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IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 61, NO. 5, MARCH 2013 1

A soft-switching blind equalization scheme via

convex combination of adaptive filters

Magno T. M. Silva, Member, IEEE and Jeronimo Arenas-Garcıa, Senior Member, IEEE

Abstract

Blind equalizers avoid the transmission of pilot sequences, allowing a more efficient use of the channel bandwidth. Normally,

after a first rough equalization is achieved, it is necessary to switch these equalizers to a decision-directed (DD) mode to reduce the

steady-state mean-square error (MSE) to acceptable levels. The selection of an appropriate MSE threshold for switching between

the blind and the DD modes is critical to obtain a good overall performance; however, this is not an easy task, since it depends on

several factors such as the signal constellation, the communication channel, or the signal-to-noise ratio. In this paper, we propose

an equalization scheme that adaptively combines a blind and a DD equalizers running in parallel. The combination is itself adapted

in a blind manner, and as a result the overall scheme can automatically switch between the component filters, avoiding the need

to set the transition MSE level a priori. The performance of our proposal is illustrated both analytically and through an extensive

set of simulations, where we show its advantages with respect to existing hard- and soft-switching equalization schemes.

Index Terms

Adaptive signal processing, blind equalizers, gradient methods, convex combination, decision-directed, tracking.

I. INTRODUCTION

ADAPTIVE equalizers are widely used in digital communications systems to remove the intersymbol interference

introduced by dispersive channels. In order to avoid the transmission of pilot sequences and use the channel bandwidth

in an efficient manner, these equalizers can be initially adapted using a blind criterion and switched to a decision-directed

(DD) mode after the blind equalization achieves a sufficiently low error (see, e.g., [1]–[5] and references therein).

The constant modulus algorithm (CMA) [6], [7] and its variant known as multimodulus algorithm (MMA) [8]–[10] are the

most popular for the blind adaptation of finite impulse response (FIR) equalizers, due to their low computational cost. MMA

was proposed to solve the phase ambiguity of CMA by minimizing the dispersion of the real and imaginary parts of the

Manuscript received September 05, 2012; accepted December 07, 2012. The associate editor coordinating the review of this manuscript and approving it

for publication was Prof. Jonathon Chambers.

Copyright (c) 2010 IEEE. Personal use of this material is permitted. However, permission to use this material for any other purposes must be obtained

from the IEEE by sending a request to [email protected].

The work of Silva was supported by FAPESP under Grant 2011/13581-6 and by CNPq under Grant 302423/2011-7. The work of Arenas-Garcıa was partly

supported by MICINN under Grants TEC2011-22480 and PRI-PIBIN-2011-1266.

Silva is with Electronic Systems Engineering Department, Escola Politecnica, Universidade of Sao Paulo, Sao Paulo, SP, Brazil, and Arenas-Garcıa is

with Department of Signal Theory and Communications, Universidad Carlos III de Madrid, Leganes, Spain; e-mails: [email protected], [email protected].

ph. +55-11-3091-5134, fax: +55-11-3091-5718.

Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/TSP.2012.2236835

2 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 61, NO. 5, MARCH 2013

equalizer output separately. Although MMA may present some advantages when compared to CMA (see, e.g., [11] and its

references), it still achieves a large mean-square error (MSE) in steady-state, mainly for nonconstant modulus signals.

The large steady-state error incurred by filters applying a blind cost function justifies the need of switching to DD mode after

an initial rough equalization. However, selecting an appropriate MSE level for hard switching between the blind and DD modes

is not an easy task, and highly depends on different factors such as the signal constellation, the communication channel, or the

signal-to-noise ratio (SNR), among others [5]. An inadequate selection of the switching MSE level has a significant impact on

the overall equalization performance, since a too large MSE value usually leads to an ill-convergence or no convergence at all

for the DD algorithm [12], [13], whereas a too small selection can result in a long delay or even failure to switch between

modes.

To overcome the need of selecting an MSE threshold for hard switching between the blind and DD modes, several soft-

switching schemes have been proposed in the literature (see e.g [14]–[17]). Additionally, blind equalization algorithms with a

good transient and steady-state performance have also been proposed in order to bypass the switching mechanism to the DD

mode (see e.g, [18] and its references). However, these schemes are typically difficult to adjust, and their performance is still

very dependent on the particular scenario in which they are applied.

A recent approach for improving the performance of blind equalizers is based on combinations of adaptive filters. The

idea consists in mixing the outputs of independently-run adaptive algorithms with different settings, so that the overall filter

behaves, at each iteration, as the best component filter or even better than any of them. Due to its relative simplicity, this

approach has received a lot of attention in the adaptive signal processing literature in the recent years [19]–[29]. Specifically

in blind equalization, combination schemes were exploited in [26]–[29]. To improve the tradeoff between the convergence rate

and the steady-state MSE, a convex combination of two CMA equalizers with different step sizes was proposed in [26]. Later,

[27] proposed a convex combination of CMA with the Shalvi-Weinstein algorithm to obtain an equalizer with better tracking

capability. Affine combinations of CMA equalizers with different step sizes were exploited in [28]. Although all these schemes

achieve a lower steady-state MSE than that of a single CMA, they are just based on blind components, so switching to a DD

mode is still necessary in most cases, and the problem of selecting an appropriate MSE level for the transition between modes

persists.

In this paper, we propose a novel blind equalization scheme based on a convex combination of adaptive filters. The idea is

to combine the output of a blind equalization algorithm with that of an equalizer operating in DD mode. The blind equalization

algorithm must mitigate the intersymbol interference during the initial convergence or when abrupt changes in the channel

occur. Then, when the steady-state MSE is sufficiently low, the overall equalizer should work in the DD mode. Since both

equalizers are continuously running in parallel and their combination is adapted to maximize the overall performance, our

scheme enables a soft switching between the blind and DD components as soon as a sufficiently low MSE level is achieved.

Therefore, it is not necessary to a priori set an MSE level for the transition between modes. A scheme similar to ours was

previously used in [29]; however, in that work the mixing parameter was computed through an error power ratio method, which

does not ensure an optimal adjustment with respect to the overall performance neither that the switching occurs at the right

time.

The proposed scheme extends previous work on convex combination of adaptive filters in different ways:

SILVA AND ARENAS-GARCIA: A SOFT-SWITCHING BLIND EQUALIZATION SCHEME VIA CONVEX COMBINATION OF ADAPTIVE FILTERS 3

• Mixtures of filters of different kinds have been previously considered. Our scheme, however, does not just consider a

mixture of two filters of different kinds, but their operation modes are also different: one of them applies a blind criterion,

while the other minimizes a supervised cost. This implies some changes in the configuration of the overall combination,

namely, the two filters are not longer independently adapted, but the overall decision is fed back to the supervised element

in the mixture.

