Selective partial update and set-membership subband adaptive filters

ARTICLE IN PRESS Signal Processing 88 (2008) 2463– 2471

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Selective partial update and set-membership subband adaptive filters Mohammad Shams Esfand Abadi a,, John Ha˚kon Husøy b a b

Department of Electrical Engineering, Shahid Rajaee Teacher Training University, Tehran, Iran Department of Electrical and Computer Engineering, University of Stavanger, N-4036, Stavanger, Norway

a r t i c l e in fo

abstract

Article history: Received 2 November 2007 Received in revised form 14 April 2008 Accepted 16 April 2008 Available online 7 May 2008

This paper presents three efficient subband adaptive filter (SAF) algorithms featuring low computational complexity. In the first algorithm, which is called selective partial update SAF (SPU-SAF), the filter coefficients are partially updated in each subband rather than the entire filter at every adaptation. In the second one, the concept of set-membership (SM) adaptive filtering is extended to the SAFs and a novel SM-SAF algorithm is presented. This algorithm exhibits superior performance with significant reduction in the overall computational complexity compared with the ordinary SAF. The third algorithm is based on the combination of the ideas in the SPU-SAF and SM-SAF algorithms. We demonstrate the usefulness of the proposed algorithms through simulations. & 2008 Elsevier B.V. All rights reserved.

Keywords: Subband adaptive filter Selective partial update Set-membership Computational complexity

1. Introduction Adaptive filtering is an important subfield of digital signal processing having numerous applications [1–3]. In some of these applications, a large number of filter coefficients are needed to achieve an acceptable performance. Therefore the computational complexity is the main problem in these applications. Several adaptive filter algorithms such as the subband adaptive filters (SAFs), the adaptive filter algorithms with selective partial updates (SPU) and the set-membership (SM) filtering have been proposed to solve these problems. The SPU adaptive algorithms update only a subset of the filter coefficients in each time iteration and consequently reduce the computational complexity. The MaxNLMS [4], the MMax-NLMS [5,6], variants of the SPU normalized least mean square (SPU-NLMS) [7,8] and the SPU transform domain LMS (SPU-TD-LMS) [9] are important examples of this family of adaptive filter algo Corresponding author. Tel.: +98 21 22970003.

E-mail addresses: [email protected] (M.S.E. Abadi), [email protected] (J.H. Husøy). 0165-1684/$ - see front matter & 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.sigpro.2008.04.014

rithms. Unfortunately, as with many other adaptive filter algorithms, the step-size determines the tradeoff between steady-state mean square error (MSE) and convergence rate. Having fast convergence, low steady-state MSE, and low computational complexity at the same time is highly desirable. The SM normalized LMS (SM-NLMS) is one of the algorithms that has these three features [10]. Based on [10], different SM adaptive algorithms have been developed. The SM affine projection algorithm (SM-APA) [11,12], and the SM binormalized data-reusing LMS (SMBNDRLMS) algorithms [13] are important examples of this family of adaptive filters. Also in [14], the SM-PU-NLMS is presented based on the combination of the partial updating and SM filtering approaches. In [15], the subband adaptive algorithm called normalized SAF (NSAF) was developed based on a constrained optimization problem. The filter update equation proposed in [15] is similar to that proposed in [16,17], where the fullband filters are updated instead of subfilters as in the conventional SAF structure [18]. Again, in the SAFs, the step-size determines the tradeoff between steady-state MSE and convergence rate [19].

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What we propose in this paper can be summarized as follows:

 The establishment of the SPU-SAF algorithm. In this algorithm the filter coefficients are partially updated in each subband rather than the entire filter at every adaptation. Extension of the SM filtering concept to the SAF, and the establishment of a novel SM-SAF algorithm. The proposed algorithm exhibits superior performance with significant reduction in the overall computational complexity compared with the ordinary SAF. Combination of the SPU-SAF and SM-SAF approaches to develop the SM–SPU-SAF algorithm.





where xðnÞ ¼ ½xðnÞ; xðn  1Þ; . . . ; xðn  M þ 1ÞT . Using the method of Lagrange multipliers to solve this optimization problem leads to the following recursion: hðn þ 1Þ ¼ hðnÞ þ

j:j

xðnÞ ¼ ½xT1 ðnÞ; xT2 ðnÞ; . . . ; xTP ðnÞT ,

2

k:k

T

T

hðnÞ ¼ ½h1 ðnÞ; h2 ðnÞ; . . . ; hP ðnÞT .

