that this equivalence between the linearized LMF-PTQ and the signed-regressor LMS algorithms will only help in simplifying the theoretical analysis. Moreover, since this equivalence has been established between an approximate (here linearized) version of the exact model of the LMF-PTQ-based equalizer and the signedregressor-LMS-based one, it should in no way be interpreted as though had the latter (i.e. signed-regressor-LMS-based) equalizer been used in our simulation work, instead of the former (i.e. exact LMF-PTQ-based) one, the same results would have been produced. It is also interesting to point out here that, in [10], the linearized version of the LMS-PTQ algorithm was also shown to be equivalent to a signed-regressor LMS with a modified step-size. Note that the linearized analysis will always lead to a signed regressor LMS equivalence for any higher (than 2) algorithm, i.e., LMp where p is greater than or equal to 2 (including our proposed). Eq. (19) can be set into the following expression:
7
v(n + 1) = I −
4
7
μ sgn x(n) xT (n) v(n) + μξ(n) sgn x(n) ,
(25)
4
where
ξ(n) = d(n) − xT (n)wopt
(26)
is the estimation error due to optimum filter coefficients and I denotes an ( N × N ) identity matrix. Next, we take the expected values of both side of (25) to obtain:
T 7 7 E v(n + 1) = E I − μ sgn x(n) x (n) E v(n) + μ E ξ(n) sgn x(n) .
4
4
(27)
By virtue of the independent assumption and the fact that x(n) is zero-mean, the second term on the right-hand side of Eq. (27) reduces then to zero, that is:
E ξ(n) sgn x(n)
= 0.
(28)
Application of Price’s theorem [27,28], to each elements of the matrix E {sgn[x(n)]x (n)} (see Appendix B), will lead to: T
E sgn x(n) x T (n) =
2 R
π σx
,
(29)
where σx and R are, respectively, the standard deviation and the autocorrelation matrix of the input signal defined previously. By substituting (29) in (27), the mean behavior for the coefficient misalignment vector of the LMF-PTQ algorithm is governed by the following recursion:
7 2 R μ E v(n + 1) = I − E v(n) .
π
4
σx
(30)
Expression (30) is the evolution of the average weight error vector as the adaptation proceeds. To simplify (30) further, the unitary similarity transformation to the autocorrelation matrix R is applied. This is given as [7,29],
R = QΛQ T ,
(31)
where Q is the N × N matrix whose columns are the eigenvectors associated with the distinct eigenvalues of R and is given by:
Q = [q0
q1
···
q N −1 ],
(32)
and Λ is the diagonal matrix consisting of the eigenvalues of R defined by:
Λ = diag(λ0 , λ1 , . . . , λ N −1 ).
(33)
Noting that QQ = I, Eq. (30) can now be written as T
7 2 Λ μ E v(n + 1) = Q I − Q T E v(n) .
π
4
σx
(34)
Premultiplying (34) by Q T and defining
Φ(n) = QT E v(n) , Eq. (34) becomes:
Φ(n + 1) = I −
7 4
(35)
2
π
μ
Λ
σx
Φ(n).
