(2. Create a periodic signal by summing shifted copies of a non-periodic signal)
 
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Therefore, <math>x[n + 2\pi] = x[n]</math>
 
Therefore, <math>x[n + 2\pi] = x[n]</math>
  
However, this still does not fulfill the requirement as <math>N = 2\pi</math> is not an integer. For the signal to become periodic, the CT waveform has to be modified to <math>x(t) = sin(2\pi t)</math> and sampled at a frequency of 1 Hz.
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However, this still does not fulfill the requirement as <math>N = 2\pi</math> is not an integer. For the signal to become periodic, the CT waveform has to be modified to <math>x(t) = sin(0.5\pi t)</math> and sampled at a frequency of 1 Hz. Upon modification, <math>x[n + 4] = x[n]</math>
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<math>x(t) = sin(0.5\pi t)</math> sampled at 1 Hz[[Image:Samp4_ECE301Fall2008mboutin.jpg]]
  
 
==Non Periodic Signal==
 
==Non Periodic Signal==
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==2. Create a periodic signal by summing shifted copies of a non-periodic signal==
 
==2. Create a periodic signal by summing shifted copies of a non-periodic signal==
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Using Cheng Chen's non periodic CT signal from HW1, y(t) = t.
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MATLAB CODING
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>> t1 = 0:0.1:5;
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>> t2 = 5.001:0.1:10;
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>> t3 = 10.001:0.1:15; 
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>> t = 0:0.1:15;
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>> y1 = (t1);
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>> y2 = (t2 - 5);                  %Referred to Wei Jean's code because I was getting an error and I was not sure     
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>> y3 = (t3 - 10);                %why I still needed to shift when the t1,t2, and t3 values were defined like they are. 
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>> y_sum = [y1 y2 y3];
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>> plot(t,y_sum)                       
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>>
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[[Image:Awsaw_ECE301Fall2008mboutin.jpg]]
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As can be seen from the graph the signal is periodic. It repeats every 5 seconds. Therefore x(t + 5) = x(t).
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Therefore when y(t) = t is copied and shifted periodically by an infinite number of times a periodic signal can be created.

Latest revision as of 09:24, 11 September 2008

1. Creating two DT signals (one periodic and one non-periodic) from a periodic CT signal

Let $ x(t) = sin (2\pi t), $ which is a periodic CT signal

$ x(t) = sin (2\pi t) $ Sin1 ECE301Fall2008mboutin.jpg


Sampling every t = 0.01 Samp0 ECE301Fall2008mboutin.jpg


Periodic Signal

Sampling every $ t = \pi $ Samp pi ECE301Fall2008mboutin.jpg

This discrete time signal was produced from a CT sine wave by sampling at a frequency of $ \frac{1}{\pi} $.

As can be seen from the graph, the values of x[n] are periodic because they repeat after every period of $ t = 2\pi $.

Therefore, $ x[n + 2\pi] = x[n] $

However, this still does not fulfill the requirement as $ N = 2\pi $ is not an integer. For the signal to become periodic, the CT waveform has to be modified to $ x(t) = sin(0.5\pi t) $ and sampled at a frequency of 1 Hz. Upon modification, $ x[n + 4] = x[n] $

$ x(t) = sin(0.5\pi t) $ sampled at 1 HzSamp4 ECE301Fall2008mboutin.jpg

Non Periodic Signal

Sampling every t = 2 Samp2 ECE301Fall2008mboutin.jpg

For this discrete time signal which was produced by sampling the same sine wave at a frequency of 0.5, the values of x[n] are non-periodic because the discrete time signal is scattered all over the place with no indication of a pattern. Therefore, $ x[n + k] \neq x[n] $

2. Create a periodic signal by summing shifted copies of a non-periodic signal

Using Cheng Chen's non periodic CT signal from HW1, y(t) = t.


MATLAB CODING

>> t1 = 0:0.1:5;

>> t2 = 5.001:0.1:10;

>> t3 = 10.001:0.1:15;

>> t = 0:0.1:15;

>> y1 = (t1);

>> y2 = (t2 - 5);  %Referred to Wei Jean's code because I was getting an error and I was not sure

>> y3 = (t3 - 10);  %why I still needed to shift when the t1,t2, and t3 values were defined like they are.

>> y_sum = [y1 y2 y3];

>> plot(t,y_sum)

>>


Awsaw ECE301Fall2008mboutin.jpg

As can be seen from the graph the signal is periodic. It repeats every 5 seconds. Therefore x(t + 5) = x(t).

Therefore when y(t) = t is copied and shifted periodically by an infinite number of times a periodic signal can be created.

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