New Approach of Performance Analysis of RLS Adaptive Filter Using a Block-wise Processing

Transcription

1 EUROEAN ACADEMIC RESEARCH Vol. V, Issue 2/ May 207 ISSN Impact Factor: (UIF) DRJI Value: 5.9 (B+) New Approach of erformance Analysis of RLS Adaptive Filter Using a Bloc-wise HAMZÉ HAIDAR ALAEDDINE HKS Laboratory, Electronics and hysics Dept. Faculty of Sciences I, Lebanese University, Lebanon ALAA A. GHAIH HKS Laboratory, Electronics and hysics Dept. Faculty of Sciences I, Lebanese University, Lebanon MOSAFA AL-SAHILI HKS Laboratory, Electronics and hysics Dept. Faculty of Sciences I, Lebanese University, Lebanon MOUENES FADLALLAH HKS Laboratory, Electronics and hysics Dept. Faculty of Sciences I, Lebanese University, Lebanon Abstract: his paper presents an efficient implementation of adaptive filtering for echo cancelers for teleconference systems during single-tal situation. he problem of echo cancellation is recurrent for all modern communication systems. he general solution consists to estimate the room impulse response and reproduce the echo signal in order to subtract it from the received signal and then reduce its effect at the farend user side and consequently improves the conversation quality. Several types of adaptive algorithms eist in the estimation process. he most popular echo canceler (EC) uses Least Mean Squares (LMS) or Recursive Least Squares (RLS) adaptive filter. In practice, the choice of an algorithm must be made bearing in mind some fundamental aspects: - Faster convergence in the presence of large variations in the far-end speech: the acoustic echo cancellation system must provide an echo reduction of about 24 db for delays lower than 25 ms and of about 40 db for delays eceeding 25 ms. 030

2 Approach of erformance Analysis of RLS Adaptive Filter Using a Bloc-wise - Reduced computational compleity algorithms. he literature shows that the RLS algorithm is more efficient than the LMS algorithm. It allows to reach an echo reduction of 46 db which is large enough than the required threshold. Moreover, it presents a faster convergence compared to the LMS algorithm during single-tal situation. However, the implementation of the RLS algorithm applied to acoustic echo cancellation system has shown the disadvantage of requiring a strong computational compleity. o reduce the computational cost of this algorithm, we propose a new version of the RLS algorithm based on the Fast Fourier ransform (FF) and on the recursive operation over a bloc of N samples instead of one sample. Simulation results show that the proposed algorithm has a faster adaptation convergence, during single-tal situation, and reduce considerably the hardware resources needed for the implementation and the eecution time in terms of N. Key words: Acoustic Echo Cancellation, Adaptive Filter, Adaptive algorithm, LMS Algorithm, RLS Algorithm.. INRODUCION he ever-increasing demand for telecommunication systems is stimulating research efforts to develop efficient systems, in order to transmit an intelligible and good quality speech signal. hese problems such as sound cancellation and noise reduction must be resolved without affecting the quality of the signal. he problem dealt consists this paper is that of acoustic echo control in teleconferences in case of simple speech. he signal emitted from a distant speaer by the loudspeaer is received after passing through the acoustic channel of the so-called local microphone and returned to this same speaer. When the acoustic echo is present in an inconvenient way, ie clearly distinct subjectively from its original signal, a specific treatment, called acoustic echo cancellation, must be EUROEAN ACADEMIC RESEARCH - Vol. V, Issue 2 / May

3 Approach of erformance Analysis of RLS Adaptive Filter Using a Bloc-wise implemented in order to preserve the quality of the communication. Adaptive filtering is proved to be an effective tool for this tas for several years: In fact, filtering is made adaptive if its parameters are modified according to a given criterion, as soon as a new value of the signal becomes available. Adaptation of the adaptive filter coefficients has been carried out for decades using the LMS (Least Mean Squares) algorithms and has also been performed with recursive type methods such as RLS (Recursive Least Squares). he RLS algorithm maes it possible to obtain efficient results when identifying the acoustic channel, that is to say with a faster convergence rate compared to the LMS algorithm. On the other hand, its implementation is more comple and costly. he choice of the optimization algorithm will be made according to two criteria, which are; he speed of convergence which will be the number of iterations necessary to converge close enough to the optimal solution and the computational compleity of the adaptive algorithm. he main objective of this paper is to propose a new version of the adaptive filtering algorithm RLS by processing it in blocs of samples. his bloc processing method has the following advantages: It processes a bloc of samples at each iteration, which reduces the calculation eecution time. It involves a series of convolution products which can be realized by the implementation of the fast Fourier transform. It allows an implementation on a fied-point processor. Its convergence rate is fairly rapid. On the other hand, this paper is organized as follows: Section 2 presents quicly the acoustic echo cancellation system and EUROEAN ACADEMIC RESEARCH - Vol. V, Issue 2 / May

