On the tracking performance of combinations of least mean squares and recursive least squares adaptive filters

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On the tracking performance of combinations of least mean squares and recursive least squares adaptive filters
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  ONTHETRACKINGPERFORMANCEOFCOMBINATIONSOFLEASTMEANSQUARESANDRECURSIVELEASTSQUARESADAPTIVEFILTERS V ´ ıtor H. Nascimento †  , Magno T. M. Silva †  , Luiz A. Azpicueta-Ruiz ‡  , and Jer ´ onimo Arenas-Garc´ ıa ‡† University of S˜ao Paulo, Brazil  ‡ Univ. Carlos III de Madrid, Spain { vitor,magno } @lps.usp.br  { lazpicueta, jarenas } @tsc.uc3m.es ABSTRACT Combinations of adaptive filters have attracted attention as a simplesolution to improve filter performance, including tracking properties.In this paper, we consider combinations of LMS and RLS filters, andstudy their performance for tracking time-varying solutions. We showthat a combination of two filters from the same family (i.e., two LMSor two RLS filters) cannot improve the performance over that of a sin-gle filter of the same type with optimal selection of the step size (orforgetting factor). However, combining LMS and RLS filters it is pos-sible to simultaneously outperform the optimum LMS and RLS fil-ters. In other words, combination schemes can achieve smaller errorsthanoptimallyadjustedindividualfilters. Experimentalworkinaplantidentification setup corroborates the validity of our results.  Index Terms —  Adaptive filters, convex combination, steady-stateanalysis, tracking performance, RLS algorithm, LMS algorithm. 1. INTRODUCTION Combinations of adaptive filters have gained considerable attentionlately, since they decrease the sensitivity of the filter to choices of pa-rameters such as the step size, forgetting factor or filter length (see,e.g., [1–5]). Using a combination of two filters with different step sizes, for example, one can obtain fast convergence and low steady-state misadjustment, or use the combination to find the optimum stepsize in a nonstationary environment [1]. In general, this combinationapproach is more robust than variable step-size schemes [5].In tracking of time-varying scenarios, combination schemes offerimproved tracking capabilities with respect to the component filters[4]. However, it has been noticed in simulations that the excess mean-square error (EMSE) obtained by the combination of two filters of thesame family [e.g., two least mean-squares (LMS) filters with differentstep sizes, or two recursive least-squares (RLS) filters with differentforgetting factors] will never be better than the performance of a singlefilter employing the optimum step size (or optimum forgetting factor)for a given nonstationary condition [1,6]. More recently, the combination of filters from different families(one LMS and one RLS) was proposed as a way to take advantage of the different tracking properties of LMS and RLS [5]. In fact, despitethe fast initial convergence provided by RLS, it was shown in [7] thatLMS may outperform RLS depending on how the optimum solutionchanges with time. Although a theoretical analysis and several simu-lations were provided in [5], it was not noticed that the combination The work of Nascimento and Silva was partly supported by CNPqunder Grants 303361/2004-2 and 302633/2008-1; and FAPESP underGrants 2008/04828-5 and 2008/00773-1. The work of Azpicueta-Ruiz andArenas-Garc´ıa was partly supported by MEC project TEC2008-02473 andCAM project S-0505/TIC/0223. Table1 . Parameters of the considered algorithms.Alg.  