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.

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 $r_{zz}[l]$