(4) (5)

The SPU-NLMS algorithm for a single block update in every iteration, minimizes following optimization problem min khj ðn þ 1Þ  hj ðnÞk2 ,

hj ðnþ1Þ

(6)

subject to (2), where j denotes the index of the block that should be updated. Again by using the method of Lagrange multipliers, the update equation for SPU-NLMS is given by hj ðn þ 1Þ ¼ hj ðnÞ þ

norm of a scalar squared Euclidean norm of a vector

(3)

where eðnÞ ¼ dðnÞ  xT ðnÞ hðnÞ, and m is the step-size that determines the convergence speed and excess MSE (EMSE). Now partition the input signal vector and the vector of filter coefficients into P blocks each of length L1 which are defined as

T

We have organized our paper as follows: In the following section we briefly review the NLMS, the SPUNLMS and the SM-NLMS algorithms. In the next section the SPU-SAF algorithm is introduced. The SM-SAF and the SM–SPU-SAF are introduced in Sections 4 and 5, respectively. Finally, we present several simulation results to demonstrate the good performances of the proposed algorithms. Throughout the paper, the following notations are adopted:

m xðnÞeðnÞ, k xðnÞk2

m xj ðnÞeðnÞ, kxj ðnÞk2

(7)

transpose of a vector or a matrix ð:ÞT Trð:Þ trace of a matrix diagð:Þ has the same meaning as the MATLAB operator with the same name: If its argument is a vector, a diagonal matrix with the diagonal elements given by the vector argument results. If the argument is a matrix, its diagonal is extracted into a resulting vector.

where j ¼ arg maxkxi ðnÞk2 for 1pipP [8]. The SM-NLMS algorithm minimizes (1) subject to hðn þ 1Þ 2 Cn where2

2. Background on NLMS, SPU-NLMS and SM-NLMS algorithms

hðn þ 1Þ ¼ hðnÞ þ

Cn ¼ fh 2 RM : jdðnÞ  xT ðnÞ h jpgg.

(8)

This aim is achieved by an orthogonal projection of the previous estimate of h onto the closest boundary of Cn [10]. Doing this, the recursion for the SM-NLMS is given by aðnÞ xðnÞeðnÞ, k xðnÞk2

(9)

where In Fig. 1 we show the prototypical adaptive filter setup, where xðnÞ, dðnÞ and eðnÞ are the input, the desired and the output error signals, respectively. hðnÞ is the M  1 column vector of filter coefficients at time n. It is well known that the NLMS algorithm can be derived from the solution of the following optimization problem: min k hðn þ 1Þ  hðnÞk2 ,

(1)

hðnþ1Þ

subject to dðnÞ ¼ xT ðnÞ hðn þ 1Þ,

(2)

d(n) x(n)

h(n)

y(n) −

+

e(n)

Fig. 1. Prototypical adaptive filter setup.

aðnÞ ¼

8 <1  :

0

g jeðnÞj

if jeðnÞj4g;

(10)

otherwise:

3. SPU-SAF algorithm Fig. 2 shows the structure of the SAF [15]. In this figure, f 0 ; f 1 ; . . . ; f N1 , are analysis filter unit pulse responses of an N channel orthogonal perfect reconstruction critically sampled filter bank system. xi ðnÞ and di ðnÞ are nondecimated subband signals. It is important to note that n refers to the index of the original sequences and k denotes the index of the decimated sequences.3 Similar to the NLMS algorithm, the SAFs can be established by the solution of the following optimization 1

Note that P ¼ M=L and is an integer. The set Cn is referred to as the constraint set, and its boundaries are hyperplanes. Also, g is the magnitude of the error bound. 3 It means that in the SAF, the filter vector update is performed each time N new samples have entered the system. 2

ARTICLE IN PRESS M.S.E. Abadi, J.H. Husøy / Signal Processing 88 (2008) 2463–2471

d0(n)

f0 .. .. .. .

d(n)

where

d0,D(k)

↓N

Xj ðkÞ ¼ ½x0;j ðkÞ; x1;j ðkÞ; . . . ; xN1;j ðkÞ,

.. .. .. . dN−1(n)

fN−1

↓N

x0(n)

ments of row m of XTj ðkÞ are consecutive samples of subband no. m, it follows that the off diagonal elements of

dN−1,D(k)

h (k)

.. .. .. .

x(n)

xN−1(n)

.. .. .. .

