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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(k)=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(k)=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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