Recursive myriad-mean filters: Adaptive algorithms and applications

Accepted Manuscript

Recursive Myriad-Mean Filters: Adaptive Algorithms and Applications Juan Marcos Ramirez, Jose Luis Paredes PII: DOI: Reference:

S0165-1684(17)30127-5 10.1016/j.sigpro.2017.03.031 SIGPRO 6443

To appear in:

Signal Processing

Received date: Revised date: Accepted date:

5 October 2016 5 March 2017 28 March 2017

Please cite this article as: Juan Marcos Ramirez, Jose Luis Paredes, Recursive Myriad-Mean Filters: Adaptive Algorithms and Applications, Signal Processing (2017), doi: 10.1016/j.sigpro.2017.03.031

This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

ACCEPTED MANUSCRIPT

Recursive Myriad-Mean Filters: Adaptive Algorithms and Applications Juan Marcos Ramirez, Jose Luis Paredes

CR IP T

Department of Electrical Engineering, Universidad de Los Andes, M´erida, 5101, Venezuela

Abstract

In this paper, a new class of recursive hybrid filtering structures is proposed for impulsive noise removal; the so-called recursive myriad-mean (RMyM) filters. More precisely, the output of the RMyM filter can be thought of as the sum of two independent weighted M-filters: the nonlinear

AN US

weighted myriad acting on a subset of input samples and the linear weighted mean acting on a subset of filter’s previous outputs. The uncoupled structure of the proposed filters takes into account the benefits of both weighted M-estimators: the robustness against impulsive noise of the myriad operator and the desired spectral response induced by the linear feedback. Least mean absolute (LMA) based adaptive algorithms are developed for designing these filtering structures

M

under the equation error formulation framework. The results of extensive simulations are shown to evaluate both the behavior of the adaptive algorithms as well as the performance of the proposed

ED

recursive filters against impulsive noise. Additionally, taking into account the uncoupled structure of the proposed recursive filters, a decision feedback equalizer (DFE) based on the RMyM filter

PT

is proposed, where its performance is compared to those yielded by various conventional DFE structures, under different conditions of impulsive noise.

CE

Keywords: Adaptive algorithm, decision feedback equalizer, equation error formulation, hybrid nonlinear filters, recursive filters.

AC

1. Introduction

Recursive filters are feedback structures that consider in the estimation of the filter’s current

output a subset of previously computed outputs and a subset of input samples. In general, the performance of a recursive filter is notably better than that yielded by its non-recursive counterpart for the same number of filter coefficients. Indeed, a desired frequency-selective filtering operation can be more accurately obtained by using a recursive structure with much fewer filter Preprint submitted to Signal Processing

March 29, 2017

ACCEPTED MANUSCRIPT

coefficients compared to the non-recursive filter designed for the same filtering task, reducing thus, the computational complexity in terms of number of operations to be performed. In this context, infinite impulse response (IIR) filters, a linear filtering approach based on the mean operator, are recursive structures that have been considered in many signal processing and communications applications [1, 2, 3, 4], where the extensive use of these recursive filters is due

CR IP T

mainly to their efficiency for mitigating Gaussian noise and the low computational complexity related to linear operations. However, it is well-known that IIR filters tend to degrade notably their preformances in presence of outliers or gross errors, leading to the need of developing robust filtering structures against impulsive noise that exploit the benefits of recursive filters.

In this sense, recursive weighted median (RWM) filters [5] and the recently proposed recursive

AN US

weighted myriad (RWMy) filters [6] have been developed as robust recursive structures for impulsive noise removal. More precisely, the outputs of the RWM filters and the RWMy filters can be thought of as the maximum likelihood (ML) estimate of location parameter when the additive noise components —that affect both the input samples and the filter’s previous outputs— follow a Laplacian and a Cauchy distribution, respectively. Furthermore, assuming that the errors on filter’s

M

previous outputs have been attenuated by the robust filtering process itself, the class of recursive hybrid myriad (RHMy) filters has been also introduced in [6], where the additive perturbations

ED

that contaminate the input samples are characterized using a Cauchy statistical model but the errors that affect the previous outputs are described by the Gaussian distribution. In general,

PT

these nonlinear recursive filters are stable structures that can be adaptively designed in order to accurately perform frequency-selective filtering operations, where the main advantage of these

CE

feedback structures is the robustness against different levels of impulsive noise [5, 6]. Nevertheless, the robust recursive filters described above are coupled structures whose outputs are obtained by solving optimization problems that enclose, in a single nonlinear cost function,

AC

both the input samples and the filter’s previous outputs, making these families of recursive filters unsuitable in applications that require estimation processes acting independently on either the input signal samples or the filter’s previous outputs. Furthermore, the outputs of these recursive filters require to minimize nonlinear cost functions whose computation times increase as the observation window size becomes larger, leading to recursive filtering structures with higher computational costs compared to that yielded by IIR filters. 2

ACCEPTED MANUSCRIPT

In this paper, we introduce a novel class of recursive myriad-mean (RMyM) filters, whose recursive hybrid structure exploits the advantages of two independent and uncoupled estimation processes: the weighted myriad filter acting on a subset of input samples and the weighted mean filter acting on a subset of filter’s previous outputs. More precisely, the RMyM filters exploit both the robustness against impulsive noise of the myriad operator and the desired spectral response

CR IP T

induced by the linear feedback operator. Furthermore, since the output of the RMyM filter is restricted to an interval which is bounded by the magnitudes of the samples inside the recursive window, we also propose a broader class of the scaled recursive myriad-mean (SRMyM) filters, where the output of this kind of recursive filters is obtained by applying a suitable scaling to the response yielded by the RMyM filter.

AN US

Adaptive algorithms for obtaining the optimal parameters of the proposed recursive filtering structures are developed using the mean absolute error (MAE) as the performance criterion to be minimized. Furthermore, these adaptive algorithms are developed under the equation error formulation framework [2], a decoupling strategy that avoids the feedback inherent to recursive filters, leading to more tractable expressions for computing the gradients required by the steepest

M

descent updates. The behavior of the adaptive algorithms and the performance of the proposed recursive filters are evaluated using extensive numerical simulations. In general, the proposed

ED

recursive hybrid structures exhibit superior performances compared to those yielded by others nonlinear filters in terms of both the accuracy of the filter response against impulsive noise and the

PT

computation time of the filter output. Furthermore, the uncoupled structure of the SRMyM filter is exploited in the development of a robust decision feedback equalizer (DFE), where the proposed

CE

equalizer outperforms others DFE structures under different conditions of impulsive noise. The organization of this paper is as follows. In Section 2, the class of recursive filters in the context of ML type estimators (M-estimators) of location is briefly described. The proposed class of

AC

recursive myriad-mean filters is introduced in Section 3, and the adaptive algorithms for designing these nonlinear recursive structures are developed in Section 4. In Section 5, we evaluate the performance of recursive myriad-mean filters under various signal processing and communications applications. Finally, Section 6 summarizes some concluding remarks.

3

ACCEPTED MANUSCRIPT

2. Preliminary Background Taken a closer examination to the recursive filtering operations under the perspective of the ML estimation of location parameter, the response of a recursive filter can be described using the structure of the so-called M-estimators of location [7, 8]. To this end, let’s consider a recursive N1 1 sliding window that captures, at time n, a subset of input samples {x[n − i]|N i=−N2 } = {xi |i=−N2 }

CR IP T

taken from a discrete-time sequence {x[n]} and a subset of previous outputs {y[n − j]|M j=1 } = {yj |M j=1 }. Furthermore, let’s denote the extended set of samples inside the recursive window as

N1 M {z` |L `=1 } = {xi |i=−N2 ; yj |j=1 }, with L = (N1 + N2 + M + 1). Therefore, given the set of samples

inside the recursive observation window, an M-estimate of a common location parameter θ is given

θˆ =

=

arg min θ

`=1

arg min θ

L X

AN US

by ρ(z` − θ)

 N1  X 

i=−N2

ρ(xi − θ) +

M X j=1

ρ(yj − θ)

  

(1)

where ρ(·) is the cost function of the M-estimator of location. In general, each sample inside the

M

recursive window is modeled as z` = θ + ν` , where ν` is an additive noise perturbation which is assumed as an independent and identically distributed (i.i.d.) random sample that follows a

ED

particular statistical distribution. Under this context, the ML estimate of location parameter can be considered as a particular case of M-estimator of location, with ρ(u) = − log f (u), where f (u) is

PT

the probability density function that describes each component of additive noise. For instance, from

CE

Gaussian and Laplacian density functions, the moving average recursive filter [9] and the recursive P 2 median filter [10, 11, 12, 13] are the optimal filtering structures that minimize L `=1 (z` − θ) and PL `=1 |z` − θ|, respectively. However, these plain estimators do not properly capture the statistical relationships nor the

AC

temporal correlations between samples inside the recursive window. An alternative approach considers each noise component as an independent sample, but not identically distributed, that follows a particular statistical model, with a common location parameter but a different scale factor. This leads to define a set of non-negative weights in the cost function of the M-estimator of location, where each weight is inversely related to the scale factor of the statistical model that characterizes each noise component [14, 15], leading, in turn, to a more general class of weighted M-estimators of location, also referred to as weighted M-smoothers, given by 4

ACCEPTED MANUSCRIPT

Table 1: Cost functions and the corresponding outputs for various recursive filtering structures.