.. .. .. .. .

+

h( k)

e0(n)

g0 e(n)

↓N

gN−1

eN−1(n)

y0,D(k) −

.. .. .. .

.. .. .. .

fN−1

(19)

and eD ðkÞ ¼ ½e0;D ðkÞ; e1;D ðkÞ; . . . ; eN1;D ðkÞT . Since the ele-

+ ..... f0

2465

↓N

the approximation XTj ðkÞXj ðkÞ  diagðdiagðXTj ðkÞXj ðkÞÞÞ ¼ Kj ðkÞ resulting in the following coefficient update equation:

yN−1,D (k)

+ + −

hj ðk þ 1Þ ¼ hj ðkÞ þ mXj ðkÞ½Kj ðkÞ1 eD ðkÞ

hj ðk þ 1Þ ¼ hj ðkÞ þ m

N 1 X

xi;j ðkÞ

i¼0

kxi;j ðkÞk2

ei;D ðkÞ.

(21)

Now, we determine which block should be updated in each subband at every adaptation. From (17) and (20), we obtain

eN−1,D(k)

j ¼ arg min khp ðk þ 1Þ  hp ðkÞk2

Fig. 2. Structure of the SAF.

1pppP

¼ arg min feTD ðkÞ½Kp ðkÞ1 eD ðkÞg,

(22)

1pppP

problem: min k hðk þ 1Þ  hðkÞk2 ,

(11)

hðkþ1Þ

subject to the set of N constraints imposed on the decimated filter output di;D ðkÞ ¼ xTi ðkÞ hðk þ 1Þ

(20)

This equation can be represented as

e0,D (k)

↑N .. .. .. . ↑N

+

XTj ðkÞXj ðkÞ are sample cross correlations between different subband signals whose values are very small. This justifies

for i ¼ 0; . . . ; N  1,

which is equivalent to ( ) N 1 X jei;D ðkÞj2 j ¼ arg min . 1pppP kxi;p ðkÞk2 i¼0

(23)

(12) 3.1. Extension to the multiple blocks

where T

xi ðkÞ ¼ ½xi ðkNÞ; xi ðkN  1Þ; . . . ; xi ðkN  M þ 1Þ .

(13)

By solving this optimization problem based on the method of Lagrange multipliers, the filter update equation for the SAF which was called NSAF, can be stated as [15] hðk þ 1Þ ¼ hðkÞ þ m

N 1 X i¼0

xi ðkÞ ei;D ðkÞ, kxi ðkÞk2

(14)

where ei;D ðkÞ ¼ di;D ðkÞ  xTi ðkÞ hðkÞ is the decimated subband error signal, and m is chosen in the range 0omo2 [15]. We are now in the position to establish the SPU-SAF algorithm. Partition xi ðkÞ for 0pipN  1 and hðkÞ into P blocks each of length L which are defined as xi ðkÞ ¼ ½xTi;1 ðkÞ; xTi;2 ðkÞ; . . . ; xTi;P ðkÞT , T

T

T

hðkÞ ¼ ½h1 ðkÞ; h2 ðkÞ; . . . ; hP ðkÞT .

