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The definition of a periodic DT signal is that there exists an integer N such that <math>x[n+N] = x[n]</math> for all <math>n</math>. | The definition of a periodic DT signal is that there exists an integer N such that <math>x[n+N] = x[n]</math> for all <math>n</math>. | ||
| − | For example, <math>x[n] = j^n</math> is periodic | + | For example, <math>x[n] = j^n</math> is periodic. One can show that: |
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| + | <math>x[1] = j\,</math> | ||
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| + | <math>x[2] = -1\,</math> | ||
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| + | <math>x[3] = -j\,</math> | ||
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| + | <math>x[4] = 1\,</math> | ||
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| + | <math>x[5] = j\,</math> | ||
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| + | <math>x[6] = -1\,</math> | ||
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| + | <math>x[7] = -j\,</math> | ||
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| + | <math>x[8] = 1\,</math> | ||
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| + | As illustrated above, the function is clearly periodic, and has 4 as the smallest period. | ||
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| + | On the other hand, <math>cos(n)</math> is not a periodic signal because there is no integer that is multple of <math>2\pi</math> and is an integer. | ||
Latest revision as of 05:13, 5 September 2008
The definition of a periodic DT signal is that there exists an integer N such that $ x[n+N] = x[n] $ for all $ n $.
For example, $ x[n] = j^n $ is periodic. One can show that:
$ x[1] = j\, $
$ x[2] = -1\, $
$ x[3] = -j\, $
$ x[4] = 1\, $
$ x[5] = j\, $
$ x[6] = -1\, $
$ x[7] = -j\, $
$ x[8] = 1\, $
As illustrated above, the function is clearly periodic, and has 4 as the smallest period.
On the other hand, $ cos(n) $ is not a periodic signal because there is no integer that is multple of $ 2\pi $ and is an integer.