(36)
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For the ith natural mode of the algorithm, we have
7 2 λi μ φi (n), φi (n + 1) = 1 −
π
4
σx
i = 0, 1 , . . . , N − 1 ,
(37)
where φi (n) is the ith element of vector Φ(n) and λi is the ith eigenvalue of the autocorrelation matrix R. The solution to the first order homogeneous equation (37) can be obtained as
φi (n) = 1 −
7
2
π
4
n
λi
μ
i = 0, 1, . . . , N − 1,
φi (0),
σx
(38)
where φi (0) is the initial value of φi (n). Since R is assumed real and positive definite, convergence to the optimum solution is guaranteed if
1 − 7 2 μ λi < 1, 4 π σx
i = 0, 1 , . . . , N − 1 ,
that is, the step-size (μ) should belong to the following range
√
0
AID:1107 /FLA
[m3G; v 1.50; Prn:6/12/2010; 13:24] P.1 (1-19) Digital Signal Processing ••• (••••) •••–•••
Contents lists available at ScienceDirect
Digital Signal Processing www.elsevier.com/locate/dsp
Channel equalization using simplified least mean-fourth algorithm Musa U. Otaru a , Azzedine Zerguine a,∗ , Lahouari Cheded b a b
MTN Nigeria Communications Ltd, 1612 Adeola Hopewell Street, Victoria Island, Lagos, Nigeria Systems Engineering Department, King Fahd University of Petroleum and Minerals, Dhahran 31261, Saudi Arabia
a r t i c l e
i n f o
a b s t r a c t
Article history: Available online xxxx
The steady flow of new research results and developments in the field of adaptive equalization that was witnessed for at least the last four decades is clearly evidenced by the many footprints of success it left behind and shows no sign of ending. The thrust of research and implementation in this field is mainly powered by the use of the well-known mean-square cost function upon which relies the ubiquitous least-mean square (LMS) algorithm. However, such an algorithm is well-known to lead to sub-optimal solutions in the real world that is largely dominated by non-Gaussian interference signals. The use of a non-mean-square cost function would successfully tackle these types of interference signals but would invariably involve a higher computational cost. To address these important practical issues, this paper proposes a new adaptive equalization technique that combines both the least-mean-fourth (LMF) algorithm, which is governed by a nonmean-square cost function, with a power-of-two quantizer (PTQ) in the coefficient update process, which greatly reduces the computational cost involved and which therefore makes the proposed technique applicable to time-varying environments. This paper not only elaborates on the basic idea behind the proposed technique but also defines the necessary assumptions and provides a thorough statistical performance analysis (including a study of the convergence behavior) of the combined algorithm LMF-PTQ that is at the core of the proposed technique. An extensive simulation work was carried out and showed that the theoretical predictions are very well substantiated. © 2010 Elsevier Inc. All rights reserved.
Keywords: Adaptive equalization LMF algorithm Power-of-two quantizer
1. Introduction Since its introduction in digital communication by Lucky [1], adaptive equalization [2] has known much progress and still remains a very popular and active field of research [3–5]. The wide applicability of this technique, as evidenced by its many footprints of success, has allowed the pace of research and developmental activity in this field to grow unabatedly. The implementation of adaptive equalization has almost exclusively relied on the use of the well-known and ubiquitous leastmean square (LMS) algorithm [6,7]. The optimality of the solution provided by all of the LMS-based equalization techniques hinges upon the assumption that the interference signals involved are Gaussian. Such an assumption is however rarely justified as the real world is largely dominated by non-Gaussian interference signals. Thus, the real-world performance of the LMS-based equalization techniques remains in effect sub-optimal. Moreover, although LMS-based equalization techniques proved successful for time-invariant channels, they did not enjoy the same level of success with time-varying channels which are both time- and frequency-dispersive and where long transmission bursts may get unacceptably corrupted if a fast tracking operation is not in place. Thus, for time-varying channels, there is a need for a mechanism to reduce the computational load of the LMS algorithm to allow it to successfully track fast variations in these channels. Several attempts
*
Corresponding author. E-mail addresses: [email protected] (M.U. Otaru), [email protected] (A. Zerguine), [email protected] (L. Cheded).