• Unlike most previous works that exploit combinations to get improved convergence, tracking, and steady-state performance,

our goal here is to provide an automatic mechanism for smoothly switching between the blind and DD equalization modes.

• A new rule for adapting the mixing parameter is proposed, which is based on the minimization of the MMA cost function.

We will illustrate that such rule provides a smaller symbol error rate (SER) than previous adaptation schemes based on

MSE.

• Finally, although our approach is valid for combinations of different kinds of adaptive filters, we will pay special attention

to the combination of the MMA equalizer and the least-mean-square (LMS) filter, presenting a theoretical analysis of the

tracking performance of the combination using energy conservation arguments [30]. Such analysis requires the evaluation

of the cross excess excess mean-square error (cross-EMSE) between the filters, which is a challenging task provided their

different operation modes.

The experiments section shows that the proposed equalizer is a very practical solution to effectively switch between blind

and DD modes. The advantages of our scheme with respect to existing hard- and soft-switching algorithms are also studied

considering different constellations, communication channels, and SNR conditions.

The paper is organized as follows. The problem is described in the next section. The convex combination of the MMA

equalizer with the LMS algorithm is introduced in Section III, where we also provide a novel MMA-based algorithm to update

the mixing parameter. Section IV contains a steady-state analysis for the combination, whose accuracy is verified through

simulations. In Section V, we present simulation results in order to compare the proposed scheme with some existing hard-

and soft-switching equalization methods in different situations. Finally, Section VI provides the main conclusions of the paper.

II. PROBLEM FORMULATION

A simplified baseband communication system [1] is depicted in Fig. 1 . The signal a(n) ∈ C, assumed to be i.i.d. (independent

and identically distributed) taken from a known constellation, is transmitted through an unknown channel, whose model is

constituted by an FIR filter with impulse response vector h and additive white Gaussian noise (AWGN). We assume an M -tap

FIR equalizer, with input regressor vector u(n) and output y(n) = uT (n)w(n) = yR(n) + jyI(n), where (·)T indicates

transposition, w(n) is the equalizer weight (column) vector, and yR(n)=Re{y(n)} and yI(n) = Im{y(n)} are the real and

imaginary parts of y(n), respectively. From the received signal u(n) and the known statistical properties of the transmitted

signal, the blind algorithm seeks to estimate a delayed version of the signal a(n) = aR(n)+ jaI(n), obtaining a(n − ∆)

at the output of the decision device, with ∆ a positive integer. The estimate a(n − ∆) is used as a desired signal in the

decision-directed mode.

We assume that the equalization algorithms are implemented in the baud rate (T ) or in the T/2 fractionally-spaced form. The

fractionally-spaced implementation is widely considered in the literature since it ensures perfect equalization in a noise-free

environment, under certain well-known conditions (see, e.g., [4], [31], [32] and the references therein).


a(n) u(n)w(n−1)

y(n) a(n−∆)channel

blindalgorithm

equalizer

devicedecision

statisticsof a(n)

Fig. 1. Simplified communication system with a blind equalizer.

III. A CONVEX COMBINATION OF ADAPTIVE EQUALIZERS

In this paper, we take advantage of recent advances in the adaptive filtering literature regarding combinations of filters to put

together the strengths of blind and DD equalization, providing an automatic and soft switching between modes. Specifically,

the overall output of the proposed scheme is given by

y(n) = λ(n)y1(n) + [1 − λ(n)]y2(n), (1)

where λ(n) ∈ [0, 1] is the mixing parameter, and yi(n) = uT (n)wi(n) are the outputs of the blind and DD equalizers,

respectively for i = 1, 2. Although the proposed configuration can be used with other kinds of equalizers, to simplify the

presentation we will consider in the sequel that y1(n) and y2(n) are the outputs of the MMA and LMS equalizers, as

illustrated in Fig. 2. The overall output is decoded by a decision device, and the decoded sequence, a(n − ∆), is used by

the LMS filter, that plays the role of a DD algorithm. Note that feeding back the overall decision to the LMS component

introduces a coupling between the component filters. Specifically, for λ(n) ≈ 1 the decoded output of the MMA component

will be used as the desired signal for the LMS filter.

u(n)

w1(n)

w2(n)

y(n)

y1(n)

y2(n)

λ(n)

1−λ(n)

e2(n)

a(n−∆)

MMA

LMS

devicedecision

Fig. 2. Convex combination of MMA with LMS. The LMS filter updates its output in DD mode following the output of the decision device.

For the good behavior of the combination, it is important that the component filters are adapted according to their own

rules, whereas λ(n) should optimize the overall performance [19]. When doing so, the combination can put together the

complementary strengths of both components, that is, the blind equalization capabilities of MMA and the smaller error of

LMS working in DD mode.

Standard operation of the MMA filter consists in the stochastic minimization of [8]–[10]

JMMA,1(n) = E{[

r − y21,R(n)

]2}+ E

{[r − y2

1,I(n)]2}

, (2)

where y1,R(n) and y1,I(n) are the real and imaginary parts of y1(n), r = E{a4R(n)}/E{a2

R(n)} = E{a4

I(n)}/E{a2

I(n)} for

symmetric constellations (symbols are assumed i.i.d.), and E{·} represents the expectation operator. Its update equation is


given by

w1(n + 1) = w1(n) + ρ c1(n)u∗(n), (3)

where (·)∗ denotes complex conjugate, ρ is a step size, and

c1(n) = c1,R(n) + jc1,I(n)

=[r − y2

1,R(n)]y1,R(n) + j

[r − y2

1,I(n)]y1,I(n). (4)

The decision-directed LMS algorithm minimizes an instantaneous approximation of the mean-square decision error defined

as

JMSE,2(n) = E{|a(n − ∆) − y2(n)|2

}. (5)

Its update equation is given by

w2(n + 1) = w2(n) + µe2(n)u∗(n), (6)

where µ is a step-size and

ei(n) = a(n − ∆) − yi(n), (7)

i = 1, 2 are decision errors. Although we use only e2(n) in (6), the computation of e1(n) with the output of the MMA

equalizer is employed in the definition of the overall error, that is,

e(n) = λ(n)e1(n) + [1 − λ(n)] e2(n)

= a(n − ∆) − y(n). (8)

A. Blind update of the mixing parameter

In order to get a fully blind equalizer, we adapt the mixing parameter with the goal to minimize a blind cost function. We

can define the MMA cost of the overall system as

JMMA(n) = E{[

r − y2R(n)

]2}+ E

{[r − y2

I(n)

]2}, (9)

and update λ(n) using stochastic gradient descent rules. Rather than directly adjusting λ(n), it is customary to update an

auxiliar parameter α(n) which is deterministically related to λ(n) via

λ(n) = ϕ[α(n)] =sgm[α(n)] − sgm[−α+]

sgm[α+] − sgm[−α+], (10)

where sgm[x] = [1 + e−x]−1

is the sigmoidal function, and α+ is the positive maximum value that α(n) can assume. This

activation function was proposed in [24] and is a scaled and shifted version of the sigmoidal function. It is important to notice

that λ(n) attains values 1 and 0 for α(n) = α+ and α(n) = −α+, respectively.