PN1

i=−N2

PM

i=−N2

··· + P N1

i=−N2

··· + RHMy

PM

i=−N2

··· +

|gi ||sgn(gi )xi − θ|

PM

j=1

|hj ||sgn(hj )yj − θ|

log[K22 + |hj |(sgn(hj )yj − θ)2 ]

log[K 2 + |gi |(sgn(gi )xi − θ)2 ] PM

j=1

|hj |(sgn(hj )yj − θ)2

θˆ = arg min θ

= arg min

j=1

hj yj

M 1 myriad(|gi | ◦ sgn(gi )xi |N i=−N2 , K1 ; |hj | ◦ sgn(hj )yj |j=1 , K2 )

M 1 RHMy(|gi | • sgn(gi )xi |N i=−N2 , K; |hj | • sgn(hj )yj |j=1 )

√ ρ ( w` (z` − θ))

`=1  N1  X



PM

 M p  X ρ ( gi (xi − θ)) + ρ , hj (yj − θ) 

i=−N2

√

(2)

j=1

ED

θ

L X

g i xi +

M 1 median(|gi |  sgn(gi )xi |N i=−N2 ; |hj |  sgn(hj )yj |j=1 )

log[K12 + |gi |(sgn(gi )xi − θ)2 ]

j=1

P N1

i=−N2

2 j=1 |hj |(sgn(hj )yj − θ)

P N1

RWM

PN1

AN US

··· +

|gi |(sgn(gi )xi − θ)2

M

IIR

RWMy

ˆ y, g, h) Output θ(x,

Cost Function

CR IP T

Filter

1 where the weights {gi |N i=−N2 } define the different levels of reliability of the input samples and the

PT

coefficients {hj |M j=1 } weigh the influence of the previous outputs in the estimation of filter’s current N1 M output; with {w` |L `=1 } = {gi |i=−N2 ; hj |j=1 }.

CE

Since the weighted M-smoothers only admit non-negative weights, this class of operators is constrained to low-pass filtering operations, limiting its potential use for signal processing applications, where highpass and bandpass filters are frequently required [7]. A more general approach has

AC

been introduced such that real-valued weights can be used, and thus, frequency selective filtering operations can be effectively implemented [7, 15, 16, 17]. More precisely, this approach consists in uncoupling the sign of each weight from its magnitude, and passing it to the corresponding sample [16]. These estimators are known as weighted M-filters, and, in the context of recursive filters, the linear IIR filter, the recursive weighted median (RWM) filter [5] and the recently proposed recursive weighted myriad (RWyM) filter [6] are the corresponding filtering structures that arise assuming each noise component, that contaminates each sample inside the recursive window, 5

ACCEPTED MANUSCRIPT

follows a Gaussian, Laplacian and Cachy distribution, respectively. The cost functions and the corresponding outputs of these recursive filters are shown in the first three rows of Table 1, where sgn(u) is the sign function defined as +1 if u ≥ 0, −1 otherwise; and  and ◦ denote the weighting operators for the RWM filter and the RWMy, respectively. The recursive hybrid myriad (RHMy) filter, shown in the last row of Table 1, has been also

CR IP T

introduced in [6], whose structure is based on the fact that the myriad output asymptotically follows a normal distribution [8, 18, 19, 20], therefore, it can be considered that the impulsive noise that contaminated the previous outputs has been sufficiently mitigated. This leads to the characterization of two different loss functions inside the entire objective cost function, where one loss function is optimal under the assumption that the additive noise that contaminates the input

AN US

samples follows a Cauchy distribution, and the other loss function characterizes the perturbations on the previous outputs using a Gaussian statistical model. The output of this recursive hybrid structure is obtained by minimizing the entire objective cost function. 3. Recursive Myriad-Mean Filters

M

In this section, the general class of the recursive myriad-mean filters is introduced and some

ED

properties are derived. 3.1. Recursive Myriad-Mean Filters

PT

N1 1 Given a set of input samples {x[n − i]|N i=−N2 } = {xi |i=−N2 } taken from a discrete-time signal

M {x[n]} and a set of previously obtained outputs {y[n − j]|M j=1 } = {yj |j=1 }; we can define a recursive

CE

window that gathers together the components of these two sample sets. Furthermore, let’s consider 1 a set of filter weights {gi |N i=−N2 } that defines the different levels of reliability of the input samples,

and a set of coefficients {hj |M j=1 } that controls the influence of the previous outputs on the current

AC

filter response. The output of the recursive myriad-mean (RMyM) filter, at time n, is given by

y[n] = P N1

"

N1 X

!

(

N1 X



2

 |gi | arg min log K + |gi |(sgn(gi )xi − θn ) θn |hj | i=−N2 i=−N2 (M )# M X X 2 + |hj | arg min |hj |(sgn(hj )yj − θr ) i=−N2

...

1

j=1

= P N1

i=−N2

|gi | + ! 1 |gi | +

PM

j=1

θr

PM

j=1

j=1

 |hj |

"

N1 X

i=−N2

!



|gi | myriad |gi | ◦

6

1 sgn(gi )xi |N i=−N2 ; K



+

M X j=1

hj y j

)

2

#

(3)

ACCEPTED MANUSCRIPT

where (N1 + N2 + 1) and M are, respectively, the number of input samples and the number of previous outputs inside the recursive window; and K > 0 is a tunable parameter that represents the impulsive noise rejection capability of the weighted myriad operator [15]. Notice in (3) that the output of the RMyM filter is obtained as the sum of two independent estimation processes, namely, the scaled weighted myriad filter [15] acting on the subset of input samples and the

CR IP T

linear weighted mean operator acting over the filter’s most recent outputs. Thus, the proposed recursive filter has two desirable characteristics. On one hand, the robustness against impulsive noise exhibited by the weighted myriad operator and, on the other, the desirable sharp-form of the spectral characteristics introduced by the recursive linear structure. Furthermore, as observed in

AN US

(3), the sum of the two independent estimation processes is scaled by a normalization factor 1/β,  P PM N1 |h | |g | + with β = j=1 j , and, as will be shown later, this scaling operation restricts i=−N2 i the output of the proposed filter to a limited range whose bounds are directly related to the

magnitudes of the samples inside the recursive observation window, leading to a stable structure under the bounded-input bounded-output (BIBO) criterion. A block diagram of the RMyM filter is schematically depicted in Fig. 1, which illustrates the architecture of the proposed recursive

M

filtering structure. Notice in Fig. 1, that the scaled weighted myriad filter is applied to a non-

ED

causal subset of input samples, where the scaling factor of this non-recursive structure is given by P 1 λg = N i=−N2 |gi |. Upon a closer examination of (3), each independent estimation process can be thought of as a

PT

ML estimate of a particular location parameter. More specifically, the weighted myriad output can 1 be considered as the ML estimation of location θn of the modified input samples {sgn(gi )xi |N i=−N2 }

