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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>.]] |
Revision as of 06:08, 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
To be completed...
