Chapter_14 - CHAPTER 14 14.1 In an adaptive equalizer the...

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300 CHAPTER 14 14.1 In an adaptive equalizer, the input signal equals the channel output and the desired response equals the channel input (i.e., transmitted signal). In a stationary environment, both of these signals are stationary with the result that the error-performance surface is fixed in all respects. On the other hand, in a nonstationary environment, the channel output (i.e., equalizer input) is nonstationary with the result that both the correlation matrix R of the input vector and the cross-correlation vector p between the input vector and desired response take on time-varying forms. Consequently, the error-performance surface is continually changing its shape and is also in a constant state of motion. 14.2 In adaptive prediction applied to a nonstationary process, both the input vector (defined by a set of past values of the process) and the desired response (defined by the present value of the process) are nonstationary. Accordingly, in such a case the error-performance surface behaves in a manner similar to that described for adaptive equalization in Problem 14.1. Specifically, the error-performance surface constantly changes its shape and constantly moves. In contrast, the error-performance surface for the adaptive prediction of a stationary process is completely fixed. 14.3 We have, by definition, We may therefore expand Invoking the assumption that w ( n ) and w o are statistically independent, we may go on to write = 0 (1) n ( ) 1 w ˆ n ( ) E w ˆ n ( ) [ ] = n ( ) 2 E w ˆ n ( ) [ ] w o = E [ n ( ) n ( ) ] 2 H 1 E w ˆ n ( ) - w ˆ n ( ) [ ] ( ) H E w ˆ n ( ) [ ] w o ( ) [ ] = E [ w ˆ H n ( ) E w ˆ n ( ) [ ] E w ˆ H n ( ) [ ] E w ˆ n ( ) [ ] = w ˆ H n ( ) w o E w ˆ H n ( ) [ ] w o ] + E w ˆ H n ( ) w o [ ] E w ˆ H n ( ) [ ] E w o [ ] + = E [ n ( ) n ( ) ] 2 H 1 E w ˆ H n ( ) [ ] E w o [ ] ( ) E w ˆ H n ( ) [ ] E w o [ ] + =
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301 From this result we immediately deduce that we also have (2) Finally, we note that where in the last line we have made use of Eqs. (1) and (2). 14.4 Invoking the low-pass filtering action of the LMS filter for small µ , we note that 1 ( n ) and 2 ( n ) are both independent of the input vector u ( n ). We may therefore write: 1.
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