(15) (16)

The SPU-SAF solves the following optimization problem: min khj ðk þ 1Þ  hj ðkÞk2 ,

hj ðkþ1Þ

(17)

subject to (12), where j denotes the index of the block. Using the method of Lagrange multipliers to solve this optimization problem leads to the following update equation: hj ðk þ 1Þ ¼ hj ðkÞ þ mXj ðkÞ½XTj ðkÞXj ðkÞ1 eD ðkÞ,

(18)

In the previous section a single block of filter coefficients in each subband is updated during every adaptation. In this section we extend this approach to multiple block update. Suppose, we want to update S blocks out of P blocks in each subband at every adaptation. Let F ¼ fj1 ; j2 ; . . . ; jS g denote the indices of the S blocks out of P blocks. In this case, the optimization problem is defined as min khF ðk þ 1Þ  hF ðkÞk2 ,

(24)

hF ðkþ1Þ

subject to (12). Again by using the Lagrange multipliers approach, the filter vector update equation is given by hF ðk þ 1Þ ¼ hF ðkÞ þ mXF ðkÞ½KF ðkÞ1 eD ðkÞ,

(25)

where XF ðkÞ ¼ ½XTj1 ðkÞ; XTj2 ðkÞ; . . . ; XTjS ðkÞT ,

(26)

diagðdiagðXTF ðkÞXF ðkÞÞÞ.

and KF ðkÞ ¼ represented as

hF ðk þ 1Þ ¼ hF ðkÞ þ m

N 1 X

xi;F ðkÞ

i¼0

kxi;F ðkÞk2

Eq. (25) can also be

ei;D ðkÞ,

(27)

where xi;F ðkÞ ¼ ½xTi;j ðkÞ; xTi;j ðkÞ; . . . ; xTi;j ðkÞT . 1 2 S Now, we determine which blocks should be updated in each subband at every adaptation. From (24) and (25),

ARTICLE IN PRESS 2466

M.S.E. Abadi, J.H. Husøy / Signal Processing 88 (2008) 2463–2471

we obtain

which is equivalent to

F ¼ arg min khF ðk þ 1Þ  hF ðkÞk2

hðk þ 1Þ ¼ hðkÞ þ XðkÞ½KðkÞ1 aðkÞeD ðkÞ,

F

¼ arg minfeTD ðkÞ½KF ðkÞ1 eD ðkÞg F 8 9 2 31 < = X ¼ arg min eTD ðkÞ4 Kj ðkÞ5 eD ðkÞ , ; F :

where aðkÞ ¼ diagða0 ðkÞ; a1 ðkÞ; . . . ; aN1 ðkÞÞ, ðdiagðXT ðkÞXðkÞÞÞ, and (28)

(35) KðkÞ ¼ diag

XðkÞ ¼ ½x0 ðkÞ; x1 ðkÞ; . . . ; xN1 ðkÞ.

(36)

j2F

which is equivalent to ( ) N 1 X jei;D ðkÞj2 F ¼ arg min . F kxi;F ðkÞk2 i¼0

5. SM–SPU-SAF algorithm (29)

The computational complexity of the exact selection of the blocks to update may be very high. Therefore we may need to use a simplified criterion as we present in the following. 3.2. Simplified SPU-SAF (SSPU-SAF) algorithm To reduce the computational complexity associated with the selection of the blocks to update, we propose two alternative simplified criteria: (1) In the first approach, we compute the following values: TrðKp ðkÞÞ ¼

N 1 X

kxi;p ðkÞk2

for 1pppP.

(30)

i¼0

hðk þ 1Þ ¼ hðkÞ þ m

N 1 X i¼0

Ak xi ðkÞ ei;D ðkÞ, kAk xi ðkÞk2

(37)

where the Ak matrix is the M  M diagonal matrix with the ILL and 0LL matrices4 on the diagonal and the positions of 1s on the diagonal determining which coefficients should be updated in each subband at every adaptation. This matrix can be represented as 2 3 ½0 or ILL 0LL  0LL 6 7 ½0 or ILL    0LL 7 6 0LL 6 7 Ak ¼ 6 . (38) 7 .. .. .. .. 6 7 . . . . 4 5 0LL 0LL    ½0 or ILL MM

The indices of the set F correspond to the indices of the S largest values of (30) [8]. (2) Another selection strategy would be to modify (23) in such a way that rather than identifying one index, we identify a set of indices, corresponding to the S smallest values. In the first approach we used the simplified form of (23) to identify the indices of F. But in the second strategy, the exact form of (23) was used. This criterion slightly increases the computational complexity but leads to somewhat better performance. Sections 6 and 7 present the computational complexity and the performance of the SSPU-SAF algorithms. 4. SM-SAF algorithm The SM-SAF minimizes (11) subject to hðk þ 1Þ 2 ðCk;0 \ Ck;1 \    \ Ck;N1 Þ,