1051-2004/$ – see front matter doi:10.1016/j.dsp.2010.11.005
©
2010 Elsevier Inc. All rights reserved.
JID:YDSPR
AID:1107 /FLA
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were made in this direction by reducing the structural complexity of the LMS algorithm [8–15]. Most notable amongst these various contributions are those that replace the standard analog-to-digital converter (ADC) with a power-of-two quantizer (PTQ). In [9], the performance of adaptive filters based on a class of correlation multipliers using a power-of-two quantizer was studied. In [10], Xue and Liu showed that an improvement in the overall performance of the LMS-based algorithm can be achieved by using a power-of-two quantizer. The key advantage to using a power-of-two quantizer is that the binary code representing the signal samples consists of only one non-zero bit, thus reducing all required multiplications to mere logical shifts. This in turn leads to an attractive increase in the processing speed of the LMS algorithm, thus rendering it capable of successfully tracking at least reasonably fast-changing channels. Although this structural simplicity enhances the computational speed of the LMS algorithm, it does not however improve its sub-optimal performance in the face of nonGaussian interference signals which can lead to an unacceptable degradation in performance levels. Several attempts were made at improving the equalizer’s performance in real-world situations by exploiting an adaptive mechanism that rely on a non-mean square error criterion. This situation was later remedied with the proposed use, in [16], of the least-mean fourth (LMF) algorithm instead of the LMS one. This proved to yield an improved performance in non-Gaussian environments. It must be pointed out here that later on, a number of equalization techniques based on higher (than 2) order statistics (HOS) were proposed to handle channels that were non-Gaussian, nonlinear and non-stationary. Although these techniques did produce some notable contributions, they nevertheless require a lot more data and suffer from a higher statistical variance of the sampled estimators used than do their second-order (SOS)-based counterparts [17]. As such, they were therefore not found suitable for use with fast-varying channels as in the case for example in wireless communications. However, while the proposed LMF algorithm of [16] led to an improved performance in non-Gaussian environments, it did so at the cost of an increase in the complexity and hence in the computational speed of the equalization technique based on it. It thus follows that any attempt at reducing the structure of the traditional LMF would yield a much-desired improvement in its computational load, and hence in its processing time, as well as in the cost of its implementation. The main contribution of this paper is therefore to introduce and carry out a thorough performance analysis of a new LMF-PTQ-based equalization algorithm which incorporates a power-of-two quantizer and which is capable of providing an improved performance for both time-invariant and time-varying channels, thus extending the improved performance of the traditional LMF algorithm to the important realm of fast time-varying channels. Due to the inherent complexity of the LMF algorithm and in order to make the performance analysis mathematically tractable, we have adopted the approach of simplification-through-linearization wherein a linearized description of the PTQ is first obtained and then used to carry out our convergence and steady-state mean-square analyses of the proposed LMF-PTQ algorithm. It is worth pointing out at this juncture that another contribution, in the context of adaptive channel equalization, using a combined scheme based on switching between the LMS and LMF algorithms was proposed in [18]. However, our approach is radically different from that in [18] in that ours is solely based on the LMF. The equalization performance of our new algorithm is tested by simulation on both time-invariant channels, using a standard transversal (FIR type) equalizer, as well as on time-varying ones using a decision feedback equalizer (DFE). The paper is organized as follows: following the introduction is Section 2 which introduces the mean-square analysis of the LMF algorithm in adaptive equalization. Section 3 introduces the new LMF-PTQ and both its defining and weight update equations. Then Sections 4 and 5 give respectively the convergence analysis and the steady-state mean-square error analysis, both based on a linearized version of this proposed algorithm. Section 6 presents and discusses several simulation results on the performance and the existence of an optimal step size of the proposed equalizer with different channels, including a time-varying one. The results substantiate very well our theoretical predictions. Finally some conclusions are given in Section 7. 2. Mean-square analysis for LMF algorithm in adaptive equalization Consider the model of a linear channel followed by an N-tap equalizer, whose input is expressed by:
x(n) =
N −1
h(i )a(n − i ) + ν (n),
(1)
i =0
where h(i ), i = 0, 1, . . . , N − 1, is the channel impulse response, a(n) is the binary input sequence drawn from {±1}, ν (n) is a non-Gaussian additive noise with power spectral density σν2 , and N represents the length of the equalizer. The estimated output, y (n), is defined as:
y (n) = w T (n)x(n),
(2)
where
T
w(n) = w (0), w (1), . . . , w ( N − 1)
is the ( N × 1) current value of the adaptive weights, superscript T denotes transpose operation, and
T
x(n) = x(n), x(n − 1), . . . , x(n − N + 1)
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represents the ( N × 1) vector of the last n samples of the equalizer input. The system output error sample at the nth moment, e (n) can be found as:
e (n) = a(n − n0 ) − y (n)
= d(n) − xT (n)w(n),
(3)
where d(n) is the desired signal. The adaptation is done by an adaptive filter whose weight vector, w(n), is updated by LMF algorithm [16] according to
w(n + 1) = w(n) + 2μe 3 (n)x(n),
(4)
where μ is the step-size constant which controls stability and rate of convergence. Let v(n) denotes the weight error vector which is defined as:
v(n) = w(n) − wopt ,
(5)
where wopt is the optimal weight vector which minimizes the mean square error and is given by
wopt = R−1 p,
(6)
where R = E [x(n)x (n)] is the autocorrelation matrix of the tap input vector, p = E [x(n)d(n)] is the cross-correlation vector between the tap input vector and the desired response, and E [·] denoting statistical expectation. It is assumed that the weight error vector v(n) and the equalizer input x(n) are statistically independent. The optimal estimation error e min is given by T
e min = d(n) − x T (n)wopt .
