Contents
Homework 5, ECE438, Fall 2013, Prof. Boutin
Harcopy of your solution due in class, Friday September 27, 2013
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.
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) Derive a formula for a band-limited interpolation of the samples (i.e., an expression for a continuous signal z(t) in terms of the samples y[n]). (Do not simply write down the formula; show how to derive it.)
b) Show that your interpolation is equal to the original signal at all sample points.
c) Under what circumstances is your interpolation equal to the original signal x(t)? Explain.
Question 2
Again, we consider a continuous-time signal x(t) and a sampling y[n]=x(nT) of that signal.
a) Write a formula for a piece-wise constant interpolation of the samples.
b) Derive the relationship between the Fourier transform of the interpolation you wrote in 2a) and the Fourier transform of x(t). (Do not simply write down the formula; show how to derive it.)
c) Is the interpolation you wrote in 2a) band-limited? Explain.
Discussion
Please discuss the homework below.
- I'm having a bit of trouble getting started on this homework? Does anyone have any suggestions on what I should read? Perhaps page numbers in the textbook or supplemental notes? Thanks.
- Which part are you having trouble with? If you are stuck at Question 1a), try to write the reconstructed signal in the frequency domain first. (Recall that you just need to low-pass-filter the ideal sampling.) Then invert the Fourier transform to get the reconstructed signal in the time-domain. -pm
