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A CT signal x(t) is called periodic if there exists an integer T > 0 such that x(t+T) = x(t). | A CT signal x(t) is called periodic if there exists an integer T > 0 such that x(t+T) = x(t). | ||
| − | x(t) = <math>e^{j*w*t}</math> can be a periodic function. The period is <math>(2 * pi) / w</math>. If <math>w / (2 * pi)</math> is a rational number then the exponential function is periodic. | + | x(t) = <math>e^{n*j*w*t}</math> can be a periodic function. The period is <math>(2 * pi) / w</math>. If <math>w / (2 * pi)</math> is a rational number then the exponential function is periodic. |
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Periodic Function: | Periodic Function: | ||
| − | <math>e^{(j * pi / 6)}</math> , has w =<math> pi / 6 </math> therefore: <math>w/(2*pi) = 1/12</math> | + | <math>e^{(n* j * pi / 6)}</math> , has w =<math> pi / 6 </math> therefore: <math>w/(2*pi) = 1/12</math> which is a rational number, thus proving that the function is periodic. |
Nonperiodic Function: | Nonperiodic Function: | ||
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| + | <math>e^{(n* j / 6)}</math> , has w =<math> 1 / 6 </math> therefore: <math>w/(2*pi) = 1/(12*pi)</math> which is not a rational number. Accordingly the function is nonperiodic. | ||
Latest revision as of 07:17, 5 September 2008
Periodic Functions
A DT signal x[n] is called periodic if there exists an integer N such that x[n+N] = x[n] for all n.
A CT signal x(t) is called periodic if there exists an integer T > 0 such that x(t+T) = x(t).
x(t) = $ e^{n*j*w*t} $ can be a periodic function. The period is $ (2 * pi) / w $. If $ w / (2 * pi) $ is a rational number then the exponential function is periodic.
Periodic Function:
$ e^{(n* j * pi / 6)} $ , has w =$ pi / 6 $ therefore: $ w/(2*pi) = 1/12 $ which is a rational number, thus proving that the function is periodic.
Nonperiodic Function:
$ e^{(n* j / 6)} $ , has w =$ 1 / 6 $ therefore: $ w/(2*pi) = 1/(12*pi) $ which is not a rational number. Accordingly the function is nonperiodic.