(7)
Combining (3), (5) and (7), it follows that
e (n) = e min − x T (n)v(n).
(8)
Now let
ave (n) = E e2 (n) ,
(9)
and
min = E e2min
(10)
denote the average output mean-squared estimation error after the nth iteration and the minimum (Wiener) mean-squared error, respectively. By employing Eq. (8) in Eq. (9), we obtained the average output mean-squared estimation error as:
2
ave (n) = E emin − xT (n)v(n)
= min + E v(n)T x(n)xT (n)v(n) − 2E e min v(n)T x(n) .
(11)
By the independent assumption, the last expectation of (11) becomes zero. Thus Eq. (11) reduces to:
ave (n) = min + tr RK(n) ,
(12)
where K(n) is the autocorrelation matrix of the coefficient error vector v(n) defined as:
K(n) = E v(n)v T (n) ,
(13)
and tr{·} denotes the trace of {·}. Eq. (12) indicates that, the average mean-square estimation error consists of two components—the minimum (Wiener) mean-square error min and a component depending on the transient behaviour of the weight-error correlation matrix, K(n), known as the excess mean square error (EMSE) [7]. The EMSE represents the MSE above min in the steady state. Finally, the minimum mean-square error, min , is obtained by substituting (7) in (10), so that
min = E d2 (n) − pT wopt
= E d2 (n) − pT R−1 p.
(14)
The next section develops the simplified algorithm—power-of-two quantizer based LMF algorithm. As it has been mentioned earlier, this algorithm will make the implementation of the LMF algorithm very simple as it will based upon binary shifts.
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3. Power-of-two quantizer based LMF algorithm Power-of-two quantizer is one of the several simplifications that have been proposed in the literature as a means of simplifying gradient-based algorithms to reduce their hardware complexity. Here, we use it for one of the two inputs of the multiplier to replace its input by a finite bit words having only a single non-trivial bit, thereby reducing the multiplication to a simple shift [9,10]. Quantized gradient algorithms can use quantization of the estimation error e (n), the regressor x(n), or both of them. However, Sethares and Johnson [15] have shown in their work that the quantization of x(n) may lead to instability of the algorithm. Therefore, in this work we have considered the LMF algorithm with power-of-two quantization of the estimation error e(n) only. A power-of-two quantizer is defined by Duttweiler [9] as:
q(u ) = 2ln |u | sgn(u ),
(15)
where u is the largest integer less than u and sgn(u ) is the sign of u defined as:
sgn(u ) =
1,
u 0,
−1, u < 0.
The quantizer defined by (15) is an infinite bit quantizer. However, in a real application, a finite bit quantizer is often used. The analysis of a finite bit power-of-two quantizer incorporated with LMS algorithm is given by Xue and Liu in [10], where they have indicated that a B-bit power-of-two quantizer converts an input u to a “one-bit” word according to:
q (u ) =
⎧ ⎨ sgn(u ), ⎩
| u | 1,
2ln |u | sgn(u ),
2− B +1 | u | < 1,
0,
| u | < 2− B +1 .
