Advance Convergence Characteristic Based on Recycling Buffer Structure in Adaptive Transversal Filter

Transcription

1 Advance Convergence Characteristic ased on Recycling uffer Structure in Adaptive Transversal Filter Gwang Jun Kim, Chang Soo Jang, Chan o Yoon, Seung Jin Jang and Jin Woo Lee Department of Computer Engineering, Chonnam National University, Doondeok-dong, Yeosu, Jeonnam, Korea {kgj, csjang, chanho, sjjang, jwlee}@jnu.ac.kr Abstract We extend the use of the least squares method to develop a recursive algorithm for the design of adaptive transversal filters such that, given the least-square estimate of a vector of a filter at iteration, we may compute the updated estimate of this vector at iteration upon the arrival of new data. In this paper, we propose a new tap-weight-updated RLS algorithm for an adaptive transversal filter with data-recycling buffer structure. We prove that the convergence speed of the learning curve of an RLS algorithm with a data-recycling buffer is faster than existing RLS algorithms at mean square error versus iteration number. Also, the resulting rate of convergence is typically an order of magnitude faster than the simple LMS algorithm. We show that the number of desired samples should be increased to converge the specified value from the three-dimensional simulation results of mean square error according to the degree of channel amplitude distortion and data-recycle buffer number. This improvement of the convergence performance is achieved at times the convergence speed of the mean square error increase in the data recycle buffer number with new proposed RLS algorithm. Keywords: Adaptive Transversal Filter, Recycling uffer Structure, Convergence Character, RLS Algorithm, Tap Weight 1. Introduction Adaptive filters are essential for meeting the need to acquire more improved performance in digital signal processing and communication systems. Adaptive filter algorithms have rapid convergence speed, lower mean squared error (MSE) and are practical for hardware implementation [1]. We begin the development of a new RLS algorithm by reviewing some basic relations that pertain to the method of least squares. Then, by exploiting the relation in matrix algebra known as the matrix inversion lemma, we develop the RLS algorithm []. An important feature of the RLS algorithm is that it utilizes information contained in the input data, extending back to the instant of time when the algorithm was initiated. In this paper, a simple and efficient technique for rapid convergence speed of a transversal filter which uses RLS algorithms is introduced. The basic idea of the technique involves the use of discarded data samples to update the tap weight vector in a sample period. This can increase the convergence speed by a factor of () without increasing the computational complexity substantially, where is the number of recycled data. 377

2 . Recycling uffer RLS Algorithm In Recursive implementations of the least squares method, we start the computation with known initial conditions and use the information contained in new data samples to update the old estimates. We therefore find that the length of the observable data is valuable. Accordingly, we express the cost function to be minimized as ( n), where is the variable length of the observable data. Also, it is customary to introduce a weighting factor into the definition of the cost function ( n) [3]. We thus write: n ( n) ( n, i) e( i) i1 (1) Where is the difference between the desired response and the output produced by a transversal filter whose tap input is equal to, as in Figure 1. u(n) w * o, 0 u(n-1) w * o,1 u(n-m+1) * w o, M1 e o (n) d(n) Figure 1. Adaptive transversal filter That is, is defined by: () where is the tap-input vector at time, defined by: (3) and is the tap-weight vector at time, defined by: (4) Note that the tap weights of the transversal filter remain fixed during the observation interval for which the cost function ( n) is defined. 378

3 International Journal of Software Engineering and Its Applications 3. Proposed Data Recycling Filter Structure We proposed the new data recycling filter structure to improve the convergence characteristic of the recycling buffer RLS algorithm. Instead of using a single to update the tap weight vector, we can use the discarded input data vectors, which are stored in finite buffers in multi-linear feedback adaptive transversal filters. The proposed structure with the buffers used for recycling data and the coefficient updating process are depicted in Figure. uffer (=3) b u(n-4) u(n-3) u(n-) u(n-1) W 1 (n-1,3) W 1 (n-1,) W 1 (n-1,1) W 1 (n-1,0) (n,3) Updating w(n,3) to w(n,) (n,) uffer (=3) a u(n-3) u(n-) u(n-1) u(n) W 1 (n-1,-1) W 0 (n-1,3) W 0 (n-1,) W 0 (n-1,1) W 0 (n-1,0) Updating w(n,) to w(n,1) (n,1) Updating w(n,1) to w(n,0) (n,0) Input data W 0 (n-1,-1) Updating w(n,0) to w(n,-1) Figure. Adaptive transversal filter structure with data-recycling buffer For Simplicity, the proposed structure including just two taps and is described. Tap and have three recycling data (=3) in each of their buffers. Firstly, the recycling data in the buffers of tap and in the buffer of tap are used to update to using error ( n,3). Secondly, is updated to using and ( n,). Finally, the data vector and coefficient vector produce. Now is copied to the TDL filter coefficient vector, which will be used to produce filter output. The output estimation of the adaptive transversal filter with the recycling data buffer that is shown in Figure, a product tap input, and tap weight vector, is calculated as follows M 1 ˆ d( n) w ( n 1) u( n) wˆ ( n 1) u( n) j1 j (5) 379

