(New page: ==Periodic Signal== Let <math>x[n] = e^{jn}</math> For x[n] to be periodic, the following must hold true: <math>e^{jn} = e^{j(n+N)}</math> ==Non Periodic Signal==) |
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| + | A discrete time signal is periodic if there exists T > 0 such that x(t + T) = x(t) | ||
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| + | A continuous time signal is periodic if there exists some integer N > 0 such that x[n + N] = x[n] | ||
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==Periodic Signal== | ==Periodic Signal== | ||
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| + | Let <math>Insert formula here</math> | ||
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| + | ==Non Periodic Signal== | ||
Let <math>x[n] = e^{jn}</math> | Let <math>x[n] = e^{jn}</math> | ||
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For x[n] to be periodic, the following must hold true: | For x[n] to be periodic, the following must hold true: | ||
| − | <math>e^{jn} = e^{j(n+N)}</math> | + | <math>e^{jn} = e^{j(n+N)}</math> for some integer N |
| − | == | + | <math>e^{jn} = e^{jn} e^{jN}</math> |
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| + | <math>1 = e^{jN}</math> | ||
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| + | 1 = cos(N) + jsin(N) | ||
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| + | This equation only holds true if <math>N = 2\pi</math> or some multiple of <math>2\pi</math> | ||
| + | |||
| + | Therefore<math>x[n] = e^{jn}</math> is not periodic because <math>2\pi</math> is not an integer. | ||
Revision as of 08:23, 4 September 2008
A discrete time signal is periodic if there exists T > 0 such that x(t + T) = x(t)
A continuous time signal is periodic if there exists some integer N > 0 such that x[n + N] = x[n]
Periodic Signal
Let $ Insert formula here $
Non Periodic Signal
Let $ x[n] = e^{jn} $
For x[n] to be periodic, the following must hold true:
$ e^{jn} = e^{j(n+N)} $ for some integer N
$ e^{jn} = e^{jn} e^{jN} $
$ 1 = e^{jN} $
1 = cos(N) + jsin(N)
This equation only holds true if $ N = 2\pi $ or some multiple of $ 2\pi $
Therefore$ x[n] = e^{jn} $ is not periodic because $ 2\pi $ is not an integer.
