Acoust. Sci. & Tech. 28, 2 (27) PAPER #27 The Acoustical Society o Japan Method estimating relection coeicients o adaptive lattice ilter and its application to system identiication Kensaku Fujii 1;, Masaaki Tanaka 1;y, Naoto Sasaoka 2;z and Yoshio Itoh 2;x 1 Department o Computer Engineering, University o Hyogo 2 Faculty o Engineering, Tottori University ( Received 27 April 26, Accepted or publication 5 August 26 ) Abstract: In this paper, we propose a method o estimating the relection coeicients o an adaptive lattice ilter. In this method, conventional adaptive algorithms, or example, the normalized least mean square (NLMS) algorithm, are used or the estimation. In general, the relection coeicients are estimated as cross-correlation coeicients between orward and backward prediction errors in each stage o the adaptive lattice ilter. Accordingly, two divisions in each stage, and eectively doubling the number o stages, are required. A problem is that the processing cost o division is higher than that o multiplication, especially in cheap digital signal processors (DSPs). Hence, the reduction o the number o divisions is strongly desired in practical use. The proposed technique can decrease the number o divisions to one, provided that the NLMS algorithm is used. Moreover, in the application o the adaptive lattice ilter, system identiication is also important. In this paper, we present a technique or the application. The technique is derived rom the proposed method. Keywords: Lattice iler, Adaptive algorithm, Division, System identiication, Linear prediction PACS number: 43.6.Mn [doi:1.125/ast.28.98] 1. INTRODUCTION Lattice ilters have an advantage o lower sensitivity to round-o error in comparison with inite impulse response (FIR) ilters [1]. This round-o error aects the perormance o ilters, and especially, this error is serious in ixed-point processing. On the other hand, system design using ixed-point processing is strongly desired as a means o reducing the production cost. The low sensitivity o lattice ilters is an attractive characteristic or such system design. A problem is that the ixed-point processing type o digital signal processor (DSP) has higher processing costs or division than or multiplication, which is signiicant, especially or a cheap DSP. Unortunately, an adaptive lattice ilter requires many such divisions or the estimation o relection coeicients, provided that conventional methods are used. The conventional methods individually estimate the orward and backward relection coeicients at each stage o adaptive lattice iltering; thereore the total number o required divisions is twice the number o stages. e-mail: ujiken@eng.u-hyogo.ac.jp y Presently with Mitsubishi Electric Co. z e-mail: sasaoka@ele.tottori-u.ac.jp x e-mail: itoh@ele.tottori-u.ac.jp Even when both relection coeicients are assumed to be equal, the number o the divisions is only halved. In this paper, we propose a method requiring only one division. This method reduces the number o divisions by employing conventional adaptive algorithms applied to the FIR ilter, or example, the normalized mean square (NLMS) algorithm [2]. This employment is made possible by noting that the purpose o an adaptive lattice ilter is to minimize the inal orward prediction error. Actually, the inal orward prediction error is calculated as the product sum o the orward relection coeicients and the backward prediction errors occurring at a sample time. The other components, the backward relection coeicients and the orward prediction error, do not act on the inal orward prediction error at the same sample time; they do at the next sample time. The structure or calculating the product sum shows that the inal orward prediction error is obtained as the response o an FIR ilter. Moreover, this states that the orward relection coeicients can be estimated using the conventional adaptive algorithms applied to the FIR ilter. For example, the application o the NLMS algorithm to the estimation can decrease the number o divisions to only one. The remaining backward relection coeicients can be obtained as replicas o the orward relection coeicients, 98
