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Homework 4, ECE438, Fall 2016, Prof. Boutin
Hard copy due in class, Wednesday September 21, 2016.
The goal of this homework is to learn two different ways to reconstruct a signal.
Question 1
Let x(t) be a continuous-time signal and let y[n]=x(nT) be a sampling of that signal with period T>0. We would like to interpolate the samples (i.e., "connect the dots") in order to try to recover x(t).
a) Write a formula for a band-limited interpolation of the samples (i.e., an expression for a continuous signal $ x_r(t) $ in terms of the samples y[n]).
b) Prove that the formula you gave in a) yields a band-limited signal $ x_r(t) $.
c) Under what circumstances is $ x_r(kT)=x(kT) $ for all integer values of k?
d) Under what circumstances is your interpolation equal to the original signal x(t)?
Question 2
Let x(t) be a continuous-time signal and consider a sampling y[n]=x(nT) of that signal.
a) Write a formula for a zero-order hold reconstruction $ x_r(t) $ of the samples.
b) Is the interpolation you wrote in 2a) band-limited? Answer yes/no and give a mathematical proof of your answer.
c) Under what circumstances is $ x_r(kT)=x(kT) $ for all integer values of k?
d) Under what circumstances is your interpolation equal to the original signal x(t)?
Hand in a hard copy of your solutions. Pay attention to rigor!
Presentation Guidelines
- Write only on one side of the paper.
- Use a "clean" sheet of paper (e.g., not torn out of a spiral book).
- Staple the pages together.
- Include a cover page.
- Do not let your dog play with your homework.
Discussion
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