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\beta cos[(1+\beta)\pi(n+5)/2]}{\pi[1-4\beta^2(n+5)^2]}+\dfrac{sin[(1-\beta)\pi(n+0.5)/2]}{\pi[(n+.5)-4\beta^2(n+.5)^3]},-\infty<n<\infty $ with $ \beta=0.5 $

$ h_1[n]=(-1)^n h_0[n] $ $ f_0[n]=h_0[n] $ $ f_1[n]=-h_1[n] $

The DTFT of the halfband filter $ h_0[n] $ above may be expressed as follows:

$ H_0(\omega)= \begin{cases} e^{j\dfrac{\omega}{2}} |\omega|<\dfrac{\pi}{4},\\ e^{j\dfrac{\omega}{2}} cos[(|\omega|-\dfrac{\pi}{4})], \dfrac{\pi}{4}<|\omega|<\dfrac{3\pi}{4} \\ 0 \dfrac{3\pi}{4}<|\omega|<\pi \end{cases} $

Consider the following input signal

$ x[n]=16\dfrac{sin(\dfrac{3\pi}{8}n)}{\pi n}\dfrac{sin(\dfrac{\pi}{8}n)}{\pi n}cos(\dfrac{\pi}{2}n) $

HINT: The solution to problem is greatly simplified if you exploit the fact that the DTFT of the input signal $ x[n] $ is such that $ X(\omega)=X(\omega-\pi) $.

(a) Plot the magnitude of the DTFT of $ x[n] $, $ X(\omega) $, over $ -\pi<\omega<\pi $. Show all work.

(b) Plot the magnitude of the DTFT of $ x_0[n] $, $ X_0(\omega) $, over $ -\pi<\omega<\pi $. Show all work.

(c) Plot the magnitude of the DTFT of $ x_1[n] $, $ X_1(\omega) $, over $ -\pi<\omega<\pi $. Show all work.

(d) Plot the magnitude of the DTFT of $ y_0[n] $, $ Y_0(\omega) $, over $ -\pi<\omega<\pi $. Show all work.

(e) Plot the magnitude of the DTFT of $ y_1[n] $, $ Y_1(\omega) $, over $ -\pi<\omega<\pi $. Show all work.

(f) Plot the magnitude of the DTFT of the final output $ y[n][n] $, $ Y(\omega) $, over $ -\pi<\omega<\pi $. Show all work.

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Problem 2. [40 pts]

(a) Let $ x[n] $ and $ y[n] $ be real-valued sequences both of which are even-symmetric: $ x[n]=x[-n] $ and $ y[n]=y[-n] $. Under these conditions, prove that
$ r_{xy}[l]=r_{yx}[l] $ for all $ l $.

(b) Express the autocorrelation sequence r_{zz}[l] for the complex-valued signal $ z[n]=x[n]+jy[n] $ where $ x[n] $ and $ y[n] $ are real-valued sequences, in terms of $ r_{xx}[l] $, $ r_{xy}[l] $, $ r_yx[l] $ and $ r_{yy}[l] $.

(c) Determine a closed-form expression for the autocorrelation sequence $ r_{xx}[l] $ for the signal $ x[n] $ below.

$ x[n]=({\dfrac{sin(\dfrac{\pi}{4}n)}{\pi n}})({1+(-1)^n}) $

(d) Determine a closed-form expression for the autocorrelation sequence $ r_yy[l] $ for the signal $ y[n] $ below

$ y[n]=(\dfrac{sin(\dfrac{\pi}{4})n}{\pi n})cos(\dfrac{\pi}{2}n) $

(e) Determine a closed-form expression for the autocorrelation sequence $ r_{zz}[l] $ for the complex-valued signal $ z[n] $ formed with $ x[n] $ and $ y[n] $ defined above as the real and imaginary parts, respectively, as defined below. You must show all work and simplify as much as possible.

$ z[n]=x[n]+jy[n] $

(f) Plot $ r_{zz}[l] $

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