NI Jin-gen(倪錦根)

(School of Electronic and Information Engineering, Soochow University, Suzhou 215006, China)

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Subband adaptive filter with variable reusing order of coefficient vectors

NI Jin-gen(倪錦根)

(School of Electronic and Information Engineering, Soochow University, Suzhou 215006, China)

To increase the convergence rate of the improved normalized subband adaptive filter, a simple but effective method is presented to change the reusing order of coefficient vectors of the adaptive filter. At the beginning of adaptation the algorithm just uses its current coefficient vector to update the adaptive filter to maintain fast convergence rate, while in steady state it employs several most recent coefficient vectors to update the adaptive filter to reduce misalignment. Simulation results show that the proposed algorithm can obtain both fast convergence rate and small steady-state misalignment.

adaptive filtering; subband adaptive filter; reusing coefficient vector; misalignment

Subband adaptive filtering algorithms can increase convergence rate for correlated input signals and/or reduce computational cost as compared to the well-known least-mean-square (LMS) and normalized LMS (NLMS) algorithms. Due to these advantages, subband adaptive filters (SAFs) have been widely used in system identification, network and acoustic echo cancellation, active noise control, and so on[1]. In the conventional SAF[2], each subband uses an individual adaptive sub-filter in its own adaptation loop, and thus their convergence rate is decreased by aliasing and band-edge effects[3]. This SAF aims to obtain low computational cost.

To address the problem of aliasing and band-edge effects in the conventional SAF, two weighted SAFs have been proposed from different points of view in Ref. [4] and Ref. [5], respectively. Based on the principle of minimum disturbance, Lee and Gan proposed a new SAF, called the normalized SAF (NSAF)[6], which can be viewed as a subband generalization of the NLMS. Its central idea is to employ the subband input signals, normalized by their respective subband input variances, to update the coefficient vectors of the adaptive filter. If the normalization term in theith subband, i.e., the squaredl2-norm of theith subband input vector, is replaced by the power of theith subband input vector, then the update equation of the NSAF coincides with the two weighted SAFs[7].

Similar to the NLMS, however, if the step-size (regularization parameter) of the NSAF is set to a large (small) value to obtain fast convergence rate, its steady-state misalignment is large[8]. Inspired by the NLMS with reusing coefficient vectors (NLMS-RC)[9], an improved NSAF (INSAF) has been proposed in Refs.[10-11] to reduce steady-state misalignment, which is derived by minimizing the sum of the squaredl2-norms of the differences between the updated coefficient vector and the past coefficient vectors, subject to constraints on the updated coefficient vector. It has been shown that the steady-state misalignment of the INSAF can be largely reduced, but its convergence rate decreases as the reusing order of coefficient vectors increases.

In this paper, a method for the INSAF to change the reusing order of coefficient vectors to address the above problem is presented. The concept of changing the reusing order of coefficient vectors was first proposed for the NLMS with reusing coefficient vector (NLMS-RC) in Ref. [12]. The proposed method in this paper is different from that one. Simulation results show that the proposed algorithm can obtain both fast convergence rate and small steady-state misalignment.

1 Preliminaries

1.1 Signal model

Fig.1 Structure of the NSAF and INSAF

Consider the desired response which arises from the modeld(n)=uT(n)wo+η(n),wherenisthetimeindex, (·)Tstandsfortransposition,u(n)=[u(n),u(n-1),…,u(n-M+1)]TrepresentstheinputvectoroflengthM,whichconsistsoftheMmostrecentinputsamples,wodenotesthecoefficientvectoroftheunknownsystemWo(z),andη(n)isthesystemnoisewhichisusuallyassumedtobezero-meanandindependentandidenticaldistributed.

1.2INSAF

(1)

subject to

(2)

whereRis the reusing order of coefficient vectors. Using the method of Lagrange multipliers, we can obtain the update equation of the INSAF

(3)

whereμisthestep-size, δistheregularizationparameter,and

(4)

IfRis set to 1, then Eq.(3) reduces to the classical NSAF in Ref. [6].

2 Proposed algorithm

2.1 Derivation of the proposed algorithm

As presented earlier, the steady-state misalignment of the INSAF decreases, but its convergence rate also decreases, as the reusing order of coefficientvectors increases. If the filter is expected to obtain both fast convergence and small steady-state misalignment, a reasonable method is to change the reusing order of coefficient vectors of the adaptive filter according to the adaptation phase. At the beginning of adaptation, the adaptive filter just uses the current coefficient vector to maintain fast convergence rate, while in steady state the adaptive filter uses several most recent coefficient vectors to reduce misalignment. To this end, we modify Eq.(3) and Eq.(4), respectively, as

(5)

and

(6)

Itisknownthatthemeansquareerror(MSE)canbeusedtoindicatetheadaptationphase.Thesteady-stateexcessMSE(EMSE)oftheINSAFwithR=1 (i.e., NSAF) can be written as[13]

(7)

(8)