4 The a priori estimation error ( n) is defined by the difference between the desired response and the actual adaptive transversal filter output. The desired recursive equation with the data recycling buffer RLS algorithm for updating the tap weight vector is: * ( ) ˆ( 1) ( ) (, ) wˆ n w n k n n i i1 and we may express the a priori estimation error ( n) as: (6) ( n) e ( n) [ wˆ ( n 1, i) w ] u( n i) 0 0 i1 i1 e ( n) ( n 1, i) u( n i) 0 i1 where is the weight error vector at time (7) 4. Tap weight updating algorithm with data recycling RLS algorithm As an index of statistical performance for the RLS algorithm, it is convenient to use the a priori estimation error ( n) to define the mean-squared error: ( ) [ ( ) ] J n E n i1 The Prime in the symbol is intended to distinguish the mean-square value of( n) from that of. Substituting Eq.(7) into Eq.(8), and then expanding terms, we get: ( ) [ 0( ) ] ( [ ( ) ( 1, ) ( 1, ) ( )] i1 J n E e n E u n i n i n i u n i E[ ( n 1, i) u( n i) e ( n)] E[ e ( n) u ( n i) ( n 1, i)] * 0 0 With the measurement assumed to be of zero mean, the first expectation on the righthand side of Eq.(9) is simply the variance of, which is denoted by. The second expectation on the right-hand side of Eq.(9) is the estimate, and therefore, the weight-error vector is independent of the tap-input vector. The latter is assumed to be drawn from a wide-sense stationary process of zero mean. ence, we may use this statistical independence together with well-known results from matrix algebra to express the second expectation on the right-hand side of Eq.(9) as follows: i1 E[( u ( n i) ( n 1, i) ( n 1, i) u( n i)] E[ tr{ u ( n i) ( n 1, i) ( n 1, i) u( n i)}] i1 (8) (9) 380

5 E[ tr{ u( n i) u ( n i) ( n 1, i) ( n 1, i)}] i1 tr{ E[ u( n i) u ( n i) ( n 1, i) ( n 1, i)]} i1 tr{ E[ u( n i) u ( n i)] E[ ( n 1, i) ( n 1, i)]} i1 i1 tr{ RK( n 1, i)} where, in the last line, we have made use of the definitions of the ensemble-averaged correlation matrix R and weight-error correlation matrix. The measurement error depends on the tap-input vector. The weight-error vector is therefore independent on both and. Accordingly, we may show that the third expectation on the right-hand side of Eq.(9) is zero by first reformulating it as follows: (10) * * 0 0 i1 i1 E[ ( n 1, i) u( n i) e ( n)] E[ ( n 1, i)] E[ u( n i) e ( n)] We now form the principle of orthogonality that all the elements of the tap-input vector orthogonal to the measurement error. We therefore have: (11) i1 E[ ( n 1, i)] u( n i) e ( n)] 0 * 0 The fourth and final expectation on the right-hand side of Eq.(9) has the same mathematical form as that just considered in point, except for a trivial complex conjugation. We may therefore set this expectation equal to zero, too: i1 E[ e ( n) u ( n i) ( n 1, i)] 0 0 Thus, recognizing that is equal to, and using the results of Eqs.(10) to (13) in (9), we get the following simple formula for the mean-squared error in the RLS algorithm. ( ) [ ( 1, )] J n tr RK n i i1 We may express the weight-error correlation matrix K 1 ( n ), 1 1 R n M n M (15) as: (1) (13) (14) 381