4 Approach of erformance Analysis of RLS Adaptive Filter Using a Bloc-wise recalls the theory of adaptive filtering. It also recalls and compares the two different adaptation algorithms, LMS and RLS. Section 3 describes the method of processing the RLS algorithm by sample blocs and highlights the value of using this method in signal processing. It also presents an implementation using the FF of this new algorithm. In the last section, numerical results of the realization of the RLS algorithm by blocs using the FF are given. 2. ACOUSIC ECHO he acoustic echo is caused by the transmission of the signal emitted by the loudspeaer and received by the microphone: this transmission is composed of a direct path and multiple reflections captured by the microphone, and consequently return to the speaer who spoe, in a distant room, his own signal. he impulse response of the echo path is in the form of a direct wave and a succession of waves reflected from the walls of a particular room. he long duration of the echo path is mainly due to the low speed of sound in the air. When the acoustic echo is present in an inconvenient way, it is necessary to remove it [Gilloire 987]. 2.. Acoustic echo cancellation elecommunication systems use an acoustic echo canceller to cancel unwanted echoes resulting from coupling between a loudspeaer and a microphone. Generally, echo cancellation is performed by modeling the impulse response of the echo path by adaptive finite impulse response filtering and subtracting an echo estimate of the microphone signal [Kazuo 990], [Farhang 998]. A typical acoustic echo canceller has been illustrated in Figure (). EUROEAN ACADEMIC RESEARCH - Vol. V, Issue 2 / May

5 Approach of erformance Analysis of RLS Adaptive Filter Using a Bloc-wise Figure. rinciple of the acoustic echo cancellation system he echo cancellation is performed by identifying the impulse response of the echo path by adaptive filtering represented by its coefficients, ˆ wˆ 0 wˆ wˆ L where represents w, the discrete time. ypically, the echo cancellation system uses two main adaptive algorithms which are: LMS (Least Mean Squares) and Recursive Least Squares (RLS) Acoustic echo cancellation algorithms Least Mean Square Algorithm (LMS) he Least Mean Square algorithm (LMS) designed in 959, is based on the gradient method which computes and updates recursively the weights. his gradient method consists of obtaining from a given vector ŵ, a vector wˆ by incrementing of the vector ŵ in the opposite direction of the gradient of the cost function : [Kazuo 990], [George 999], [aleologu 200]: ˆ 2 wˆ w where is the adaptation step of the gradient algorithm which controls the convergence of the adaptive filter. he cost function, which will minimize the mean squared error (MSE), is defined by: E e 2 () (2) EUROEAN ACADEMIC RESEARCH - Vol. V, Issue 2 / May

6 Approach of erformance Analysis of RLS Adaptive Filter Using a Bloc-wise he error e is a quadratic form of the weights, and intuitively, the optimal solution is obtained by stepwise correcting the weighting vector in the direction of the minimum. e ˆ ˆ d y d w (3) [ X X X L - ] is the input sequence. - ŷ is the echo estimation that can be obtained by filtering the input sequence with the coefficients ŵ of the filter as follow: yˆ L n wˆ X n wˆ n 0 (4) Adaptation of the coefficients of the adaptive filter ŵ according to the LMS algorithm is given by: wˆ wˆ e (5) he computational compleity of the LMS algorithm is nown: each iteration consists (2L + ) multiplications and (2L) additions. It can be seen that the LMS algorithm has an etremely low computational compleity. On the other hand, for non-stationary signals, it is difficult to follow the variations of the input signal by adapting the coefficients of the adaptive filter by this algorithm, which gives a slow convergence. o solve this problem, we use the RLS algorithm Recursive Least Squares Algorithm (RLS) o compensate the problem of the non-stationarity of the signal, we use the Recursive Least Squares (RLS) algorithm. In this method, instead of minimizing a statistical criterion established on the error committed by estimating a signal d, we minimize, at each iteration, the weighted sum of the squares of the errors committed from the initial moment. In this case the cost function, is given by: [Ljung 983], [Benesty 20] EUROEAN ACADEMIC RESEARCH - Vol. V, Issue 2 / May

