Research Article Subband Adaptive Filtering with -Norm...

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2013, Article ID 601623, 7 pageshttp://dx.doi.org/10.1155/2013/601623

Research ArticleSubband Adaptive Filtering with 𝑙1-Norm Constraint forSparse System Identification

Young-Seok Choi

Department of Electronic Engineering, Gangneung-Wonju National University, Gangneung 210-702, Republic of Korea

Correspondence should be addressed to Young-Seok Choi; [email protected]

Received 27 September 2013; Revised 26 November 2013; Accepted 26 November 2013

Academic Editor: Yue Wu

Copyright © 2013 Young-Seok Choi. This is an open access article distributed under the Creative Commons Attribution License,which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

This paper presents a new approach of the normalized subband adaptive filter (NSAF) which directly exploits the sparsity conditionof an underlying system for sparse system identification.The proposed NSAF integrates a weighted 𝑙

1-norm constraint into the cost

function of the NSAF algorithm. To get the optimum solution of the weighted 𝑙1-norm regularized cost function, a subgradient

calculus is employed, resulting in a stochastic gradient based update recursion of the weighted 𝑙1-norm regularized NSAF. The

choice of distinct weighted 𝑙1-norm regularization leads to two versions of the 𝑙

1-norm regularized NSAF. Numerical results clearly

indicate the superior convergence of the 𝑙1-norm regularized NSAFs over the classical NSAF especially when identifying a sparse

system.

1. Introduction

Over the past few decades, the relative simplicity and goodperformance of the normalized least mean square (NLMS)algorithm have made it a popular tool for adaptive filter-ing applications. However, its convergence performance issignificantly deteriorated in case of correlated input signals[1, 2]. As a popular solution, adaptive filtering in the subbandhas been recently developed, which is referred to as subbandadaptive filter (SAF) [3–7]. Its distinct feature is based on theproperty that the LMS-type adaptive filters converge faster forwhite input signals than colored ones [1, 2]. Thus, carryingout a prewhitening on colored input signals, it results in theaccelerated convergence compared to the LMS-type adaptivefilters. Recently, the use of multiple-constraint optimizationcriteria into formulation of a cost function has resulted in thenormalized SAF (NSAF) with its computational complexityclose to that of the NLMS algorithm [6, 7].

In the context of a system identification, the unknownsystem to be identified is sparse in common scenarios, suchas echo paths [8] and digital TV transmission channels [9].Namely, the unknown system consists of many near-zerocoefficients and a small number of large ones. However, theadaptive filtering algorithms suffer from poor convergence

performance in case of identifying the sparse system [8].Indeed, the capability of the NSAF is faded in a sparse systemidentification scenario. To deal with this issue, a variety ofproportionate adaptive algorithms have been presented forNSAF, which utilize proportionate step sizes to distinct filtertaps [10–12]. However, these algorithms have not exploitedthe sparsity condition of an underlying system.

Recently, motivated by compressive sensing framework[13, 14] and the least absolute shrinkage and selection oper-ator (LASSO) [15], a number of adaptive filtering algorithmswhich make use of the sparsity condition of an underlyingsystem have been developed [16–20]. The core idea behindthis approach is to incorporate the sparsity condition ofunderlying system by imposing an a sparsity-inducing con-straint term. Adding the sparsity constraint using 𝑙

0or 𝑙1-

norm constraint to the cost function makes the least relevantweights of the filter shrink to zeros. However, to the best ofthe author’s knowledge, adaptive filtering in subband whichexploits the sparsity condition has not been studied yet.

With regard, this paper presents a novel approach of thesparsity-regularized NSAFs, which incorporates the sparsitycondition of the system directly into the cost function viaa sparsity-inducing constraint term. This is carried out byregularizing a weighted 𝑙

1-norm of the filter weights estimate

2 Mathematical Problems in Engineering

u(n)

w∘

H0(z)

H1(z)

HN−1(z)

H0(z)

↓N

↓N

↓N

↓N

↓N

↓N

d0(n) d0(k)�(n)

d(n)d1(n) d1(k)

H1(z)

......