CE

that are corrupted by additive noise, where each noise entry is assumed as a realization of a Cauchy distribution with a particular scale factor. On the other hand, the output of the weighted mean can be thought of as the ML estimate of location θr of the filter’s previous outputs {yj |M j=1 } whose

AC

samples are contaminated with additive noise that follows a zero mean Gaussian distribution, where the j-th noise component has a particular variance σj2 . This assumption is based on the fact that the sample myriad asymptotically tends to follow a normal distribution [8, 18, 19], therefore, the Gaussian distribution becomes a suitable model for describing the additive contamination that affects the previous filter outputs. Furthermore, unlike the recursive hybrid myriad filter where both cost functions are jointly minimized [6]; the proposed recursive filter performs the minimization 7

ACCEPTED MANUSCRIPT

gN 2 x[n + N2 ]

z +1

g1

x[n + 1] g0

{x[n]}

1 β

λg Myriad

z −1

+

g−1

h1

x[n − 1]

+

z −1

CR IP T

z +1

g−N1

AN US

+

z −1

y[n − 1]

h2

x[n − N1 ]

y[n]

hM

z −1

y[n − 2]

z −1 y[n − M ]

M

Figure 1: A non-causal structure of the proposed recursive myriad-mean filter.

process separately on input samples and previous outputs, respectively, leading to a more flexible

ED

filtering structure that can be suitably implemented in a wide variety of applications. From the properties of the scaled weighted myriad filters, which are derived in [15], some underlying features of the proposed recursive filter can be easily obtained. First, it is straightforward

PT

to see that the RMyM filter depends on (N1 + N2 + M + 2) filter parameters, that correspond to the (N1 + N2 + M + 1) filter weights in the recursive window {g−N2 , . . . , g0 , . . . , gN1 , h1 , . . . , hM }

CE

and the linearity parameter K of the weighted myriad operator. Therefore, the linearity parameter K should be optimally determined in order to properly design the RMyM filter.

AC

Furthermore, since the response of the proposed filter can be seen as the sum of two independent estimation processes, and taking into account the linear property of the scaled weighted myriad filter [15], it is easy to show that the output of the RMyM filter, as K → ∞, is given by y[n]K→∞ =

PN1

i=−N2 gi xi PN1 i=−N2 |gi |

+ +

PM

j=1

hj yj

j=1

|hj |

PM

,

(4)

that can be considered as the response of a normalized version of a linear IIR filter. Furthermore, since the normalization factor is the sum of magnitudes of the filter weights, it is straightforward to 8

ACCEPTED MANUSCRIPT

see that the right-hand side of (4) is restricted to the interval of magnitudes of the samples inside the recursive window, where this constrain can be easily extended for finite values of the linearity parameter. Hence, the RMyM filter is a stable system under BIBO criterion. Interestingly, as K decreases, the non-recursive part of the proposed filter becomes more robust against impulsive noise, mitigating thus, the effects of the impulsive noise over the subset of input samples. Indeed,

CR IP T

when K → 0, the output of the RMyM filter reduces to the sum of the weighted mean of the previous outputs and the outlier-resistant sample mode of the subset of input samples [21, 22]. 3.2. Scaled Recursive Myriad-Mean Filters

Since the proposed RMyM filter is unable to amplify the dynamic range of the samples inside

AN US

the recursive observation window, a broader recursive filtering structure is proposed, where the magnitude of the filter output can be adjusted by applying a suitable scaling over the response of the RMyM filter. To be more precise, the output of the proposed filter is obtained by removing the normalization factor that is included in the definition of the RMyM filter, where this proposed structure can be considered as the sum of two scaled operators: the scaled weighted myriad filter

M

[15] acting on a subset of samples of the input signal and the unnormalized weighted mean filter that acts over the filter’s most recent outputs. Namely, the output of the scaled recursive myriad-mean (SRMyM) filter is defined as

!

M   X 1 hj yj . |gi | myriad |gi | ◦ sgn(gi )xi |N ; K + i=−N2

ED

y[n] =

N1 X

i=−N2

(5)

j=1

PT

Some properties of this scaled recursive filter can be directly derived by applying a similar analysis to that developed for the respective normalized version. On the other hand, since the

CE

recursive part of the proposed filter is a linear structure, the z-transform can be determined as PM −j H(z) = j=1 hj z , and this feedback structure can be considered as an all-pole filter whose

transfer function reduces to 1/(1 − H(z)). Therefore, in order to determine the stability of the

AC

SRMyM filter, under BIBO criterion, it is necessary to evaluate if the zeros of 1 − H(z) (the poles of its inverse) fall inside the unit circle [2]. This condition should be carefully considered in the development of the adaptive algorithm. 4. Adaptive Algorithms In this section, the adaptive algorithms for designing the proposed recursive filtering structures are developed. To this end, let’s consider a general recursive filter with feedforward and feedback 9

ACCEPTED MANUSCRIPT

M 1 weights given by {gi |N i=−N2 } and {hj |j=1 }, respectively; and let the filter output, at time n, be

denoted by y[n]. The adaptive algorithms are developed considering the mean absolute error (MAE) between the filter output y[n] and a desired response d[n] as the performance criterion to be minimized. This minimization criterion has been widely used in the design of nonlinear filtering

the best filter parameters that minimize the cost function J(g, h) = E{|y[n] − d[n]|},

CR IP T

structures [5, 6, 14, 16, 23]. To be more precise, the proposed adaptive algorithms aim at finding

(6)

where E{·} denotes the statistical expectation, g = [g−N2 , . . . , g0 , . . . , gN1 ]> and h = [h1 , . . . , hM ]> . Since direct minimization of J(g, h) does not lead to a closed-form solution, we resort to the steepest descent approach in an attempt to iteratively converge to the global minimum of the cost

according to =

hj [n + 1]

=



 ∂ J(g, h) , i = −N2 , . . . , N1 , ∂gi   ∂ hj [n] + µ J(g, h) , j = 1, . . . , M, ∂hj

gi [n] + µ

(7) (8)

M

gi [n + 1]

AN US

function given by (6). In essence, the weights of the proposed recursive filter are iteratively updated

where gi [n] and hj [n] are, respectively, the i-th and j-th filter weights at time n; µ is the step-size of the update; and the gradient of the cost function is given by 

 ∂y E sgn(y[n] − d[n]) , i = −N2 , . . . , N1 , ∂gi   ∂y E sgn(y[n] − d[n]) , j = 1, . . . , M . ∂hj

ED = =

PT

∂ J(g, h) ∂gi ∂ J(g, h) ∂hj

(9) (10)

In practical applications, however, the statistics of the error sequence are generally unknown

CE

or the input signals are non-stationary, thus, the statistical expectation in (9) and (10) can not be evaluated. Like in [2, 24], to overcome this apparent limitation, the proposed adaptive algorithms

AC

are designed such that they minimize the instantaneous estimate of the cost function J(g, h). This is, the proposed adaptive algorithms attempt to minimize L(g, h) = |y[n] − d[n]| [2, 24]. Furthermore, the derivative of y[n] with respect to each filter weight becomes an intractable

problem as a direct consequence of the implicit feedback in the recursive filters. To overcome this drawback, we resort to the equation error formulation1 approach as an uncoupling strategy for 1

Equation error formulation was initially proposed for designing IIR filters [2]. However, this approach has been

successfully applied in the design of nonlinear recursive filters [5, 6].

