Steady-state mean-square deviation analysis of improved normalized subband adaptive filter

Signal Processing 106 (2015) 49–54

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Steady-state mean-square deviation analysis of improved normalized subband adaptive filter Jae Jin Jeong a, Keunhwi Koo a, Gyogwon Koo a, Sang Woo Kim a,b,n a

Department of Electrical Engineering, Pohang University of Science and Technology (POSTECH), Gyungbuk, Republic of Korea Department of Creative IT Excellence Engineering and Future IT Innovation Laboratory, Pohang University of Science and Technology (POSTECH), Gyungbuk, Republic of Korea b

a r t i c l e i n f o

abstract

Article history: Received 20 January 2014 Received in revised form 2 May 2014 Accepted 26 June 2014 Available online 4 July 2014

A new minimization criterion for the normalized subband adaptive filter (NSAF), which is called improved NSAF (INSAF), was introduced recently to improve the performance of the steady-state mean-square deviation (MSD). However, the steady-state MSD analysis of the INSAF was not studied. Therefore, this paper proposes a general solution of steady-sate MSD analysis of the INSAF algorithm, which is based on the substitution of the past weight error vector in the weight error vector. The simulation shows that our theoretical results correspond closely with the computer simulation results in various environments. & 2014 Elsevier B.V. All rights reserved.

Keywords: Adaptive filter Normalized subband adaptive filter (NSAF) Steady-state analysis Mean-square deviation (MSD)

1. Introduction Adaptive filters have been used in many applications such as in system identification, channel estimation, inverse system modeling, active noise control, and echo cancelation [1–3]. The normalized least-mean-square (NLMS) algorithm is a widely used adaptive filter algorithm owing to its low computational complexity and ease in implementation. Recently, a new minimization criterion was introduced to improve the steady-state misalignment in the NLMS algorithm [4]. This criterion minimizes the sum of the squared Euclidean norms of the differences between the updated weight vector and the past weight vectors, subject to the constraint on the updated weight vector of the adaptive filter. This algorithm is called the NLMS with reused coefficient-vector (NLMS-RC) [5]. However, similar to the NLMS algorithm, the NLMS-RC algorithm still suffers from a slow convergence rate for n

Corresponding author. Tel.: þ 82 54 279 5018; fax: þ82 54 279 2903. E-mail addresses: [email protected] (J.J. Jeong), [email protected] (K. Koo), [email protected] (G. Koo), [email protected] (S.W. Kim). http://dx.doi.org/10.1016/j.sigpro.2014.06.026 0165-1684/& 2014 Elsevier B.V. All rights reserved.

correlated input signals. To overcome this drawback, the normalized subband adaptive filter (NSAF) algorithm was introduced to improve the convergence rate of the NLMS algorithm for highly correlated input signals [6]. The NSAF algorithm divides the input signal into multiple subbands to whiten the input signal in each subband. To improve the steady-state misalignment, the minimization criterion was extended to the subband adaptive filter in [7], called the improved NSAF (INSAF). The mean-square deviation (MSD) analysis of adaptive filters has been extensively studied [5,8–10]. In [5], the steady-state MSD of the NLMS-RC algorithm was presented, which was based on the implementation of energy conservation. This analysis result showed good agreement with the simulation results. However, this analysis is not a general solution, i.e., this analysis only covers a special case when the number of reusing coefficient vectors is two and three. In addition, analysis of the INSAF algorithm is rarely addressed. In the present paper, we propose the first general solution for the steady-state MSD analysis of the INSAF algorithm for highly correlated input signals. This theoretical analysis is based on replacing the past weight error vector in the weight

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error vector. The simulations illustrated that the theoretical results matched the computer simulation results in various environments consisting of the signal-to-noise ratio (SNR), number of subbands, and tap length. The remainder of this paper is organized as follows. Section 2 briefly introduces the INSAF algorithm. Section 3 provides a theoretical steady-state MSD analysis of the INSAF algorithm. The theoretical results and the computer simulation results are compared in Section 4, as obtained from the simulation results. Finally, Section 5 presents our conclusions. 2. Review of improved normalized subband adaptive filter

dðnÞ ¼ u ðnÞwopt þ ηðnÞ;

H i ðzÞ ¼ ∑ hi ðjÞz  j ; j¼0

i ¼ 0; 1; …; N  1;

^ þ 1Þ  wðk ^  pÞ‖2 ∑ ‖wðk

^ subject to dD ðkÞ ¼ UT ðkÞwðkþ 1Þ;