7 Approach of erformance Analysis of RLS Adaptive Filter Using a Bloc-wise n0 d yˆ n 2 n he signal estimation d using the least-squares method and ˆ an impulse response w, is obtained when the cost function is minimized. By analogy with the LMS algorithm, the estimated impulse response of the RLS filter is therefore to be modified with each new iteration. o limit the number of computations, we use a recursive equation given by: wˆ wˆ G e (7) e ˆ ˆ (6) d y d w (8), he column vector G with size L, is called the Kalman gain, which can be defined as follows: G - is a weighting factor that always taes a positive value: 0<<. his factor is also called forgetting factor because it is used to forget data that corresponds to a distant past. - is the inverse auto-correlation matri of size (L L), which can be calculated recursively as follows: G (9) (0) 0. I L, is a positive constant I L is an identity matri of size (L L) he details of the calculations necessary to arrive at the formulation of the RLS algorithm which is represented by equations (8), (9) and (0) is given in the Appendi. he computational compleity of the RLS algorithm is proportional to 2 L, the number of additions/subtractions, EUROEAN ACADEMIC RESEARCH - Vol. V, Issue 2 / May

8 Approach of erformance Analysis of RLS Adaptive Filter Using a Bloc-wise multiplications/divisions implemented for the RLS algorithm at each iteration is respectively of the order 2L 2, and 3L 2 7L. hen the RLS algorithm requires many more operations than the LMS algorithm erformance comparison of LMS and RLS algorithms Numerical simulations were made in the single tal situation to evaluate the performance of the two adaptive algorithms LMS and RLS using MatLab programming software. he two algorithms are evaluated by a speech signal and an echo path impulse response respectively presented in the two figures (2-a) and (2-b). In our study, we are interested in the case of a simple speech (presence of a distant signal only). he problem that arises is that of choosing an optimization algorithm. his choice will be determined by the convergence speed of the filter and the computational compleity. he optimal algorithm is one that satisfies these two criteria. he attenuation of the echo can be measured by the evolution of the convergence Nm of the adapted filter given by the two algorithms LMS and RLS. he measurement is carried out in a conventional manner using the L successive samples, that is to say, at the inde time : 2 wwˆ N m 0log 0 w 2 () Where w and ŵ denote respectively the impulse response and the estimated impulse response of the echo path. he results of the simulations are obtained by choosing the length of the adaptive filter, equal to 28 (the acoustic echo is therefore present for an overall transmission delay of 6 ms), EUROEAN ACADEMIC RESEARCH - Vol. V, Issue 2 / May

9 Approach of erformance Analysis of RLS Adaptive Filter Using a Bloc-wise the adaptation step , the forgetting factor and Figure 2. (a) Input Signal (b) he impulse response of the echo path An echo cancellation system must provide echo attenuation of the order of 24 db. he same system must be able to provide 40 db echo attenuation for delays eceeding 25 ms. [Gilloire 994] Figure (3) shows the evolution of the convergence of two adaptive algorithms, LMS and RLS as a function of the number of samples. his figure shows that the RLS (discontinuous line curve) algorithm has a faster convergence rate (-46 db) than the LMS algorithm (-38.7 db). he difference between the two convergence speeds can reach 7 db. he choice of the value of the coefficients is only an eample to demonstrate the performance of the RLS algorithm with respect to the LMS algorithm. Any other choice of the coefficients can only show that the RLS algorithm remains the most efficient in terms of maing the residual echo less audible at the output of the acoustic echo cancellation system. he objective observations, such as the measurement of the convergence of the coefficients of the adaptive filter which remain insufficient to properly evaluate the acoustic echo cancellation system. For this purpose, a listening test was carried out to obtain a subjective assessment of the quality of listening to the residual echo issued from both echo cancellation EUROEAN ACADEMIC RESEARCH - Vol. V, Issue 2 / May