ρ i  M  − 1 i  ( n ) LMS  µ i  I  RLS  1  b R  i ( n ) = n X l =1 λ n − li  u ( l ) u T  ( l ) may actually obtain a smaller EMSE than its component filters evenwhen the optimum step size is used for LMS and the optimum forget-ting factor is used for RLS. In other words, the combination of filtersof different families may obtain a performance that would not be pos-sible with a combination of two filters of the same family, or with analgorithm that chooses the optimum stepsize forLMS(orthe optimumforgetting factor for RLS). In this paper we illustrate these facts boththrough theoretical analysis and simulations.The paper is organized as follows. In the next section, we presentthe data model and introduce the notation that will be used throughoutthe paper, reviewing also some results for the tracking performanceof LMS and RLS filters. Then, in Sec. 3 we recall some theoreticalresults regarding the performance of convex and affine combinationof adaptive filters in nonstationary environments. We also prove thatcombinations of filters of the same family cannot improve the perfor-mance over that of a single filter with optimum selection of the param-eters (i.e., step size or forgetting factor), and that this limitation can beovercome when filters of different families are combined. Several ex-amples that validate the analysis are provided in Sec. 4. Finally, Sec. 5presents the main conclusions of our work. 2. PROBLEMFORMULATION In this paper, we consider combinations of two LMS, two RLS or oneRLS and one LMS filters. The update laws for LMS and RLS may bewritten as w i ( n ) = w i ( n − 1) +  ρ i M  i ( n ) u ( n ) e i ( n ) ,  (1)where the index  i  = 1 , 2  refers to each filter in the combination, w i ( n )  ∈  R M  is the coefficient vector of each filter at time  n ,  ρ i is a step-size parameter,  u ( n )  ∈  R M  is the input regressor vector,and  e i ( n )  is the estimation error. For RLS, M  i ( n )  ∈  R M  × M  is anestimate of the inverse of the regressor autocovariance matrix,  R   =E { u ( n ) u T  ( n ) } , and can be computed in an efficient way using thematrix inversion lemma or, if lattice algorithms are used, its explicitevaluation may be avoided [8]. Table 1 lists the values of the differentparameters in (1) for each class of the considered filters.The estimation error is given by e i ( n ) =  d ( n ) − y i ( n ) ,  (2)  Table 2 . Analytical expressions for the steady-state EMSEs of LMSand RLS.Alg.  ζ  LMS  µ i σ 2 v  Tr { R  } +  µ − 1 i  Tr { Q } 2 RLS  ν  i σ 2 v M   +  ν  − 1 i  Tr { QR  } 2 Table 3 . Optimum tracking parameters ( µ o  and  ν  o ) and steady-stateEMSEs ( ζ  o ) for LMS and RLS.Alg.  µ o ,ν  o  ζ  o LMS s   Tr { Q } σ 2 v  Tr { R  } p  σ 2 v  Tr { R  } Tr { Q } RLS s  Tr { QR  } σ 2 v M  p  σ 2 v M   Tr { QR  } where  y i ( n ) =  w i ( n − 1) T  u ( n ) ,  i  = 1 , 2  are the filter outputs, and d ( n )  is the desired response. As it is well-known, there exists a linearregression model relating  d ( n )  and u ( n ) , such that d ( n ) = w T  o u ( n ) +  v ( n ) ,  (3)where  v ( n )  is white and zero-mean measurement noise with variance σ 2 v , uncorrelated with u ( n ) , and w o  provides the optimum linear leastmean-squares estimate of   d ( n )  given  u ( n )  [8]. We assume in thispaper, as it is common in studies of adaptive filters, that  u ( n )  and v ( n )  are stationary zero-mean processes, and moreover that  v ( n )  isindependent of  u ( m )  for all  m,n .Define also the  a priori  errors  e a,i ( n ) = [ w o − w i ( n − 1)] T  u ( n ) .Under the stated assumptions, the mean-square error  E { e 2 i ( n ) }  maybe shown to be [8] E { e 2 i ( n ) }  =  ζ  i ( n ) +  σ 2 v , where  ζ  i ( n ) = E { e 2 a,i ( n ) }  is the so-called excess mean-square error(EMSE) of each filter.The optimum weight vector w o  is usually not constant in practice.A common approach to model its variations is through the brownianmotion model below, which allows a tractable analysis w o ( n ) = w o ( n − 1) + q ( n ) ,  (4)where { q ( n ) } is a sequence of i.i.d. vectors with zero mean and auto-correlation matrix Q = E { q ( n ) q T  ( n ) } .Using this model and (1)–(3), and assuming sufficiently small stepsize and forgetting factor sufficiently close to 1, it can be shown thatthe steady-state EMSE [i.e.,  ζ  i  = lim n →∞ ζ  i ( n ) ] of LMS and RLSareasgiveninTable2[5,7,8]. Forconvenience, wewilloftenuse ν  i  =1 − λ i . Since  ν  i  plays in the expressions for RLS a similar role to thatof   µ i  for LMS, we will also refer to  ν  i  as a “step size”. Differentiatingthe expressions in Table 2 with respect to either  µ i  or  ν  i , one cancompute the optimum step sizes  µ o  and  ν  o  for a given environment(i.e., values of  Q , R  and noise variance  σ 2 v ). These optimum values,along with the resulting optimum EMSEs for each filter, are given inTable 3 [7,8]. Through these expressions, it was shown in [7] that, despite itsslow initial convergence, LMS may present better tracking perfor-mance than RLS, depending on the values of  Q and R  . In particular,if   Q  is proportional to  R  , the optimum steady-state EMSE of LMSwill be smaller than that of RLS. On the contrary, if  Q is proportionalto  R  − 1 , RLS will present better performance. When  Q  is propor-tional to I   (the identity matrix), both algorithms will present similarbehavior [7].In the next section we introduce combinations of adaptive filters,and prove that the tracking performance of the combination of twoLMS or two RLS filters is lower bounded by the values of Table 3, butthe combination of one RLS with one LMS algorithm may achieve abetter performance. 3. CONVEXCOMBINATIONSANDOPTIMALTRACKING One promising way of increasing the performance of adaptive filters isto run two or more filters in parallel, and combine their outputs con-structing an overall output given by y ( n ) =  η ( n − 1) y 1 ( n ) + [1 − η ( n − 1)] y 2 ( n ) . Up to now, good results have been obtained with both a convex com-bination model, in which the mixing parameter  η ( n )  is constrained toremain in the interval  [0 , 1]  [1], and an affine combination model, inwhich  η ( n )  may be any real number [3]. In the first case, the mixingparameter is usually updated using an auxiliary variable  a ( n ) , accord-ing to a ( n ) =  a ( n − 1) + ˜ µ a ( n ) e ( n )∆ e ( n )  η ( n − 1) [1 − η ( n − 1)] ,η ( n ) = 11 + exp[ − a ( n )] , in which  ˜ µ a ( n )  is a (possibly normalized) step size,  e ( n ) =  d ( n ) − y ( n )  is the overall estimation error, and  ∆ e ( n ) =  e 2 ( n ) − e 1 ( n ) . Ingeneral, to avoid slow adaptation close to  η  = 1  or  η  = 0 ,  a ( n )  is con-strained (by simple saturation) to the interval  [ − a + ,a + ] . A commonchoice for  a +  is 4.For affine combinations, one possible method for updating themixing parameter is through the recursion η ( n ) =  η ( n − 1) +  µ a ( n ) e ( n )∆ e ( n ) . In both cases, it is convenient to choose a normalized step size  ˜ µ a ( n ) using an estimate  p ( n )  of   E { ∆ e 2 ( n ) } , such that  p ( n ) =  λ p  p ( n − 1) + (1 − λ p )∆ e 2 ( n ) , where  λ p  is a forgetting factor, and  ˜ µ a ( n ) =  µ a / [ δ   +  p ( n )] ,  δ   beinga small regularization term [6,9]. It can be shown that the optimum mixing parameter in steady stateis given by [1,3,6] η ∗  =  ζ  2  − ζ  12 ζ  1  − 2 ζ  12  +  ζ  2 ,  (5)where  ζ  12  = lim n →∞ E { e a, 1 ( n ) e a, 2 ( n ) }  is the steady-state cross-EMSE between both filters in the combination, given in Table 4 forthe different combination possibilities considered in this paper [5]. Forconvex combinations  η ∗  is given by (5) only if the value falls in theinterval  [0 , 1] , otherwise  η ∗  = 0  (resp. 1), if (5) is negative (resp.larger than 1).The EMSE of the combination, using the optimum  η ∗ , is given by ζ  ∗  =  ζ  1 ζ  2  − ζ  212 ζ  1  − 2 ζ  12  +  ζ  2 (6)  Table 4 . Analytical expressions for the steady-state cross-EMSE of the considered combinations.Combination  ζ  12 µ 1 -LMS and  µ 2 -LMS  µ 1 µ 