10

ACCEPTED MANUSCRIPT

designing the proposed recursive filtering structures [2]. More precisely, under the equation error formulation approach, the adaptive algorithms replace the previous filter outputs by the previous desired responses, i.e. y[n − j] = d[n − j] for j = 1, . . . , M , leading to a two-input-single-output structure that does not depend on the filter’s previous outputs, and therefore, the filter does not introduce feedback. Furthermore, since each input sample x[n] and the desired response d[n] are

CR IP T

not functions of the filter coefficients, the derivative of y[n] with respect to each filter weight is no longer a computation issue. Learning the best filter coefficients —under the equation error formulation framework— is particularly suitable in the context of developing channel equalization techniques, whose learning phase is executed by using pilot signals that are previously known by the receiver. The expressions of the derivatives and, therefore, the corresponding adaptive algorithms

AN US

for designing the proposed recursive filtering structures, are developed next. 4.1. Adaptive RMyM Filter

As shown above, the adaptive algorithm first requires the evaluation of the derivatives of y[n] with respect to each filter coefficient. To this end, the output of the proposed recursive filter

M

is redefined and denoted as yˆ[n], where the previous filter outputs {yj |M j=1 } are replaced by the

corresponding previous desired responses {dj |M j=1 }. In other words, the output of the RMyM filter,

yˆ[n] = P N1

1

 |h | j j=1

PM

"

CE

N1 X

i=−N2

PT

i=−N2 |gi | +

ED

under the equation error formulation approach, at time n, is given by !

|gi | arg min 

+

θn

M X j=1

(



N1 X

i=−N2

)  2  2 log K + |gi |(sgn(gi )xi − θn )

|hj | arg min θr

 M X 

j=1

  |hj |(sgn(hj )dj − θr )2  . (11) 

In order to obtain the derivatives with respect to the filter parameters, the expression of the RMyM filter output is rewritten as follows h

AC

i M 1 Q(gi |N ; K) + T (h | ) j j=1 i=−N 2  , yˆ[n] = P PM N1 i=−N2 |gi | + j=1 |hj |

where

1 Q(gi |N i=−N 2 ; K)

T (hj |M j=1 )

=

N1 X

i=−N2

!

|gi | arg min θn

(

N1 X

i=−N2

)  2  log K + |gi |(sgn(gi )xi − θn )2

    M M X  X =  |hj | arg min |hj |(sgn(hj )dj − θr )2 .  θr  j=1

j=1

11

(12)

(13)

(14)

ACCEPTED MANUSCRIPT

Notice that (13) and (14) are two independent estimation processes. Furthermore, the nota1 tion Q(gi |N i=−N 2 ; K) is included in order to underline the dependency of the function on the non-

M 1 recursive filter weights {gi |N i=−N2 } and the linearity parameter K. Similarly, T (hj |j=1 ) indicates

the dependency of T (·) on the recursive filter weights {hj |M j=1 }. Thus, the derivative of the modified RMyM filter output with respect to each non-recursive filter weight gi , for i = −N2 , . . . , N1 , can N1 i=−N2

|gi | +

h i M 1 − Q(gi |N ; K) + T (h | ) sgn(gi ) j j=1 i=−N 2 , i = −N2 , . . . , N1 , (15)  2 PM N1 i=−N2 |gi | + j=1 |hj |

PM

with

j=1 |hj | P

∂Q = ∂gi



N1 X

i=−N2

∂Q ∂gi

!

|gi |

where the expressions of the derivatives −

∂ ˆ θ n = hP N1 ∂gi

h

∂ ˆ θn + sgn(gi )θˆn , i = −N2 , . . . , N1 , ∂gi

AN US

∂ yˆ [n] = ∂gi

P

CR IP T

be obtained by evaluating the derivative of the quotient with respect to gi , i.e.,

∂ ˆ ∂gi θn

for i = −N2 , . . . , N1 are given by

ˆn −sgn(gi )xi ) K sgn(gi )(θ ˆn )2 ]2 [K 2 +|gi |(sgn(gi )xi −θ 2

2

i

ˆ

2

K −|gi |(sgn(gi )xi −θn ) ˆ )2 ]2 i=−N2 |gi | [K 2 +|g |(sgn(g )x −θ

The evaluation of the derivatives

i

∂ ˆ ∂gi θn ,

(16)

i

i

n

i , i = −N2 , . . . , N1 .

(17)

for i = −N2 , . . . , N1 , follows a similar approach to

M

that developed for obtaining the derivatives of the non-recursive weighted myriad filter output with respect to each filter weight [15]. Nevertheless, for the sake of completeness, the steps for

ED

determining these derivatives are outlined in Appendix A. On the other hand, the derivatives of yˆ[n] with respect to each feedback coefficient hj , for j = 1, . . . , M , can be obtained as |gi | +

i h  N1 ∂T M ; K) + T (h | ) sgn(hj ) − Q(g | |h | j i j j=1 i=−N 2 j=1 ∂hj , j = 1, . . . , M, P  2 PM N1 i=−N2 |gi | + j=1 |hj |

PM

PT

where

N1 i=−N2

CE

∂ yˆ [n] = ∂hj

P

∂T = dj , j = 1, . . . , M. ∂hj

(18)

(19)

AC

As described in Section 3.1, the RMyM filter depends on both the (N1 + N2 + M + 1) filter

weights and the linearity parameter K of the weighted myriad operator. Therefore, the adaptive algorithm of the RMyM filter requires —in addition to update the filter weights— the computation of the optimal value of K. To do so, however, some considerations must be set. First, since the

linearity parameter is, in general, given by its squared value, the adaptive algorithm continually updates this squared value, in other words, the algorithm optimizes K = K 2 . Secondly, the linearity parameter, in the definition of the weighted myriad operator, is strictly a positive parameter, i.e. 12

ACCEPTED MANUSCRIPT

K > 0, therefore, at each iteration, the proposed adaptive algorithm is designed to return the magnitude of K. In summary, K is iteratively updated according to

  ∂ yˆ K[n + 1] = K[n] − µ sgn(e[n]) [n] . ∂K ∂ yˆ ∂K [n]

can be obtained from (12) as follows ∂Q ∂K

∂ yˆ [n] = P N1 ∂K

i=−N2

with

N1 X

∂Q = ∂K

where the expression for computing

∂ ˆ ∂K θn

|gi | +

i=−N2

!

|gi |

PM

j=1

, |hj |

CR IP T

where the expression for the derivative

(20)

∂ ˆ θn , ∂K

is given by

AN US

PN1 |gi |(θˆn −sgn(gi )xi ) i=−N2 [K+|gi |(sgn(gi )xi −θˆn )2 ]2 ∂ ˆ θn = P . ˆ 2 N1 ∂K |g | K−|gi |(sgn(gi )xi −θn ) i=−N2

(21)

(22)

(23)

i [K+|g |(sgn(g )x −θˆ )2 ]2 n i i i

Appendix B summarizes the steps for obtaining the derivative

∂ ˆ ∂K θn .

Finally, since the RMyM

filter is a stable system under BIBO criterion, the adaptive algorithm does not need to satisfy

M

additional design constrains, unlike the SRMyM filter, where, as will be shown next, the poles of

ED

the recursive part of this scaled version must lie inside the unit circle. 4.2. Adaptive SRMyM Filter

given by

!

|gi | arg min

CE

y¯[n] =

N1 X

PT

The output of the SRMyM filter under the equation error formulation approach, at time n, is

AC

i=−N2

θn

(

N1 X

i=−N2



2

2

log K + |gi |(sgn(gi )xi − θn ) 

+

M X j=1





)

|hj | arg min θr

 M X 

j=1

|hj |(sgn(hj )dj − θr )2

By differentiating (24) with respect to the filter coefficients, we obtain ∂ y¯ [n] ∂gi

=

∂ y¯ [n] ∂hj

=

∂ y¯ [n] ∂K

N1 X

i=−N2

=

|gi |

!

∂ ˆ θn + sgn(gi )θˆn , i = −N2 , . . . , N1 , ∂gi

dj , j = 1, . . . , M, N1 X

i=−N2

|gi |

!

  

, (24)

(25) (26)

∂ ˆ θn , ∂K

13

(27)

ACCEPTED MANUSCRIPT

where the expressions for obtaining

∂ ˆ ∂gi θn

and

∂ ˆ ∂K θn

have been already derived in (17) and (23),

respectively. Furthermore, in order to ensure the stability of the SRMyM filter, the adaptive algorithm should evaluate the zeros of 1 − H(z) at each iteration. If the system is not stable, i.e. one or more zeros of 1 − H(z) (poles of its inverse) are updated outside the unit circle, the adaptive algorithm projects the corresponding unstable roots inside the stability region [2], keeping their

CR IP T

former phases, but with magnitudes fixed in 0.98, following the constrains described in [25]. The feedback filter weights are obtained then from the inverse of the stable system.