10 Approach of erformance Analysis of RLS Adaptive Filter Using a Bloc-wise systems, successively adapted by the LMS and the RLS. he echo cancellation system wors well in the case of simple speech. We observed that the residual echo provided by the RLS algorithm becomes less audible compared to that provided by the LMS algorithm. On the other hand, the implementation of the RLS algorithm presented a significant global computational load compared to the LMS algorithm. herefore, problems of eecution time and compleity of computation always arise when we try to implement this algorithm in a real time processor. For these reasons, it is then necessary to reduce the costs of these operations to be processed. o remedy to this problem, a new technique to treat the RLS algorithm has been proposed in this article. his technique consists of processing the RLS algorithm by blocs of samples instead of processing it sample by sample. Figure 3. Convergence of filter coefficients by the two algorithms 3. Recursive Least Square by Bloc Structure he main idea of our wor is to propose a new and more efficient version of the RLS algorithm. In fact, instead of adapting the coefficients to each new sample, that meant that adapting the filter coefficients 6000 times per second for a sampling frequency f s 6 Hz. he proposed algorithm, "Bloc RLS (BRLS)", consists of adapting the filter coefficients, only each N samples, N> designating the size of the adaptation EUROEAN ACADEMIC RESEARCH - Vol. V, Issue 2 / May

11 Approach of erformance Analysis of RLS Adaptive Filter Using a Bloc-wise bloc. his implies a reduction of the eecution time and a reduction of the calculation compleity by a factor N, at least for the adaptation part. In this section, we describe the bloc adaptive filtering procedure. he mathematical formulation of this procedure is obtained in vector form by processing the N samples of the interval N; NN at each iteration, denoting the inde of the bloc ( N). Let ŵ the adaptive filter of length L designed to estimate the acoustic feedbac path and updated at each iteration. he estimated echo using a bloc-processed algorithm is given by: [Clar 98], [Alaeddine 202]: XN XN XNL wˆ 0 N N N L wˆ X X X ~ ~ y 2 R. wˆ X X X N N N N N N L wˆ 2 L yˆ N yˆ N yˆ NN (2) Where R ~ is the oeplitz matri of size (N L). he principle of bloc processing is described by the figure (4). his figure shows that the mathematical formulation of the procedure for processing the RLS algorithm by blocs is [ X X X L] obtained by replacing the input vector by the oeplitz matri in the adaptive equations of the filter. his algorithm will be called oeplitz-brls. he error vector (residual echo) of length N is given by: en en enn ~ d where dn dn dnn signal of length N. (3) is the vector of the echo EUROEAN ACADEMIC RESEARCH - Vol. V, Issue 2 / May

12 Approach of erformance Analysis of RLS Adaptive Filter Using a Bloc-wise Figure 4. Adaptive filtering by bloc structure From the equations (2 and 3), it can be seen that bloc processing allows us to calculate N elements at each iteration, unlie sample processing. he implementation of the oeplitz-brls algorithm which is based on matri computation is very complicated from a computational point of view. o remedy this problem, we propose to perform this calculation by the circular discrete convolution which can be made less epensive and does not modify the treatment to be carried out. he elements of the ~ estimated echo signal, y, will be calculated so that the property of the circular convolution can be used as follows: [Khong 2005] L N m wˆ i0 yˆ inmi wˆ m m (4) where represents the circular convolution and 0 m N. ~, Similarly, the matri product, R in both equations (7) and (9) of the adaptive filter coefficient update can be calculated by the discrete convolution as follows: ~ N N R e j. j jn (5) ~ R, corresponds to a cross-correlation whose coefficients i are given by: EUROEAN ACADEMIC RESEARCH - Vol. V, Issue 2 / May

13 Approach of erformance Analysis of RLS Adaptive Filter Using a Bloc-wise N enm m0. Nmi, 0 i L (6) If we put n N m, the i can be written in the form of a convolution in the following way: NN i en. ni en * nn i (7) From the equations (4) and (7), it can be seen that the adaptive filtering by the proposed algorithm "Bloc-RLS (BRLS)" consists in calculating the circular convolution between the vectors ŵ ~ and, of respective length N and N+L-, and the vectors ~ and, where ~ being the sequence of the input signal defined by: ~ [ X N L XNL2 XNN ] (8) We can conclude that with our proposition, replacing the matri calculation, by the convolution product, the computational 2 compleity will be proportional to L instead of L. he adaptation of the coefficients of the adaptive filter by the proposed BRLS algorithm can be represented according to the equation: p wˆ ~. ~ p ~ wˆ * (9) Following our proposal, the estimated echo is given by: ~ y wˆ ~ * (20) Figure (5) shows the advantage of using the bloc processing method to reduce the eecution time and in a particular case where the length of the bloc is equal to N = 2. his figure demonstrates the interest of our proposal at the eecution time: Indeed, if the length of the bloc is equal to N=2, the bloc processing method allows us, for 3 iterations, to treat 6 new samples compared with the other method (Figure EUROEAN ACADEMIC RESEARCH - Vol. V, Issue 2 / May