ð3Þ where P denotes number of the reusing coefficient vectors and dD ðkÞ ¼ ½d0;D ðkÞ; d1;D ðkÞ; …; dN  1;D ðkÞT ;

ð4Þ

UðkÞ ¼ ½u0 ðkÞ; u1 ðkÞ; …; uN  1 ðkÞ:

ð5Þ

^ ðk þ1Þ ¼ w

1 P 1 ^ ðk  pÞ ∑ w Pp¼0 þ μUðkÞΦ

ð1Þ

where the input vector is defined as uðnÞ ¼ ½uðnÞ; uðn  1Þ; …; uðn  M þ 1ÞT . wopt ¼ ½w0 ; w1 ; …; wM  1 T is the tap-weight vector of an unknown system to be estimated using an adaptive filter. ηðnÞ denotes the measurement noise and the length of the adaptive filter is M. ðÞT represents the transpose of a matrix or vector. Signals dðnÞ; uðnÞ; ηðnÞ, and filter output y(n) are divided into N subbands using the passing analysis filter given as L1

P 1

min

^ ðk þ 1Þ p ¼ 0 w

The update equation of the INSAF algorithm is [7]

The NSAF structure is shown in Fig. 1. The system identification for input signal uðnÞ and desired signal d(n), which originates from the system, is expressed as T

state misalignment [4]. The optimization criterion is extended to improve the performance of the conventional NSAF algorithm under a low SNR [7]:

1

! 1 P 1 ^ ðk  pÞ ; ðkÞ dD ðkÞ  UT ðkÞ ∑ w Pp¼0 ð6Þ

where dD ðkÞ ¼ UT ðkÞwopt þ ηD ðkÞ;

ð7Þ

ηD ðkÞ ¼ ½η0 ðkÞ; η1 ðkÞ; …; ηN  1 ðkÞT ;

ð8Þ

ΦðkÞ ¼ diag½‖u0 ðkÞ‖2 ; ‖u1 ðkÞ‖2 ; …; ‖uN  1 ðkÞ‖2 :

ð9Þ

ð2Þ

where N is the number of subbands and L represents the length of the prototype filter. di;D ðkÞ and yi;D ðkÞ are derived by critically decimating di(n) and yi(n), respectively. The ith ^ decimated subband output is defined as yi;D ðkÞ ¼ uTi ðkÞwðkÞ, where ui ðkÞ ¼ ½ui ðkNÞ; ui ðkN  1Þ; …; ui ðkN  M þ 1ÞT and ^ wðkÞ is estimation of the unknown system. n represents the time index of the original signal, and k represents the index of the decimated signal. For the case of a low SNR, a new minimization criterion for the NLMS algorithm is proposed to improve the steady-

3. Steady-state analysis of the INSAF algorithm By subtracting wopt by (6), we can rewrite (6) as follows in terms of the weight error vector, which is defined by ~ ^ wðkÞ 9 wopt  wðkÞ: ~ ðk þ1Þ ¼ w

1 P 1 ~ ðk  pÞ ∑ w Pp¼0  μUðkÞΦ

1

! 1 P 1 ~ ðkÞ U ðkÞ ∑ w ðk  pÞ þ ηD ðkÞ : Pp¼0 T

ð10Þ To make the analysis tractable, we use the following assumptions: Assumption 1. ηðnÞ is a zero-mean signal white Gaussian distribution with variance σ 2η . ~ are statistically Assumption 2. ηi;D ðkÞ, ui ðkÞ, and wðkÞ independent. Assumption 3. The ith subband input signal is close to a white signal, i.e., Efui ðkÞuTi ðkÞg  σ 2ui ðkÞIM and EfuTi ðkÞ ui ðkÞg  Mσ 2ui ðkÞ, where IM is an M  M identity matrix. Assumption 4. The fluctuation in the subband input signal energy from one iteration to the next iteration is small. The matrix form of (10) is expressed as 1 ~ ~ ~ þ1Þ ¼ WðkÞb wðk  μUðkÞΦ ðkÞðUT ðkÞWðkÞb þ ηD ðkÞÞ;

Fig. 1. Structure for the normalized subband adaptive filter.

