Communicates & Signal Process (CS)
Question 2: Signal Processing
August 2011
Problem 1. [60 pts]
In the system below, the two analysis filters, $ h_0[n] $ and $ h_1[n] $, and the two synthesis filters, $ f_0[n] $ and $ f_1[n] $,form a Quadrature Mirror Filter (QMF). Specially,
$ h_0[n]=\dfrac{2\betacos[(1+\beta)\pi(n+5)/2]}{\pi[1-4\beta^2(n+5)^2]}+\dfrac{sim[(1-\beta)\pi(n+0.5)/2]}{\pi[(n+.5)-4\beta^2(n+.5)^3]},-\infty<n<\infty with \beta=0.5 $
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 $.
(a) Let x[n] be a discrete-time rectangular pulse of length $ L=12 $ as defined below:
(i) $ X_{N}(k) $ is computed as a 16-point DFT of x[n] and used in Eqn (1) with N=16. Write a closed-form expression for resulting reconstructed spectrum $ X_{r}(\omega) $.
(ii) $ X_{N}(k) $ is computed as a 12-point DFT of x[n] and used in Eqn (1) with N=12. Write a closed-form expression for the resulting reconstructed spectrum $ X_{r}(\omega) $.
(iii) $ X_{N}(k) $ is computed as an 8-point DFT of x[n] ans used in Eqn (1) with N=8. That is, $ X_{N}(k) $ is obtained by sampling the DTFT of x[n] at 8 equi-spaced frequencies between 0 and 2$ \pi $. Write a closed-form expression for the resulting reconstructed spectrum $ X_{r}(\omega) $.
(b) Let x[n] be a discrete-time sinewave of length L=12 as defined below. For all sub-parts of part (b), $ X_{N}(k) $ is computed as a 12-pt DFT of x[n] and used in Eqn (1) with N=12.
(i) Write a closed-form expression for the resulting reconstructed spectrum $ X_{r}(\omega) $.
(ii) What is the numerical value of $ X_{r}(\frac{\pi}{3}) $? The answer is a number and you do not need a calculator to determine the value; this also applies to the next 2 parts.
(iii) What is the numerical value of $ X_{r}(\frac{5 \pi}{3}) $?
(iv) What is the numerical value of $ X_{r}(\frac{\pi}{2}) $?
Problem 2. [50 pts]
Consider a finite-length sinewave of the form below where $ k_{o} $ is an interger in the range $ 0 \leq k_{o} \leq N-1 $.
In addition, h[n] is a causal FIR filter of length L, where L < N. In this problem $ y[n]=x[n] \star h[n] $ is the linear convolution of the causal sinewave of length N in Equation (1) with a causal FIR filter of length L, where L < N.
(a) The region $ 0 \leq n \leq L-1 $ corresponds to partial overlap. The covolution sum can be written as:
Determine the upper and lower limits in the convolution sum above for $ 0 \leq n \leq L-1 $
(b) The region $ L \leq n \leq N-1 $ corresponds to full overlap. The convolution sum is:
(i) Determine the upper and lower limits in the convolution sum for $ L \leq n \leq N-1 $.
(ii) Substituting x[n] in Eqn (1), show that for this range y[n] simplifies to:
where $ H_{N}(k) $ is the N-point DFT of h[n] evaluated at $ k = k_{o} $. To get the points, you must show all work and explain all details.
(c) The region $ N \leq n \leq N+L-2 $ corresponds to partial overlap. The convolution sum:
Determine the upper and lower limits in the convolution sum for $ N \leq n \leq N+L-2 $.
(d) Add the two regions of partial overlap at the beginning and end to form:
(i) Determine the upper and lower limits in the convolution sum above.
(ii) Substituting x[n] in Eqn (1), show that for this range z[n] simplifies to:
where $ H_{N}(k) $ is the N-point DFT of h[n] evaluated at $ k = k_{o} $ as defined previously.
(e) $ y_{N}[n] $ is formed by computing $ X_{N}(k) $ as an N-pt DFT of x[n] in Enq (2), $ H_{N}(k) $ as an N-pt DFT of h[n], and then $ y_{N}[n] $ as the N-pt inverse DFT of $ Y_{N}(k) = X_{N}(K)H_{N}(k) $. Write a simple, closed-form expression for $ y_{N}(k) $. Is $ z[n]=y_{N}[n] + y[n+N] \ \ for \ 0 \leq n \leq N-1 $?
