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One can take a signal that would be periodic in continuous time and turn it into a signal that is <b>not</b> periodic in discrete time. Consider the continuous time signal <math>x(t)=sin(t)</math>. Plotting this signal yields a smooth waveform that repeats itself with period <math>T=2\pi</math>. | One can take a signal that would be periodic in continuous time and turn it into a signal that is <b>not</b> periodic in discrete time. Consider the continuous time signal <math>x(t)=sin(t)</math>. Plotting this signal yields a smooth waveform that repeats itself with period <math>T=2\pi</math>. | ||
| − | [[Image:hw2a1a_blaskows_ECE301Fall2008mboutin.jpg|frame|center|The continuous-time signal <math>x(t)=sin(t)</math> is periodic.]] | + | [[Image:hw2a1a_blaskows_ECE301Fall2008mboutin.jpg|300px|frame|center|The continuous-time signal <math>x(t)=sin(t)</math> is periodic.]] |
Sampling this signal at every integer time yields something altogether different. | Sampling this signal at every integer time yields something altogether different. | ||
| − | [[Image:hw2a1b_blaskows_ECE301Fall2008mboutin.jpg|frame|center | + | [[Image:hw2a1b_blaskows_ECE301Fall2008mboutin.jpg|300px|frame|center|Sampling the continuous-time signal <math>x(t)=sin(t)</math> at integer times yields something like this. Note that the new discrete-time function <math>x[n]=sin(n)</math> is not periodic. Here we have shown five cycles of the formerly-periodic continuous time function.]] |
The new discrete time function looks like this on its own. | The new discrete time function looks like this on its own. | ||
| − | [[Image:hw2a1c_blaskows_ECE301Fall2008mboutin.jpg|frame|center | + | [[Image:hw2a1c_blaskows_ECE301Fall2008mboutin.jpg|300px|frame|center|The non-periodic discrete-time function <math>x[n]=sin(n)</math>.]] |
For the signal to be periodic, there must exist an integer N such that <math>x[n]=x[n+N]</math>. For the signal defined as it is here, no such integer N exists. | For the signal to be periodic, there must exist an integer N such that <math>x[n]=x[n+N]</math>. For the signal defined as it is here, no such integer N exists. | ||
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Suppose our sampling frequency, instead of being 1, was <math>\frac{\pi}{8}</math>. Then the newly sampled function overlaid with the continuous function would look something like | Suppose our sampling frequency, instead of being 1, was <math>\frac{\pi}{8}</math>. Then the newly sampled function overlaid with the continuous function would look something like | ||
| − | [[Image:hw2a1d_blaskows_ECE301Fall2008mboutin.jpg|frame|center | + | [[Image:hw2a1d_blaskows_ECE301Fall2008mboutin.jpg|300px|frame|center|The periodic discrete-time function <math>x[n]=sin(\frac{\pi}{8}n)</math> overlaid with its continuous time equivalent.]] |
| − | [[Image:hw2a1e_blaskows_ECE301Fall2008mboutin.jpg|frame|center | + | [[Image:hw2a1e_blaskows_ECE301Fall2008mboutin.jpg|300px|frame|center|The periodic discrete-time function <math>x[n]=sin(\frac{\pi}{8}n)</math>.]] |
==Part 2== | ==Part 2== | ||
| − | + | Consider the non-periodic function <math>f(t)=e^{-0.2t}*sin(10t)</math>. | |
| + | |||
| + | [[Image:Nonperiodic_blaskows_ECE301Fall2008mboutin.jpg|frame|center|Non-periodic function <math>f(t)=e^{-0.2t}sin(10t)</math>.]] | ||
| + | |||
| + | If we run the following MATLAB code, we can create a signal with an arbitrary period. Presented here is a signal with period 5. | ||
| + | |||
| + | <pre> | ||
| + | %Set up the environment | ||
| + | clear | ||
| + | clc | ||
| + | |||
| + | %Define parameters for shifiting the function | ||
| + | delta=0.0001; %step size | ||
| + | period=5; %period of function to produce | ||
