ECE Ph.D. Qualifying Exam in "Communication, Networks, Signal, and Image Processing" (CS)
Question 5, August 2011, Part 1
- Part 1,2]
$ \color{blue}\text{Consider the following discrete space system with input } x(m,n) \text{ and output } y(m,n). $
$ \color{blue} y(m,n) = \sum_{k=-\infty}^{\infty}{\sum_{l=-\infty}^{\infty}{x(m-k,n-l)h(k,l)}}. $
$ \color{blue} \text{For parts a) and b) let} $
$ \color{blue} h(m,n)=sinc(mT,nT) $
$ \color{blue} \text{where } T\leq1. $
$ \color{blue} \text{For parts c), d), and e) let} $
$ \color{blue} h(m,n)=sinc\left( \frac{(n+m)T}{\sqrt[]{2}},\frac{(n-m)T}{\sqrt[]{2}} \right) $
$ \color{blue} \text{where } T\leq1. $
$ \color{blue}\text{a) Calculate the frequency response, }H \left( e^{j\mu},e^{j\nu} \right). $
$ \color{blue}\text{Solution 1:} $
$ H(e^{j\mu},e^{j\nu}) = \frac{1}{T^2} rect(\frac{\mu}{T},\frac{\nu}{T}) $
$ \color{blue}\text{Solution 2:} $
here put sol.2
$ \color{blue}\text{b) Sketch the frequency response for } |\mu| < 2\pi \text{ and } |nu| < 2\pi \text{ when } T = \frac{1}{2} $
$ \color{blue}\text{Solution 2:} $
sol2 here
$ \color{blue}\text{c) Calculate the frequency response, }H \left( e^{j\mu},e^{j\nu} \right). $
$ \color{blue}\text{Solution 1:} $
$ H(e^{j\mu},e^{j\nu}) = \frac{1}{T^2} rect \left ( \frac{(\mu + \nu)}{\sqrt{2}T},\frac{(\nu - \mu)}{\sqrt{2}T} \right ) $
$ \color{blue}\text{Solution 2:} $
sol2 here
$ \color{blue}\text{d) Sketch the frequency response for } |\mu| < 2\pi \text{ and } |nu| < 2\pi \text{ when } T = \frac{1}{2} $
$ \color{blue}\text{Solution 1:} $
$ \color{blue}\text{Solution 2:} $
sol2 here
$ \color{blue}\text{e) Calculate } y(m,n) \text{ when } x(m,n)=1. $
$ \color{blue}\text{Solution 1:} $
$ Y(e^{j\mu},e^{j\nu}) = \delta(e^{j\mu},e^{j\nu}) \cdot H(e^{j\mu},e^{j\nu}) $
$ = \frac{1}{T^2} rect (0,0) = 4 $
$ \Rightarrow y(m,n) = 4\delta(m,n) $
$ \color{blue}\text{Solution 2:} $
sol2 here
"Communication, Networks, Signal, and Image Processing" (CS)- Question 5, August 2011
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- Part 1: solutions and discussions
- Part 2: solutions and discussions
