In this problem, we explore an equalization method used to avoid intersymbol

Chapter 8, Problem 8.44

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In this problem, we explore an equalization method used to avoid intersymbol interference caused in PAM ~ystems by the channel having nonlinear phase over its bandwidth. When a PAM pulse with zero-crossings at integer multiples of the symbol spacing T 1 is passed through a channel with nonlinear phase, the received pulse may no longer have zero-crossings at times that are integer multiples ofT 1 Therefore, in order to avoid intersymbol interference, the received pulse is passed through a zeroforcing equalizer, which forces the pulse to have zero-crossings at integer multiples of T1 This equalizer generates a new pulse y(t) by summing up weighted and shifted versions of the received pulse x(t). The pulse y(t) is given by N y(t) = L GtX(t- lTJ), 1=-N where the a1 are all real and are chosen such that y(kT1) ~ { 6: k = 0 k = 1, 2, 3, ... , N. (P8.44-1) (a) Show that the equalizer is a filter and determine its impulse response. (b) To illustrate the selection of the weights a1, let us consider an example. If x(OTJ) = 0.0, x(-TJ) = 0.2, x(TJ) = -0.2, and x(kT1) = 0 for lkl > 1, determine the values of a0 , a 1, and a_ 1 such that y(TJ) = 0.