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=[[HW5ECE38F15|Homework 5]] Solution, [[ECE438]], [[2015_Fall_ECE_438_Boutin|Fall 2015]], [[user:mboutin|Prof. Boutin]]=
 
=[[HW5ECE38F15|Homework 5]] Solution, [[ECE438]], [[2015_Fall_ECE_438_Boutin|Fall 2015]], [[user:mboutin|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'''
 +
 +
<math>
 +
\mathcal{Y}(\omega) =\frac{1}{4} \sum_{k=0}^{3} \mathcal{X} \left (\frac{\omega-k2\pi}{4} \right )
 +
</math>
 +
 +
b) What is the relationship between the DT Fourier transform of x[n] and that of
 +
 +
<math>z[n]=\left\{ \begin{array}{ll}
 +
x[n/4],& \text{ if } n \text{ is a multiple of } 4,\\
 +
0, & \text{ else}.
 +
\end{array}\right.</math>
 +
 +
(Give the mathematical relation and sketch an example.)
 +
 +
'''Solution'''
 +
 +
<math>
 +
\mathcal{Z}(\omega) = \mathcal{X}(4\omega)
 +
</math>
 +
----
 +
==Question 2==
 +
'''Downsampling and upsampling'''
 +
 +
Let <math>x_1[n]=x(Tn)</math> be a sampling of a CT signal <math>x(t)</math>. Let D be a positive integer.
 +
 +
a) Under what circumstances is the downsampling <math>x_D [n]= x_1 [Dn]</math> equivalent to a resampling of the signal with a new period equal to DT (i.e. <math>x_D [n]= x(DT n)</math>)?
 +
 +
b) Under what circumstances is it possible to construct the sampling <math>x_3[n]= x(\frac{T}{D} n) </math> directly from <math>x_1[n]</math> (without reconstructing x(t))?
 +
----
 +
==Question 3==
 +
Define System 1 as the following LTI system
 +
 +
<math> 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)
 +
</math>
 +
 +
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 <math>H_1(\omega)</math> that would make the following system equivalent to System 1.
 +
 +
<math> 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)
 +
</math>
 +
----
 +
==Question 4==
 +
Define System 2 as the following LTI system
 +
 +
<math> x[n]\rightarrow 
 +
\left[ \begin{array}{ccc} & & \\
 +
&  H_1(\omega) & \\
 +
& & \end{array}\right]
 +
\rightarrow y[n]
 +
</math>
 +
 +
where the frequency response <math>H_1(\omega)</math> 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".)
 +
 +
<math> 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]
 +
</math>
 +
 +
----
 +
==Question 5==
 +
Define System 3 as the following LTI system
 +
 +
<math> 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]
 +
</math>
 +
 +
where the frequency response <math>H_1(\omega)</math> 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".)
 +
 +
<math> 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]
 +
</math>

Revision as of 22:05, 16 October 2015


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] $

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