Problem

It is often of interest to zoom in on a region of a DFT of a signal to examine it in mor...

It is often of interest to zoom in on a region of a DFT of a signal to examine it in more detail. In this problem, you will explore two algorithms for implementing this process of obtaining additional samples of X (e^jω) in a frequency region of interest.

Suppose X_N [k] is the N-point DFT of a finite-length signal x[n]. Recall that X_N [k] consists of samples of X (e^jω) every 2π/N in ω. Given X_N [k], we would like to compute N samples of X (e^jω) between ω = ω_c − Δω and ω = ωc + Δω with spacing 2 Δω/N, where

This is equivalent to zooming in on X (e^jω) in the region ω_c −Δω < ω < ωc + Δω. One system used to implement the zoom is shown in Figure P10.50-1. Assume that x_z[n] is zero-padded as necessary before the N-point DFT and h[n] is an ideal lowpass filter with a cutoff frequency Δω.

(a) In terms of k_Δ and the transform length N, what is the largest (possibly noninteger) value of M that can be used if aliasing is to be avoided in the downsampler?

(b) Consider x[n] with the Fourier transform shown in Figure P10.50-2. Using the maximum value of M from part (a), sketch the Fourier transforms of the intermediate signals and x_z[n] when ω_c = π/2 and Δω = π/6. Demonstrate that the system provides the desired frequency samples.

Another procedure for obtaining the desired samples can be developed by viewing the finite-length sequence X_N[k] indexed on k as a discrete-time data sequence to be processed as shown in Figure P10.50-3. The impulse response of the first system is

and the filter h[n] has the frequency response

The zoomed output signal is defined as

for appropriate values of kc and kΔ. Assume that kΔ is chosen so that M is an integer in the following parts.

(c) Suppose that the ideal lowpass filter h[n] is approximated by a causal Type I linear phase filter of length 513 (nonzero for 0 ≤ n ≤ 512). Indicate which samples of provide the desired frequency samples.