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==Periodic Function== | ==Periodic Function== | ||
| + | The definition of a periodic function is as follows: | ||
| + | x[n] is periodic iff there exists an integer N such that x[n+N] = x[n]. The value of N is called the "Period." | ||
| − | + | An example of a periodic function is <math>f(t) = e^{2\pi j}</math>. | |
| + | To prove this, we do the following: | ||
| − | + | <math>x[n+N] = x[n]</math> | |
| − | + | ||
| + | <math>e^{2\pi j (n+N)} = e^{2\pi j n}</math> | ||
| + | |||
| + | |||
| + | <math>{e^{2\pi j n} e^{2\pi j N}} = e^{2\pi j n}</math> | ||
| + | |||
| + | |||
| + | <math>e^{2\pi j N} = 1</math> | ||
| + | |||
| + | |||
| + | <math>\cos(2\pi N) + j\sin(2\pi N) = 1</math> | ||
| + | |||
| + | Which is true if | ||
| + | |||
| + | <math>2\pi N = 2\pi </math> | ||
| + | |||
| + | at some point. | ||
| + | |||
| + | Since N = 1 yields <math> 2\pi = 2\pi </math>, we can conclude that in fact, <math>f(t) = e^{2\pi j}</math> is periodic. | ||
==Non-Periodic Function== | ==Non-Periodic Function== | ||
Revision as of 13:35, 3 September 2008
Periodic Function
The definition of a periodic function is as follows: x[n] is periodic iff there exists an integer N such that x[n+N] = x[n]. The value of N is called the "Period."
An example of a periodic function is $ f(t) = e^{2\pi j} $. To prove this, we do the following:
$ x[n+N] = x[n] $
$ e^{2\pi j (n+N)} = e^{2\pi j n} $
$ {e^{2\pi j n} e^{2\pi j N}} = e^{2\pi j n} $
$ e^{2\pi j N} = 1 $
$ \cos(2\pi N) + j\sin(2\pi N) = 1 $
Which is true if
$ 2\pi N = 2\pi $
at some point.
Since N = 1 yields $ 2\pi = 2\pi $, we can conclude that in fact, $ f(t) = e^{2\pi j} $ is periodic.
Non-Periodic Function
The function shown is $ f(t) = {\sin(t)\over t} $.
Note that the function never repeats itself. It changes constantly over its entire expanse.
The function $ f(t) = {\sin(t)\over t} $ from 0 to 15.
