Viput SubharngkasenAbstracti. IntroductionInputOutputInputOutputCHANNEL/SYSTEM IDENTIFICATION USING TOTAL LEAST MEAN SQUARES ALGORITHMbyViput SubharngkasenFinal ProjectinECE 539Introduction to Artificial Neural Networkand Fuzzy systemsABSTRACTThe Total Least Mean Squares (TLMS) algorithm is recursive technique used to identify transfer function of unknown systems by using input and output data, which is corrupted by noise. This paper evaluates the accuracy and stability of the TLMS algorithm. The TLMS algorithm demonstrates a better performance over the Least Mean Square (LMS) and Total Least Square (TLS) algorithms. It provides accurate results and faster computational time as compared to conventional approaches.2TABLE OF CONTENTSI. INTRODUCTION ………………………………………………………… 1II. REVIEW OF LITERATURE ……………………………………………... 4Least Mean Squares (LMS) algorithm ……………………………. 4Total Least Mean Squares (TLMS) algorithm ……………………. 8III. METHODOLOGY ………………………………………………………... 12Part I ………………………………………………………………..12Part II ……………………………………………………………… 13IV. RESULTS …………………………………………………………………. 15Part I ………………………………………………………………..15Part II ……………………………………………………………… 19V. DISCUSSION ……………………………………………………………...22VI. CONCLUSION …………………………………………………………….25VII. REFERENCE ………………………………………………………………26APPENDIX A ……………………………………………………………………... 27APPENDIX B ……………………………………………………………………... 313I. INTRODUCTIONIn the adaptive signal processing, the adaptive linear combiner is the simplest and the most widely used method. The basic properties of the adaptive linear combiner system are time varying and self-adjusting performance. Thus, the adaptive linear combiner systems are adjustable, and their adaptations usually depend upon the average values of finite-duration of input rather than upon the instantaneous values of the input. The adaptive system can be used in various applications such as the prediction application, the inverse modeling application, the interference canceling application, the channel equalization application, and the system identification application.In the system identification application, the Least Mean Squares (LMS) algorithmis widely used to determine the transfer function of an unknown system. By using inputs and outputs of that system, the LMS algorithm is applied in an adaptive process based on the minimum mean squares error. The LMS algorithm is the unsupervised learning algorithm, which is used to extract some features from data. The LMS algorithm can be used with both stationary and non-stationary signal processing. In the situation of havingno interference in both inputs and outputs or having interference only in the outputs of the unknown system, the LMS algorithm’s results can always achieve the optimal solution. However, if the interference exists in both the input and the output of the unknown system, the LMS algorithm’s results can obtain only the sub-optimal solution for that unknown system.1The Total Least Squares (TLS) algorithm, proposed in 1901, is also used in the adaptive signal processing to identify the impulse response of the unknown system. The solutions of the TLS algorithm can be determined by minimizing the Frobenius norm of the cascaded matrix between the input interference matrix and output interference matrix.The results of this minimization can be achieved from the singular value decomposition (SVD) of that combined matrix. The results of the TLS algorithm are based on the assumption that the interference in the input and output are independent of each other. The computation time of the TLS algorithm is substantially time consuming; for example, the N-by-N matrix is 6N3 per iteration. The TLS algorithm’s performance, which is the same as the LMS algorithm’s performance, cannot be used to find the global optimal solution for the situation if the interference exists in both the input and the outputsignals of that system.One of the ways to find the optimal solution of the impulse response of the unknown system when it has the interference presented in both the input and the output isby using the Total Least Mean Squares (TLMS) algorithm. Instead of basing the approach on the minimum mean squares error as the LMS algorithm does, the TLMS algorithm is based on the total least mean squares or the minimum Raleigh quotient approach. Like the LMS algorithm, the TLMS algorithm is the unsupervised learning algorithm. The TLMS algorithm was derived from Oja and Xu’s learning algorithm, which is used for extracting only the minor features from the data sequence, unlike the 2LMS algorithm. By extracting only the subsidiary information, the effect of the interference can be eliminated. Moreover, the TLMS algorithm has also an advantage over the Total Least Squares (TLS) algorithm in the computation time. The TLMS algorithm’s computation time for N-by-N matrix is 4N per iteration, whereas the TLS algorithm’s is 6N3 per iteration.In this paper, I developed the methodology for making comparisons between the LMS and TLMS algorithm. These comparisons will help to prove the accuracy of the two methods. In addition, it will also show the different results for both methods from the ideal solution.3II. REVIEW OF LITERATURELeast Mean Squares (LMS) algorithmThe LMS algorithm is very useful and easy to compute. The LMS algorithm will perform well, if the adaptive system is an adaptive linear combiner, as well as, if both the n-dimensional input vector X (k) and the desire output d (k) are available in each iteration,where X (k) is)()()()(21kxkxkxkXnand the n-dimensional corresponding set of adjustable weights W(k) is)()()()(21kwkwkwkWn4By having the input vector X (k), the estimated output y (k), can be computed as a linear combination of the input
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