QE2010 CS-2 ECE538 - Rhea

Communicates & Signal Process (CS)

Question 2: Signal Processing

August 2017

Problem 1. [50 pts]
Equation 1 below is the formula for reconstructing the DTFT, $X(\omega)$ , from $$ N $$ equi-spaced samples of the DTFT over $0 \leq \omega \leq 2\pi$ . $X_{N}(k) = X(\frac{2\pi k}{N},k=0,1,...,N-1)$ is the N-pt DFT of x[n], which corresponds to N equi-spaced samples of the DTFT of x[n] over $0 \leq \omega \leq 2\pi$ .

X_{r}(\omega)=\sum_{k=0}^{N-1} X_{N}(k) \frac{sin[\frac{N}{2}(\omega - \frac{2 \pi k}{N})]}{N sin[\frac{1}{2} (\omega -\frac{2 \pi k}{N})]} e^{-j\frac{N-1}{2}(\omega - \frac{2 \pi k}{N}) }

(1)

(a) Let x[n] be a discrete-time rectangular pulse of length $$ L=12 $$ as defined below:

x[n] = \{-1,-1,-1,-1,1,1,1,1,1,1,1,1 \}

(i) $X_{N}(k)$ is computed as a 16-point DFT of x[n] and used in Eqn (1) with N=16. Write a close-form expression for resulting reconstructed spectrum $X_{r}(\omega)$ .
(ii)

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