| + | nperiod=5; %number of periods to reproduce | ||
| + | |||
| + | %define an x-vector for the inputs to the function | ||
| + | x=-2*period:delta:(nperiod+2)*period; | ||
| + | |||
| + | %intermediate results -- this gets shifted and added as necessary | ||
| + | f=exp(-0.2.*x).*sin(10.*x); | ||
| + | |||
| + | %preallocate the y-vector for speed (MATLAB's suggestion) | ||
| + | y=zeros(size(x)); | ||
| + | |||
| + | %for each period we are to reproduce, add the appropriate section of the | ||
| + | %f-vector (shifted horizontally) to the output vector. | ||
| + | for count=1:period/delta:(nperiod+8)*period/delta | ||
| + | y(1,count:size(y,2))=y(1,count:size(y,2))+f(1,1:size(y,2)-count+1); | ||
| + | end | ||
| + | |||
| + | %plot the results | ||
| + | plot(x,y) | ||
| + | grid | ||
| + | axis([0,nperiod*period,min(y),max(y)]); | ||
| + | </pre> | ||
| + | |||
| + | Running this code produces this figure. | ||
| + | |||
| + | [[Image:HW2a1f_blaskows_ECE301Fall2008mboutin.jpg|frame|center|Non-periodic function <math>f(t)=e^{-0.2t}sin(10t)</math> shifted and added several times to produce a periodic function.]] | ||
Latest revision as of 10:07, 10 September 2008
Part 1
Changing a Periodic Continuous Time Signal to a Non-Periodic Discrete Time Signal
One can take a signal that would be periodic in continuous time and turn it into a signal that is not periodic in discrete time. Consider the continuous time signal $ x(t)=sin(t) $. Plotting this signal yields a smooth waveform that repeats itself with period $ T=2\pi $.
The continuous-time signal $ x(t)=sin(t) $ is periodic.
Sampling this signal at every integer time yields something altogether different.
Sampling the continuous-time signal $ x(t)=sin(t) $ at integer times yields something like this. Note that the new discrete-time function $ x[n]=sin(n) $ is not periodic. Here we have shown five cycles of the formerly-periodic continuous time function.
The new discrete time function looks like this on its own.
The non-periodic discrete-time function $ x[n]=sin(n) $.
For the signal to be periodic, there must exist an integer N such that $ x[n]=x[n+N] $. For the signal defined as it is here, no such integer N exists.
Changing a Periodic Continuous Time Signal to a Periodic Discrete Time Signal
Suppose our sampling frequency, instead of being 1, was $ \frac{\pi}{8} $. Then the newly sampled function overlaid with the continuous function would look something like
The periodic discrete-time function $ x[n]=sin(\frac{\pi}{8}n) $ overlaid with its continuous time equivalent.
The periodic discrete-time function $ x[n]=sin(\frac{\pi}{8}n) $.
Part 2
Consider the non-periodic function $ f(t)=e^{-0.2t}*sin(10t) $.
Non-periodic function $ f(t)=e^{-0.2t}sin(10t) $.
If we run the following MATLAB code, we can create a signal with an arbitrary period. Presented here is a signal with period 5.
%Set up the environment
clear
clc
%Define parameters for shifiting the function
delta=0.0001; %step size
period=5; %period of function to produce
nperiod=5; %number of periods to reproduce
%define an x-vector for the inputs to the function
x=-2*period:delta:(nperiod+2)*period;
%intermediate results -- this gets shifted and added as necessary
f=exp(-0.2.*x).*sin(10.*x);
%preallocate the y-vector for speed (MATLAB's suggestion)
y=zeros(size(x));
%for each period we are to reproduce, add the appropriate section of the
%f-vector (shifted horizontally) to the output vector.
for count=1:period/delta:(nperiod+8)*period/delta
y(1,count:size(y,2))=y(1,count:size(y,2))+f(1,1:size(y,2)-count+1);
end
%plot the results
plot(x,y)
grid
axis([0,nperiod*period,min(y),max(y)]);
Running this code produces this figure.
Non-periodic function $ f(t)=e^{-0.2t}sin(10t) $ shifted and added several times to produce a periodic function.