2  Tr { R  } σ 2 v  + Tr { Q } µ 1  +  µ 2 λ 1 -RLS and  λ 2 -RLS  ν  1 ν  2 Mσ 2 v  + Tr { QR  } ν  1  +  ν  2 λ 1 -RLS and  µ 2 -LMS  µ 2 ν  1  σ 2 v  Tr ˘ Σ ¯ + Tr ˘ Q Σ ¯ , where Σ  ( ν  1 I  +  µ 2 R  ) − 1 R  .We remark that the optimality here is with respect to the choice of   η only, which we denote by the subscript ∗ .We will now search for the optimum  ζ  ∗  with respect to the stepsizes for the particular case of a combination of two filters of the samefamily. Consider first the combination of two LMS filters. Assumingthat the optimal  η ∗  is selected in steady state (which is usually a goodapproximation for the considered recursions [6, 9]), the steady-state EMSE of the combination will be given by (6) with  ζ  1  and  ζ  2  givenby the first row in Table 2, and  ζ  12  given by the first row in Table 4.Differentiating  ζ  ∗  with respect to  µ 2 , and after some manipulations,we obtain d ζ  LMS ∗ d µ 2 =  − 12(Tr { Q }− µ 21 σ 2 v  Tr { R  } ) 2 (Tr { Q }− µ 22 σ 2 v  Tr { R  } )( µ 1 µ 2 σ 2 v  Tr { R  } + Tr { Q } ) 2 ( µ 1  +  µ 2 ) 2  , where we have used the superscript  LMS to emphasize that we are con-sidering a combination of two LMS filters. From this expression onecan see that  d ζ  LMS ∗  / d µ 2  = 0  if   µ 2  =  µ o  (notice the factor depend-ing on  µ 2  in the numerator), so the minimum value of   ζ  LMS ∗  is attainedwhen one of the component filters has optimum step size µ o . Note that,duetothesymmetryofthecombination, wewouldreachthesamecon-clusion if we had differentiated with respect to  µ 1  (this is easily seenif one replaces  η  by  1 − η  in all expressions).Furthermore, if we substitute  ζ  2  =  ζ  LMSo  and  µ 2  =  µ o  in theexpressions for  η ∗  from (5) and for  ζ  12  from Table 4, we conclude that η ∗ o  = 0 . This means that ζ  LMS ∗ o  =  ζ  LMSo  , where  ζ  LMS ∗ o  is the optimal EMSE of the combination, both with respectto the mixing parameters and the step sizes of the constituent filters. Inother words, the smallest EMSE obtainable with a combination of twoLMS algorithms is exactly equal to that obtained with a single LMSfilter with optimum step size  µ o .Just the same conclusion is obtained for a combination of two RLSfilters. The only difference is that for two RLS filters, the derivative d ζ  RLS ∗  / d ν  2  reads d ζ  RLS ∗ d ν  2 =  − 12(Tr { QR  }− ν  21 σ 2 v M  ) 2 (Tr { QR  }− ν  22 σ 2 v M  )( ν  1 ν  2 σ 2 v M   + Tr { QR  } ) 2 ( ν  1  +  ν  2 ) 2  , and again we see that  ν  2  =  ν  o  minimizes the overall steady-state errorof the combination, and  ζ  RLS ∗ o  =  ζ  RLSo  .The above results allow us to conclude that, although a combina-tion of two LMS (or two RLS) filters can improve the tracking per-formance when the degree of nonstationarity is not known  a priori or time-varying, the steady-state EMSE of the combination is lowerbounded by the optimal EMSE of an individual filter from the same 00.5110310.30.1 −35−34−33−32 µ 2  /  µ o α    E   M   S   E ,   [   d   B   ] 00.5110310.30.1−35−34−33−32 µ 2  /  µ o α    E   M   S   E ,   [   d   B   ] Fig. 1 . Steady-state EMSE of a combination of two LMS filters forvarying  α  and  µ 2 , and  µ 1  = 0 . 3 µ o  (left) and  µ 1  = 3 µ o  (right).family. In the next section, we will illustrate that this is not the case fora heterogeneous combination of one LMS and one RLS filters. Sincefor the combination of one RLS and one LMS filters the expressionsbecome too complex for an analytical approach, we will proceed toshow via examples that for this case it is possible to obtain an over-all steady-state EMSE strictly smaller that the minimum of   ζ  LMSo  and ζ  RLSo  . 