14 Approach of erformance Analysis of RLS Adaptive Filter Using a Bloc-wise 5.a), which processes only 3 new samples for the same number of iterations. So we can see that our proposal to process the RLS algorithm by blocs of samples allows us to reduce the eecution time as a function of the length N of the bloc. Figure 5. Representation of the input sequence by the two adaptation methods A circular convolution in the time domain is equivalent to a term-to-term multiplication in the domain of the Fast Fourier ransform (FF), which maes it possible to reduce the computational compleity and the eecution time in the algorithms using the FF. 3.. Bloc-RLS by the Fast Fourier ransform (FF- BRLS) he circular convolution presented in the BRLS algorithm can be calculated in the FF domain with reduced computational compleity. In this way, we tae advantage of the circular convolution properties of the discrete Fourier transform (DF) and the speed of computation by the Fast Fourier ransform (FF). he BRLS algorithm consists of calculating the circular EUROEAN ACADEMIC RESEARCH - Vol. V, Issue 2 / May

15 Approach of erformance Analysis of RLS Adaptive Filter Using a Bloc-wise convolution between vectors ~. ŵ and, ~ and vectors and Note that Y ~ is the product of the circular convolution between ~ and ŵ : ~ Y wˆ 0 ~ M L 0 M N L (2) FF FF wˆ 0 M L FF ~ 0 M N L Where FF and FF denote respectively the direct and 0 inverse Fourier transforms. i j is a null matri of size i j. For the calculation of the FF, it is necessary to choose M power of 2. In general, we choose M N L (in our case N L ) if this value is a power of 2, if not it is necessary to complete the vectors by a necessary samples of zero coefficients in order to give them a size equal to a power of 2, as shown in equation (2). At each iteration, the adaptation of the adaptive filter is then performed by calculating the circular convolution in the domain of the FF following this epression: wˆ wˆ 0 ~. M N 0 M N L (22) wˆ. FF FF 0 M L FF ~ 0 M N L p Where ~ ~, and p ~ [ X N N XNN 2 XNL]. EUROEAN ACADEMIC RESEARCH - Vol. V, Issue 2 / May

16 Approach of erformance Analysis of RLS Adaptive Filter Using a Bloc-wise is a matri of size LM which maes it possible to eliminate the first M L vector components introduced during the inverse transformation. It is defined by: 0 L M L IL (23) I L is the identity matri of size LL. By computing, at each iteration, the inverse autocorrelation matri according given the equation: ~ G. (24) 0.I M Where IM is the identity matri of size MM. By using of the FF, the calculation of the error and the update of the coefficients of the bloc filter has become cheaper than in the case of temporal BRLS. 4. RESULS OF SIMULAIONS OF HE FF IMLEMENAION OF HE ROOSED BRLS ALGORIHM he BRLS algorithm that we proposed was developed to allow its implementation using the Fast Fourier ransform (FF). Indeed, an implementation of this algorithm, based on the cyclic convolution property, will then lead to the use of the FF of length M 256 for an impulse response of L 28 coefficients and for a sample bloc of length N 64. Several simulations have been carried out to evaluate the performance of the BRLS algorithm. For these tests, the FF-BRLS algorithm is compared with the other algorithms, RLS and oeplitz-brls. EUROEAN ACADEMIC RESEARCH - Vol. V, Issue 2 / May

17 Approach of erformance Analysis of RLS Adaptive Filter Using a Bloc-wise he values of the various parameters used in the simulation are given by: et 8. N 64 is the bloc size of the input sequence. he simulations are performed using the same input signal and the same echo path impulse response used later (Figure 2). 4.. Convergence of adaptive filter coefficients One of the performance criteria used in this simulation is the convergence, N m, of adaptive filter coefficients: N m 0log 0 wwˆ w 2 2 (25) Where w and ŵ denote respectively the impulse response and the estimated impulse response of the echo path. his criterion of convergence of the coefficients of the adaptive filter is illustrated in figure (6). his figure shows that the two algorithms, FF-BRLS (dashed linear curve) and oeplitz-brls (continuous linear curve) have the same result in terms of the convergence rate of the coefficients of the adaptive filter and consequently in terms of having the same residual echo at the output of the acoustic echo cancellation system. Moreover, the simulations do not reveal any noted difference in terms of convergence between the two adaptation methods (sample processing represented by the RLS algorithm and bloc processing). he difference between the two methods lies rather in the compleity of computation and the eecution time. EUROEAN ACADEMIC RESEARCH - Vol. V, Issue 2 / May