HN−1(z)dN−1(n) dN−1(k)

u0(n) y0(n) y0(k)

u1(n) y1(n) y1(k)

uN−1(n) yN−1(n yN−1(k))w(k)

w(k)

w(k)...

......

G0(z)e0(n) e0(k)

↑N

G1(z)e1(n) e1(k)

↑N

GN−1(z)eN−1(n) eN−1(k)

↑N

e(n)...

...

+

+

+

+

+

−

−

−

Figure 1: Subband structure with the analysis filters and synthesis filters and the subband desired signals, subband filter outputs, and subbanderror signals.

to the cost function. Considering the two choices of theweighted 𝑙

1-norm regularization, two stochastic gradient-

based 𝑙1-norm regularized NSAF algorithms are derived.

First, the 𝑙1-norm NSAF (𝑙

1-NSAF) is obtained by uti-

lizing the identity matrix as a weighting matrix. Second,the reweighted 𝑙

1-norm NSAF (𝑙

1-RNSAF) which uses the

current estimate of the system as a weighted 𝑙1-norm is

developed. Through numerical simulations, the resultantsparsity-regularized NSAFs have proven their superiorityover the classical NSAFs, especially when the sparsity of theunderlying system becomes severe.

The remainder of the paper is organized as follows.Section 2 introduces the classical NSAF, followed by thederivation of the proposed sparsity-regularized NSAFs inSection 3. Section 4 illustrates the computer simulationresults and Section 5 concludes this study.

2. Conventional NSAF

Consider a desired signal 𝑑(𝑛) that arises from the systemidentification model

𝑑 (𝑛) = u (𝑛)w∘ + V (𝑛) , (1)

where w∘ is a column vector for the impulse response of anunknown system that we wish to estimate, V(𝑛) accounts formeasurement noise with zeromean and variance𝜎2V , and u(𝑛)denotes the 1 ×𝑀 input vector,

u (𝑛) = [𝑢 (𝑛) 𝑢 (𝑛 − 1) ⋅ ⋅ ⋅ 𝑢 (𝑛 −𝑀 + 1)] . (2)

Figure 1 shows the structure of the NSAF, where thedesired signal 𝑑(𝑛) and input signal 𝑢(𝑛) are partitioned into

𝑁 subbands by the analysis filters𝐻0(𝑧),𝐻

1(𝑧), . . . , 𝐻

𝑁−1(𝑧).

The resulting subband signals, 𝑑𝑖(𝑛) and 𝑦

𝑖(𝑛) for 𝑖 =

0, 1, . . . , 𝑁 − 1, are critically decimated to a lower samplingrate commensurate with their bandwidth. Here, the variable𝑛 to index the original sequences and 𝑘 to index the decimatedsequences are used for all signals. Then, the decimatedfilter output signal at each subband is defined as 𝑦

𝑖,𝐷(𝑘) =

u𝑖(𝑘)w(𝑘), where u

𝑖(𝑘) is 1 ×𝑀 row vector, such that

u𝑖(𝑘) = [𝑢

𝑖(𝑘𝑁) , 𝑢

𝑖(𝑘𝑁 − 1) , . . . , 𝑢

𝑖(𝑘𝑁 −𝑀 + 1)] , (3)

andw(𝑘) = [𝑤0(𝑘), 𝑤

1(𝑘), . . . , 𝑤

𝑀−1(𝑘)]𝑇 denotes an estimate

for w∘ with length 𝑀. Thus the decimated subband errorsignal is given by

𝑒𝑖,𝐷

(𝑘) = 𝑑𝑖,𝐷

(𝑘) − 𝑦𝑖,𝐷

(𝑘) = 𝑑𝑖,𝐷

(𝑘) − u𝑖(𝑘)w (𝑘) , (4)