Due the nonlinear nature of the weighted myriad operator, which acts on the subset of input samples, the convergence analysis of the proposed adaptive algorithms becomes intractable, therefore, the exact bounds of the step size parameter µ are not available. Thus, we include an

AN US

adjustable step size parameter in the iterative update of the filter’s parameters, whose value decreases as the steepest descent algorithm progresses. This approach has been widely used in least mean absolute (LMA) based adaptive algorithms for designing nonlinear filters [5, 6, 15, 16, 23].

5. Results and Discussion

M

5.1. Behavior of the adaptive algorithms

ED

To evaluate the behavior of the proposed adaptive algorithms, the RMyM filter and the SRMyM filter are designed for a bandpass-type application. Each filter design consist of 120-tap structure, where 80 coefficients weigh the subset of input samples and the 40 remaining ones weigh the

PT

subset of previous outputs. The cut-off frequencies of the desired filter are given by (ω1 , ω2 ) = (0.075, 0.125), with the sampling frequency normalized to one. At the training stage, a Bernoulli

CE

random sequence, i.e. x[n] = ±1 with probability p = 0.5, is used as input signal [24], and the desired signal is obtained by passing the binary sequence through a linear FIR filter with

AC

120 taps, whose coefficients are computed using the MATLAB’s fir1 function for the desired cut-off frequencies. The initial filter weights are set to identical and normalized values, gi [0] =

hj [0] = 1/120 for i = −39, . . . , 40 and j = 1, . . . , 40. Furthermore, the initial value of the linearity parameter is set to one, i.e. K[0] = 1.00. Finally, the adjustable step size parameter decreases its value according to µ[n] = µ0 exp(−n/1000), with µ0 = 0.0025. To observe the behavior of the adaptive RMyM filter, the iterative updates of two randomly

selected filter coefficients are depicted in Fig. 2(a). Note that for a single realization of the adaptive 14

ACCEPTED MANUSCRIPT

·10−2

0.4 Single Realization Ensemble Average 0.3

2 gi (Single Trial) gi (Ensemble Average) hj (Single Trial) hj (Ensemble Average)

0

MAE

0.1

−2 −4

0.2

0

1,000

2,000

3,000

0

4,000

0

(a)

0.2

MAE

AN US

0.3

MAE

3,000

4,000

RMyM SRMyM

Single Realization Ensemble Average

0.2 0.1

2,000

3,000

Iteration [n]

0

1,000

2,000

3,000

4,000

Iteration [n] (d)

ED

(c)

0.1

0

4,000

M

1,000

2,000

(b)

0.4

0

1,000

Iteration [n]

Iteration [n]

0

CR IP T

Weight value

4

Figure 2: (a) Tap update for two randomly selected filter coefficients of the RMyM filter. MAE learning curves: (b)

PT

RMyM filter, (c) SRMyM filter, (d) Ensemble average for the RMyM filter and the SRMyM filter.

algorithm, the weight update follows a noisy exponential behavior. However, the ensemble average

CE

curves —by averaging 1000 independent trials— have a smoother form, where the noisy behavior of a single trial has been significantly reduced. Since the adaptive algorithms are developed under MAE performance criterion, Fig. 2(b) shows

AC

the MAE learning curves yielded by the adaptive RMyM filter as the training algorithm progresses. As can be seen in this figure, the learning curve of a single realization of the adaptive algorithm exhibits a very noisy pattern. This noisy pattern is attributable to the approximation of the cost function using the corresponding instantaneous estimate [2]. Nevertheless, as can be seen in this figure, the MAE decreases its value as the number of iterations of the adaptive algorithm increases. Furthermore, the learning curve of the ensemble average has a smoother behavior, where the MAE 15

ACCEPTED MANUSCRIPT

decreases as the adaptive algorithm progresses. Further, the MAE learning curves for the scaled RMyM filter is shown in Fig. 2(c), where these curves follow a similar behavior than those shown in Fig. 2(b). For comparison purposes, the ensemble average of the MAE learning curves of the RMyM filter as well as the SRMyM filter are shown in Fig. 2(d). Since the scaled version suitably amplifies its output to minimize the instantaneous error between the filter output and the desired

CR IP T

signal, the resulting SRMyM filter exhibits lower error as the adaptive algorithm progresses. 5.2. Testing the designed bandpass filter

For testing the designed bandpass filter, a chirp signal is used as the filter input signal. Figure 3(a) depicts the input test signal, where its instantaneous frequency varies from 0 Hz to 400 Hz

AN US

in one second, with a sampling rate of 2 kHz. For comparison purposes, Fig. 3(b) shows the output of the linear FIR filter, where the filter parameters of this linear structure are the same to those used to yield the desired signal at the training stage. Furthermore, the output of a 120-tap linear IIR filter is depicted in Fig. 2(c), where the coefficients of this recursive linear filter are obtained using the MATLAB’s yulewalk2 function for the same bandpass of interest. The output

M

of the non-recursive weighted myriad (WMy) filter is shown in Fig. 3(d), where the optimization algorithm to design this filter has been implemented following the guidelines given in [15].

ED

For comparative purposes, the output of the dual weighted iterative truncated mean (DWITM) filter [27] and the output of the recursive weighted median (RWM) filter [5] are shown in Fig. 3(e) and 3(f), respectively, where the coefficients of the DWITM filter are the same to those obtained

PT

for the FIR filter, and the weights of the RWM filter are determined using the fast LMA algorithm proposed in [5]. Furthermore, the outputs of the recursive weighted myriad (RWMy) filter and

CE

the recursive hybrid myriad (RHMy) filter, recently proposed by the authors in [6], are shown in Fig. 3(g) and 3(h), respectively, where the correponding design algorithms follow the procedures

AC

reported in [6]. Finally, the responses of the proposed filters, the RMyM filter and the SRMyM filter, are depicted in figures 3(i) and 3(j), respectively. Notice that the outputs of the proposed filters outperforms to those yielded by others filtering

structures, following the shape of the desired signal while the rejecting bands are properly attenuated. As mentioned in the caption of Fig. 3, the MAE values yielded by the proposed filtering 2

The yulewalk function designs a linear IIR filter for a defined frequency response using the modified Yule-Walker

method developed in [26].

16

CR IP T

ACCEPTED MANUSCRIPT

(b)

AN US

(a)

M

(c)

(f)

PT

ED

(e)

(d)

(h)

AC

CE

(g)

(i)

(j)

Figure 3: Bandpass type filtering outputs: (a) test input signal, (b) desired signal, (c) linear IIR filter with MAE = 0.0746, (d) non-recursive WMy filter with MAE = 0.1385, (e) DWITM filter with MAE = 0.0443, (f) RWM filter with MAE = 0.0762, (g) RWMy filter with MAE = 0.0429, (h) RHMy filter with MAE = 0.0429, (i) RMyM filter with MAE = 0.0425, and (h) SRMyM filter with MAE = 0.0102.

17

CR IP T

ACCEPTED MANUSCRIPT

(b)

AN US

(a)

M

(c)

(f)

PT

ED

(e)

(d)

(h)

AC

CE

(g)

(i)

(j)

Figure 4: Response against impulsive noise: (a) test input signal, (b) FIR filter with MAE = 6.4556, (c) IIR filter with MAE = 7.2222, (d) non-recursive WMy filter with MAE = 0.1381, (e) DWITM filter with MAE = 0.0573, (f) RWM filter with MAE = 0.0728, (g) RWMy filter with MAE = 0.0542, (h) RHMy filter with MAE = 0.0541, (i) RMyM filter with MAE = 0.0538, and (j) SRMyM filter with MAE = 0.0361.

18

ACCEPTED MANUSCRIPT

structures are the lowest compared to those yielded by the others filters. Furthermore, since the SRMyM filter applies a suitable scaling to the response generated by the normalized version, the output of the SRMyM filter exhibits a closer fit to the desired signal compared to that produced by the RMyM filter, yielding lowest value of the MAE. The performance of the proposed recursive filters against impulsive noise is also evaluated. To

CR IP T

this end, the chirp signal is contaminated with additive symmetric α-stable (SαS) noise (α = 0.75, γ = 0.01) [28, 29], as shown in Fig. 4(a). Furthermore, for comparison purposes, figures 4(b)-(h) depict the outputs of the linear FIR filter, the linear IIR filter, the non-recursive WMy filter, the DWITM filter, the RWM filter, the RWMy filter and the RHMy filter, respectively. Notice that each filter response is truncated such that all plots have the same scale in the vertical axis. The

AN US

outputs of the RMyM and SRMyM filters are shown in figures 4(g) and 4(h), respectively. As can be observed in Fig. 4, the outputs of the proposed recursive filters outperform those yieded by the others filtering structures in the presence of impulsive noise, where the output of each proposed filtering structure keeps closer the shape with respect to the desired signal, attenuating properly the rejecting bands.