18 Approach of erformance Analysis of RLS Adaptive Filter Using a Bloc-wise Figure 6. Convergence of the adaptive filter coefficients using the sample bloc processing method 4.2. Calculation Compleity his section is devoted to the study of the computational compleity of three adaptive algorithms: RLS, oeplitz-brls and FF-BRLS. We will present the number of operations (additions, multiplications, subtractions and divisions) necessary to implement the three algorithms in question. As mentioned in paragraph 2.2.2, the number of additions/subtractions and multiplications/divisions implemented for the RLS algorithm at each iteration, is respectively of the order 2L 2, and 3L 2 7L. Consequently, for processing the echo signal of our study, which is composed of samples, this algorithm needs iterations. he FF-BRLS algorithm requires two convolution products as shown in the two equations (2) and (22). he computation of each convolution product requires the computation of three transforms, two direct (FF) and one inverse FF. he number of additions and multiplications implemented for a FF or for one FF is of the order of M log2 M, M log 2 M and 2 where M is the size of the transform. EUROEAN ACADEMIC RESEARCH - Vol. V, Issue 2 / May

19 Approach of erformance Analysis of RLS Adaptive Filter Using a Bloc-wise he figure (7) provides a comparison in terms of the number of additions and multiplications necessary for the implementation of the two adaptation methods, RLS and FF-BRLS as a function of N. his comparison is carried out from a sequence of length M 256. Figure 7. Computation compleity of two RLS and FF-BRLS algorithms as a function of N and for L = 28 he curves show that, as N increases, the number of calculations decreases for the FF-BRLS algorithm, and the more the benefit of FF implementation will be mared. If N =, this corresponds to the RLS algorithm. In this particular case where N =, the FF-BRLS algorithm has the same number of operations as that of the RLS algorithm Eecution ime he performance of the bloc-processed RLS algorithm is intended to compare this time the eecution time of both the RLS and the FF-BRLS algorithms to highlight the advantage of using the bloc processing method using the Fast Fourier transform (FF). In this comparison, the length of the blocs varies between N= (this is the particular case that corresponds to the RLS algorithm) and N=024. he figure (8) shows an eample of this comparison. his figure shows that the eecution time of the FF- BRLS algorithm depends on the values of N. It also shows that EUROEAN ACADEMIC RESEARCH - Vol. V, Issue 2 / May

20 Approach of erformance Analysis of RLS Adaptive Filter Using a Bloc-wise the reduction of the eecution time is observed when N increases. On the other hand, the eecution time represented by the RLS algorithm is generally the same, is equal to t = 29.5 seconds in order to process samples (number of samples of the echo signal). Figure 8. Eecution time of the FF-BRLS algorithm as a function of N 5. CONCLUSION he wor carried out and described in this article provides the first elements to evaluate the possibility of developing an acoustic echo cancellation system. First, we have presented the origin of the acoustic echo and in what situation this acoustic echo is present in an annoying way. he two adaptive algorithms, LMS and RLS, used in the acoustic echo cancellation system and their comparison at the level of the convergence of their coefficients have been described. his comparison led us to conclude that the RLS algorithm has a better acoustic echo attenuation compared to the LMS algorithm. On the other hand, the implementation of the RLS algorithm presents a significant global computational compleity compared to the LMS algorithm. o remedy this problem, we proposed to treat it in blocs of samples instead of treating it sample by sample. his process involves a series of convolution products that can be realized with reduced EUROEAN ACADEMIC RESEARCH - Vol. V, Issue 2 / May