(16)
To take care of the cubic error power in the LMF recursion formula of Eq. (4), we substitute u = e 3 in Eq. (16), and considering such substitution as bijective i.e. a one-to-one mapping of the two quantities in either direction, we arrived at a modified power-of-two quantizer as:
3
q e
=
⎧ sgn(e ), ⎪ ⎨ ⎪ ⎩
|e | 1,
23 ln |e| sgn(e ),
2
− B +1 3
|e | < 1,
|e | < 2
0,
− B +1 3
(17)
.
Combination of Eqs. (4) and (17) result in a power-of-two quantizer based LMF algorithm (LMF-PTQ). The updating algorithm (4) is then replaced by the following recursion formula:
w(n + 1) = w(n) + 2μq e 3 (n) sgn x(n) .
(18)
Since the input considered here is bipolar binary, and in order to carry out our analysis along lines similar to those in [9– 12], we replaced the regressor with its signed version as this would not change its values. Fig. 1 illustrates the input–output characteristics of such quantizer with B = 4 bits. 4. Convergence analysis of LMF-PTQ algorithm In the ensuing analysis, we will make use of the following assumptions in the derivation of the convergence behaviour of the LMF-PTQ algorithm. These assumptions are valid in most practical applications and have been successfully used in the literature [19–23]: A1 The input x(n) is real, zero mean, independent and identically distributed Gaussian process with a positive definite covariance matrix. A2 The additive noise ν (n) is a zero mean, stationary non-Gaussian random process and independent of the input signal x(n). A3 The input x(n) is independent of the weight error vector. Assumption A3 is the so-called independence assumption [7], which has been extensively used in the literature as it leads to tractable convergence analysis [23–25]. It is interesting to note that the effect of the independence assumption is even less pronounced for independent binary inputs, as in the case of channel equalization. It was demonstrated [26] that, for small values of step-size, the independence assumption is quite accurate in predicting the steady-state performance of the LMS algorithm. Consequently, for values of error raised to the third power, e 3 (n), it will be more accurate for a small step-size and an absolute error satisfying: |e (n)| < 1. Therefore, it is also applicable to LMF algorithm. By subtracting wopt , from both sides of (18) and using (5) accordingly, we have
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Fig. 1. Input–output characteristics of a 4-bit power-of-two quantizer.
v(n + 1) = v(n) + 2μq e 3 (n) sgn x(n) .
(19)
Squaring both sides of (19) and taking the expected values, noting that sgn[x(n)]2 = N, and defining
2 θ(n) = E v(n) ,
(20)
we obtain:
θ(n + 1) = θ(n) + 4N μ2 E q2 e 3 (n) + 4μ E q e 3 (n) vT (n) sgn x(n) .
(21)
As expected, Eq. (21) resulted in a highly nonlinear equation. To study the behavior of this algorithm according to this recursion, very highly complex mathematical tools must be used. As a remedy to this obstacle, only a linearized version of this recursion will be analyzed. 4.1. Linearized analysis The linearized analysis involves approximation of the quantizer function q[e 3 (n)] to a linear function. This is done by drawing a straight line passing through the center of each step of the quantizer transfer characteristic. Such line is shown dotted in Fig. 1. Although, the approach may give less accurate results, but it will surely render the analysis more tractable. Geometrical analysis of Fig. 1 (see Appendix A) leads to the following approximation:
q e 3 (n) ≈
7 8
e (n).
(22)
4.2. Convergence in the mean On using approximation (22), Eq. (18) becomes
w(n + 1) = w(n) +
7 4
μe(n) sgn x(n) .