By taking the derivative of (9) with respect to α(n) and approximating expectations by their instantaneous values, we obtain

the following stochastic gradient descendent rule for the update of α(n):

α(n)=α(n−1)+ρα(n)Re{c(n)[y1(n)−y2(n)]∗}ϕ′[α(n−1)], (11)


where, similarly to (4), c(n) =[r − y2

R(n)

]yR(n) + j

[r − y2

I(n)

]yI(n), ϕ′[α(n− 1)] = dλ(n−1)

dα(n−1) , and ρα(n) is a step size. In

practice, α(n) is restricted by saturation to a symmetric interval [−α+, α+], since factor ϕ′[α(n− 1)] in (11) would virtually

stop adaptation if |α(n)| were allowed to grow too much. A common choice in the literature is α+ = 4 [19], [20], [24].

Although it is possible to use a constant ρα(n), a more robust behavior can be obtained from a power-normalized update

rule. Reinterpreting the combination as a “second-layer” adaptive filtering [20] and noticing that [y1(n) − y2(n)] plays the

role of the input signal for this second-layer [see (11)], we can set ρα(n) = ρα

p(n) , with p(n) a rough estimate of the power of

[y1(n)−y2(n)], i.e,

p(n) = η p(n − 1) + (1 − η)|y1(n) − y2(n)|2. (12)

The normalized rule is easier to adjust than its non-normalized counterpart, as observed in [20], [28]. Additionally, the selection

of the forgetting factor η is not critical for the appropriate performance of the algorithm, and we will simply set it to 0.9.

The benefits of using the activation function for the mixing parameter are twofold. Apart from keeping λ(n) in the desired

interval [0, 1], we can see that the derivative ϕ′[α(n−1)] that appears in (11) takes a reduced value whenever λ(n) approaches

its upper or lower limiting values, thus minimizing the adaptation speed and gradient noise in these important situations1.

B. An illustrative example

To illustrate the performance of the proposed combination scheme, we show next a simulation example for Scenario I

described in Table II (see Section V), in which a 256-QAM (quadrature amplitude modulation) signal is transmitted through a

noisy channel that changes abruptly at iteration n=1.5 × 104. Fig. 3-(a) shows the MSE of the MMA and LMS components

and of their combination using the above derived rule. It is worth mentioning that an isolated LMS filter would not be able to

converge in this situation without an initial training phase using a pilot sequence. In our scheme, it is the MMA component

which provides a first rough equalization of the communication channel, with λ(n) ≈ 1 –see the evolution of the mixing

parameter in Fig. 3-(c) for a single realization. Once the decoded sequence is good enough to allow LMS convergence in a

DD mode, the combination automatically switches to the LMS component, with λ(n) ≈ 0. A similar behavior occurs after the

change at n=1.5 × 104, showing the ability of our scheme to revert to the blind component if necessary.

The use of blind adaptation for the mixing parameter plays a crucial role in the performance of the proposed equalizer.

To see this, we have studied the performance of a similar system that uses instead an MSE criterion to adapt λ(n), i.e, an

extension of the rule in [19] for complex input signals:

α(n) = α(n − 1) +µα

p(n)Re{e(n)[y1(n) − y2(n)]∗}ϕ′[α(n − 1)], (13)

where µα is the step size for the adaptation of α(n), and p(n) is given by (12).

To obtain an appropriate behavior from such combination, (13) needs to be used with very large µα. Fig 3-(b) shows its

average MSE performance for µα = 100, while the evolution of λ(n) for a single run is depicted in 3-(c). During the initial

convergence, the decoded signal does not provide a good reference neither for the update of the LMS component, nor for the

mixing parameter. As a consequence, and due to the use of a large step size µα, the mixing parameter oscillates continuously

between 0 and 1, and the overall scheme behaves similarly to the MMA component. However, as soon as the MSE achieves

a sufficiently low level, the combined filter very rapidly switches to a pure DD mode.

1For a more detailed discussion about the advantages of the sigmoid activation in combination schemes, the interested reader is referred to [19], [24].


−20

0

0.5 1.5 2.5 3.51 2 3 4

20

10

−10

MMA component

LMS component

Combination using (11)

MS

E(d

B)

(a)

−20

0

0.5 1.5 2.5 3.51 2 3 4

20

10

−10

MMA component

LMS component

Combination using (13)

MS

E(d

B)

(b)

0

0.5

0.5 1.5 2.5 3.5

1

1 2 3 4

Update with (11)

Update with (13)

Iterations ×104

(c)

λ(n

)

Fig. 3. MSE performance of combination schemes for Scenario I of Table II: (a) MSE of MMA, LMS, and of the proposed convex combination, estimated

from an ensemble-average of 1000; (b) MSE of MMA, LMS, and of their convex combination using (13); (c) Mixing parameters considering one run of the

algorithms.

In spite of very similar performances in terms of MSE, analysis of the steady-state symbol error rate (SER) achieved by both

combination schemes shows a clear advantage of the MMA criterion. Fig. 4 depicts curves of steady-state SER as a function

of SNR, assuming the first channel of Scenario I (Table II) for MMA and LMS components, as well as for their combinations

using (11) and (13) to update the mixing parameter. These error rates were estimated after the convergence of the algorithms,

by counting the number of errors when comparing the transmitted sequence with the sequence obtained at the output of the

decision device. We disregarded 2× 105 symbols due to the convergence and used 107 symbols to estimate the SER for each

SNR. The curve of SER for the Wiener solution obtained with a delay of ∆ = 13 samples is also shown in the figure for

comparison. It is clear that MMA adaptation of λ(n) makes the combination behave exactly as the DD component, which is

relatively close to Wiener solution. On the other hand, the MSE adaptation introduces a significant degradation in terms of

SER, which can be observed in the performance of the combination, as well as in that of the DD component. Probably, this

occurs due to the large value required for µα.