The positions of the identity matrices at every adaptation are determined by F from (29) or from the simplified procedure associated with (30). Now, from the previous section we obtain the SM–SPU-SAF algorithm: hðk þ 1Þ ¼ hðkÞ þ

N 1 X i¼0

ai ðkÞ

Ak xi ðkÞ ei;D ðkÞ, kAk xi ðkÞk2

(39)

where the ai ðkÞ values for 0pipN  1 are obtained from (34). To differentiate the full implementation of the SM–SPU-SAF algorithm in (29) from its simplified version that uses (23) or (30), we will refer to the latter as the SM simplified SPU-SAF (SM–SSPU-SAF) algorithm. This algorithm combines the features of the SM-SAF and SPU-SAF algorithms. In this algorithm we determine which filter coefficients in which subband should be updated and then we partially update the filter coefficients.

(31) 6. Computational complexity

where Ck;i ¼ fh 2 RM : jdi;D ðkÞ  xTi ðkÞ h jpgg.

(32)

This aim is obtained by an orthogonal projection of the previous estimate of h onto the closest boundary of Ck;i in each subband. Doing this, the filter vector update equation for SM-SAF can be stated as hðk þ 1Þ ¼ hðkÞ þ

N 1 X i¼0

ai ðkÞ

xi ðkÞ ei;D ðkÞ, kxi ðkÞk2

(33)

where ai ðkÞ ¼

In this section, we combine the approaches in SPU-SAF and SM-SAF to develop the SM–SPU-SAF algorithm. Eq. (27) can be written in the form of a full update equation

8 <1  :

0

g jei;D ðkÞj

if jei;D ðkÞj4g; otherwise;

In [15], it has been shown that the computational complexity of SAF for each input sampling period is approximately 3M þ 3NK, where K is the length of the channel filters of the analysis filter bank. But from [15] we obtain that the exact computational complexity of SAF is 3M þ 3NK þ 1 multiplications and 1 division. By comparing (14) and (27), it can be shown that the SSPU-SAF based on the first criterion needs 2M þ SL þ 3NK þ 1 multiplications, 1 division, and OðPÞ þ Plog2 S comparisons when using the heapsort algorithm [20]. The SSPU-SAF based on the second criterion slightly increases the

(34) 4

I is the identity matrix and 0 is the zeros matrix.

ARTICLE IN PRESS M.S.E. Abadi, J.H. Husøy / Signal Processing 88 (2008) 2463–2471

Table 1 Computational complexity of the SAF, SSPU-SAF, SM-SAF, and SM– SSPUSAF algorithms Algorithm

Multiplications

Divisions

SAF SSPU-SAF (based on the first criterion) SSPU-SAF (based on the second criterion) SM-SAF SM–SSPU-SAF (based on the first criterion) SM–SSPU-SAF (based on the second criterion)

