ECE Ph.D. Qualifying Exam in Communication Networks Signal and Image processing (CS), Question 5, August 2011
Question
Part 1. 50 pts
$ \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), \text{where} T\leq1. $
$ \color{blue}\text{a) Calculate the frequency response, }H \left( e^{j\mu},e^{j\nu} \right). $
$ \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{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{c) Sketch the frequency response for } |\mu| < 2\pi \text{ and } |nu| < 2\pi \text{ when } T = \frac{1}{2} $
$ \color{blue}\left( \text{d} \right) \text{Find } f_{Y}\left(y|z \right ). $
$ \color{blue}\left( \text{e} \right) \text{Find } f_{XY}\left(x,y|z \right ). $
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Part 2. 25 pts
$ \color{blue} \text{Show that if a continuous-time Gaussian random process } \mathbf{X}(t) \text{ is wide-sense stationary, it is also strict-sense stationary.} $
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