WhentheMSEreachitssteady-statevalueJss(k), the filter operates in steady state. According to the statement at the beginning of this subsection, we set the reusing order of coefficient vectors to

(9)

(10)

and

(11)

(12)

(13)

ri(k)=λri(k-1)+(1-λ)ui(k)ei,D(k)

(14)

and

(15)

BycomparingtheproposedINSAFwithvariablereusingorderofcoefficientvectors(INSAF-VRC)totheNLMS-RCwithvariablereusingorderofcoefficientvectors(NLMS-VRC)inRef. [12],itcanbeseenthatthemethodsforvaryingthereusingorderaredifferentfromeachother.First,theNLMS-VRCvariesthereusingorderintherangebetween1andthemaximumreusingorder,whiletheINSAF-VRCjustswitchesitbetween1andthemaximumreusingorder,whichcanreducetheimplementationcomplexity.Second,theNLMS-VRCdeterminesthereusingorderbycomparingtheerrorsignaltothesteady-stateMSE,whiletheINSAF-VRCdeterminesitbycomparingthepoweroftheerrorsignaltothesteady-stateMSE,whichavoidstheeffectoferrorsignalfluctuationswhendeterminingthereusingorder.Third,theNLMS-VRCassumesthesystemnoisepowertobeknown,whiletheINSAF-VRCnotonlytakesintoaccountthesystemnoisepowerestimationbutalsoconsiderstheperiodicupdateofthereusingorder,whichwillbepresentedinthenextsubsection.

2.2Periodicupdateofthereusingorder

ThereusingorderoftheproposedINSAF-VRCisnotverysensitivetotheupdatedelaycausedbyestimationofpowersofthesubbandsystemnoisesinsteadystate,andthereforewecanusethemethodproposedinRef. [17]toestimatethepowersperiodically,i.e.,

(16)

where

(17)

ri(k)=

(18)

andβ(k)=mod(k, P)withPapositiveinteger,whichwecallupdateperiod.

R(k)=

(19)

Ithasbeenshownthatthecomputationalcostofestimatingthepowersofallsubbandsystemnoisesisabout3Mmultiplications[16], while the computational cost of the INSAF is about 4Mmultiplications. Using the periodic update method, the computational cost of estimating the powers of all subband system noises reduces to about 3M/P, which can be ignored as compared to the computational cost of the INSAF ifP>4, and therefore the computational cost of the INSAF-VRC is approximately equal to that of the INSAF.

3 Simulation results

Fig.2 Impulse response of the unknown system to be estimated

Fig.3comparestheperformanceoftheNLMS-VRCandINSAF-VRC.Forthesakeoffairness,weassumethatthepowerofthesystemnoiseisknownforbothalgorithms[12].ThemaximumreusingordersoftheNLMS-VRCandINSAF-VRCarebothsettoRmax=6.TheforgettingfactorandmodifiedfactoroftheINSAF-VRC, λandφ,aresetto0.98and1.1,respectively.OnecanseethattheINSAF-VRCconvergesfasterthantheNLMS-VRCiftheirsteady-statemisalignmentissettothesamevalue.

Fig.3 Misalignment curves of the NLMS-VRC and INSAF-VRC

Fig.4comparestheperformanceoftheNSAF,INSAF,andINSAF-VRC.Allstep-sizesofsimulatedalgorithmsaresetto0.5,thereusingorderoftheINSAFandthemaximumreusingorderoftheINSAF-VRC,RandRmax,aresetto6,respectively.Theparametersλandφaresetto0.993and1.1,respectively.ItisseenthattheNSAFsuffersfromlargesteady-statemisalignmentandtheINSAFsuffersfromslowconvergencerate,whiletheINSAF-VRCwithupdateperiod1≤P≤4canobtainbothfastconvergencerateandsmallsteady-statemisalignment.

Fig.4 Misalignment curves of the INSAF-VRC with different update periods 1≤P≤4

4 Conclusion

Thispaperproposedanimprovedsubbandadaptivefilterwithvariablereusingorderofcoefficientvectors.Atthebeginningofadaptation,theproposedalgorithmjustutilizesthemostrecentcoefficientvectoroftheadaptivefiltertomaintainfastconvergencerate,whileinsteady-stateitemploysseveralrecentcoefficientvectorstoreducemisalignment.Thus,theproposedalgorithmswitchesbetweentheNSAFandINSAFtoobtainbothfastconvergencerateandsmallsteady-statemisalignment.Moreover,aperiodicupdateofthereusingorderwasalsodiscussedtoreducecomputationalcost.Simulationresultsweregiventoverifythesuperiorperformanceoftheproposedalgorithm.

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(Edited by Cai Jianying)

10.15918/j.jbit1004-0579.201524.0314

TN 911.72 Document code: A Article ID: 1004- 0579(2015)03- 0375- 06

Received 2015- 01- 20

Supported by the National Natural Science Foundation of China (61471251; 61101217); the Natural Science Foundation of Jiangsu Province of China (BK20131164)

E-mail: jni@suda.edu.cn