Periodic CT and DT Signals

Part 1

Background

When given a CT signal, DT signals can be produced by sampling the CT at certain intervals. For Example: Given the periodic signal $ 4cos(\pi t) $, The following 2 signals can be produced, one periodic and one not.

CT Periodic Signal

Eat21 ECE301Fall2008mboutin.png

DT Periodic Signal

This function has a sampling frequency of 1 unit.

Eat ECE301Fall2008mboutin.png

DT Non-Periodic Signal

If we now sample this function at $ \frac{1}{\pi} $, the equation would turn into $ 4cos[n] $ , giving

Eat3 ECE301Fall2008mboutin.png

which is not periodic.

Part 2

Background

A function which is not periodic can be made to be periodic if it is concatenated with itself over a given range. The equation for this is given as $ \sum_{k}^\infty x(t+kT) $. For the equation $ y=4t^2 $, when $ T=5 $, the equation can be periodic.

Matlab Code

%Coding for part 2, citing help from Wei Jian Chan's homework posting

%Creating a 3,99999 size vector
t=[0.001:0.001:5;5.001:0.001:10;10.001:0.001:15];

%setting the three variables, with their offsets    
y=2*t(1,:).^2;
y2=2*(t(2,:)-5).^2;
y3=2*(t(3,:)-10).^2;

%concatenating them, ie putting them after another
y2=[y,y2,y3];

%making the new time axis for the whole thing
t=0.001:0.001:15;

%plotting
plot(t,y4);

Periodic Function

Eat4 ECE301Fall2008mboutin.png

Which is now a periodic version of that algebraic function.

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