We are interested in obtaining 256 equally spaced samples of the z-transform of xw[n]. x...

We are interested in obtaining 256 equally spaced samples of the z-transform of x_w[n]. x_w[n] is a windowed version of an arbitrary sequence x[n] where x_w[n] = x[n]w[n] and w[n] = 1, 0 ≤ n ≤ 255 and w[n] = 0 otherwise. The z-transform of x_w[n] is defined as

The samples X_w[k] that we would like to compute are

We would like to process the signal x[n] with a modulated filter bank, as indicated in Figure P10.27. Each filter in the filter bank has an impulse response that is related to the prototype causal lowpass filter h₀[n] as follows:

Each output of the filter bank is sampled once, at time n = N_k, to obtain X_w[k], i.e.,

X _w[k] = v_k[N_k].

Determine h₀[n], ω_k and N_k so that