| Line 4: | Line 4: | ||
<math>x[n] = x[n+N] = e^{jw_{o}(n+N)} = e^{jw_{o}n}e^{jw_{o}N}</math> | <math>x[n] = x[n+N] = e^{jw_{o}(n+N)} = e^{jw_{o}n}e^{jw_{o}N}</math> | ||
to be periodic | to be periodic | ||
| − | <math>e^{jw_{o}N} = 1 = e^{j2\pi k | + | <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 w_{o}/2\pi = K/N \rightarrow</math>Rational number | <math>\rightarrow w_{o}/2\pi = K/N \rightarrow</math>Rational number | ||
<math>\therefore w_{o} / 2\pi</math> shold be a rational number | <math>\therefore w_{o} / 2\pi</math> shold be a rational number | ||
Revision as of 16:14, 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 w_{o}/2\pi = K/N \rightarrow $Rational number $ \therefore w_{o} / 2\pi $ shold be a rational number