(23)
Interestingly, Eq. (23) reveals the equivalence between the linearized LMF-PTQ algorithm and the signed-regressor LMS algorithm [23] with a modified step-size of:
7
μ = μ. 4
(24)
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However, it should be noted here that this equivalence between the linearized LMF-PTQ and the signed-regressor LMS algorithms will only help in simplifying the theoretical analysis. Moreover, since this equivalence has been established between an approximate (here linearized) version of the exact model of the LMF-PTQ-based equalizer and the signedregressor-LMS-based one, it should in no way be interpreted as though had the latter (i.e. signed-regressor-LMS-based) equalizer been used in our simulation work, instead of the former (i.e. exact LMF-PTQ-based) one, the same results would have been produced. It is also interesting to point out here that, in [10], the linearized version of the LMS-PTQ algorithm was also shown to be equivalent to a signed-regressor LMS with a modified step-size. Note that the linearized analysis will always lead to a signed regressor LMS equivalence for any higher (than 2) algorithm, i.e., LMp where p is greater than or equal to 2 (including our proposed). Eq. (19) can be set into the following expression:
7
v(n + 1) = I −
4
7
μ sgn x(n) xT (n) v(n) + μξ(n) sgn x(n) ,
(25)
4
where
ξ(n) = d(n) − xT (n)wopt
(26)
is the estimation error due to optimum filter coefficients and I denotes an ( N × N ) identity matrix. Next, we take the expected values of both side of (25) to obtain:
T 7 7 E v(n + 1) = E I − μ sgn x(n) x (n) E v(n) + μ E ξ(n) sgn x(n) .
4
4
(27)
By virtue of the independent assumption and the fact that x(n) is zero-mean, the second term on the right-hand side of Eq. (27) reduces then to zero, that is:
E ξ(n) sgn x(n)
= 0.
(28)
Application of Price’s theorem [27,28], to each elements of the matrix E {sgn[x(n)]x (n)} (see Appendix B), will lead to: T
E sgn x(n) x T (n) =
2 R
π σx
,
(29)
where σx and R are, respectively, the standard deviation and the autocorrelation matrix of the input signal defined previously. By substituting (29) in (27), the mean behavior for the coefficient misalignment vector of the LMF-PTQ algorithm is governed by the following recursion:
7 2 R μ E v(n + 1) = I − E v(n) .
π
4
σx
(30)
Expression (30) is the evolution of the average weight error vector as the adaptation proceeds. To simplify (30) further, the unitary similarity transformation to the autocorrelation matrix R is applied. This is given as [7,29],
R = QΛQ T ,
(31)
where Q is the N × N matrix whose columns are the eigenvectors associated with the distinct eigenvalues of R and is given by:
Q = [q0
q1
···
q N −1 ],
(32)
and Λ is the diagonal matrix consisting of the eigenvalues of R defined by:
Λ = diag(λ0 , λ1 , . . . , λ N −1 ).
(33)
Noting that QQ = I, Eq. (30) can now be written as T
7 2 Λ μ E v(n + 1) = Q I − Q T E v(n) .
π
4
σx
(34)
Premultiplying (34) by Q T and defining
Φ(n) = QT E v(n) , Eq. (34) becomes:
Φ(n + 1) = I −
7 4
(35)
2
π
μ
Λ
σx
Φ(n).
(36)
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For the ith natural mode of the algorithm, we have
7 2 λi μ φi (n), φi (n + 1) = 1 −
π
4
σx
i = 0, 1 , . . . , N − 1 ,
(37)
where φi (n) is the ith element of vector Φ(n) and λi is the ith eigenvalue of the autocorrelation matrix R. The solution to the first order homogeneous equation (37) can be obtained as
φi (n) = 1 −
7
2
π
4
n
λi
μ
i = 0, 1, . . . , N − 1,
φi (0),
σx
(38)
where φi (0) is the initial value of φi (n). Since R is assumed real and positive definite, convergence to the optimum solution is guaranteed if
1 − 7 2 μ λi < 1, 4 π σx
i = 0, 1 , . . . , N − 1 ,
that is, the step-size (μ) should belong to the following range
√
0
