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Homework 5, ECE438, Fall 2016, Prof. Boutin
Hard copy due in class, Wednesday September 28, 2016.
The goal of this homework are to
- understand the effect of downsampling and upsampling (in the Fourier domain);
- understand under what circumstances decimating and interpolating are equivalent to resampling;
- go over the relationship between DT signal processing and CT signal processing (for a simple filter) once more, this time with a focus on the effect of changing the sampling frequency.
Downsampling and upsampling viewed in the Fourier domain
Question 1
Let x[n] be a DT signal. Let z[n]=x[2n] be a downsampling of x[n]. Let y[n] be an upsampling of x[n], namely $ y[n]=\left\{ \begin{array}{ll} x[n/5],& \text{ if } n \text{ is a multiple of } 5,\\ 0, & \text{ else}. \end{array}\right. $
a) What is the relationship between the DTFT of x[n] and the DTFT of z[n]? (Write a mathematical expression for the relationship.)
b) Sketch an example of the relationship you obtained in a).
c) What is the relationship between the DTFT of x[n] and the DTFT of y[n]? (Write a mathematical expression for the relationship.)
d) Sketch an example of the relationship you obtained in c).
Decimation and interpolation versus resampling
Question 2
Let $ x_1[n]=x(Tn) $ be a sampling of a CT signal $ x(t) $. Let D be a positive integer.
a) Draw a diagram for a downsampling by a factor D of the DT signal $ x_1[n] $.
b) Under what circumstances is the downsampling you described in a) equivalent to a resampling of the signal with a new period equal to DT (i.e. $ x_D [n]= x(DT n) $)?
c) Draw a diagram for a decimation by a factor D of the DT signal $ x_1[n] $. Point out the difference between this diagram and the one you drew in a)
b) Under what circumstances is the decimation you described in c) equivalent to a resampling of x(t) with a new period equal to DT (i.e. $ x_D [n]= x(DT n) $)?
DT processing after decimation and interpolation
Question 3
A continuous-time signal x(t) is such that its CTFT X(f) is zero when when |f|>1000 Hz. You would like to low-pass-filter the signal x(t) with a cut off frequency of 900Hz and a gain of 7. Let's call this desired filtered signal y(t).
a) Assume that you are only given a sampling of x(t), specifically a sampling obtained by taking 5000 samples per second (samples equally spaced in time). Can one process this sampling in such a way that a band-limited interpolation of the processed (output) DT signal would be the same as y(t)? Answer yes/no. If you answered yes, explain how. If you answered no, explain why not.
b) Now assume that the sampling from Part a) is downsampled by a factor 2. Can one process this downsampled signal in such a way a band-limited interpolation of the processed (output) DT signal would be the same as y(t)? Answer yes/no. If you answered yes, explain how. If you answered no, explain why not.
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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