Latest revision as of 10:10, 16 September 2016

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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