3M þ 3NK þ 1 2M þ SL þ 3NK þ 1

1 1

2M þ SL þ 3NK þ 2

2

3M þ 3NK þ 1 2M þ SL þ 3NK þ 1

2 2

2M þ SL þ 3NK þ 2

3

computational complexity but leads to somewhat better performance. In such applications as network and acoustic echo cancellation, the adaptive filter may be required to have a large number of coefficients in order to model the underlying physical system with sufficient accuracy. Therefore, the reduction in the computational complexity will be M  SL which is large for these applications. In Section 7, we present several simulation results to show the performance of SSPU-SAF algorithm. In SAF, the filter vector adaptation needs 2M þ 1 multiplications and 1 division [15] and all the coefficients in each subband are updated at every adaptation. For the SM-SAF, this adaptation in each subband is related to the condition in (34). This relation determines that the filter coefficients in which subband should be updated at every adaptation. If the condition in (34) always becomes true (which in practice it does not), then the computational complexity of SM-SAF is 3M þ 3NK þ 1 multiplications and 2 divisions which is similar to the complexity of SAF. But the gains of applying the SM-SAF algorithm comes through the reduced number of required updates, which cannot be accounted for a priori, and an increased performance as compared to the SAF. In Section 7, we present several applications to show the ability of SM–SAF to decrease the overall computational complexity. In SM–SSPU-SAF algorithm, the filter coefficients are partially updated in each subband which again leads to additional reduction in the computational complexity. The adaptation in this algorithm is also related to the condition in (34). The computational complexity of SM–SSPU-SAF algorithms is similar to the SSPU-SAF for both selected criteria. This algorithm needs 2M þ SL þ 3NK þ 1 multiplications and 2 divisions based on the first criterion and 2M þ SL þ 3NK þ 2 multiplications and 3 divisions based on the second criterion. The number of comparison operations for both algorithms is OðPÞ þ P log2 S comparisons. Table 1 summarizes the computational complexity of the proposed algorithms.

7. Simulation results We demonstrate the performance of the proposed algorithms by several computer simulations in a system identification scenario. The unknown systems have 32 and 64 taps and are selected at random. The input signal, xðnÞ,

2467

is a fourth order autoregressive (AR(4)) signal generated according to5 xðnÞ ¼ 0:6617xðn  1Þ þ 0:3402xðn  2Þ þ 0:5235xðn  3Þ  0:8703xðn  4Þ þ wðnÞ,

(40)

where wðnÞ is a zero mean white Gaussian signal. The measurement noise, vðnÞ, with s2v ¼ 103 was added to the T noise free desired signal generated through dðnÞ ¼ ht xðnÞ, where ht is the true unknown filter vector. The adaptive filter and the unknown filter vector are assumed to have the same number of taps. For M ¼ 32 and 64, the eigenvalue spreads of the input signal signals are 1497 and 2456, respectively. The filter bank used in the SAFs was the four subband extended lapped transform (ELT) [22]. In all the simulations, the simulated learning curves were obtained by ensemble averaging over 200 independent trials. For M ¼ 32, the number of blocks (P) was set to 4 and for M ¼ 64, this parameter was set to 8. Also the pffiffiffiffiffiffiffiffi value of g was set to 5s2v [13]. Figs. 3 and 4 show the learning curves of SAF [15] and SSPU-SAF algorithms with M ¼ 32 for the two proposed selection criteria for the blocks. For SAF, we set m ¼ 0:5. To make the comparison fair, the step-sizes of SSPU-SAF were chosen to get approximately the same steady-state MSE as the SAF. Different values for the S (S ¼ 1; 2; 3) were employed in SSPU-SAF algorithm. By increasing the S parameter, the performance of SSPU-SAF will be close to the ordinary SAF. Also using the second criterion of Section 3. B leads to better performance especially for S ¼ 1. Selecting S ¼ 1 and using the second criterion corresponds to the exact SPU-SAF algorithm. Figs. 5 and 6 show the learning curves of SAF and SSPU-SAF algorithms for M ¼ 64 and for the two proposed selection criteria. For SAF, we set m ¼ 0:5 and for SSPU-SAF, different values of the S (S ¼ 1; 2; 4; 6) were employed. These figures show the better performance based on the second criterion especially for low values of S. Fig. 7 shows the learning curves of SAF [15] and SM-SAF algorithms for M ¼ 32. For the SAF algorithm, the step-size is set to m ¼ 1 (the same step-size that was used in [15]) and 0.1, respectively. As we can see, the SM-SAF algorithm has both fast convergence similar to that of SAF and a significantly lower steady-state MSE than ordinary SAF. Furthermore, the average numbers of updates in SM-SAF for each subband were 285, 129, 169 and 157, respectively, instead of 2000 for each subband in SAF algorithm. Fig. 8 shows the results for M ¼ 64. For the SAF algorithm, the step-size is set to 0.1 and 1. Again, the SM-SAF algorithm has both fast convergence similar to that of SAF and a significantly lower steady-state MSE than ordinary SAF. The average numbers of updates in SM-SAF for each subband were 692, 305, 408 and 388, respectively, instead of 5000 for each subband in SAF algorithm. Fig. 9 shows the learning curves of SSPU-SAF and SM–SSPU-SAF algorithms for M ¼ 32. The S parameter was set to 2 and the second block selection criterion was used. For the SSPU-SAF algorithm, the same values for the stepsizes (m ¼ 0:1; 1) were used. Again, the SM–SSPU-SAF has 5

The same type of input that was used in [21].