where 𝑑𝑖,𝐷(𝑘) = 𝑑

𝑖(𝑘𝑁) is the decimated desired signal at

each subband.In [6], the authors have formulated the Lagrangian-

based multiple-constraint optimization problem, which isformulated as

𝐽NSAF (𝑘) = ‖w (𝑘 + 1) − w (𝑘)‖2

+

𝑁−1

∑𝑖=0

𝜆𝑖[𝑑𝑖,𝐷

(𝑘) − u𝑖(𝑘)w (𝑘 + 1)] ,

(5)

where 𝜆𝑖for 𝑖 = 0, 1, . . . , 𝑁 − 1 denote the Lagrange multipli-

ers. Solving the cost function (5), the update recursion of theNSAF algorithm is given by [6, 7]. Consider

w (𝑘 + 1) = w (𝑘) + 𝜇𝑁−1

∑

𝑖=0

u𝑇𝑖(𝑘)

u𝑖 (𝑘)2𝑒𝑖,𝐷

(𝑘) , (6)

where 𝜇 is the step-size parameter.

Mathematical Problems in Engineering 3

3. Weighted 𝑙1-Norm Regularized NSAF

3.1. Derivation of the Proposed Algorithm. To reflect thesparsity condition of the true system, that is, w∘, a weighted𝑙1-norm of the filter weight estimate is regularized on the cost

function of the NSAF, which is given by

𝐽𝑙1−NSAF (𝑘) = ‖w (𝑘 + 1) − w (𝑘)‖

2

+

𝑁−1

∑

𝑖=0

𝜆𝑖[𝑑𝑖,𝐷

(𝑘) − u𝑖(𝑘)w (𝑘 + 1)]

+𝛾‖Πw (𝑘 + 1)‖1,

(7)

where ‖Πw(𝑘 + 1)‖ accounts for the weighted 𝑙1-norm of the

filter weight vector w(𝑘 + 1) and is written as

‖Πw (𝑘 + 1)‖1 =𝑀−1

∑

𝑚=0

𝜋𝑚

𝑤𝑚 (𝑘 + 1) , (8)

where Π is a 𝑀 × 𝑀 weighting matrix whose diagonalelements are 𝜋

𝑚and other elements are equal to zero, and

𝑤𝑚(𝑘 + 1) denotes the 𝑚th tap weight of w(𝑘 + 1), for 𝑚 =

0, 1, . . . ,𝑀 − 1. In addition, 𝛾 is a positive value parameterwhich plays a role in compromising the error related term andthe weighted 𝑙

1-norm regularization in the right-hand side of

(7).To find the optimal weight vector w(𝑘 + 1) which

minimizes the cost function (7), the derivative of (7) withrespect to w(𝑘 + 1) is taken and set to zero. Note that theweighted 𝑙

1-norm regularization term, that is, ‖Πw(𝑘 + 1)‖

1,

is not differentiable at any point in case 𝑤𝑚(𝑘 + 1) = 0. To

address this issue, a subgradient calculus [21] is carried out.Thus, taking the derivative of (7) with respect to the

weight vector w(𝑘 + 1) and letting the derivative be equal tozero, it leads to

w (𝑘 + 1) = w (𝑘) + 12

𝑁−1

∑

𝑖=0

𝜆𝑖u𝑇𝑖(𝑘) −

𝛾

2∇𝑠

w‖Πw (𝑘 + 1)‖1,

(9)

where ∇𝑠w𝑓(⋅) denotes a subgradient vector of a function 𝑓(⋅)with respect to w(𝑘 + 1). An available subgradient vector∇𝑠

w‖Πw(𝑘 + 1)‖1 is obtained as [21]. Consider

∇𝑠

w‖Πw (𝑘 + 1)‖1 = Π𝑇 sgn (Πw (𝑘 + 1))