6 Next, substituting Eq.(15) in Eq.(14), we get for : M J( n), n M 1 nm 1 (16) ased on this result, we may make the following deduction. The ensemble-averaged learning curve of the recycling buffer RLS algorithm converges more rapidly by a factor of if the proposed structure has number of data recycling buffers., denoted in Eq.(16), means the number of taps in the transversal filter. This means that the rate of convergence of the RLS algorithm is typically an order of magnitude faster than that of the LMS algorithm. As the number of iterations,, approaches infinity, the mean-squared error approaches a final value equal to the variance of the measurement error. In other words the RLS algorithm, in theory, produces zero excess mean-squared error when operating in a stationary environment. Convergence of the RLS algorithm in the mean square is independent of the eigenvalue of the ensembleaveraged correlation matrix R of the input vector. 5. Computer Simulation Results and Analysis For our computer simulation, we use the recycling buffer RLS algorithm to reject interference of channel which is produced in the adaptive equalization of a linear dispersive communication channel. The block diagram of the system used in the study is depicted in Figure. Two independent random-number generators are used. One, denoted by, is used for probing the channel, and the other, denoted by, is for simulating the effect of additive white noise at the receiver input. The sequence is a ernoulli sequence with and has zero mean and unit variance. The second sequence has no zero mean, and its variance is determined by the desired signal to noise ratio. The equalizer has 11 taps. The impulse response of the channel is defined by: h n 1 1 cos ( n ), n 1,,3 W 0, otherwise (17) where W controls the amount of amplitude distortion produced by the channel. Equivalently, the parameter W controls the eigenvalue spread of the correlation matrix of the tap inputs in the equalizer, with the eigenvalue spread increasingly with. The Sequence, produced by the second random generator has zero mean and variance. The first tap input of the equalizer at time equals: n u( n) h a( n k) v( n) k1 k where all the parameters are real. ence, the correlation matrix R of the tap inputs of the equalizer,, is a symmetric 11 by 11 matrix. Also, Since the impulse response has nonzero values only for n=1,,3 and the noise process is white with zero mean and variance, the correlator matrix R is quantdiagonal. That is, the only nonzero elements of are on the main diagonal and the four diagonals directly above and below it, and two on either side, as shown by the special structure. (18) 38

7 Mean-squared error International Journal of Software Engineering and Its Applications r(0) r(1) r() 0 0 r(1) r(0) r(1) r() 0 r() r(1) r(0) r(1) 0 R 0 r() r(1) r(0) r(0) (19) where,, and. The variance is, hence are determined by the value assigned to the parameter in Eq.(17). We have obtained values for the autocorrelation function with Eq.(17) and (19) for and the smallest eigenvalue,, the largest eigenvalue,, and the eigenvalue spread. Computer simulation with the recycling buffer RLS algorithm can use time variance of channel that provides input to the adaptive transversal filter. The impulse response of the channel is used in Eq.(18). The adaptive transversal filter with proposed recycling data buffer RLS algorithm has 11 taps, and the additive white Gaussian noise variance,, is =0 =1 =4 = Number of iterations Figure 3. MSE learning curves of RLS Algorithm with number of taps M=11, standard deviation parameter, eigenvalue spread and recycling data uffer The result of the experiment for a fixed eigenvalue spread and varying the number of buffers are presented in Figure 3. The four parts of that figure correspond to the parameters, respectively. The result presented in Figure 3 clearly show that the superior rate of convergence of the recycling buffer RLS algorithm with buffer over the RLS algorithm without buffer. The recycling buffer RLS algorithm converges much faster by a factor of compared to the RLS algorithm. Convergence of the recycling buffer RLS algorithm is attained in about 134, 67, 3, 15 iterations, corresponding to the four different values of the buffer,, respectively. Finally, the new recycling buffer RLS algorithm in the proposed structure is proven to be efficient in controlling of channel interference from the computer simulation above. 383

8 Figure 4. Three-dimensional simulation of MSE learning curves of RLS algorithm with number of taps M=11, standard deviation parameter, varying amplitude distortion w and recycling data buffer The three-dimension simulation shown in Figure 4 (a) was a typical result that was applied to the bufferless case in the adaptive transversal filter with recycling data buffer structure. This result, obtained from the state of convergence of the mean square error, corresponds to the distortion of channel which is controlled by the tap weight vector with error, which is the difference between the actual estimation value and the desired response. The x-axis and y- axis presented in Figure 4 represent the number of iterations of a sample and the degree of amplitude distortion, respectively. For the computer simulation, the number of iterations is set to 300, and the amplitude distortion is established from 1 to 4. The z-axis in Figure 4 represents the log value of the mean square error. The simulation shown in Figure 4(b) and (c) present the convergence of the MSE varying eigenvalue spread corresponding to amplitude distortion, W. From the simulation results, we show that the value of MSE increases proportionally to increasing amplitude distortion. Also, the simulation results in Figure 4(b) and (c) present the cases of the number of data recycling buffers equal to 4 and 9, with equivalent previously mentioned parameters, number of taps, white Gaussian noise variance and eigenvalue spread. The results of the computer simulation demonstrate that the simulation results converge at times the convergence speed in accordance with recycling data buffer, for controlling the tap weight using the recycling buffer RLS algorithm in the proposed structure. 384