A curve of SER as a function of the number of received symbols is provided in Fig. 5 for Scenario I (Table II), assuming

SNR = 30 dB and an abrupt change in the channel at iteration n = 4 × 104. These curves were obtained by fixing the

coefficients of the component filters as well as those of the combination at each iteration and transmitting 5 × 105 symbols

to compute the SER. Again, we can observe that the combination using (11) presents a clear advantage with respect to the

combination using (13), which does not converge properly to a fixed value of SER. Therefore, we consider only the update


MMA componentLMS component using (11)LMS component using (13)

Combination using (13)Combination using (11)

Wiener

20 22.5 27.5 32.525 30 35

100

10−1

10−2

10−3

10−4

10−5

10−6

SER

SNR (dB)

Fig. 4. Steady-state SER as a function of the SNR for the MMA and LMS components, as well as of the combinations using (11) and (13) for the adaptation

of the mixing parameter, assuming the first channel of Scenario I of Table II.

(11) henceforward.

100

10−0.4

10−0.8

10−1.2

10−1.6

10−2

1 2 3 4 5 6 7 8

Iterations

MMA component

×10

04

SE

R

LMS component using (11)LMS component using (13)

Combination using (13)Combination using (11)

Fig. 5. SER along the iterations for Scenario I and parameters of the algorithms as in Table II, assuming SNR = 30 dB. Average of 100 runs, where for

each iteration and run, 5 × 105 symbols were transmitted and decoded to estimate the SER.

IV. PERFORMANCE ANALYSIS

In this section, we obtain an analytical expression for the steady-state EMSE of the combination of MMA with LMS in a

nonstationary environment. The EMSE of the overall filter depends on the EMSE of the component filters and also on their

cross-EMSE [19]. Since the literature contains analytical expressions for the steady-state EMSE of LMS and MMA, we only

obtain an expression for the cross-EMSE, using the energy conservation arguments of [30, Ch. 21] and the results of the

steady-state analysis of LMS [30], CMA [4], [27], [30], [32], [33], and MMA [18].

We start by describing the simplifying assumptions used in the analysis and defining the performance indicators. Then, we

revisit the analytical expression for the EMSE of the convex combination of adaptive filters, obtained in [19, Sec. III]. In the

sequel, we obtain an analytical expression for the cross-EMSE of the combination of MMA with LMS. We close this section

with some simulation results to verify the accuracy of the steady-state analysis.


A. Assumptions and performance indicators

Our analysis is based on the following simplifying assumptions:

A1. the transmitted signal a(n − ∆) is related to u(n) via

a(n − ∆) = uT (n)wo(n) + v(n), (14)

where wo(n) is the optimal solution and v(n) = vR(n) + jvI(n) plays the role of a complex disturbance, assumed to

be i.i.d., zero-mean, and with variance σ2v = E{|v(n)|2} = 2E{v2

R(n)} = 2E{v2

I(n)}. In order to make the performance

analysis more tractable, the sequences {u(n)} and {v(n)} are assumed stationary and we will use the common assumption

that v(n) is independent of {u(ℓ)}, ℓ ≤ n (not only uncorrelated).

The model (14) is most commonly used in the context of system identification being refereed as linear regression model [30],

but it can also be used in the analysis of adaptive equalization algorithms as shown in [34]. For a T/2 fractionally-spaced

equalizer in the absence of noise, v(n) ≡ 0 and the optimum filter achieves perfect equalization [31], [32].

A2. in a nonstationary environment, the optimal solution follows a random-walk model [30], that is,

wo(n + 1) = wo(n) + q(n), (15)

where q(n) is an i.i.d. vector with positive-definite autocorrelation matrix Q=E{q∗(n)qT (n)}, assumed to be independent

of the initial conditions {wo(0),wi(0), α(0)}, i = 1, 2 and of {u(ℓ), a(ℓ − ∆)} for all ℓ ≤ n.

In the case of a baud-rate equalizer, wo(n) is given by the Wiener solution. On the other hand, for a T/2 fractionally-spaced

equalizer, wo(n) represents the zero-forcing solution [31], [32].

A3. ‖u(n)‖2 is independent of c1(n) and e2(n) in steady-state, where ‖ · ‖ represents the Euclidean norm. In steady-state

analyses of supervised adaptive filters, this assumption is usually referred as separation principle [30]. A similar assumption

was adopted in steady-state analyses of CMA [32], [33] and MMA [18].

To measure the performance of the combined scheme, it is convenient to previously define the steady-state EMSE of the

component filters and their steady-state cross-EMSE:

ζi , limn→∞

E{|ea,i(n)|2}, i = 1, 2 (16)

and

ζ12 , limn→∞

E{ea,1(n)e∗a,2(n)} = limn→∞

E{e∗a,1(n)ea,2(n)}, (17)

where

ea,i(n) = uT (n)wi(n), i = 1, 2 (18)

are the a priori errors of the MMA and LMS filters and

wi(n) = wo(n) − wi(n), i = 1, 2 (19)

are their weight-error vectors2.

2In Subsection IV-C, we will see from our analysis that the cross-EMSE ζ12 is real-valued in steady-state, which justifies the definition given in (17).


Now, noticing from (1) that the overall filter can be seen as a filter with weights w(n) = λ(n)w1(n) + [1 − λ(n)]w2(n),

the a priori error of the combination can be expressed as

ea(n) = uT (n) [wo(n) − w(n)]

= λ(n)ea,1(n) + [1 − λ(n)]ea,2(n) (20)

from which the overall steady-state EMSE can be defined as

ζ , limn→∞

E{|ea(n)|2}. (21)

To make our analysis more tractable, we also assume that

A4. in steady state, the mixing parameter λ(n) is independent of the a priori errors ea,i(n) of both component filters and has

a small variance [19].

B. Steady-state EMSE of the combination

Under Assumption A4 and using (20), the steady-state EMSE of the overall filter can be estimated as [19, Sec. III]

ζ ≈E{λ2(∞)

}ζ1 + E

{[1 − λ(∞)]2

}ζ2

+ 2E {λ(∞)[1 − λ(∞)]} ζ12. (22)

Under the assumption of zero variance for λ(∞), the combination of MMA with LMS is nearly universal in the mean-square

error sense since it performs as well as the best of its components, and under certain conditions, it may outperform both of

them. Thus, when MMA outperforms LMS in steady-state, λ(∞) = 1 and the behavior of the combination will be close to that

of MMA, i.e, ζ ≈ ζ1. On the other hand, when LMS is superior to MMA in steady-state, λ(∞) = 0 and ζ ≈ ζ2. Moreover,

there are situations where ζ12 < ζi, i = 1, 2 and the combination will outperform both components. In this case, the EMSE

of the overall filter can be estimated as

ζ ≈ ζ12 +∆ζ1∆ζ2

∆ζ1 + ∆ζ2, (23)

where ∆ζi = ζi − ζ12, i = 1, 2. This expression was obtained in [19, Eq.(33)] for the combination of two LMS filters with

different step-sizes. However, as pointed out in [19], (23) also holds for the convex combination of other adaptive algorithms,

as the one proposed here.