= Π sgn (w (𝑘 + 1)) ,(10)

sinceΠ is assumed as a diagonal matrix with positive-valuedelements, where sgn(⋅) is a componentwise sign functiondefined by

sgn (𝑥) ={

{

{

𝑥

|𝑥|, 𝑥 ̸= 0

0, elsewhere.(11)

Substituting (10) into (9) and assuming sgn[w(𝑘 + 1)] ≈sgn[w(𝑘)], it is given by

w (𝑘 + 1) = w (𝑘) + 12

𝑁−1

∑

𝑖=0

𝜆𝑖u𝑇𝑖(𝑘) −

𝛾

2Π sgn (w (𝑘)) . (12)

Substituting (12) into the multiple constraints of theNSAF, that is, 𝑑

𝑖,𝐷(𝑘) = u

𝑖(𝑘)w(𝑘 + 1), 𝑖 = 0, 1, . . . , 𝑁 − 1

and rewriting as a matrix form, it leads to

Λ = 2[U (𝑘)U𝑇 (𝑘)]−1

e𝐷(𝑘)

+𝛾[U (𝑘)U𝑇 (𝑘)]−1

U (𝑘)Π sgn (w (𝑘)) ,(13)

where Λ = [𝜆0, 𝜆1, . . . , 𝜆

𝑁−1]𝑇 is the𝑁 × 1 Lagrange vector,

U (𝑘) = [[[

u0(𝑘)

...u𝑁−1

(𝑘)

]]

]

, e𝐷(𝑘) =

[[

[

𝑒0,𝐷

(𝑘)

...𝑒𝑁−1,𝐷

(𝑘)

]]

]

. (14)

By neglecting the off-diagonal elements ofU(𝑘)U𝑇(𝑘) [6],the components of Λ in (13) can be simplified to

𝜆𝑖= 2

𝑒𝑖,𝐷

(𝑘)

u𝑖 (𝑘)2+ 𝛾

u𝑖(𝑘)

u𝑖 (𝑘)2Π sgn (w (𝑘)) , (15)

for 𝑖 = 0, 1, . . . , 𝑁 − 1.Consequently, combining (12) and (15), the update recur-

sion of the sparsity-regularized NSAF is given by

w (𝑘 + 1) = w (𝑘)

+ 𝜇

𝑁−1

∑

𝑖=0

[u𝑇𝑖(𝑘)


(𝑘)

+1

2𝛾

u𝑖(𝑘)

u𝑖 (𝑘)2Π sgn (w (𝑘)) u𝑇

𝑖(𝑘)]

−𝜇𝛾

2Π sgn (w (𝑘)) ,

(16)

where 𝜇 is the step-size parameter.

3.2. Determination of the Weighted 𝑙1-Norm Regularization.

Here, by choosing the weighting matrix Π, two versions ofthe sparsity-regularized NSAF are developed. First, the use ofthe identity matrix as the weighting matrix, that is, Π = I

𝑀,

results in the following update recursion:

w (𝑘 + 1) = w (𝑘)

+ 𝜇

𝑁−1

∑

𝑖=0

[u𝑇𝑖(𝑘)


(𝑘)

+1

2𝛾

u𝑖(𝑘)

u𝑖 (𝑘)2sgn (w (𝑘)) u𝑇

𝑖(𝑘)]

−𝜇𝛾

2sgn (w (𝑘)) ,

(17)

which is referred to as the 𝑙1-norm NSAF (𝑙

1-NSAF) as

an unweighted case. The 𝑙1-NSAF uniformly attracts the


Table 1: Computational complexity.

NSAF 𝑙1-NSAF 𝑙

1-RNSAF

Multiplications 3𝑀 + 3𝑁𝐿 6𝑀 + 3𝑁𝐿 7𝑀 + 3𝑁𝐿Divisions 1 2 2 +𝑀/𝑁(𝑀: filter length,𝑁: number of subbands, and 𝐿: length of the analysis filtersand synthesis filters).

tap coefficients of w(𝑘) to zero. The zero attraction processleads to the improved convergence of the 𝑙

1-NSAF when the

majority of entries of a system are zero; that is, a system issparse.