21 Approach of erformance Analysis of RLS Adaptive Filter Using a Bloc-wise computational compleity by implementing the Fast Fourier ransform (FF). he results of the simulations reveal no noted difference in terms of convergence between the two adaptation methods (sample processing represented by the RLS algorithm and bloc processing). he only difference between the two methods resides in the computational compleity and the eecution time, which was reduced by the new FF-BRLS algorithm. REFERENCES. [Gilloire 987] Gilloire A., and Julien J.. (987). L acoustique des salles dans les télécommunications. L écho des recherches, n 27, p [Kazuo 990] Kazuo M, Shigeyui U, and Fumio A (990). Echo Cancellation and Applications. IEEE Communications Magazine, p [Farhang 998] Farhang-Boroujeny B (998). Adaptive Filters. heory and Apllications. Wiley & Sons, ISBN [George 999] George-Othon G, Kostas B, and Sergios (999). Efficient Least Squares Adaptive Algorithms for FIR ransversal Filtering. IEEE Signal Magazine, p [aleologu 200] aleologu C, Benesty J, and Ciochina S (200). Sparse Adaptive Filters for Echo Cancellation. Morgan & Claypool ublishers, 6. [Ljung 983] Ljung L and Söderström (983). heory and ractice of Recursive Identification. M.I..ress. 7. [Benesty 20] Benesty J, aleologu. C, and Ciochina S (20). Regularization of the RLS algorithm. IEICE rans. Fundamentals, vol. E94-A, p EUROEAN ACADEMIC RESEARCH - Vol. V, Issue 2 / May

22 Approach of erformance Analysis of RLS Adaptive Filter Using a Bloc-wise 8. [Gilloire 994] Gilloire A, and Vetterli M (994). erformance evaluation of acoustic echo controls: required values and measurement procedures. Annales des élécommunications. vol 49, n 7, p [Clar 98] Clar G.A, Mitra S.K, and arer S.R (98). Bloc implementation of adaptive digital filters. IEEE rans. on Circuits and Systems. vol. CAS-28, p [Alaeddine 202] Alaeddine H, Bazzi O, Alaeddine A, Mohanna Y and Burel G (202). Fast convolution using generalized sliding Fermat number transform with application to digital filtering. Journal of he Institute of Electronics, Information and Communication Engineers (IEICE), E95-A (6), p [Khong 2005] Khong A, Benesty J, and Naylor.A (2005). An improved proportionate multi-delay bloc adaptive filter for pacet-switched networ echo cancellation, in roc. EUSICO Actes du colloque Sémio 2005, Antalya, urey. AENDIX: RECURSIVE LEAS SQUARES (RLS) ALGORIHM We will develop a recursive algorithm which, using the coefficients of, the filter at the instant will estimate these coefficients at the instant using the new available data. roblem Statement Our objective is to estimate the parameters least squares criterion: n 2 d nwˆ n n0 ŵ using the following (A.) With is a weighting factor that always taes a positive value: 0. his factor is also called forgetting factor because it is used to EUROEAN ACADEMIC RESEARCH - Vol. V, Issue 2 / May

23 Approach of erformance Analysis of RLS Adaptive Filter Using a Bloc-wise forget data that corresponds to a distant past. he particular case corresponds to an infinite memory. he normal equations he problem is to determine the vector of the coefficients ŵ which minimizes. he solution is obtained by calculating the derivatives of the cost function n0 Let: n d wˆ n n dn n n0 n0 It then comes: Q wˆ n n 0 n and by equating them to zero: L n n wˆ (A.2) (A.3) (A.4) ˆ (A.5) w Q With: Q n0 n0 n n n d n n n he above equations can be computed recursively (A.6) (A.7) (A.8) We have also: Q Q d RLS algorithm It is recalled that the correlation matri is: (A.9) (A.0) he inversion lemma can be used to compute the inverse of, A, B C and D, let: EUROEAN ACADEMIC RESEARCH - Vol. V, Issue 2 / May

24 Approach of erformance Analysis of RLS Adaptive Filter Using a Bloc-wise EUROEAN ACADEMIC RESEARCH - Vol. V, Issue 2 / May hus we obtain the following recursive equation for the inverse of the correlation matri: 2 (A.) Let: and G (A.2) hen the recursive equation for can again be written: G (A.3) he vector G is called Kalman gain, which can be rearranged: G (A.4) Now, we need to compute the filter coefficients recursively. We have: d Q Q Q w ˆ (A.5) By replacing in the first term of the preceding epression by equation (A.3), we obtain: ˆ ˆ ˆ w d G w d Q G Q w (A.6) Finally, we have: e G w w ˆ ˆ (A.7) Where ˆ w d e (A.8) is the a priori error that is different from the posterior error: w d e ˆ (A.9)