In summary, the EMSE of the combination of MMA with LMS can be estimated by the minimum of the values computed

by analytical expressions for ζ1, ζ2, and (23).

C. Steady-state cross-EMSE

The update equations of MMA and LMS can be rewritten as a function of their respective weight-error vectors. Thus,

subtracting both sides of (3) and (6) from wo(n + 1) and using (19) and (15), we arrive at

w1(n + 1) = w1(n) − ρ c1(n)u∗(n) + q(n) (24)

w2(n + 1) = w2(n) − µ e2(n)u∗(n) + q(n). (25)


To obtain a theoretical expression for ζ12, we multiply w1(n + 1) from the left by the complex-conjugate transpose of

w2(n + 1), denoted as wH

2 (n + 1). Taking expectations of both sides of the resulting expression, using (18) and Assumption

A2, after some algebra we arrive at

E{wH

2 (n + 1)w1(n + 1)} = E{wH

2 (n)w1(n)}

− ρE{c1(n)e∗a,2(n)} − µE{ea,1(n)e∗2(n)}

+ ρµE{c1(n)e∗2(n)‖u(n)‖2} + E{qH(n)q(n)}. (26)

Assuming that the filters have achieved the steady-state, i.e.,

E{wH

2 (n + 1)w1(n + 1)} ≈ E{wH

2 (n)w1(n)}, for n → ∞

and under Assumption A3, (26) can be rewritten for n → ∞ as

−Tr(Q) = − ρE{c1(n)e∗a,2(n)} − µE{ea,1(n)e∗2(n)}

+ ρµTr(R)E{c1(n)e∗2(n)}, (27)

where Tr(·) stands for the trace of a matrix and R , E{u∗(n)uT (n)} represents the autocorrelation matrix of the input vector.

To proceed further, we need to find relations between c1(n) and ea,1(n), and between e2(n) and ea,2(n). Using (18), (19),

and (14) (Assumption A1), we can rewrite yi(n), i = 1, 2 as a function of ea,i(n), that is,

yi(n) = uT (n)wi(n)

= uT (n)[wo(n) − wi(n)]

= a(n − ∆) − ea,i(n) − v(n), i = 1, 2. (28)

From (28), a relation between e2(n) and ea,2(n) is obtained straightforwardly using (7) and assuming that a(n−∆) = a(n−∆)

in steady-state, which leads to

e2(n) = ea,2(n) + v(n). (29)

This is a very well-known relation for supervised adaptive filters as LMS [30].

On the other hand, to obtain a relation between c1(n) and ea,1(n) for MMA is not so simple and requires additional

assumptions. Based on previous works on steady-state analysis of CMA [27], [32], [33] and MMA [18], we replace (28) in (4)

and assume that terms depending on eka,1,R(n) and ek

a,1,I(n) are sufficiently small for k ≥ 2 when compared to linear terms

on ea,1,R(n) and ea,1,I(n) (see appendix). Thus, c1(n) can be approximated in steady-state by

c1(n) ≈ [γR(n)ea,1,R(n) + βR] + j [γI(n)ea,1,I(n) + βI(n)] , (30)

where

γR(n), 3a2R(n−∆)−r−6vR(n)aR(n−∆)+3v2

R(n) (31)

and

βR(n) , r aR(n−∆)−a3R(n−∆)−3aR(n−∆)v2

R(n)

+vR(n)[3a2

R(n−∆)−r

]+v3

R(n). (32)


Expressions for γI(n) and βI(n) can be obtained by replacing subscript ‘R’ by subscript ‘I’ in the above expressions. The

main assumptions used in the derivation of this model are summarized in the appendix. Since we did not disregard terms

depending on vkR

for k ≥ 2 in the definitions (31) and (32) (similarly for the imaginary counterpart), this model is an extension

of previous models used in the steady-state and transient analysis of CMA and MMA (see, e.g., [18], [27], [30], [32]).

Thus, using (29), (30), Assumption A1, and the results from the appendix, we obtain the following steady-state approximations

E{c1(n)e∗a,2(n)} ≈ γζ12, (33)

E{ea,1(n)e∗2(n)} ≈ ζ12, (34)

and

E{c1(n)e∗2(n)}≈ γζ12+E{c1(n)v∗(n)}, (35)

where γ , E{γR(n)}=E{γI(n)} is a constant that depends on higher-order statistics of the transmitted sequence and is given

by (40) (see appendix). Under Assumption A7 from the appendix

E{γR(n)ea,1,R(n)v∗(n)} = E{γI(n)ea,1,I(n)v∗(n)} ≈ 0.

Consequently, the last term of the right-hand side of (35) can be approximated by

E{c1(n)v∗(n)} ≈ E{β(n)v∗(n)}

≈ γσ2v + E{|v(n)|4} − 0.5σ4

v . (36)

Replacing (36), (34) and (33) in (27), we arrive at

ζ12 ≈ρµTr(R)[γσ2

v+E{|v(n)|4}−0.5σ4v ]+Tr(Q)

ργ + µ − ρµγTr(R). (37)

For γσ2v ≫ E{|v(n)|4}−0.5σ4

v , (37) reduces to

ζ12 ≈ρµTr(R)γσ2

v+Tr(Q)

ργ + µ − ρµγTr(R). (38)

In passing, we note that, as previously stated, the cross-EMSE between MMA and LMS is real valued.

Analytical expressions for the steady-state EMSE of MMA (ζ1) in a nonstationary environment were obtained previously in

the literature (see, e.g., [18], [27], [33]). In [18], for instance, a normalized version of MMA was analysed in steady-state using

energy conservation arguments and simplifying assumptions similar to those considered here. The EMSE expression obtained

in [18] can be straightforwardly extended to the non-normalized version of MMA and is shown in Table I. For convenient

reference, this table also contains an expression for the EMSE of LMS (ζ2) [30] in conjunction with expression (38) for the

cross-EMSE (ζ12).