ð11Þ

J.J. Jeong et al. / Signal Processing 106 (2015) 49–54

~ The cross term between the weight error matrix WðkÞ and the measurement noise vector is neglected by applying Assumptions 1 and 2. By applying Assumptions 3 and 4, we derive [11]

where ~ ~ ~ 1Þ; …; wðk ~ P þ1Þ; WðkÞ ¼ ½wðkÞ; wðk

ð12Þ

1 b ¼ ½1; 1; …; 1T : P

ð13Þ

We can rearrange (11) as follows: ~ þ 1Þ ¼ ðIM  μUðkÞΦ wðk  μUðkÞΦ

1

1

~ ðkÞU ðkÞÞWðkÞb T

ðkÞηD ðkÞ:

51

  2μN μ2 N T EfΛ ðkÞΛðkÞg  1  þ IM ; M M

ð17Þ

σ 2ηi;D : 2 i ¼ 0 M σ ui ðkÞ

ð18Þ

ð14Þ EfηTD ðkÞΦ

~ ~ ðkÞwðkÞg. By taking the MSD is defined as MSDðkÞ 9 Efw squared Euclidean norm and mathematical expectation of both sides of the equation, we can obtain T

1

ðkÞηD ðkÞg þ Cross term;

N1

ðkÞηD ðkÞg ¼ ∑

Substituting (15) by (17) and (18), we obtain

~ ~ T ðkÞΛT ðkÞΛðkÞWðkÞbg MSDðk þ1Þ ¼ Efb W T

þ μ2 EfηTD ðkÞΦ

1

  o 2μN μ2 N T n ~ T ~ þ b E W ðkÞWðkÞ b MSDðk þ 1Þ ¼ 1  M M

ð15Þ

σ 2ηi;D : 2 i ¼ 0 M σ ui ðkÞ

where

N1

ΛðkÞ ¼ IM  μUðkÞΦ

1

ðkÞUT ðkÞ:

þ μ2 ∑

ð16Þ

ð19Þ

−5

NMSD (dB)

−10

−15

−20

−25 0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0.7

0.8

0.9

1

0.7

0.8

0.9

1

Step−size −5

NMSD (dB)

−10 −15 −20 −25 −30 0.1

0.2

0.3

0.4

0.5

0.6

Step−size −15

NMSD (dB)

−20 −25 −30 −35 −40 0.1

0.2

0.3

0.4

0.5

0.6

Step−size Fig. 2. Steady-state NMSD curves of the INSAF for the white input signals as a function of the step size: (a) SNR ¼ 5 dB, (b) SNR ¼10 dB, and (c) SNR ¼20 dB [M ¼ 512, N¼ 4].

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J.J. Jeong et al. / Signal Processing 106 (2015) 49–54

~ T ðkÞWðkÞg ~ The calculation of EfW is needed to obtain the MSD. T ~ ~ EfW ðkÞWðkÞg is expressed as 82 ~ T ðkÞwðkÞ ~ w > > > >6 T < 6 w ~ ~ ðk 1ÞwðkÞ T ~ ðkÞWðkÞg ~ EfW ¼E 6 6 > ⋮ > 4 > > : ~T ~ w ðk  P þ 1ÞwðkÞ

 μUðkÞΦ

~ T ðkÞwðk ~ 1Þ w

⋯

⋯ ⋯

1

) T ~ ðk  1Þ : ðkÞηD ðkÞ w

ð22Þ

39 > > > 7> ~ T ðk  1Þwðk ~  P þ 1Þ 7= w 7 : 7> 5> > > ~ T ðk  P þ 1Þwðk ~ P þ1Þ ; w ~ T ðkÞwðk ~ P þ1Þ w

ð20Þ

By applying Assumptions 1 and 2, we can obtain

The diagonal term of the matrix is expressed as ~ T ðkÞWðkÞg ~ diag½EfW ¼ diag½MSDðkÞ; MSDðk  1Þ; …; MSDðk P þ 1Þ:

(

ð21Þ

~ 1Þg ¼ AE ~ T ðkÞwðk Efw

)

P 1

~ T ðk  1  pÞwðk ~  1Þ ∑ w

p¼0

The upper diagonal entries of the matrix are derived as follows: ( 1 P 1 ~ T ðkÞw ~ T ðk  1Þg ¼ E Efw ΛðkÞ ∑ w~ ðk  1  pÞ Pp¼0

~ ðk  2Þwðk ~  1Þg ¼ AðMSDðk  1Þ þ Efw T

~ 1ÞgÞ; ~ ðk  PÞwðk þ⋯ þEfw T

ð23Þ

NMSD (dB)

−7

−14

−21

−28 0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0.7

0.8

0.9

1

0.7

0.8

0.9

1

Step−size

NMSD (dB)

−7

−14

−21

−28 0.1

0.2

0.3

0.4

0.5

0.6

Step−size

NMSD (dB)

−7

−14

−21

−28 0.1

0.2

0.3

0.4

0.5

0.6

Step−size Fig. 3. Steady-state NMSD curves of the INSAF for the correlated input signals as a function of the step size: (a) N ¼ 4, (b) N ¼8, and (c) N ¼ 16 [M ¼ 512, SNR ¼20 dB, Input: Gaussian AR(1), pole at 0.9].