M

The computation of the filter’s outputs based on the sample myriad operator, by direct minimization of its objective cost function, is a highly expensive computational task. Furthermore,

ED

the number of local minima in the objective cost functions increase as the linearity parameter decreases, increasing significantly the computational cost of the estimation problem. Therefore,

PT

several fast computation algorithms have been developed for obtaining the outputs of different filtering structures based on the sample myriad with a high degree of accuracy and low computa-

CE

tional costs [6, 14, 30, 31]. More precisely, the outputs of the robust filters based on the sample myriad (WMy filter, RWMy filter, RHMy filter and the non-recursive part of the RMyM filter) are obtained using fixed point search methods [6, 14] that start with the best sample as a initial

AC

point and, subsequently, the filter output is refined by iteratively updating a fixed-point iteration algorithm.

We compare the computation speed by measuring the filtering time of the RMyM filter and

compare it with respect to other nonlinear structures using an Intel Core i3 CPU 3.1 GHz, RAM 4 GB, and Scientific Fedora 21 operating system. The normalized filtering times —with respect to the filtering time taken by the RMyM filter— versus the number of filter coefficients are shown in 19

ACCEPTED MANUSCRIPT

−10

RMyM WMy RWMy RHMy DWITM

6

−20 −30 −40

4

−50

2 0

20

40

60

80

100

−60

120

Window size

RMyM SRMyM WMy

CR IP T

8

10 log10 |y[n]2 |

Normalized computing time

10

0.075

0.125

Normalized frequency

AN US

Figure 5: (Left) Normalized filter time against the number of filter weights. The filtering times are normalized with respect to the values yielded by the proposed RMyM filter. (Right) Estimated frequency responses of various nonlinear bandpass filters.

Fig. 5(left). Each computation time is obtained by ensemble averaging over the time results of 500 realizations, where, for each trial, all filters act on the same clean chirp signal with 2000 samples.

M

As can be seen in Fig. 5(Left), the RMyM filter is much faster than the other nonlinear filtering structures.

ED

Finally, the frequency response of the designed passband filters are estimated. To this end, a zero-mean SαS noise (α = 1.00, γ = 0.025) is used as the input signal to the various nonlinear

PT

filtering structures, where the frequency response of each filter is estimated using the Welch method [32]. Figure 5(Right) shows the frequency characteristics of the various nonlinear filters, which are

CE

obtained by averaging the results of 500 realizations. As can be observed in Fig. 5(Right), the proposed recursive filtering structures exhibit more accurate frequency responses compared to that outputted by the non-recursive weighted myriad filter. Furthermore, the frequency response of the

AC

two proposed filtering structures are similar in the passband, observing the amplification effect provided by the SRMyM filter. 5.3. Image denoising The performances of the proposed recursive filters are evaluated under an image denoising context. More precisely, the responses of the proposed recursive structures are compared to those yielded by the iterative truncate arithmetic mean (ITM) filter [33], the iterative trimmed and 20

ACCEPTED MANUSCRIPT

(c)

(e)

(f)

(g)

(d)

CR IP T

(b)

(h)

AN US

(a)

Figure 6: (a) Original image and (b) image with salt and pepper noise. Image denoising yielded by (c) ITM filter, (d) ITTM filter, (e) RWMy filter, (f) RHMy filter, (g) RMyM filter, (h) SRMyM filter.

truncated mean (ITTM) filter [34], and the recently proposed RWMy and RHMy filters. To this

M

end, the “House” image of size 256 × 256 (see Fig. 6(a)) is corrupted with salt and pepper noise, with a contamination level  = 10% (Fig. 6(b)). The window size for each filtering operation is

ED

5 × 5. Furthermore, the 64 × 64 bottom right parts of the original image and the noisy image are used as the desired image and the input image, respectively, for the various optimization adaptive

figures 6(c)-(h).

PT

algorithms [5, 27]. The filtered images outputted by the various filtering structures are shown in

CE

Table 2 shows the values of MAE, mean square error (MSE), peak signal-to-noise ratio (PSNR) and filtering time yielded by the various filtering approaches, where each value is obtained as the

AC

ensemble average of 100 experiments, and for each trial, a different realization of the salt and pepper noise is generated. For each performance metric, underlined and bold font value highlights the best value, while the second best value is in font bold. The SRMyM filter exhibits a superior performance yielding the best values for the MSE and PSNR performance criteria. Furthermore, the proposed recursive structures (RMyM and SRMyM filters) are the fastest filters, with the lowest values of filtering time, compared to the ones yielded by the other nonlinear filtering structures.

21

ACCEPTED MANUSCRIPT

Table 2: MAE, MSE, PSNR and filtering time for various nonlinear filtering structures applied on the noisy “House” image of size 256 × 256 pixels. Noise level  = 10%. Window size: ws = 5 × 5.

MAE

MSE

PSNR[dB]

Filtering time (s)

ITM

4.4102

64.560

30.032

25.119

ITTM

4.5755

70.024

29.679

15.650

RWMy

2.5104

38.621

32.272

RHMy

4.5738

62.068

30.203

RMyM

4.3996

43.741

31.724

SRMyM

3.3077

37.084

32.441

CR IP T

Filter

21.165

20.089

9.8152

AN US

9.5538

5.4. Decision feedback equalizer based on SRMyM filter

Data transmission in wireless communication channels are subject to intersymbol interference

M

(ISI), which can be modelled using a linear FIR filter and an additive noise source that contaminates the transmitted signal. In general, ISI is attributed to both the presence of multipath distortion

ED

and the limited bandwidth of the communication channel, inducing bit errors at the receiving end [35]. In this context, adaptive equalizers attempt to recover a reliable version of the transmitted signal at the receiver end. To be more precise, an adaptive equalizer can be thought of as an

PT

adaptive filtering structure that estimate the optimal filter coefficients in order to compensate the distortion introduced by the communications channel.

CE

On the other hand, the designs of most adaptive equalizers are developed under the assumption that the underlying noise follows a Gaussian distribution. However, in real communications chan-

AC

nels, the environmental noise, obtained from measurements, is known to be non-Gaussian with the presence of outliers or gross errors, and, therefore, it can be better described by using distributions with heavier-than-Gaussian tails [29, 36]. Furthermore, it is well known that traditional equalizers degrade severely their performance in presence of impulsive noise. Therefore, it is necessary the development of equalizers that minimize the effects of impulsive noise in the characterization of the communications channels. Taking into account the uncoupled structure of the proposed recursive filters, a decision feedback 22

ACCEPTED MANUSCRIPT

gN 2 x[n + N2 ]

z +1

g1

z +1

g0

{x[n]}

λg

Myriad z −1

+

g−1

detector h1

x[n − 1]

+

z −1

CR IP T

x[n + 1]

g−N1

z −1

s[n − 2]

AN US

+

z −1

s[n − 1]

h2

x[n − N1 ]

s[n]

hM

z −1 s[n − M ]

M

Figure 7: Structure of the decision feedback equalizer based on the SRMyM filter.

equalizer based on the SRMyM filter (SRMyM-DFE) is developed next, whose general structure is

ED

shown in Fig. 7. As can be observed in this figure, the proposed equalizer consists of a weighted myriad filter acting on a subset of input samples, where its filter coefficients are updated during the training stage in order to mitigate the effects of the impulsive noise on the received signal and

PT

to compensate the distortion induced by the communication channels. Furthermore, the proposed decision feedback equalizer (DFE) contains a linear feedback weighted mean filter that is driven

CE

by a subset of previously detected symbols, where its filter weights are tuned to mitigate the intersymbol interference caused by the communication channel. In other words, the output of the

AC

DFE based on the SRMyM filter is given by ! y[n] =

N1 X

i−N2

M   X 1 |gi | myriad |gi | ◦ sgn(gi )x[n − i]|N ; K + hj s[n − j] i=−N2

(28)

j=1

where s[n] denotes the detected symbol at time n. The performance of the proposed DFE is compared with respect to those yielded by the least mean square based DFE (LMS-DFE) with µ = 0.01 [35], the recursive least square based DFE (RLS-DFE) with λ = 0.99 [35], the recursive least p-norm based DFE (RLP-DFE) with λ = 0.99

and p = 1 [37] and the SRMyM filter, described in Section 3.2. This latter structure can be 23

ACCEPTED MANUSCRIPT

thought of as a nonlinear recursive equalizer, where the detector is placed after the filter output. Furthermore, it should be pointed out that the recently introduced filtering structures [6] can not be adapted to the application at hand, since these nonlinear recursive filters are coupled structures, where both the input samples and the previous outputs are included in a single cost function to be minimized. In contrast, the uncoupled structure of the proposed recursive myriad-mean filter

process independently feedforward and feedback parts.