Second, to approximate the actual sparsity condition of anunderlying system, that is, 𝑙

0-norm of the system, the weights

of Π are chosen inversely proportional to magnitude of theactual coefficients of the system as given by

𝜋𝑚={

{

{

1𝑤𝑚

, 𝑤𝑚

̸= 0

∞, 𝑤𝑚= 0,

(18)

where 𝑤𝑚denotes the 𝑚th coefficients of the system, that

is, w∘. However, since the actual coefficients of the systemare unavailable, the estimate of the current filter weights isutilized instead of the actual weights, which is referred to asthe reweighting scheme [22], as follows:

𝜋𝑚(𝑘) =

1𝑤𝑚 (𝑘)

+ 𝜖for 𝑚 = 0, 1, . . . ,𝑀 − 1, (19)

where 𝑤𝑚(𝑘) denotes the𝑚th tap weight of the w(𝑘) and 𝜖 is

a small positive value to avoid singularity when |𝑤𝑚(𝑘)| = 0.

Then, the weighting matrix Π consists of the values of 𝜋𝑚(𝑘)

as the𝑚th diagonal elements and has a time-varying feature.Finally, the update recursion is given by

w (𝑘 + 1) = w (𝑘)

+ 𝜇

𝑁−1

∑

𝑖=0

[u𝑇𝑖(𝑘)


(𝑘)

+1

2𝛾

u𝑖(𝑘)

u𝑖 (𝑘)2sgn (w (𝑘)) u𝑇

𝑖(𝑘)]

−𝜇𝛾

2

sgn (w (𝑘))|w (𝑘)| + 𝜖

,

(20)

where u𝑖(𝑘) = u

𝑖(𝑘)Π and the vector division operation in

last term accounts for a componentwise division. Then, thisrecursion is called the reweighted 𝑙

1-normNSAF (𝑙

1-RNSAF)

Table 1 lists the number of multiplications and divisionsof the NSAF [6], 𝑙

1-NSAF, and 𝑙

1-RNSAF per iteration. As

shown in Table 1, the use of 𝑙1-norm constraint leads to an

acceptable increase in computation.

4. Numerical Results

Theperformance of the proposed sparsity-regularizedNSAFsis validated by carrying out computer simulations in a system

identification scenario in which the unknown channel israndomly generated. The lengths of the unknown system are𝑀 = 128 and 512 in experiments where 𝑆 of them arenonzero. The nonzero filter weights are positioned randomlyand their values are taken from a Gaussian distributionN(0, 1/𝑆). Here, 𝑆 = 4 is used in the simulations exceptFigure 5 in which various 𝑆 values are considered. Theadaptive filter and the unknown system are assumed to havethe same number of taps. The input signals are obtainedby filtering a white, zero-mean, Gaussian random sequencethrough a first-order system𝐺(𝑧) = 1/(1−0.9𝑧−1).The signal-to-noise ratio (SNR) is calculated by

SNR = 10 log10(𝐸 [𝑦2(𝑛)]

𝐸 [V2 (𝑛)]) , (21)

where 𝑦(𝑛) = u(𝑛)w∘. The measurement noise V(𝑛) is addedto𝑦(𝑛) such that SNR= 10, 20, and 30 dB. In order to comparethe convergence performance, the normalized mean squaredeviation (MSD),

NormalizedMSD = 𝐸[w∘− w(𝑘)

2

‖w∘‖2] , (22)

is taken and averaged over 50 independent trials.The cosine-modulated filter banks [23] with the subband number of𝑁 =4 are used in the simulations. The prototype filter of length𝐿 = 32 is used. For comparison purpose, the proportionateNSAF (PNSAF) [12] is considered, which has been developedfor sparse system identification.The step-size is set to 𝜇 = 0.5for SAF algorithms except the PNSAFwhere the step sizes𝜇 =0.6 (Figure 2) and 𝜇 = 0.65 (Figure 6) are chosen to achievesimilar steady-state MSD with the 𝑙

1-RNSAF for comparison

purpose. For the 𝑙1-RNSAF, 𝜖 = 0.01 is chosen. In addition,

𝜌 = 0.05 is used for the PNSAF. The 𝛾 values are obtained byrepeated trials to minimize the steady-state MSD.

