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Chapter_13

# Chapter_13 - CHAPTER 13 13.1 The analog(innite-precision...

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290 CHAPTER 13 13.1 The analog (infinite-precision) form of the LMS algorithm is described by where is the tap-weight vector estimate at time n , u ( n ) is the tap-input vector, d ( n )is the desired response, and µ is the step-size parameter. The digital (finite-precision) counterpart of this update may be expressed as where and the use of subscripts q signifies the use of finite-precision arithmetic. Let where the quantizing noise vector v ( n ) is determined by the manner in which the term is computed. Hence, (1) The quantized value of the estimation error e ( n ) may be expressed as where denotes residual error. The quantized value of u ( n ) equals Hence, we may express , ignoring second-order effects, as follows w ˆ n +1 () w ˆ n () µ u n dn w T n u n [] , n + 012 ,,, == w ˆ n w ˆ q n +1 w ˆ q n Q µ u q n e q n + = en w T n u n = Q µ u q n e q n µ u q n e q n v n + = u q n e q n w ˆ q n +1 w ˆ q n u q n e q n v n ++ = e q n () ∆ u T n w n u T n ()∆ w n ζ n + = ζ n u q n u n u n + = u q n e q n u q n e q n u n u n u n u T n w ˆ n ( ) + = u n u T n w ˆ n u n ζ n +

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291 We may therefore rewrite Eq. (1) as (2) But Hence, from Eq. (2) we deduce that where Note that 13.2 Building on the solution to Problem 13.1, assume that is stationary. Then the expectation of is zero because the expectation of t ( n ) is zero. 13.3 We note that w ˆ q n +1 () w ˆ q n () µ u q n e q n () µ∆ u n en ++ = µ u n ()∆ u T n w ˆ n u n u T n w ˆ n u n ζ n + v n + w ˆ q n +1 w ˆ n +1 w ˆ n +1 + = w ˆ q n w ˆ n () ∆ w ˆ n + = w ˆ n +1 w ˆ n u n + = w n +1 F n w n t n + = F n I µ u n u T n = t n µ∆ u n u n w ˆ T n u n u n ζ n v n = u T n w ˆ n w ˆ T n u n = w n w n y I n w i un i i = y II n w iq i =
292 Hence, The mean-square value of ε ( n ) is (1) Assuming that we may simplify Eq. (1) as 13.4 (a) The digital residual error is With 12-bit quantization, the least significant bit is We are given Hence, the digital residual error is ε n ()

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Chapter_13 - CHAPTER 13 13.1 The analog(innite-precision...

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