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30 20 10

(a) SAF [15] (b) SPU-SAF, S = 1

20

(d) SSPU-SAF, P = 8, S = 4

10

Input: Gaussian AR (4) (c) SSPU-SAF, S = 2 (d) SSPU-SAF, S = 3

(b) SPU-SAF, P = 8, S = 1 (c) SSPU-SAF, P = 8, S = 2

MSE in dB

MSE in dB

(b) SSPU-SAF,S = 1

0

30

(a) SAF[15] (b) SSPU-SAF, P = 4,S = 1 (c) SSPU-SAF, P = 4,S = 2 (d) SSPU-SAF, P = 4,S = 3

(e) SSPU-SAF, P = 8, S = 6 Input:Gaussian AR (4)

0

(c) SSPU-SAF, S = 2

−10

−10

−20

−20

(d) SSPU-SAF, S = 4 (e) SSPU-SAF, S = 6

(a) SAF

(a) SAF

−30

−30 0

500

1000

1500

0

2000

1000

×4 Sample Number Fig. 3. Learning curves of SAF and SSPU-SAF algorithms with M ¼ 32 according to block selection criterion no. 1. (Input: Gaussian AR(4).)

30

0

(c) SSPU-SAF, S = 2 (d) SSPU-SAF, S = 3

10

−10

−20

−20 (a) SAF

0

Input: Gaussian AR (4) (a) SAF,  = 0.1

0

−10

−30

(a) SAF [15],  = 0.1 (b) SAF [15],  = 1 (c) SM-SAF

20

MSE in dB

MSE in dB

(b) SPU-SAF, S = 1

(c) SM-SAF

(b) SAF,  = 1

−30 500

1000

1500

2000

0

500

×4 Sample Number Fig. 4. Learning curves of SAF and SSPU-SAF algorithms with M ¼ 32 according to block selection criterion no. 2. (Input: Gaussian AR(4).)

1000 1500 ×4 Sample Number

2000

Fig. 7. Learning curves of SAF and SM-SAF algorithms with M ¼ 32. (Input: Gaussian AR(4).)

30

30 (b) SSPU-SAF, S = 1

20

(a) SAF [15] (b) SSPU-SAF, P = 8, S = 1

MSE in dB

(d) SSPU-SAF, P = 8, S = 4

10

(e) SSPU-SAF, P = 8, S = 6 Input: Gaussian AR (4)

0 −10

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Input: Gaussian AR (4)

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Fig. 6. Learning curves of SAF and SSPU-SAF algorithms with M ¼ 64 according to block selection criterion no. 2. (Input: Gaussian AR(4).)

(a) SAF [15] (b) SPU-SAF, P = 4,S = 1 (c) SSPU-SAF, P = 4, S = 2 (d) SSPU-SAF, P = 4, S = 3

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(a) SAF

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Fig. 5. Learning curves of SAF and SSPU-SAF algorithms with M ¼ 64 according to block selection criterion no. 1. (Input: Gaussian AR(4).)

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4000

5000

Fig. 8. Learning curves of SAF and SM-SAF algorithms with M ¼ 64. (Input: Gaussian AR(4).)

ARTICLE IN PRESS M.S.E. Abadi, J.H. Husøy / Signal Processing 88 (2008) 2463–2471

30

30 (a) SSPU-SAF, P = 4, S = 2,  = 0.1 (b) SSPU-SAF, P = 4, S = 2,  = 1 (c) SM-SSPU-SAF, P = 4, S = 2

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500

1000

1500

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(a) SAF

−30 0

×4 Sample Number Fig. 9. Learning curves of SSPU-SAF and SM–SSPU-SAF algorithms with M ¼ 32 according to block selection criterion no. 2. (Input: Gaussian AR(4).)

