Homework 5 Solution, ECE438, Fall 2015, Prof. Boutin

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.)

Solution

$ \mathcal{Y}(\omega) =\frac{1}{4} \sum_{k=0}^{3} \mathcal{X} \left (\frac{\omega-k2\pi}{4} \right ) $

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.)

Solution

$ \mathcal{Z}(\omega) = \mathcal{X}(4\omega) $

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] $