Figure 2 shows the normalizedMSD curves of the NLMS,NSAF, 𝑙

1-NSAF, and 𝑙

1-RNSAF, in cases of𝑁 = 4 and SNR =

30 dB. For the 𝑙1-NSAF and 𝑙

1-RNSAF, 𝛾 = 3×10−5 is chosen.

As shown in Figure 2, the not only 𝑙1-RNSAFoutperforms the

conventional NLMS, NSAF, PNSAF, and 𝑙1-NSAF, but also

the 𝑙1-NSAF has better performance than other conventional

algorithms, in terms of the convergence rate and the steady-state misalignment.

In Figure 3, to verify the effect of 𝛾 on convergenceperformance, the normalized MSD curves of the 𝑙

1-RNSAF

for different 𝛾 values are illustrated, in case of 𝑁 = 4 andSNR = 30 dB. For different 𝛾 values (𝛾 = 1×10−4, 1×10−5, 5×10−5, and 1 × 10−6), the 𝑙

1-RNSAF is not excessively sensitive

to 𝛾.The analysis of an optimal 𝛾 value remains a future work.Next, the performance of the proposed 𝑙

1-norm regular-

izedNSAFs is compared to the original NSAF under differentSNR conditions. Figure 4 depicts the normalizedMSD curvesof the NSAF, 𝑙

1-NSAF, and 𝑙

1-RNSAF under SNR = 10

and 20 dB, respectively. The 𝛾 value for the 𝑙1-NSAF and 𝑙

1-

RNSAF is set to 5 × 10−5. It is clear that both the 𝑙1-NSAF

and 𝑙1-RNSAF are superior to the NSAF under several SNR

cases. Furthermore, the 𝑙1-RNSAF performs well compared

to 𝑙1-NSAF.


0 200 400 600 800 1000 1200 1400

−35

−30

−25

−20

−15

−10

−5

0

5

Number of iterations

Nor

mal

ized

mea

n sq

uare

dev

iatio

n (d

B)

NLMSNSAF (N = 4)PNSAF (N = 4)

l1-NSAF (N = 4)l1-RNSAF (N = 4)

Figure 2: Normalized MSD curves of the NLMS, NSAF, PNSAF, 𝑙1-

NSAF, and 𝑙1-RNSAF (𝑁 = 4).

0 200 400 600 800 1000 1200 1400

−35

−30

−25

−20

−15

−10

−5

0


Nor

mal

ized

mea

n sq

uare

dev

iatio

n (d

B)

𝛾 = 1 × 10−3

𝛾 = 1 × 10−4

𝛾 = 1 × 10−5

𝛾 = 5 × 10−5

𝛾 = 7 × 10−5

Figure 3: Normalized MSD curves of the 𝑙1-RNSAF for various 𝛾

values (𝑁 = 4).

In Figure 5, the convergence properties of the NSAF and𝑙1-RNSAF are compared under various sparsity conditions of

an underlying system. With the same length of the system,that is, 𝑀 = 128, different sparsity conditions (𝑆 = 4, 8, 16,and 32) are considered under SNR = 30 dB. The value of 𝛾is set to 3 × 10−5 for the 𝑙

1-RNSAF. Figure 5 shows that the

NSAF is independent of the sparsity condition. On the otherhand, the results indicate that the more sparse the underlyingsystem, the better the 𝑙

1-RNSAF.

