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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}{c} \text{ LPF} \\ \text{ no gain} \\ \text{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) corresponds to 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_1(\omega) $ that would make the following system equivalent to System 1.
$ x(t) \rightarrow \left[ \begin{array}{c} \text{LPF} \\ \text{ no gain }\\ \text{ cutoff at 1000Hz} \end{array}\right] \rightarrow \left[ \begin{array}{c} \text{C/D Converter} \\ \text{6000 samples per second} \end{array}\right] \rightarrow \left[ \begin{array}{c} H_1(\omega) \end{array}\right] \rightarrow \left[ \begin{array}{c} \text{D/C Converter} \\ \text{6000 samples per second} \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. 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. (Hint: the first two parts of the system correspond to an "interpolator".)
$ 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] $
Question 5
Define System 3 as the following LTI system
$ x[n] \rightarrow \left[ \begin{array}{c} \text{LPF} \\ \text{ no gain }\\ \text{ cutoff at} \frac{\pi}{2} \end{array}\right] \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 3 as follows? Answer yes/no. If you answered yes, sketch the graph of the required LPF2 and frequency response H3. If you answered no, explain why not. (Hint: the last two parts of the system correspond to an "interpolator".)
$ x[n] \rightarrow \left[ \begin{array}{c} \text{LPF} \\ \text{ no gain }\\ \text{ cutoff at} \frac{\pi}{2} \end{array}\right] \rightarrow \left[ \begin{array}{ccc} & & \\ & \text{Downsample by factor 2} & \\ & & \end{array}\right] \rightarrow \left[ \begin{array}{ccc} & & \\ & H_3(\omega) & \\ & & \end{array}\right] \rightarrow \left[ \begin{array}{ccc} & & \\ & \text{Upsample by factor 2} & \\ & & \end{array}\right] \rightarrow \left[ \begin{array}{c} \text{LPF2} \end{array}\right] \rightarrow y([n] $
Hand in a hard copy of your solutions. Pay attention to rigor!
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