4. EXAMPLES In this section we include several experiments for the identification of a time-varying system. Three sets of experiments have been consid-ered: the first one consists of a combination of two LMS filters withdifferent step sizes; two RLS filters with different forgetting factors arecombined in the second group of simulations; and the last experimentimplements a combination of one LMS and one RLS filters, both of them with optimal step sizes (i.e.,  µ o  and  ν  o ).In all cases, the unknown plant  w o , of length  M   = 7 , was ini-tialized with random values from interval  [ − 1 , 1] , being  w o (0) =[ . 9003 , − . 5377 ,. 2137 , − . 028 ,. 7826 ,. 5242 , − . 0871] T  . Then, the so-lutionischanged ateach iterationaccording totherandom-walk model(3), with a covariance matrix of  q ( n )  given by Q =  γ  » α  R  Tr ( R  ) + (1 − α )  R  − 1 Tr ( R  − 1 ) – ,  (7)where constant  γ   has been selected to be  γ   = 10 − 5 , so that Tr ( Q ) = γ  , and  α  ∈  [0 , 1]  is a control parameter that allows to tradeoff betweena situation with  Q  ∝  R   (for  α  = 1 ), for which  ζ  LMSo  < ζ  RLSo  , and Q ∝ R  − 1 ( α  = 0 ), in which the reverse situation occurs.The input signal is the output of a first-order AR model with trans-ferfunction [1 − a 2 ] / (1 − az  − 1 ) using a  = 0 . 8 , fedwithi.i.d. Gaussiannoise with variance  σ 2 u  =  17 , so that Tr ( R  ) = 1 . The output additivenoise is i.i.d. Gaussian with zero-mean and variance  σ 2 v  = 10 − 2 .Regarding the adjustment for the combinations, we have used convexcombinations with fixed step size  µ a  = 100 , while the step sizes of the constituent filters are selected as explained below.All estimated steady-state EMSEs have been obtained by averag-ing  25000  runs of the algorithms once the filters have completely con-verged, and  100  independent runs.To start with, we will consider the combination of two LMS filters.Note that in this case  µ o  = √  10 − 3 independently of the value of   α .Fig.1depictsthesteady-stateEMSEofthecombinationfordifferent α and  µ 2 , and for two different values of the step size for the first compo-nent,  µ 1  = 0 . 3 µ o  and  µ 1  = 3 µ o . In both subfigures, we observe a flatregion for which the combination inherits the performance of the filterwith step size  µ 1 . As predicted by our analysis in the previous section,  00.5110310.30.1 −38−34−30−26  ν 2  /   ν o α    E   M   S   E ,   [   d   B   ] 00.5110310.30.1−37−35−33−31  ν 2  /   ν o α    E   M   S   E ,   [   d   B   ] Fig. 2 . Steady-state EMSE of a combination of two RLS filters forvarying  α  and  ν  2 , and  ν  1  =  . 001  (left) and  ν  1  =  . 0186  (right).the optimal behavior of the combination is observed when  µ 2  =  µ o .Therefore, the combination performs in this situation similarly to anLMS filter with optimal step size.Similar conclusions can be extracted for the convex combinationof two RLS filters. Fig. 2 illustrates the behavior of such a combinationscheme for different values of   α  and  ν  2 . In this case,  ν  o  changes with α . Therefore, in this situation we have selected  ν  1  = 0 . 001  (left panelof Fig. 2) and  ν  1  = 0 . 0186  (right panel), respectively smaller andlarger than the optimal step sizes for  α  = 0  and  α  = 1 . For each  α ,we explore values for the step size of the second filter in the range from ν  o / 10  to  10 ν  o . Again, as predicted by the analysis, the best behaviorresults for  ν  2  =  ν  o , showing that  ζ  RLS ∗ o  =  ζ  RLSo  .We conclude the section considering the more interesting case of acombination including oneLMSandoneRLSfilters. We willillustratethat it is possible for this heterogeneous combination to outperform thesmallest EMSEs that could be achieved by any individual LMS or RLSfilters. To this end, let us select for each  α  the optimal step sizes for theLMS and RLS components according to Table 3. The upper panel of Fig. 3 displays the theoretical steady-state EMSEs of both constituentfilters and of their combination, and shows good agreement with thereal curves obtained through simulation (intermediate