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<math>e^{jw_{o}N} = 1 = e^{j2\pi k}</math> | <math>e^{jw_{o}N} = 1 = e^{j2\pi k}</math> | ||
<math>\therefore w_{o}N = 2\pi k</math> | <math>\therefore w_{o}N = 2\pi k</math> | ||
| − | <math>\Rightarrow frac{w_{o}}{2\pi} = frac{K}{N} \Rightarrow</math>Rational number | + | <math>\Rightarrow \frac{w_{o}}{2\pi} = \frac{K}{N} \Rightarrow</math>Rational number |
| − | <math>\therefore {w_{o}}{2\pi}</math> shold be a rational number | + | <math>\therefore \frac{w_{o}}{2\pi}</math> shold be a rational number |
Revision as of 16:17, 30 June 2008
(a) Derive the condition for which the discrete time complex exponetial signal x[n] is periodic.
$ x[n] = e^{jw_{o}n} $ $ x[n] = x[n+N] = e^{jw_{o}(n+N)} = e^{jw_{o}n}e^{jw_{o}N} $ to be periodic $ e^{jw_{o}N} = 1 = e^{j2\pi k} $ $ \therefore w_{o}N = 2\pi k $ $ \Rightarrow \frac{w_{o}}{2\pi} = \frac{K}{N} \Rightarrow $Rational number $ \therefore \frac{w_{o}}{2\pi} $ shold be a rational number