K. FUJII et al.: METHOD ESTIMATING REFLECTION COEFFICIENTS OF ADAPTIVE LATTICE FILTER Fig. 1 Coniguration o adaptive lattice ilter. reerring to the Burg method [1] guaranteeing that the relection coeicients are less than unity. In this paper, we next present a technique or applying the adaptive lattice ilter to system identiication. Conventional methods, such as the time-update recursion method, estimate the relection coeicients so as to minimize not the identiication error but the prediction error. Thereore, the application o adaptive lattice ilters is essentially limited to the linear prediction system. However, the lattice ilter has another advantage that the stability discrimination is easy. The technique, proposed in this paper, is applicable or the synthesis o an inverse ilter. 2. ADAPTIVE LATTICE FILTER Figure 1 depicts a coniguration o an adaptive lattice ilter. In this coniguration, we can see a unit consisting o one sample time delay z 1, two multipliers (designated as, ðmþ), and two adders. The multipliers, j ðmþ and b j ðmþ, are also named the orward and backward relection coeicients, which are estimated every sample time j so as to minimize the orward and backward prediction errors, j ðmþ and b j ðmþ, respectively. The adaptive lattice ilter is ormed cascading the units as many as needed. So ar, various methods o estimating the relection coeicients have been proposed [1]. However, all the methods are common in the policy o individually minimizing the orward and backward prediction errors at each stage and also in the principle o minimizing them by removing their cross-correlation components (designated as, or example, j ðm 1Þ and b j 1 ðm 1Þ). The time-update recursion method represents such methods estimating the relection coeicients [1]. The other methods can be regarded as variations o the timeupdate recursion method. This time-update recursion method estimates the orward and backward relection or example, j ðmþ and b j coeicients, or example, j ðmþ and b j j ðmþ ¼R jðmþ=p b j ðmþ ðmþ, as ð1þ and b j ðmþ ¼R jðmþ=p j ðmþ; where the denominators and the numerator are calculated as P j ðmþ ¼Pj 1 ðmþþ 2; jðm 1Þ ð3þ P b j ðmþ ¼Pb j 1 ðmþþ 2; b j 1ðm 1Þ ð4þ and R j ðmþ ¼R j 1 ðmþþ j ðm 1Þb j 1 ðm 1Þ using the orward and backward prediction errors, j ðm 1Þ and b j 1 ðm 1Þ, and is a constant smaller than unity. Here, note that the time-update recursion requires giving the previously estimated values as the initial values, that is, P ðmþ, Pb ðmþ and Pb 1 ðmþ. As recognized rom Eqs. (1) and (2), the time-update recursion individually estimates the orward and backward relection coeicients as the cross-correlation coeicients between the orward and backward prediction errors at each stage. This individual estimation characterizes the adaptive lattice ilter; however, it increases the number o divisions. Since division entails ar higher processing cost than multiplication, the use o many divisions is undesirable in practical use. This linear prediction can also be perormed using an FIR ilter. In this case, conventional adaptive algorithms become available. For example, using the NLMS algorithm decreases the number o divisions to one. However, the FIR ilter involves the problem that its conversion into an ininite impulse response (IIR) ilter that works stably is troublesome. 3. APPLICATION OF NLMS ALGORITHM In this paper, we derive a new method o estimating the relection coeicients. This method can apply conventional adaptive algorithms to the estimation; thereby, in the case o using the NLMS algorithm, the number o divisions ð2þ ð5þ 99