D. Accuracy of the analysis

To verify the accuracy of the steady-state analysis, we assume the transmission of a 64-QAM signal through channel

h = [ 0.2258 0.5161 0.6452 −0.5161 ]T in the absence of noise [35, Eq. (29)]. We also assume a baud-rate equalizer

with M = 12 coefficients and the convex combination of MMA (ρ = 10−6) and LMS (different step sizes) using (11) with

ρα = 5× 10−3. Fig. 6-(a) shows the theoretical curves of the EMSE for MMA and LMS, and of their cross-EMSE, predicted


TABLE I

ANALYTICAL EXPRESSIONS FOR THE STEADY-STATE EMSE AND CROSS-EMSE OF THE MMA AND LMS FILTERS IN A NONSTATIONARY ENVIRONMENT.

ζ1 (MMA) ζ2 (LMS) ζ12

ρ σ2

βTr(R) + Tr(Q)/ρ

2γ − ρ¯γTr(R)

µ σ2vTr(R) + Tr(Q)/µ

2 − µTr(R)

ρµTr(R)γσ2v+Tr(Q)

ργ + µ − ρµγTr(R)

σ2

β ≈ 2E{a6

R(n) − r2a2

R(n)} + σ2

vE{3a4

R(n) + r2}

γ ≈ 1.5(σ2

a + σ2

v) − r

¯γ ≈ 1.5(r+9σ2

v

)σ2

a+r2 − 3rσ2

v

σ2

v ≈ E{|a(n)|2} − wHo (0)Rwo(0)

by the expressions of Table I, assuming a stationary environment [Tr(Q) = 0]. The experimental values, estimated through an

ensemble-average of 1000 runs, are also shown in the figure. Since we use a fixed step-size for MMA, its steady-state EMSE

does not vary, being approximately equal to −10 dB. We can observe that the experimental results agree with our analysis for

all range of the LMS step size considered. The steady-state EMSE of combination is not shown in the figure, since in this

case the combination always provides a soft switching to the LMS filter, presenting the same steady-sate EMSE.

Assuming now a nonstationary environment with Q = 8 × 10−9I, we obtain the theoretical EMSE curves as well as their

experimental values, shown in Fig. 6-(b). Again, as we consider a fixed step size for MMA (ρ = 10−6), its EMSE does not

vary. On the other hand, the EMSE curve for LMS varies with µ and presents a minimum that occurs approximately for

µ = 7 × 10−5. As we expected, the combination of MMA and LMS performs at least as the best of the two component filters,

even outperforming both of them simultaneously for a certain interval of the LMS step size. Again, the experimental results

agree with our analysis for all range of the LMS step size considered.

It is important to mention that equally accurate results are also obtained by varying the MMA step size and considering a

fixed step size for LMS.

V. SIMULATION RESULTS

In this section, we evaluate the performance of the proposed scheme in different simulation scenarios and compare them

with the algorithms of [14]–[16], and [18]. We named these algorithms with the surname of one of the authors (Picchi, Kassam,

Castro, and Mendes, respectively). The Picchi algorithm [14] is based on a stop-and-go operation mode for the DD algorithm.

At each iteration, a flag inhibits adaptation if the reliability of the current self-decided output error is not large enough to

warrant its use in the updating. The Kassam algorithm [15] is a dual-mode CMA, where a rule decides at each iteration

whether CMA or the radius directed equalization algorithm, which is a decision-directed-like method, must be employed. The

Castro algorithm [16] combines concurrently CMA with the DD algorithm. At each iteration, an auxiliary variable, based on

the estimate provided by CMA, is used to decide if the weights of the DD algorithm must be updated or not. Finally, the

Mendes algorithm, recently proposed in [18], modified the MMA error function to allow the blind algorithm to converge in

the mean to the Wiener solution, thus bypassing the switching mechanism between the blind and DD modes.

For comparison, we also consider a hard switching from the blind equalization algorithm to the DD algorithm used as

component filters in the combination scheme. The moment of switching followed an MSE threshold, computed to ensure a

symbol error rate lower than 10−1, as suggested in [5, pp.88–89]. In this case, the running MSE was estimated using an


MMA model

MMA simu

LMS model

LMS simu

cross−EMSE model

cross−EMSE simu

(a)

−60

−50

−40

−30

−20

−10

−3−4−5−6−7

1010101010

EM

SE

(dB

)

LMS step size (µ)

MMA model

MMA simu

LMS model

LMS simu

cross−EMSE model

cross−EMSE simu

combination model

combination simu

(b)

−30

−20

−10

−15

−35

−25

−5

−3−4−5−6−7

1010101010

EM

SE

(dB

)

Fig. 6. Theoretical and experimental EMSE as a function of the LMS step size (µ); ρ = 10−6; 64-QAM, Channel H2(z) of [35, Eq. (29)], absence of

noise, baud-rate equalizer, M = 12, ensemble-average of 1000 runs; (a) stationary environment and (b) nonstationary environment with Q = 8 × 10−9I.

exponential window, i.e., ξ(n) = η ξ(n − 1) + (1 − η)|a(n−∆) − y(n)|2, where ξ(0) = 1 and 0 ≪ η < 1 is a forgetting

factor. We set η = 0.9 in all simulations and assume that the hard switching to the DD mode occurs if ξ(n) is less than

the MSE threshold in a given iteration n. Since this threshold depends on several factors such as the signal constellation, the

communication channel, or the signal-to-noise ratio, it was independently adjusted for each simulation scenario.

The simulation scenarios were characterized by different constellations, SNRs, component filters, and communication

channels. For ease of reproduction, the communication channels considered here have already been used in previously published

works. The parameters of the simulation scenarios are summarized in Table II. In Scenario I, we assume the transmission of a

256-QAM signal through the telephonic channel of [14, Fig.2] that changes abruptly to channel h = [ 0.3 1 0.3 ]T of [18] at

iteration n = 2.5 × 105. We compare the baud-rate equalization schemes assuming SNR = 40 dB. In Scenario II, a 64-QAM

signal is transmitted through noisy microwave channels (chan6 and chan9), obtained from the database available in [36]. Again,

an abrupt change occurs in the channel (from chan6 to chan9) at iteration n = 4 × 104. In this case, T/2 fractionally-spaced

equalization and SNR = 40 dB are assumed. Finally, Scenario III goes back to baud rate equalization and assumes an abrupt

change from the channel of [17, Table 2] to the channel of [35, Eq. (29)] at n = 105, with SNR = 30 dB. Additionally, the

transmitted signal is taken from a V.29 constellation, specified in ITU-T recommendation for 9600-bits/s transmission over

wire-line channels [37].


