Consider a class of DFT-based algorithms for implementing a causal FIR filter with impul...

Consider a class of DFT-based algorithms for implementing a causal FIR filter with impulse response h[n] that is zero outside the interval 0 ≤ n ≤ 63. The input signal (for the FIR filter) x[n] is segmented into an infinite number of possibly overlapping 128-point blocks x_i[n], for i an integer and −∞ ≤ i ≤∞, such that

where L is a positive integer.

Specify a method for computing

y _i[n] = x_i[n] ∗ h[n]

for any i. Your answer should be in the form of a block diagram utilizing only the types of modules shown in Figures PP9.42-1 and PP9.42-2. A module may be used more than once or not at all. The four modules in Figure P9.42-2 either use radix-2 FFTs to compute X[k], the N-point DFT of x[n], or use radix-2 inverse FFTs to compute x[n] from X[k]. Your specification must include the lengths of the FFTs and IFFTs used. For each “shift by n₀” module, you should also specify a value for n0, the amount by which the input sequence is to be shifted.