Communicates & Signal Process (CS)

Question 5: Image Processing

August 2017

Problem 1. [50 pts]
Consider the following 2D system with input $x(m,n)$ and output $y(m,n)$ for $\lambda>0$.
$y(m,n)=x(m,n)+\lambda(x(m,n)-\dfrac{1}{9}\sum_{k=-1}^{1} \sum_{l=-1}^{1} x(m-k,n-l))$.
a) Is this a linear system? Is this a space invariant system?
b) Calculate and sketch the psf, $h[n]$, for $\lambda=0.5$.
c) Is this a separable system?
d) Calculate the frequency response, $H(e^{j\mu},e^{jv})$. (Express your result in simplified from.)
e) Describe what ths filter does and how the output changes as $\lambda$ increases.

Problem 2. [50 pts]
Let $x(t)=sinc^2(t/a)$ for some $a>0$, and let $y(n)=x(nT)$ where $f_s=1/T$ is the sampling frequency of the system.
a) Calculate and sketch $X(f)$, the CTFT of $x(t)$.
b) Calculate $Y(e^{j\omega})$, the DTFT of $x(t)$.
c) What is the minimum sampling frequency, $f_s$, that ensures perfect reconstruction of the signal?
d) Sketch the function $Y(e^{j\omega})$ on the interval $[-2\pi,2\pi]$ when $T=a/2$.
e) Sketch the function $Y(e^{j\omega})$ on the interval $[-2\pi,2\pi]$ when $T=a$.