View source for HW5ECE38F15 - Rhea




[[Category:ECE438Fall2015Boutin]]
[[Category:ECE438]]
[[Category:ECE]]
[[Category:fourier transform]]
[[Category:homework]]

=Homework 5, [[ECE438]], Fall 2015, [[user:mboutin|Prof. Boutin]]=
Hard copy due in class, Wednesday September 30, 2015.
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The goal of this homework is to get an intuitive understanding on how to DT signals with different sampling frequencies in an equivalent fashion.
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==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

<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.)
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==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))?
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==Question 3==
Define System 1 as the following LTI system

<math> 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)
</math>

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 <math>H_d(\omega)</math> that would make the following system equivalent to System 1. 

<math> 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)
</math>
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==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.

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. 

<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 
\left[ \begin{array}{ccc} & & \\
& \text{D/C Converter} & \\
& & \end{array}\right]
\rightarrow y([n]
</math>



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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.
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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
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