Parameters for the benchmark algorithms were adjusted to get a good tradeoff between convergence rate and steady-state

performance, and are given in Table II using the same symbols as in the original papers. In the simulations, all the equalizers

(including the DD component) were initialized with the typical center spike [4], [5]. As for the mixing parameter, it was

initialized to λ(0) = 1, so that the combination initially follows the blind component. However, using any other initial value

in the range [0, 1] would not significantly affect the performance of the combination, since λ(n) will rapidly evolve towards 1

when the LMS component has not converged, as it will also be the case after abrupt changes in the channel. As performance

measure, we considered the MSE(n) = E{|a(n − ∆) − y(n)|2

}, estimated along the iterations through an ensemble-average

of 1000 runs.

TABLE II

SIMULATION SCENARIOS AND PARAMETERS OF THE ALGORITHMS.

Parameters Scenario I Scenario II Scenario III

Constellation 256-QAM 64-QAM V.29 [37]

Channels Channel of [14, Fig.2] chan6 Channel of [17, Table 2]

and h = [ 0.3 1 0.3 ]T and chan9 of [36] and H2(z) of [35, Eq. (29)]

SNR 40 dB 40 dB 30 dB

Processing rate Baud-rate (T ) Fractionally-spaced (T/2) Baud-rate (T )

Equalizer length M =21 M =6 M =35

LMS µ=5 × 10−5 µ=5 × 10−4 –

MMA ρ=10−7 ρ=10−6 –

EF-LSL [38] – – λ=0.999, δ=104,

DM-LSWA [39] ǫ = 10−13, µp =5 × 10−4

Combination (11) ρα =5 × 10−4 ρα =5 × 10−3 ρα =5 × 10−4

Picchi [14] α=5 × 10−6, β=14 α=5 × 10−4, β=6 –

Kassam [15] α=10−7, d=0.4 α=5 × 10−6, d=0.7 –

Castro [16] ηv =5 × 10−5, ηw =10−8 ηv =2 × 10−4, ηw =10−6 ηv =10−4, ηw =10−6

Mendes [18] µ=5 × 10−4 – –

Hard switching MSE threshold: 0.1 MSE threshold: 0.5 MSE threshold: 0.25

A. Scenario I: 256-QAM

Returning to the simulation results of Fig. 3 (Section III), which was performed for the same scenario, we now compare

the performance of the proposed schemes with those of some existing soft-switching blind equalization techniques. The MSE

along the iterations of all considered schemes are shown in Fig. 7. We can observe that the convex combination scheme of the

MMA equalizer with the LMS algorithm using (11) outperforms all other soft-switching schemes in terms of convergence rate.

Furthermore, it achieves a steady-state MSE close to that of the Wiener solution with a delay of ∆=13 samples, independently

of the SNR, as observed in the SER curves of Fig. 4. The Castro algorithm takes more iterations to converge than the convex

combination and is not able to achieve the same steady-state MSE. This occurs due to CMA, which is never disregarded in that

concurrent algorithm. The Mendes algorithm is slower than Castro and achieves a steady-state MSE slightly lower than that of

the combination scheme. Besides slower, this algorithm seems to converge to local minima more often than the others (1.1%

of the runs of this algorithm were disregarded to obtain the curve shown in the figure). Finally, the Kassam algorithm shows


the slowest convergence and Picchi algorithm presents a long delay before converging to its final steady-state MSE level.

Picchi [14]Picchi [14]

Castro [16]Castro [16]

Kassam [15]

Kassam [15]

Mendes [18]Combination

Iterations

MS

E(d

B)

−30

−20

−10

0

0 1 2 3 4 5

10

20

×105

Fig. 7. MSE along the iterations for Scenario I and parameters of the algorithms as in Table II.

B. Scenario II: 64-QAM

Different from the previous simulations, we now use a T/2 fractionally-spaced equalizer with only M = 6 coefficients. Fig. 8

shows the MSE along the iterations. The Picchi algorithm is not shown in the figure since its performance is undistinguishable

from that of the Kassam algorithm. Before the abrupt change in the channel, the convex combination of one MMA with one

LMS using (11) and the hard switching present close performances, followed by the Kassam and Castro algorithms. After

the abrupt change in the channel, the combination converges faster than the Kassam algorithm, achieving a steady-state MSE

slightly superior. The hard switching presents the slowest convergence and the Castro algorithm an intermediate behavior

between Kassam and hard switching. Comparing to the results of Fig. 7, we can observe that the performance of the soft-

switching schemes considered in the comparison are very dependent on the simulation scenario. For instance, the Kassam

algorithm presents the worst performance in terms of convergence rate for Scenario I and a performance close to that of

the combination after the channel change in Scenario II. This dependence on the scenario is not observed for the convex

combination, which outperforms all other considered schemes, independently of the constellation, channel, and processing rate.

0

0 1 2 3 4

5

5 6 7 8

10

−5

−10

−15

Kassam [15]

Kassam [15] Castro [16]

Castro [16]

Combination

Hard switching

Combination / Hard switching

Iterations

MS

E(d

B)

×104

Fig. 8. MSE along the iterations for Scenario II and parameters of the algorithms as in Table II.

C. Scenario III: V.29

Our approach can also be applied to different kinds of algorithms. To illustrate this fact, we consider now the convex

combination of the dual-mode lattice Shalvi-Weinstein algorithm (DM-LSWA) [39] with the modified error feedback least-

squares lattice (EF-LSL) algorithm [38]. The modified EF-LSL is a supervised algorithm that belongs to the fast recursive


least-squares (RLS) family and presents reliable numerical properties. DM-LSWA, in its turn, can be interpreted as a “blind

version” of the modified EF-LSL algorithm. As shown in [39], DM-LSWA can avoid divergence when there is inconsistency

in the nonlinear estimate of the transmitted signal, being numerically well behaved even in finite precision.

The MSE curves are shown in Fig. 9, where we compare the combination with the Castro algorithm. The hard-switching

technique is not shown in the figure since its performance is undistinguishable from that of the combination. Furthermore,

we do not considerer the other soft-switching schemes here since they were proposed for QAM signals. The combination of

EF-LSL with DM-LSWA using (11) converges much faster than the Castro algorithm and both achieve the same steady-state

MSE. Again, it is important to remark that, independently of the scenario, our approach presents a good performance.

−20

0

0 0.5 1.51 2

5

15

10

−5

−10

−15

Castro [16]

Combination

Iterations

MS

E(d

B)

×105

Fig. 9. MSE along the iterations for Scenario III and parameters of the algorithms as in Table II.

