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.

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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 $.

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