The comparison of performance of the NSAF, 𝑙1-NSAF,

and 𝑙1-RNSAF with a long system, here, the filter length𝑀 =

0 200 400 600 800 1000 1200 1400−25

−20

−15

−10

−5

0

5

10


Nor

mal

ized

mea

n sq

uare

dev

iatio

n (d

B)

(a) NSAF (SNR = 10 dB)(b) l1-NSAF (SNR = 10 dB)(c) l1-RNSAF (SNR = 10 dB)

(d) NSAF (SNR = 20 dB)(e) l1-RNSAF (SNR = 20 dB)(f) l1-RNSAF (SNR = 20 dB)

Figure 4: Normalized MSD curves of the NSAF, 𝑙1-NSAF, and 𝑙

1-

RNSAF under various SNR conditions (SNR = 10, 20, and 30 dB,𝑁 = 4).

0 200 400 600 800 1000 1200−35

−30

−25

−20

−15

−10

−5

0


Nor

mal

ized

mea

n sq

uare

dev

iatio

n (d

B)

NSAF (S = 4)l1-RNSAF (S = 4)NSAF (S = 8)l1-RNSAF (S = 8)

NSAF (S = 16)l1-RNSAF (S = 16)NSAF (S = 32)l1-RNSAF (S = 32)

Figure 5: Normalized MSD curves of the NSAF and 𝑙1-RNSAF

under various sparsity conditions (𝑆 = 4, 8, 16, and 32,𝑁 = 4).

512, is presented in Figure 6. For the 𝑙1-NSAF and 𝑙

1-RNSAF,

𝛾 = 5×10−5 is chosen. A similar result of Figure 2 is observed

in Figure 6.Finally, the tracking capabilities of the algorithms of a

sudden change in the system are tested for 𝑁 = 4 andSNR = 30 dB. Figure 7 shows the results when the unknownsystem is right-shifted for 20 taps. Same value of 𝛾 in Figure 2is used. As can be seen, the 𝑙

1-NSAF and 𝑙

1-RNSAF keep track

of weight change without losing the convergence rate nor


0 1000 2000 3000 4000 5000−40

−35

−30

−25

−20

−15

−10

−5

0


Nor

mal

ized

mea

n sq

uare

dev

iatio

n (d

B)

NLMSNSAF (N = 4)PNSAF (N = 4)

l1-NSAF (N = 4)l1-RNSAF (N = 4)

Figure 6: Normalized MSD curves of the NLMS, NSAF, PNSAF, 𝑙1-

NSAF, and 𝑙1-RNSAF for long system of𝑀 = 512 (𝑁 = 4).

0 500 1000 1500 2000 2500 3000−40

−35

−30

−25

−20

−15

−10

−5

0

5


Nor

mal

ized

mea

n sq

uare

dev

iatio

n (d

B)

NSAFPNSAF

l1-NSAFl1-RNSAF

Figure 7: Normalized MSD curves of the NSAF, PNSAF, 𝑙1-NSAF,

and 𝑙1-RNSAF in case of a time-varying unknown system (𝑁 = 4).

The system is right-shifted for 20 taps at 1500 iterations.

the steady-state misalignment compared to the conventionalNLMS, NSAF, and PNSAF. To be specific, the 𝑙

1-RNSAF

achieves better performance than the 𝑙1-NSAF in terms of

both convergence rate and steady-state misalignment.

5. Conclusion

A new family of the NSAFs which takes into accountthe sparsity condition of an underlying system has beenpresented by incorporating a weighted 𝑙

1-norm constraint of

filter weights in the cost function. The update recursion isobtained by employing subgradient calculus on the weighted𝑙1-norm constraint term. Subsequently, two sparsity regular-

ized NSAFs, that is, the unweighted 𝑙1-NSAF and 𝑙

1-RNSAF

have been developed. The numerical results indicate that theproposed 𝑙

1-NSAF and 𝑙

1-RNSAF achieve highly improved

convergence performance over the conventional algorithmsfor sparse system identification.

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper.

Acknowledgment

This research was supported by new faculty research pro-gram funded by Gangneung-Wonju National University(2013100162).

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