Acoust. Sci. & Tech. 28, 2 (27) Fig. 2 Structure o j ðmþ yielded at sample time j. necessary or the estimation decreases to one. We derive the new method by taking notice o the structure yielding the inal orward prediction error j ðmþ. As seen rom the coniguration o the lattice ilter shown in Fig. 1, j ðmþ can be expressed as j ðmþ ¼ j ðþ XM m¼1 j ðmþb j 1ðm 1Þ: This expression shows that the components used or the synthesis o j ðmþ at sample time j are only the orward relection coeicients and the backward prediction errors. The other components, the backward relection coeicients and the orward prediction errors, do not act on the inal orward prediction error j ðmþ at the same sample time; it is the next sample time when they contribute to the synthesis. Figure 2 shows the coniguration o the ilter ormed by the components directly related to the synthesis o j ðmþ. Clearly, this coniguration drawn with the solid lines gives an FIR ilter, which can be regarded as a prediction error ilter. Moreover, the second term o Eq. (6), P M m¼1 j ðmþb j 1ðm 1Þ, is a prediction value o j ðþ. In this coniguration, the orward relection coeicients can be estimated using a conventional adaptive algorithm. The orward relection coeicients are obtained as the coeicients minimizing the inal orward prediction error j ðmþ. For example, the mth orward relection coeicient j ðmþ can be estimated using jþ1ðmþ ¼j ðmþþ jðmþb j 1 ðm 1Þ ; ð7þ X M 2 b j 1 ði 1Þ where is a constant called step size. Reerring to the Burg method [1] guaranteeing that the relection coeicients are less than unity, we assume that the backward relection coeicients are approximate to the orward relection coeicients. Then, the backward relection coeicients can be obtained as replicas o the orward i¼1 ð6þ Prediction Error (db) 1 2 3 (1) FIR (NLMS) (2) Lattice (NLMS) (3) Lattice (Time update) 1 2 Fig. 3 Convergence properties provided by proposed and conventional methods: (1) FIR ilter using NLMS algorithm, (2) Proposed prediction method, (3) Timeupdate recursion method. relection coeicients estimated using Eq. (7). Figure 3 shows the convergence properties obtained by applying the proposed and conventional methods. In this example, the prediction errors shown in the vertical axis is calculated under the ollowing conditions: (1) The input signal x j (¼ j ðþ) is generated by eeding white Gaussian noise, n j, into a ilter whose transer unction is expressed as 1 XðzÞ ¼ ; ð8þ 1 2 cos z 1 þ 2 z 2 where ¼ :9, and ¼ =4, corresponding to the resonance requency o 1 khz when the sampling requency is 8 khz. Incidentally, this ilter is modeled on the noise o a jet an discharging exhaust gas to prevent it rom illing in a tunnel. (2) The weight is given as ¼ 1 =M; consisting o the step size provided to the NLMS algorithm,, and the number o stages, M. (3) ¼ :1 and M ¼ 64. ð9þ 1
K. FUJII et al.: METHOD ESTIMATING REFLECTION COEFFICIENTS OF ADAPTIVE LATTICE FILTER (4) The prediction error is shown as the average o K samples, d n ¼ 1 log 1 e j n j g 2 2 x j ; ð1þ where K ¼ 512, e j ¼ jðmþ (lattice ilter) ; ð11þ x j y j (FIR ilter) and y j is the output o the FIR ilter. This simulation result shows that the proposed method can provide almost the same convergence property as the conventional time-update recursion method does and higher convergence speed than the FIR ilter using the NLMS algorithm. Here, note that the conventional time-update recursion requires the prior estimation o P j ðmþ and Pb j ðmþ. An inappropriate estimation may diverge the prediction error. We next veriy the validity o the proposed method by applying it to speech analysis. Figure 4 shows the accumulated prediction errors calculated as D k ¼ 1 log 1 X k e j n j g 2 j¼1 X k 2 x j j¼1 ; ð12þ applying the proposed and conventional methods to the speech signal depicted above, where M ¼ 128 is selected Accumulated Prediction Error (db) 5 1 Speech Signal (1) FIR (NLMS) (2) Lattice (NLMS) (3) Lattice (Time recursion) 15 2 4 Number o Samples (k) Fig. 4 Accumulated prediction errors obtained by applying the proposed and conventional methods to a speech signal: (1) FIR ilter using the NLMS algorithm, (2) the proposed method, (3) time-update recursion method. Accumulated Prediction Error (db) 5 1 (2) Lattice (NLMS) 15 2 4 Number o Samples (k) as the desirable number o stages or the noise suppression system [3], and