VI. CONCLUSIONS

We proposed a soft-switching blind equalization scheme based on the convex combination of a blind equalization algorithm

and a DD algorithm. Our approach outperforms hard- and soft-switching schemes in terms of convergence rate and does not

depend on the constellation or other characteristics of the simulation scenario. We proposed an easy-to-adjust adaptive algorithm

for updating the mixing parameter and also provided a theoretical model for the steady-state EMSE of the combination. This

model presents a good agreement with the simulation results and can predict situations in which the proposed scheme can

achieve a better performance, being useful for the designer. Although we particularized most of our results to the combination

of the MMA equalizer with the LMS algorithm, our approach can be applied to different kinds of adaptive algorithms as shown

in the simulations.

Possibilities for further extensions of the proposed scheme include exploitation of the combination of filters to manage the

length of the adaptive filters, similarly to [21], or even to find a convenient delay for the equalization. The use of SER-oriented

schemes (see, e.g., [40]) could also be considered as a replacement for the DD equalizer, with potential advantages with respect

to MSE-oriented schemes, specially when channel coding is used.

APPENDIX

To obtain (30), we assume that

A5. in steady-state, terms depending on eka,1,R(n) and ek

a,1,I(n), k ≥ 2 can be disregarded since they are sufficiently small

when compared to terms depending on ea,1,R(n) and ea,1,I(n), respectively. In other words, MMA can not achieve perfect


equalization, but sufficiently mitigate the intersymbol interference introduced by the channel. A similar assumption was

used in the CMA analysis [27], [32], [33].

Thus, replacing the real part of (28) for i = 1 in the real part of (4) and using A5, after some algebraic manipulations, we get

(30).

Since we use (30) to estimate the expected values in (33) and (35), we need to compute the first moment of γR(n) and

γI(n). Additionally, the analytical expression for the steady-state of MMA (see Table I) also depends on the second moments

of γR(n), γI(n), and β(n) = βR(n)+ jβI(n). In the sequel, we obtain analytical expressions for the first and second moments

of these random variables. For this purpose, we also assume that

A6. E{akR(n)} = E{ak

I(n)} = 0, for all positive odd integer k, and for complex data E{a2(n)} = 0 (circularity condition). In

other words, the constellation is symmetric (zero-mean) as is the case for most constellations used in digital communications

[1]. Additionally, we assume that v(n) also obeys the circularity condition.

A7. a(n−∆), v(n) and ea,1(n) are mutually independent in steady-state. This assumption requires the steady-state fluctuations

of ea,1(n) to be insensitive to the actual transmitted symbols a(n−∆) [32], [33]. Additionally, the independence assumption

between ea,1(n) and v(n) is commonly used in the analysis of supervised adaptive filters [30]. An immediate consequence

of this assumption is that γR(n), γI(n), and β(n) are independent of ea,1(n) in steady-state.

A8. The real parts of a(n), ea,1(n), and v(n) are independent of their respective imaginary parts in steady-state.

Using A6-A8 and the definitions (31) and (32), we find

E{β(n)} = 0

σ2β = 2E{β2

R(n)}=2E{β2

I(n)}

≈ 2E{a6R(n) − r2a2

R(n)} + σ2

vE{3a4R(n) + r2},

+ 18E{a2R(n)}E{v4

R(n)}

+ 4E{v4R(n)}E{3a2

R(n) − r} + 2E{v6

R(n)}, (39)

γ , E{γR(n)} = E{γI(n)} = 1.5(σ2a + σ2

v) − r, (40)

and

¯γ , E{γ2R(n)} = E{γ2

I(n)}

≈ 1.5(r+9σ2

v

)σ2

a+r2 − 3rσ2v + 9E{v4

R(n)}. (41)

where σ2a = 2E{a2

R(n)} and σ2

v = 2E{v2R(n)}.

The variance of v(n) can be theoretically estimated by

σ2v ≈ σ2

a − wH

o Rwo, (42)

where wo , E{wo(n)} = wo(0). When the optimum filter achieves perfect equalization wH

o Rwo = σ2a and σ2

v = 0.

Furthermore, when wo is the Wiener solution, i.e., wo = wWIE = R−1p∆, σ2v = σ2

a−wT

WIEp∗

∆, where p∆ = E{a(n−∆)u∗(n)}

represents the cross-correlation between the transmitted sequence and the input vector.


Closed-form expressions for the fourth and sixth moments of v(n) as a function of σ2a, wo, R, and p∆ are complicated.

Additionally, when v(n) is relatively small in steady-state, terms depending on the fourth and sixth moments of v(n) can be

disregarded in (39) and (41), leading to simpler expressions, i.e.,

σ2β ≈ 2E{a6

R(n) − r2a2

R(n)} + σ2

vE{3a4R(n) + r2}, (43)

and

¯γ ≈ 1.5(r+9σ2

v

)σ2

a+r2 − 3rσ2v . (44)

Different from [18], here we did not disregard terms depending on vkR

and vkI

for k ≥ 2 in the definitions of γR(n), βR(n),

γI(n), and βI(n) [see Eqs. (31) and (32)]. Therefore, even disregarding terms depending on the fourth and sixth moments of

v(n), expressions (40) and (44) are slightly different from Eqs. (40) and (41) of [18], respectively. The model obtained here

is more accurate than that of [18]. This accuracy becomes evident when v(n) is small enough to disregard terms depending

on its fourth and sixth moments but not so small to disregard terms depending on its second order moment.

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Magno T. M. Silva (M’05) was born in Sao Sebastiao do Paraıso, Brazil. He received the B.S. degree in 1999, the M.S. degree

in 2001, and the Ph.D. degree in 2005, all in Electrical Engineering from Escola Politecnica, University of Sao Paulo, Sao Paulo,

Brazil. From February 2005 to July 2006 he was an Assistant Professor at Mackenzie Presbyterian University, Sao Paulo. Since

August 2006, he has been with the Department of Electronic Systems Engineering at Escola Politecnica, University of Sao Paulo,

where he is currently an Assistant Professor. From January to July 2012, he worked as a Postdoctoral Researcher at the Universidad

Carlos III de Madrid, Leganes, Spain. His research interests include linear and nonlinear adaptive filtering.

Jeronimo Arenas-Garcıa (S’00–M’04–SM’12) received the Telecommunication Engineer degree (honors) from Universidad Politecnica

de Madrid, Madrid, Spain, in 2000, and the Ph.D. degree in Telecommunication Technologies (honors) from Universidad Carlos III

de Madrid, Leganes, Spain, in 2004. After a postdoctoral stay at the Technical University of Denmark, Lyngby, he returned to the

Universidad Carlos III de Madrid, where he is currently a Lecturer of digital signal and information processing with the Department

of Signal Theory and Communications. His current research interests include statistical learning theory, particularly in adaptive

algorithms and advanced machine learning techniques, as well as their applications.

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