Let x[n] = cos(17?n 64 ) + 2sin(23?n 32 ) for 0 ? n ? 255, and 0otherwise. The signal x[n] is passed through an LTI system withtransfer function H(z) = (1? 1 2z?1)(1? 1 3z?1). Denote the signalat the output of the system by y[n]. We want to recover x[n] bypassing y[n] through an inverse system.
(a) Find (analytically) an impulse response g[n] of the inversesystem, G(z) = 1 H(z). (Note: the inverse system is an IIRsystem.)
(b) Apply a rectangular window of length M = 8, w[n] =( 1 0 ? n? 7, 0 otherwise, to obtain an FIR ?lter r[n] = g[n]·w[n]. Pleaseplot the frequency response of the FIR ?lter. Compare it with thefrequency response of the IIR inverse system Is M = 8 anappropriate choice?
(c) Compute ˆx[n] = r[n]?y[n]. Plot x[n], y[n], and ˆx[n]?x[n].