30 (a) SSPU-SAF, P = 8,S= 2,  = 0.1 (b) SSPU-SAF, P = 8,S = 2,  = 0.5 (c) SM-SSPU-SAF, P = 8, S = 2

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(c) SM-SSPU-SAF

0

1000

2000

3000

4000

5000

×4 Sample Number Fig. 10. Learning curves of SSPU-SAF and SM–SSPU-SAF algorithms with M ¼ 64 according to block selection criterion no. 2. (Input: Gaussian AR(4).)

30 (a) SAF[15], µ = 0.1 (b) SM-SSPU-SAF, P = 4,S = 2 (c) SM-SSPU-SAF, P = 4, S = 3 (d) SM-SAF

MSE in dB

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(a) SAF,  = 0.1

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−10 Input: Gaussian AR (4)

−20 −30

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1000 1500 ×4 Sample Number

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Fig. 12. Learning curves of SAF, and SPU-SAF algorithms with M ¼ 32. (b) SPU-SAF with S ¼ 1, where the filter blocks are selected based on selection criterion no. 2 and (c) SSPU-SAF with S ¼ 2, where the filter blocks are randomly selected (Input: Gaussian AR(4).)

both fast convergence and low steady-state MSE features compared with ordinary SSPU-SAF. Also, the average numbers of updates in SM–SSPU-SAF for each subband were 413, 182, 247 and 213, respectively, instead of 2000 for each subband in SSPU-SAF algorithm. Furthermore in this algorithm, the filter coefficients are partially updated which again leads to additional reduction in the computational complexity. Fig. 10 shows the learning curves of SSPU-SAF and SM–SSPU-SAF algorithms for M ¼ 64. For the SSPU-SAF algorithm, the values for the step-sizes were set to m ¼ 0:1 and 0.5, respectively. As we can see, the SM–SSPU-SAF has both fast convergence and low steadystate MSE features. The average numbers of updates in SM–SSPU-SAF for each subband were 1033, 470, 621 and 559, respectively, instead of 5000 for each subband in SSPU-SAF algorithm. Fig. 11 shows the learning curves of SM-SAF and SM–SSPU-SAF algorithms with M ¼ 32 and for different values for S. As we can see, for S ¼ 3, the performance of the SM–SSPU-SAF will be very close to the SM-SAF. Fig. 12 presents the results for the random selection of the coefficient blocks to update. As we can see, the results based on the second criterion and with S ¼ 1 is better than the results based on the randomly selection of the filter coefficient blocks with S ¼ 2. Finally, we have presented Figs. 13 and 14. These figures show the number of coefficients updated in SMSAF and SM–SSPU-SAF algorithms with M ¼ 32 in each subband versus sample number for a single realization. Fig. 13 shows the results for SM-SAF and Fig. 14 shows the results for SM–SSPU-SAF with S ¼ 2. These figures show that when the filter coefficients in each subband (i ¼ 0; 1; 2; 3) will be updated during the adaptation.

(d) SM-SAF

0

500

1000

1500

2000

×4 Sample Number Fig. 11. Learning curves of SAF, SM-SAF and SM–SSPU-SAF algorithms with M ¼ 32 and for different values of S. (Input: Gaussian AR(4).)

8. Conclusions In this paper, the concepts of the selective partial updates and set-membership adaptive filtering were extended to the subband adaptive filters and the novel

ARTICLE IN PRESS M.S.E. Abadi, J.H. Husøy / Signal Processing 88 (2008) 2463–2471

No. of coeff. in update (i = 3)

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No. of coeff. in update (i = 2)

No. of coeff. in update (i = 1)

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Fig. 13. Number of coefficients updated in SM-SAF algorithm with M ¼ 32 in each subband versus sample number in a single realization (Input: Gaussian AR(4).)

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0

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Fig. 14. Number of coefficients updated in SM–SSPU-SAF algorithm with M ¼ 32 and S ¼ 2 in each subband versus sample number in a single realization. (Input: Gaussian AR(4).)

ARTICLE IN PRESS M.S.E. Abadi, J.H. Husøy / Signal Processing 88 (2008) 2463–2471

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