panel). We cansee that for all values of   α  other than  α  = 0  and  α  = 1 , the combinedLMS-RLS scheme reduces the individual EMSEs of both components,thus leading to the interesting result that  ζ  LMS-RLS ∗ o  <  min [ ζ  LMSo  ,ζ  RLSo  ] ,i.e., this combined scheme is able to improve the tracking capabilitiesof optimal LMS and RLS filters.It is also interesting to pay attention to the optimal values of themixing parameter (bottom panel of Fig. 3). In first place, we see thatfor  α  ∈  [0 , 1]  the optimal mixing parameter lies in interval  [0 , 1] , i.e.,affine combinations can be expected to work equally well –but notbetter– than convex combinations for the considered scenario. It isalso important to remark that no gains over the tracking performanceof optimal LMS or RLS can occur when Q  ∝  R   ( α  = 1 ) or Q  ∝ R  − 1 ( α  = 0 ), since in these cases optimal selections of the mixingparameters are  η  = 0  and  η  = 1 , respectively. 5. CONCLUSIONS In this paper we have studied the tracking performance of combina-tions of LMS and RLS filters. We have provided theoretical and em-pirical evidence that the steady-state EMSE of a combination of twofilters of the same family is lower bounded by the optimal EMSE of a filter of the same family. However, heterogeneous combinations of one LMS and one RLS filters have been shown to simultaneously re-duce the EMSEs of both optimal LMS and RLS filters, thus providinga way to obtain filters with superior tracking capabilities. 0 0.2 0.4 0.6 0.8 1−38−36−34−32 α    E   M   S   E ,   d   B   µ o − LMS  ν o − RLSCombination   0 0.2 0.4 0.6 0.8 1−38−36−34−32 α    E   M   S   E ,   d   B   µ o − LMS  ν o − RLSCombination   0 0.2 0.4 0.6 0.8 100.20.40.60.81 α      η   TheoreticalExperimental Fig. 3 . Steady-state performance of an adaptive combination of oneLMS and one RLS filters with optimal step sizes ( µ o  and  ν  o , respec-tively). The figure displays, from top to bottom, the theoretical EMSEof both constituent filters and of the combination for different valuesof   α , the observed EMSEs obtained through simulation, and the theo-retical and simulated steady-state values of   η ( n ) . 6. REFERENCES [1] J. Arenas-Garc´ıa, A. R. Figueiras-Vidal, and A. H. Sayed, “Mean-squareperformance of a convex combination of two adaptive filters,”  IEEE Trans. Signal Process. , vol. 54, no. 3, pp. 1078–1090, Mar. 2006.[2] Y. Zhang and J. Chambers, “Convex combination of adaptive filters for avariable tap-length LMS algorithm,”  IEEE Signal Process. Lett. , vol. 13,no. 10, pp. 628–631, Oct. 2006.[3] N. J. Bershad, J. C. M. Bermudez, and J.-Y. Tourneret, “An affine com-bination of two LMS adaptive filters—transient mean-square analysis,”  IEEE Trans. Signal Process. , vol. 56, no. 5, pp. 1853–1864, May 2008.[4] J. Arenas-Garc´ıa, A. R. Figueiras-Vidal, and A. H. Sayed, “Trackingproperties of a convex combination of two adaptive filters,” in  Proc.2005 IEEE Workshop on Stat. Signal Process. , Bourdeaux, France.[5] M. T. M. Silva and V. H. Nascimento, “Improving the tracking capabilityof adaptive filters via convex combination,”  IEEE Trans. Signal Process. ,vol. 56, no. 7, pp. 3137–3149, July 2008.[6] R. Candido, M. T. M. Silva, and V. H. Nascimento, “Affine combinationsof adaptive filters,” in  Conf. Rec. of the 42nd Asilomar Conf. on Sign.,Syst. & Comp. , 2008.[7] E. Eweda, “Comparison of RLS, LMS, and sign algorithms for trackingrandomly time-varying channels,”  IEEE Trans. Signal Process. , vol. 42,no. 11, pp. 2937–2944, Nov. 1994.[8] A. H. Sayed,  Fundamentals of Adaptive Filtering , Wiley-Interscience,2003.[9] L. A. Azpicueta-Ruiz, A. R. Figueiras-Vidal, and J. Arenas-Garc´ıa, “A normalized adaptation scheme for the convex combination of two adap-tive filters,” in  Proc., ICASSP 2008  , Las Vegas, NV, USA, pp. 3301–3304.
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