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Homework 5, ECE438, Fall 2015, Prof. Boutin
Hard copy due in class, Wednesday September 30, 2015.
The goal of this homework is to get an intuitive understanding on how to DT signals with different sampling frequencies in an equivalent fashion.
Question 1
Downsampling and upsampling
a) What is the relationship between the DT Fourier transform of x[n] and that of y[n]=x[4n]? (Give the mathematical relation and sketch an example.)
b) What is the relationship between the DT Fourier transform of x[n] and that of
$ z[n]=\left\{ \begin{array}{ll} x[n/4],& \text{ if } n \text{ is a multiple of } 4,\\ 0, & \text{ else}. \end{array}\right. $
(Give the mathematical relation and sketch an example.)
Question 2
Downsampling and upsampling
Let $ x_1[n]=x(Tn) $ be a sampling of a CT signal $ x(t) $. Let D be a positive integer.
a) Under what circumstances is the downsampling $ x_D [n]= x_1 [Dn] $ equivalent to a resampling of the signal with a new period equal to DT (i.e. $ x_D [n]= x(DT n) $)?
b) Under what circumstances is it possible to construct the sampling $ x_3[n]= x(\frac{T}{D} n) $ directly from $ x_1[n] $ (without reconstructing x(t))?
Question 3
Define System 1 as the following LTI system
$ x(t)\rightarrow \left[ \begin{array}{ccc} & & \\ & \text{ LPF cutoff at 1000Hz} & \\ & & \end{array}\right] \rightarrow \left[ \begin{array}{ccc} & & \\ & H(f) & \\ & & \end{array}\right] \rightarrow y(t) $
where the frequency response H(f) is a band-pass filter with no gain and cutoff frequencies f1=200Hz and f2=600Hz.
a) Sketch the graph of the frequency response H(f) of System 1.
b) Sketch the graph of the frequency response $ H_d(\omega) $ that would make the following system equivalent to System 1.
$ x(t) \rightarrow \left[ \begin{array}{ccc} & & \\ & \text{ LPF cutoff at 1000Hz} & \\ & & \end{array}\right] \rightarrow \left[ \begin{array}{ccc} & & \\ & \text{C/D Converter} & \\ & & \end{array}\right] \rightarrow x[n]=x(\frac{t}{6000}) \rightarrow \left[ \begin{array}{ccc} & & \\ & H_1(\omega) & \\ & & \end{array}\right] \rightarrow \left[ \begin{array}{ccc} & & \\ & \text{D/C Converter} & \\ & & \end{array}\right] \rightarrow y(t) $
Question 4
Define System 2 as the following LTI system
$ x[n]\rightarrow \left[ \begin{array}{ccc} & & \\ & H_1(\omega) & \\ & & \end{array}\right] \rightarrow y[n] $
where the frequency response $ H_1(\omega) $ is the one you obtained in Question 3.
a) Is it possible to implement System 2 as follows? Answer yes/no. If you answered yes, sketch the graph of the required LPF1 and frequency response H2. If you answered no, explain why not.
$ x[n] \rightarrow \left[ \begin{array}{ccc} & & \\ & \text{Upsample by factor 2} & \\ & & \end{array}\right] \rightarrow \left[ \begin{array}{ccc} & & \\ & \text{LPF1 } & \\ & & \end{array}\right] \rightarrow \left[ \begin{array}{ccc} & & \\ & H_2(\omega) & \\ & & \end{array}\right] \rightarrow \left[ \begin{array}{ccc} & & \\ & \text{Downsample by factor 2} & \\ & & \end{array}\right] \rightarrow y([n] $
Hand in a hard copy of your solutions. Pay attention to rigor!
Presentation Guidelines
- Write only on one side of the paper.
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- Staple the pages together.
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Discussion
You may discuss the homework below.
- I have been asked for clarification so let me try to rephrase. In each part, I am asking you to describe the digital filter that is equivalent to the given analog filter. In other words, how can you process the signal in the discrete-time domain, instead of the continuous time domain. Does that help? Let me know. -Prof. Mimi
- Write question/comment here.
- answer will go here
