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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{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{e) Calculate } y(m,n) \text{ when } x(m,n)=1. $

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Part 2. 50 pts

 $ \color{blue}\text{Consider an image } f(x,y) \text{ with a forward projection} $

$ \color{blue} p_{\theta}(r) = \mathcal{FP}\left \{ f(x,y) \right \} $

$ \color{blue}  = \int_{-\infty}^{\infty}{f \left ( r cos(\theta) - z sin(\theta),r sin(\theta) + z cos(\theta) \right )dz}.  $

$ \color{blue} \text{Let } F(\mu,\nu) \text{ be the continuous-time Fourier transform of } f(x,y) \text{ given by} $
$ \color{blue} F(u,v) = \int_{-\infty}^{\infty}{\int_{-\infty}^{\infty}{f(x,y)e^{-j2\pi(ux,vy)}dx}dy} $

$ \color{blue} \text{and let } P_{\theta}(\rho) \text{ be the continuous-time Fourier transform of } p_{\theta}(r) \text{ given by} $
$ \color{blue} P_{\theta}(\rho) = \int_{-\infty}^{\infty}{p_{\theta}(r)e^{-j2\pi(\rho r)}dr}. $

$ \color{blue}\text{a) Calculate the forward projection }p_{\theta}(r) \text{, for } f(x,y) = \delta(x,y). $

$ \color{blue}\text{b) Calculate the forward projection }p_{\theta}(r) \text{, for } f(x,y) = \delta(x-1,y-1). $

$ \color{blue}\text{c) Calculate the forward projection }p_{\theta}(r) \text{, for } f(x,y) = rect \left(\sqrt[]{x^2+y^2} \right). $

$ \color{blue}\text{d) Calculate the forward projection }p_{\theta}(r) \text{, for } f(x,y) = rect \left(\sqrt[]{(x-1)^2+(y-1)^2} \right). $

$ \color{blue}\text{e) Describe in precise detail, the steps required to perform filtered back projection (FBP) reconstruction of } f(x,y). $

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