Question

7.31. Suppose that we have used the Parks-McClellan algorithm to design a causal FIR linear phase lowpass filter. The system function of this system is denoted H(z). The length of the impulse response is 25 samples, i.e., h[n] 0 for n < 0 and for n > 24, and hol?0. The desired response and weighting function used were In each case below, determine whether the statement is true or false or that insufficient information is given. Justify your conclusions. (a) hin 12-h112-nl or hln + 12|--hl12-nl for-oo < n < oo (b) The system has a stable and causal inverse. (e) We know that H(-1)0 (d) The maximum weighted approximation error is the same in all approximation bands (e) If zo is a zero of H(z), then 1/zo is a pole of H(z) (f) The system can be implemented by a network (flow graph) that has no feedback paths. (g) The group delay is equal to 24 for 0< (h) If the coefficients of the system function are quantized to 10 bits each, the system is stll optimum in the Chebyshev sense for the original desired response and weighting function. still guaranteed to be a linear-phase filter become unstable. (i) If the coefficients of the system function are quantized to 10 bits each, the system is (j) If the coefficients of the system function are quantized to 10 bits each, the system may

Answer 1

(a) as filter low pass, hence it's impluse response is symmetrical.

So h(n)=h(12-n)

(b) as Hd(e^jw) is abosolutly summamable hence stable, and also casual.

(C) as Hd(e^jw)= 0 , 0.4?<|w|<?

Hence Hd(-1)=0; so H(-1)=Hd(-1)e^(j6)=0;

(e) if Zo is zero then 1/Zo is also a zero;