also ¼ :1 is given considering the perormance o tracking the change in phonemes. This result shows that the proposed method provides almost the same perormance as the time-update recursion method, on average, and is superior to the FIR ilter using the NLMS algorithm. Figure 5 also shows the initial part o the accumulated prediction errors plotted in Fig. 4. Clearly, the time-update recursion method provides more prediction error than the other methods, especially beore 1, samples. This is caused by inappropriate initial values ðmþ essential or the time-update recursion method. This indicates that the selection o the initial values is important in the case o applying the timeupdate recursion method to speech signal. given or the P j ðmþ and Pb j Speech Signal (3) Lattice (Time recursion) (1) FIR (NLMS) Fig. 5 Initial part o the accumulated prediction errors plotted in Fig. 4, where (1) FIR ilter using the NLMS algorithm (dot line), (2) the proposed method (solid line), (3) time-update recursion method (short dash line). 4. APPLICATION TO SYSTEM IDENTIFICATION Conventional methods estimate the relection coeicients so as to remove the auto-correlation components rom the input signal. The adaptive lattice ilter is accordingly used or the linear prediction system and hardly applied to the system identiication modeling an adaptive ilter on an unknown system. On the other hand, the lattice ilter has the advantage that the veriication o its stable perormance is easy. This advantage is useul or applications requiring the ormation o an inverse ilter. We present a technique or applying such a lattice ilter to the system identiication. As mentioned in the preceding chapter, the inal orward prediction error j ðmþ can be regarded as the 11
Acoust. Sci. & Tech. 28, 2 (27) Fig. 6 Modiication or adapting lattice ilter to system identiication. response o the FIR ilter. This structure yielding the inal orward prediction error gives a hint or applying the adaptive lattice ilter to the system identiication. This application is started with the expansion o the response o a FIR ilter as X M m¼ HðmÞX j ðmþ ¼HðÞX j ðþþ XM m¼1 HðmÞX j ðmþ; ð13þ where HðmÞ is the mth coeicient o the FIR ilter, and X j ðmþ is the mth tap output o the FIR ilter. On the other hand, the inal orward prediction error given in Eq. (6) can be rewritten as j ðmþ ¼ j ðþþ XM m¼1 j ðmþgb j 1ðm 1Þ: ð14þ A comparison o Eqs. (13) and (14) shows that the dierence between the FIR and lattice ilters is only between their irst terms, namely, the presence o HðÞ. The structures o the two responses completely correspond when ð Þ j ðþ is inserted into the irst stage, as shown in Fig. 6. Here, note that the signs o the adders used in Figs. 1 and 6 are dierent. By inserting ð Þ j ðþ and using j ðþ ¼b j 1 ð 1Þ; ð15þ the inal prediction error shown in Fig. 6 can be rewritten as j ðmþ ¼ XM m¼ ð Þ j ðmþb j 1 ðm 1Þ: ð16þ Then, the structures o Eqs. (13) and (16) become completely equivalent. This indicates that the system identiication using the lattice ilter becomes possible by estimating the relection coeicients so as to minimize the dierence between the response o an unknown system, g j, and the inal orward prediction error j ðmþ given in Eq. (16). For example, using the NLMS algorithm, the relection coeicients can be estimated as where jþ1ðmþ ¼j ðmþþ y j j ðmþ bj 1 ðm 1Þ ; ð17þ X M 2 b j 1 ði 1Þ y j ¼ g j þ n j ð18þ is the response o the unknown system, corrupted by the additive noise n j. As mentioned above, the backward relection coeicients can also be obtained as replicas o the orward relection coeicients. Figure 7 shows the convergence properties obtained by applying the NLMS algorithm to the adaptive lattice and FIR ilters, where the identiication error shown on the vertical axis is calculated under the ollowing conditions: (1) Two kinds o reerence signal are used: white Gaussian noise and the output obtained by inputting white Gaussian noise into a ilter whose transer unction is expressed as i¼1 1 HðzÞ ¼ : ð19þ 1 :8z 1 (2) The impulse response samples o the unknown system are given as normal random numbers with exponential decay. (3) The number o the impulse response samples is also equal to the number o taps o the adaptive ilter, M ¼ 64. (4) ¼ :1. (5) The power ratio o the reerence signal to the additive noise is 4 db. (6) The identiication error is calculated using where K ¼ 1. E n ¼ 1 log 1 g j j ðmþg 2 2 g j ; ð2þ 12