9 6. Conclusion We have presented an efficient method for rapidly adjusting tap weight control with the recycling data buffer in the proposed structure. We began the development of the RLS algorithm by reviewing some basic relations that pertain to the method of least squares. Then, by exploiting a relation in matrix algebra known as the matrix inversion lemma, we developed the RLS algorithm. An important feature of the RLS algorithm is that it utilizes information contained in the input data, extending back to the instant of time when the algorithm was initiated. In this paper, we proposed a new tap-weight-updated RLS algorithm for an adaptive transversal filter with a data-recycling buffer structure. We proved that the convergence speed of the learning curve of the RLS algorithm with the data-recycling buffer was faster than existing RLS algorithms for mean square error versus iteration number. The results of the computer simulation demonstrate that the simulation results converge at times the convergence speed in accordance with the recycling data buffer, for controlling the tap weight using the recycling buffer RLS algorithm in the proposed structure. Acknowledgements This research was financially supported by the Ministry of Education, Science Technology (MEST) and National Research Foundation of Korea(NRF) through the uman Resource Training Project for Regional Innovation References [1] M. G. ellanger, Computational complexity and accuracy issues in fast least squares algorithms adaptive filtering, Proc. IEEE ISCAS, (1988), pp , Finland. [] J. M. Cioffi, "The fast ouseholder filters RLS adaptive filter", Proc. IEEE ICASSP, (1990) April, pp , Albuquerque. [3] S. aykin, Adaptive Filter Theory, 3 rd Edition, Prentice all, (1996). [4]. abadi, N. Kalouptsidis and V. Tarokh, Comparison of SPARLS and RLS algorithms for Adaptive Filtering, 009 IEEE Sarnoff Symposium, (009) March 30-April 1, Princeton, NJ. [5] J. Chen and R. Priener, An Inequality by Which to Adjust the LMS Algorithm Step-Size, IEEE Trans. Commun., (1995) February, vol. 43, no., pp Authors Gwang-Jun Kim received the.e, M.E and Ph. D. degrees in computer engineering from Chosun University in 1993, 1995 and 000, respectively. e joined the department of computer engineering, Yeosu National University, in 003 and became an Assistant Professor in 005. Since 009, he has been an Associate Professor in computer engineering at Chonnam National University. During , he was a researcher in the department of electrical and computer engineering at university of California, Irvine. is current research interest lie in the area adaptive signal processing, real-time communication, TCP/IP network, spread spectrum system, ATM network and various kinds of communication systems. e is a member of ISS of Korea, ICS of Korea, ITE of Korea and IMICS of Korea. 385

10 Chang-Soo Jang received the.e degree in electronic engineering from the University of Chosun in 1979, the M.E degree in electrical engineering from the University of Gunkook in 198, and the Ph. D. degree in computer engineering from the University of Sogang in e joined the Department of Computer Engineering, Yosu National University, in 1984 and became an Assistant Professor, Associate Professor in 1989 and 1993, respectively. Since 1998, he has been a Professor in the computer engineering at Chonnam National University. is research interests lie in the area of digital signal processing, high performance computing, parallel computing and various kinds of communication systems. e is a member of ISS of Korea, ICS of Korea, ITE of Korea and IMICS of Korea. Chan-o Yoon received the.e degree in computer engineering from the University of onam in 1997, the M.E degree in computer engineering from the University of Chosun in 000. e is currently completing the Ph. D. degree in computer engineering at University of Chonnam, since 008. is research interests lie in the area of digital signal processing, real-time communication, ATM network, multimedia communication and various kinds of communication systems. e is a member of ISS of Korea, ICS of Korea, ITE of Korea and IMICS of Korea. Seung-Jin Jang received the.e degree in computer engineering from the University of Chonnam in 013. e is currently completing the M.E degree in computer engineering at University of Chonnam, since 013. is research interests lie in the area of digital signal processing, real-time communication, ATM network, multimedia communication and various kinds of communication systems. e is a member of ISS of Korea, ICS of Korea, ITE of Korea and IMICS of Korea. Jin-Woo Lee received the.e degree in computer and information security from the University of Daejeon in 007. e is currently completing the M.E degree in computer engineering at University of Chonnam, since 010. is research interests lie in the area of digital signal processing, real-time communication, ATM network, multimedia communication and various kinds of communication systems. e is a member of ISS of Korea, ICS of Korea, ITE of Korea and IMICS of Korea. 386