J.J. Jeong et al. / Signal Processing 106 (2015) 49–54

The remaining terms are derived using the same procedure. Therefore, we can drive the following:

where   1 μN 1 : A¼ P M

2

ð24Þ

1 6 A 6 T 6 1  ðP  1ÞA ~ ð1ÞW ~ ð1Þg ¼ 6 EfW 6 ⋮ 4

By substituting the weight error vector by the past weight error vector, the term at the right-hand side of (23) is calculated as ( T

3 7

A 7 1  ðP  1ÞA 7

⋯ ⋯

1

7MSDð1Þ: 7 5

In steady state, we derive   N1 σ 2ηi;D 1 1 MSDð1Þ þ μ2 ∑ ; MSDð1Þ ¼ B 2 P 1 ðP  1ÞA i ¼ 0 M σ ui ð1Þ

T

~  1  PÞgÞ ~ T ðk 2Þwðk þ ⋯ þ Efw

~  1Þg ¼ AE ~ T ðk PÞwðk Efw

A 1  ðP  1ÞA

ð26Þ

~ 3Þg ~ ðk  2Þwðk ¼ AðMSDðk  2Þ þEfw

(

⋯

)

P1

p¼0

⋮

A 1  ðP  1ÞA

A 1  ðP  1ÞA

~ T ðk 2Þwðk ~  2  pÞ ∑ w

~ ðk 2Þwðk ~  1Þg ¼ AE Efw

ð27Þ where limk-1 σ

)

P 1

~ T ðk PÞwðk ~  2  pÞ ∑ w

B ¼ 1

p¼0

~ ðk PÞwðk ~  2Þg ¼ AðEfw T

2 ui ðkÞ ¼

σ

2 ui ð1Þ

and

2μN μ N þ : M M 2

ð28Þ

Finally, the steady-state MSD equation is obtained as

~  3Þg ~ ðk  PÞwðk þEfw T

MSDð1Þ ¼

~ 1 PÞgÞ: ~ ðk  PÞwðk þ⋯ þEfw T

Using the above procedure and assumption, i.e., MSDð1Þ ¼ MSDðkÞ ¼ MSDðk  1Þ ¼ ⋯ ¼ MSDðk  P þ 1Þ in steady state, we can obtain

N1

∑

ð29Þ

This section presents the steady-state MSD of the INSAF algorithm through computer simulations for the system identification application. The unknown system is randomly generated with 512 or 1024 taps (M¼512 or M¼1024). The length of the adaptive filter is the same as that of the unknown system. uðnÞ is generated by filtering

þ ðP  1ÞA2 MSDð1Þ þðP  1Þ2 A3 MSDð1Þ þ ⋯ A MSDð1Þ; 1 ðP  1ÞA

σ 2ηi;D : Pð1  PAþ AÞ  B i ¼ 0 Mσ 2ui ð1Þ μ2 Pð1  PA þ AÞ

4. The simulation results

~ T ðkÞw ~ T ðk 1Þg ¼ AMSDð1Þ Efw



53

ð25Þ

where limk-1 MSDðkÞ ¼ MSDð1Þ.

NMSD (dB)

−8

−16

−24

−32 0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0.7

0.8

0.9

1

Step−size

NMSD (dB)

−8

−16

−24

−32 0.1

0.2

0.3

0.4

0.5

0.6

Step−size Fig. 4. Steady-state NMSD curves of the INSAF for the correlated input signals as a function of the step size: (a) M ¼ 512 and (b) M ¼ 1024 [SNR ¼20 dB, N ¼8, Input: Gaussian AR(2)].