K. FUJII et al.: METHOD ESTIMATING REFLECTION COEFFICIENTS OF ADAPTIVE LATTICE FILTER Identiication Error (db) 2 Lattice 4 FIR 6 1 2 (a) White Gaussian noise Identiication Error (db) 2 4 Random Copy 6 5 1 Fig. 8 Convergence properties calculated under condition that backward relection coeicients are ixed to regular random numbers. Identiication Error (db) 2 4 FIR Lattice 6 2 4 (b) First order IIR ilter output Fig. 7 Convergence properties provided by proposed (Lattice) and conventional methods (FIR). This result, which is expressed as the average o E n obtained by 2 trials, shows that the application o the lattice ilter to the system identiication becomes possible by adopting the proposed method. As mentioned above, the results shown in Fig. 7 are obtained using replicas o the orward relection coeicients as the backward relection coeicients. Here, we examine the eect o the backward relection coeicients on the system identiication. Figure 8 shows the convergence properties calculated using regular random numbers as the backward relection coeicients. This result indicates that arbitrary constants can be given to the adaptive lattice ilter as the backward relection coeicients. We have hence veriied the convergence property obtained when the constants reducing the auto-correlation o the reerence signal are given as the backward relection coeicients. The simplest example o such a reerence signal is the output o the irst-order IIR ilter deined by Eq. (19). Figure 9 shows the convergence property calculated by ixing only the backward relection coeicient j b ð1þ to.8, corresponding to the coeicient o the irstorder IIR ilter. In this result, the convergence speed is improved. These results, shown in Figs. 7 and 9, indicate that the proposed method can provide perormance Identiication Error (db) 2 4 FIR Lattice 6 2 4 Fig. 9 Convergence properties calculated under condition that backward relection coeicients are ixed to auto-correlation coeicient o reerence signal. equivalent or superior to that o the conventional methods using the FIR ilter. 5. CONCLUSION In this paper, we have proposed a method that can apply conventional adaptive algorithms to the estimation o relection coeicients. For example, when the NLMS algorithm is applied, the number o divisions thereby decreases to one. The computer simulation results presented in this paper show that the proposed method provides almost the same prediction perormance as the conventional time-update recursion method. Moreover, we have presented a technique or applying the lattice ilter to system identiication and have shown that almost the same perormance as that o the conventional method. The proposed method has some advantages other than those mentioned in this paper [4,5]. Hence, our subsequent studies will ocus on a noise reduction system [3] using the proposed method and the application o the technique to a system requiring an inverse ilter. 13
Acoust. Sci. & Tech. 28, 2 (27) REFERENCES [1] S. Haykin, Introduction to Adaptive Filters (Macmillan Publishing Co., New York, 1984), Chaps. 3 and 4. [2] J. Nagumo and A. Noda, A learning method or system identiication, IEEE Trans. Autom. Control, AC-12, 282 287 (1967). [3] T. Amitani, S. Miyata, K. Maeda, K. Fujii and Y. Itoh, Experimental veriication o the proposed microphone array system, Tech. Rep. IEICE, EA24-121 (25). [4] M. Tanaka, K. Fujii, N. Sasaoka, Y. Itoh and Y. Fukui, Fast convergence o system identiication using adaptive lattice ilter, Tech. Rep. IEICE, EA23-145 (24). [5] M. Tanaka, K. Fujii and Y. Hata, A system identiication method using relection wave, Tech. Rep. IEICE, EA25-18 (25). 14