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FIGURE 3.14The three-tap equalizer response using zero forcing technique.We denote the desired and actual equalizer’s output by z d ðtÞ and z a ðtÞrespectively. So the MMSE technique is expressed as [3]e ¼ Ef½z a ðtÞ z d ðtÞŠ 2 g ð3:35Þwhere Efg denotes the expectation of fg. Like (3.29), the actual output isz a ðtÞ ¼ PNn¼a n z c ðt nDÞ ð3:36ÞNwhere, in this instance, the tap spacing D does not correspond to the samplingtime. Our task is to set a sufficient condition to minimize the tap weights a n .Since e is a concave function of a n , then a sufficient condition for errorminimization is@e¼ 0 ¼ 2E ½z@a a ðtÞnz d ðtÞŠ @z aðtÞ@a n; n ¼ 0; 1; ...; N ð3:37ÞUpon substituting (3.36) in (3.37) and differentiating the resultant, we obtainthe conditionsEf½z a ðtÞ z d ðtÞŠz c ðt nDÞg ¼ 0; n ¼ 0; 1; ...; N ð3:38ÞAlternatively,Ef½z d ðtÞŠz c ðt nDÞg ¼ Ef½z a ðtÞŠz c ðt nDÞg ð3:39ÞLet’s pause to emphasize what we have obtained in terms of data autocorrelation.By definition, cross correlation of signals x and y isR xy ðtÞ ¼EfxðtÞyðt þ tÞgð3:40aÞCopyright © 2002 by Marcel Dekker, Inc. All Rights Reserved.