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Problem 1. [50 pts] <br> | Problem 1. [50 pts] <br> | ||
| + | Consider the following 2D system with input <math>x(m,n)</math> and output <math>y(m,n)</math> for <math>\lambda>0</math>.<br> | ||
| + | <math>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))</math>.<br> | ||
| + | a) Is this a linear system? Is this a space invariant system?<br> | ||
| + | b) Calculate and sketch thw psf, <math>h[n]</math>, for <math>\lambda=0.5</math>.<br> | ||
| + | c) Is this a separable system?<br> | ||
| + | d) Calculate the frequency response, <math>H(e^{j\mu},e^{jv})</math>. (Express your esult in simplified from.)<br> | ||
| + | e) Describe what ths filter does and how the output changes as <math>\lambda</math> increases.<br> | ||
| + | |||
:'''Click [[ECE_PhD_QE_CNSIP_2015_Problem1.1|here]] to view student [[ECE_PhD_QE_CNSIP_2015_Problem1.1|answers and discussions]]''' | :'''Click [[ECE_PhD_QE_CNSIP_2015_Problem1.1|here]] to view student [[ECE_PhD_QE_CNSIP_2015_Problem1.1|answers and discussions]]''' | ||
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| − | + | ---- | |
| − | + | Problem 2. [50 pts]<br> | |
| + | Let <math>x(t)=sinc^2(t/a)</math> for some <math>a>0</math>, and let <math>y(n)=x(nT)</math> where <math>f_s=1/T</math> is the sampling frequency of the system.<br> | ||
| + | a) Calculate and sketch <math>X(f)</math>, the CTFT of <math>x(t)</math>.<br> | ||
| + | b) Calculate <math>Y(e^{j\omega})</math>, the DTFT of <math>x(t)</math>.<br> | ||
| + | c) What is the minimum sampling frequency, <math>f_s</math>, that ensures perfect reconstruction of the signal?<br> | ||
| + | d) Sketch the function <math>Y(e^{j\omega})</math> on the interval <math>[-2\pi,2\pi]</math> when <math>T=a/2</math>.<br> | ||
| + | e) Sketch the function <math>Y(e^{j\omega})</math> on the interval <math>[-2\pi,2\pi]</math> when <math>T=a</math>.<br> | ||
[[ECE_PhD_Qualifying_Exams|Back to ECE QE page]] | [[ECE_PhD_Qualifying_Exams|Back to ECE QE page]] | ||
Revision as of 12:40, 18 February 2019
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 thw 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 esult in simplified from.)
e) Describe what ths filter does and how the output changes as $ \lambda $ increases.
- Click here to view student answers and discussions
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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