CR IP T

makes it a suitable alternative in applications, such as decision feedback equalizers, that require to

The performances of the various DFEs are evaluated under α-stable noise for different values of the characteristic exponent α. For a specific value of α, a pilot signal, with 2000 symbols, is contaminated using SαS noise with dispersion γ = 0.025. At the training stage, the linearity

AN US

parameters of the SRMyM-DFE and the SRMyM filter are adjusted using an approximation of the α − K relationship reported in [22]. More precisely, for a specific set of parameters of the SαS noise, the value of K is computed according to r α K= γ 1/α (2 + ε) − α

(29)

2.

M

where the parameter ε = 0.0001 is included in order to avoid the division by zero when α = Figure 8 shows the bit error rate (BER) of a BPSK system versus the geometric signal

ED

to noise ratio (GSNR) [38], for different values of α, where for this experiment, the number of feedforward and feedback coefficients for all DFEs are N1 + N2 + 1 = 8 and M = 4, re-

PT

spectively; and the communication channel is characterized using a transfer function given by H(z) = 0.3482 + 0.8704z −1 + 0.3482z −2 [37].

CE

As can be observed in Fig. 8, the proposed DFE based on SRMyM structure outperforms the others DFE techniques for the various values of α. More precisely, when the underlying noise is of impulsive nature, the equalizers based on the SRMyM filters exhibit a much better performances

AC

than those yielded by the other approaches, as the GSNR increases. Indeed, at these impulsiveness levels, when the GSNR = 20 dB, the channel equalizers based on the proposed recursive filters have at least 10 times lower BER values than the other equalization techniques (LMS-DFE, RLSDFE and RLP-DFE). Furthermore, for all values of α, the SRMyM-DFE structure outperforms the SRMyM filter, where the difference in the performance is attributed to the fact that the DFE based on the SRMyM filter has the information about the previously detected symbols, and therefore, the ISI induced on the subsequent symbols can be more accurately subtracted [35]. Note that 24

ACCEPTED MANUSCRIPT

SRMyM-DFE SRMyM RLS-DFE LMS-DFE RLP-DFE

10−2 0

5

10 GSNR [dB]

15

10−3

20

SRMyM-DFE SRMyM RLS-DFE LMS-DFE RLP-DFE

10−2

0

5

10−1

10−1

BER

10−2 SRMyM-DFE

10−4

0

5

10 GSNR [dB]

10−5

15

10−6

20

15

20

SRMyM-DFE SRMyM RLS-DFE LMS-DFE RLP-DFE

10−4

SRMyM RLS-DFE LMS-DFE RLP-DFE

10−3

10−3

10 GSNR [dB]

AN US

BER

10−2

CR IP T

10−1

BER

BER

10−1

0

2

4

6 8 GSNR [dB]

10

12

Figure 8: BER of the equalizers under test versus the GSNR of the input signal, for a fixed-value of the characteristic

M

exponent (α) (top-left) α = 1.00, (top-right) α = 1.50, (bottom-left) α = 1.75, (bottom-right) α = 2.00.

ED

for α = 2, i.e. additive white Gaussian noise, the proposed DFE equalizer exhibits a competitive

6. Conclusions

PT

performance compared to those yielded by LMS-DFE, RLS-DFE and RLP-DFE.

CE

In this paper, the new class of recursive myriad-mean (RMyM) filters is proposed. More precisely, the output of the proposed recursive structure is obtained as the sum of two uncoupled and independent estimation processes: the weighted myriad acting on the subset of input samples

AC

and the weighted sum of the filter’s previous outputs. This class of filter has been motived by the ML estimation analysis under the assumption that the perturbations acting on input samples and previous outputs can be modelled using Cauchy and Gaussian distributions, respectively. By suitably scaling the RMyM filter, the class of scaled recursive myriad-mean (SRMyM) filters is also proposed. Adaptive algorithms for determining the optimal filter parameters are developed under the equation error formulation, which is an uncoupling strategy that leads to more tractable expressions for computing the instantaneous gradient of the steepest descent algorithm. 25

ACCEPTED MANUSCRIPT

Extensive simulation results provided for various signal and image processing applications, upon different impulsive noise scenarios, show the superior performances of the proposed recursive filters compared to those yielded by other nonlinear filtering structures in terms of accuracy of the filter output and computation time for obtaining the filter output. Finally, a decision feedback equalizer based on the SRMyM filter is developed, whose description exploits the uncoupled structures of

CR IP T

the proposed recursive filters. The performance of the proposed DFE is evaluated in presence of symmetric α-stable noise, where the SRMyM-DFE outperforms various conventional channel equalizers, for a wide range of impulsiveness levels of the additive noise. Appendix A. Evaluation of

∂ ˆ θ ∂gi n

AN US

From (3) in Section 3.1, it can be observed that θˆn is the output of the weighted myriad estimator acting on the subset of input samples, i.e. θˆn = arg minθn H(θn ), where H(θn ) is a non-convex loss function with multiple local minima, given by H(θn ) =

N1 X

i=−N2

∂ ˆ ∂gi θn ,

(A.1)

let θˆn be one of the local minima of H(θn ), thus,

M

In order to evaluate the derivative

  log K 2 + |gi |(sgn(gi )xi − θn )2 .

H0 (θˆn ) = 0. By differentiating the loss function with respect θn , we have N1 X

|gi |(θˆn − sgn(gi )xi ) h i = 0, 2 ˆ 2 i=−N 2 K + |gi |(sgn(gi )xi − θn )

ED

I(θˆn , gi ) = H0 (θˆn ) = 2

(A.2)

PT

where I(θˆn , gi ) is used to show the implicit dependence of θˆn with respect to the weights of the

∂I ∂ θˆn

∂ ˆ θn = − ∂gi

∂I ∂gi ∂I ∂ θˆn

(A.3)

is obtained as

AC

where

CE

weighted myriad operator. By applying the implicit differentiation on (A.2), it follows that

N1 X K 2 − |gi |(sgn(gi )xi − θˆn )2 ∂I =2 |gi | ∂ θˆn [K 2 + |gi |(sgn(gi )xi − θˆn )2 ]2 i=−N2

and taking into account that |gi |sgn(gi ) = gi , the derivative

∂I ∂gi

(A.4)

can be determined as

∂I 2K 2 sgn(gi )(θˆn − sgn(gi )xi ) = . ∂gi [K 2 + |gi |(sgn(gi )xi − θˆn )2 ]2 Finally, by replacing (A.4) and (A.5) in (A.3), we obtain the derivative 26

(A.5) ∂ ˆ ∂gi θn ,

given in (17).