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J.J. Jeong et al. / Signal Processing 106 (2015) 49–54

a zero-mean white Gaussian random sequence using the following models: G1 ð z Þ ¼

1 ; 1  0:9z  1

G2 ðzÞ ¼

1 : 1  0:1z  1  0:8z  2

ð30Þ

The SNR is defined as SNR ¼ 10 log10 ðEfy2 ðnÞg=Efη2 ðnÞgÞ, where yðnÞ ¼ uT ðnÞwopt . ηðnÞ is added to the system output y(n), resulting in an SNR of 5, 10, or 20 dB. The number of subbands N is 4, 8, and 16 for prototype filter length L ¼32, 64, and 128, respectively. The simulation results are obtained according to different step-sizes in the range from 0.1 to 1.0, which guarantees stability. The normalized ~ T ðkÞwðkÞg=ð ~ MSD (NMSD), which is defined as 10 log10 ðEfw wTopt wopt ÞÞ, is evaluated by averaging over 10 independent trials. Fig. 2 shows the steady-state NMSD curves of the theoretical and simulation results under various SNRs such as 5, 10, and 20 dB. The length of the adaptive filter is 512. The number of subbands is 4 and the length of the prototype filter is 32. Through this simulation, we can see that the theoretical results exhibit a good agreement with the simulation results under different SNRs. Fig. 3 shows the comparison of the steady-state NMSD curves of the theoretical and simulation results for various numbers of subbands (N¼4, 8, 16). The input signals are generated by filtering a zero-mean white Gaussian signal through G1 ðzÞ. The adaptive filter tap-length is 512. As can be seen, the theoretical results match the simulation results for different N. However, the small difference between the theoretical and the simulation result exists in Fig. 3(a). The subband input signal is not close to the white signal because the number of subbands is not sufficiently large. Fig. 4 illustrates that the steady-state NMSD curve of the theoretical and simulation results for different M. The input signals are calculated by passing a zero-mean white Gaussian signal through G2 ðzÞ. The number of subbands is 8 and the length of the prototype filter is 64. In this simulation, the theoretical results are very close to the simulation results for various tap lengths. 5. Conclusion In this paper, we proposed a general solution for the steady-state MSD analysis of the INSAF algorithm by

substituting the past weight error vector in the weight error vector. The theoretical results of the INSAF algorithm are presented through computer simulation for system identification applications. The simulation results match the theoretical results in terms of the steady-state MSD in various environments.

Acknowledgment This research was supported by the MSIP (Ministry of Science, ICT and Future Planning), Korea, under the “IT Consilience Creative Program” (NIPA-2014-H0201-141001) supervised by the NIPA (National IT Industry Promotion Agency) and by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (NRF2013R1A1A2058975). References [1] A.H. Sayed, Fundamentals of Adaptive Filtering, Wiley, Hoboken, NJ, 2003. [2] S. Haykin, Adaptive Filter Theory, Prentice-Hall, New Jersey, 2002. [3] K.A. Lee, W.S. Gan, S.M. Kuo, Subband Adaptive Filtering: Theory and Implementation, Wiley, Chichester, UK, 2009. [4] H. Cho, C.W. Lee, S.W. Kim, Deviation of a new normalized least mean squares algorithm with modified minimization criterion, Signal Process. 89 (4) (2009) 692–695. [5] S. Kim, J. Lee, W. Song, Steady-state analysis of the NLMS algorithm with reusing coefficient vector and a method for improving its performance, in: ICASSP, 2011, pp. 4120–4123. [6] K.A. Lee, W.S. Gan, Improving convergence of the NLMS algorithm using constrained subband updates, IEEE Signal Process. Lett. 11 (September (9)) (2004) 736–739. [7] J. Ni, Improved normalised subband adaptive filter, Electron. Lett. 48 (March (6)) (2012) 320–321. [8] H.C. Shin, A.H. Sayed, Mean-square performance of a family of affine projection algorithms, IEEE Trans. Signal Process. 52 (January (1)) (2004) 90–102. [9] K.A. Lee, W.S. Gan, S.M. Kuo, Mean-square performance analysis of the normalized subband adaptive filter, in: Proceedings of the Asilomar Conference on Signals, System and Computers, vol. 40, 2006, pp. 248–252. [10] W. Yin, A.S. Mehr, Stochastic analysis of the normalized subband adaptive filter algorithm, IEEE Trans. Circuits Syst. II: Reg. Pap. 58 (May (5)) (2011) 1020–1033. [11] J.J. Jeong, K. Koo, G.T. Choi, S.W. Kim, A variable step size for normalized subband adaptive filters, IEEE Signal Process. Lett. 19 (December (12)) (2012) 906–909.