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Problem 2. [40 pts] <br> | Problem 2. [40 pts] <br> | ||
| − | (a) Let <math>x[n]</math> and <math>y[n]</math> be real-valued sequences both of which are even-symmetric: <math>x[n]=x[-n]</ | + | (a) Let <math>x[n]</math> and <math>y[n]</math> be real-valued sequences both of which are even-symmetric: <math>x[n]=x[-n]</math> and <math>y[n]=y[-n]</math>. Under these conditions, prove that |
| − | <math>r_{xy}[l]=r_{yx}[l]</ | + | <math>r_{xy}[l]=r_{yx}[l]</math> for all <math>l</math>.<br> |
| − | (b) Express the autocorrelation sequence r_{zz}[l] for the complex-valued signal <math>z[n]=x[n]+jy[n]</ | + | (b) Express the autocorrelation sequence r_{zz}[l] for the complex-valued signal <math>z[n]=x[n]+jy[n]</math> where <math>x[n]</math> and <math>y[n]</math> are real-valued sequences, in terms of <math>r_{xx}[l]</math>,<math>r_{xy}[l]</math>,<math>r_yx[l]</math> and <math>r_{yy}[l]</math>.<br> |
:'''Click [[ECE_PhD_QE_CNSIP_2015_Problem1.1|here]] to view student [[ECE_PhD_QE_CNSIP_2015_Problem1.1|answers and discussions]]''' | :'''Click [[ECE_PhD_QE_CNSIP_2015_Problem1.1|here]] to view student [[ECE_PhD_QE_CNSIP_2015_Problem1.1|answers and discussions]]''' | ||
Revision as of 23:00, 17 February 2019
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] $.
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