ACCEPTED MANUSCRIPT

Appendix B. Evaluation of

∂ ˆ θ ∂K n

In this section, the steps to obtain the derivative of the weighted myriad output with respect to the squared linearity parameter are outlined, where K = K 2 . To this end, note that θˆn is one

of the local minima of H(θn ), therefore, H0 (θˆn ) = 0, where H(θn ) is the non-convex loss function G(θˆn , K) = H0 (θˆn ) = 2

N1 X

i=−N 2

h

CR IP T

of the weighted myriad operator defined in (A.1). Therefore, θˆn satisfies

|gi |(θˆn − sgn(gi )xi ) i = 0, K + |gi |(sgn(gi )xi − θˆn )2

(B.1)

where G(θˆn , K) indicates the dependence of θˆn with respect to the parameter K. By applying the implicit differentiation on (B.1) with respect to K, and after some algebraic manipulations, we have

where the derivative

∂ ˆ ∂K θn

is obtained once

expressions are derived

i=−N2

= −2

|gi |

N1 X

ED

∂G ∂K

= 2

N1 X

∂G ∂K

and

∂G ∂ θˆn

i=−N2

K − |gi |(sgn(gi )xi − θˆn )2 [K + |gi |(sgn(gi )xi − θˆn )2 ]2

|gi |(θˆn − sgn(gi )xi ) [K + |gi |(sgn(gi )xi − θˆn )2 ]2

By substituting (B.3) and (B.4) in (B.2), the derivative

∂ ˆ ∂K θn

(B.3)

(B.4)

is obtained, which is depicted in

PT

(23).

(B.2)

are evaluated. From (B.1), the following

M

∂G ∂ θˆn

∂G ∂K ∂G ∂ θˆn

AN US

∂ ˆ θn = − ∂K

CE

Acknowledgements

This work was supported in part by the Fondo Nacional de Ciencia, Tecnolog´ıa e Investigaci´on

AC

- FONACIT-Venezuela under grant Nro. 2013001663.

References

[1] D. George, R. Bowen, J. Storey, An adaptive decision feedback equalizer, IEEE Transactions on Communication Technology 19 (3) (1971) 281–293. [2] J. J. Shynk, Adaptive IIR filtering, IEEE ASSP Magazine 6 (2) (1989) 4–21. [3] P. Regalia, Adaptive IIR filtering in signal processing and control, Vol. 90, CRC Press, 1994.

27

ACCEPTED MANUSCRIPT

[4] A. P. Liavas, P. A. Regalia, Acoustic echo cancellation: do iir models offer better modeling capabilities than their fir counterparts?, IEEE Transactions on Signal Processing 46 (9) (1998) 2499–2504. [5] G. R. Arce, J. L. Paredes, Recursive weighted median filters admitting negative weights and their optimization, IEEE Transactions on Signal Processing 48 (3) (2000) 768–779. [6] J. M. Ramirez, J. L. Paredes, Recursive weighted myriad based filters and their optimizations, IEEE Transactions on Signal Processing 64 (15) (2016) 4027–4039.

[8] P. J. Huber, Robust statistics, Wiley, New York, NY, USA, 1981.

CR IP T

[7] G. R. Arce, Nonlinear signal processing: a statistical approach, Wiley, New York, 2005.

[9] S. W. Smith, The Scientist and Engineer’s Guide to Digital Signal Processing, California Technical Publishing, San Diego, CA, USA, 1997.

[10] G. R. Arce, Statistical threshold decomposition for recursive and nonrecursive median filters, IEEE Transactions on Information Theory 32 (2) (1986) 243–253.

Information Theory 34 (4) (1988) 669–679.

AN US

[11] G. R. Arce, N. C. Gallagher, Stochastic analysis for the recursive median filter process, IEEE Transactions on

[12] G. Qiu, An improved recursive median filtering scheme for image processing, IEEE Transactions on Image Processing 5 (4) (1996) 646–648.

[13] A. Burian, P. Kuosmanen, Tuning the smoothness of the recursive median filter, IEEE Transactions on Signal Processing 50 (7) (2002) 1631–1639.

M

[14] S. Kalluri, G. R. Arce, Fast algorithms for weighted myriad computation by fixed-point search, IEEE Transactions on Signal Processing 48 (1) (2000) 159–171.

[15] S. Kalluri, G. R. Arce, Robust frequency-selective filtering using weighted myriad filters admitting real-valued

ED

weights, IEEE Transactions on Signal Processing 49 (11) (2001) 2721–2733. [16] G. R. Arce, A general weighted median filter structure admitting negative weights, IEEE Transactions on Signal Processing 46 (12) (1998) 3195–3205.

PT

[17] T. C. Aysal, K. E. Barner, Meridian filtering for robust signal processing, IEEE Transactions on Signal Processing 55 (8) (2007) 3949–3962.

CE

[18] J. G. Gonzalez, G. R. Arce, Statistically-efficient filtering in impulsive environments: weighted myriad filters, EURASIP Journal on Advances in Signal Processing 2002 (1) (2002) 1–17. [19] H.-S. Lim, T.-C. Chuah, H.-T. Chuah, On the optimal alpha-k curve of the sample myriad, IEEE Signal

AC

Processing Letters 14 (8) (2007) 545–548. [20] A. A. Roenko, V. V. Lukin, I. Djurovic, Two approaches to adaptation of sample myriad to characteristics of SαS distribution data, Signal Processing 90 (7) (2010) 2113–2123.

[21] J. G. Gonzalez, Robust techniques for wireless communications in non-gaussian environments, Ph.D. thesis, University of Delaware (1997). [22] J. G. Gonzalez, G. R. Arce, Optimality of the myriad filter in practical impulsive-noise environments, IEEE Transactions on Signal Processing 49 (2) (2001) 438–441. [23] S. Kalluri, G. R. Arce, Adaptive weighted myriad filter algorithms for robust signal processing in α-stable noise environments, Signal Processing, IEEE Transactions on 46 (2) (1998) 322–334.

28

ACCEPTED MANUSCRIPT

[24] S. Haykin, Adaptive Filter Theory (3rd Ed.), Prentice-Hall, Inc., Upper Saddle River, NJ, USA, 1996. [25] C.-C. Tseng, S.-C. Pei, Stable iir notch filter design with optimal pole placement, IEEE Transactions on Signal Processing 49 (11) (2001) 2673–2681. [26] B. Friedlander, B. Porat, The modified yule-walker method of arma spectral estimation, IEEE Transactions on Aerospace and Electronic Systems AES-20 (2) (1984) 158–173. [27] Z. Miao, X. Jiang, Weighted iterative truncated mean filter, IEEE Transactions on Signal Processing 61 (16)

CR IP T

(2013) 4149–4160.

[28] G. Samoradnitsky, M. S. Taqqu, Stable non-Gaussian random processes: stochastic models with infinite variance, Vol. 1, CRC Press, 1994.

[29] C. L. Nikias, M. Shao, Signal processing with alpha-stable distributions and applications, Wiley-Interscience, 1995.

[30] B. M. K. Goh, H. S. Lim, Sequential algorithms for sample myriad and weighted myriad filter, IEEE Trans.

AN US

Signal Process. 60 (11) (2012) 6047–6052.

[31] R. C. Nunez, J. G. Gonzalez, G. R. Arce, J. P. Nolan, Fast and accurate computation of the myriad filter via branch-and-bound search, IEEE Trans. Signal Process. 56 (7) (2008) 3340–3346. [32] J. G. Proakis, D. K. Manolakis, Digital Signal Processing (4th Edition), Prentice-Hall, Inc., Upper Saddle River, NJ, USA, 2006.

[33] X. Jiang, Iterative truncated arithmetic mean filter and its properties, IEEE Transactions on Image Processing

M

21 (4) (2012) 1537–1547.

[34] Z. Miao, X. Jiang, Additive and exclusive noise suppression by iterative trimmed and truncated mean algorithm, Signal Processing 99 (2014) 147–158.

ED

[35] T. S. Rappaport, et al., Wireless communications: principles and practice, Vol. 2, Prentice Hall PTR New Jersey, 1996.

[36] D. Middleton, Non-gaussian noise models in signal processing for telecommunications: new methods an results

PT

for class a and class b noise models, IEEE Transactions on Information Theory 45 (4) (1999) 1129–1149. [37] A. T. Georgiadis, B. Mulgrew, Adaptive bayesian decision feedback equaliser for alpha-stable noise environments,

CE

Signal Processing 81 (8) (2001) 1603–1623. [38] J. G. Gonzalez, J. L. Paredes, G. R. Arce, Zero-order statistics: A mathematical framework for the processing

AC

and characterization of very impulsive signals, IEEE Transactions on Signal Processing 54 (10) (2006) 3839–3851.

29