Signals in Natural Domain
Chapter 2 : Simple Operations and properties of Sequences
 
Periodicity properties of sinusoidal signals:
Let us consider the signal. We see that if we replace $ w_0$ by $ (w_0+2\pi)$ we get the same signal. In fact the signal with frequency $ w_0 \pm 2\pi\, ,w_0 \pm 4 \pi$ and so on. This situation is quite different from continuous time signal $ \{ \cos w_0 t, -\infty < t < \infty\}$ where each frequency is different. Thus in discrete time we need to consider frequency interval of length 2π only. As we increase $ w_0$ to π signal oscillates more and more rapidly. But if we further increase frequency from π to 2π the rate of oscillations decreases. This can be seen easily by plotting signal $ \{\cos w_0 n\}$ for several values of.
The signal $ \{\cos w_0 n\}$ is not periodic for every value of. For the signal to be periodic with period N > 0, we should have
                           $\displaystyle \{\cos w_0 n\}=\{cos w_0 (n+N)\}$

that is $ w_0 N$ should be some multiple of 2π.
         $\displaystyle w_0 N=2\pi m$
or     $\displaystyle \frac{w_0}{2\pi}=\frac{m}{N}$
Thus signal $ \{\cos w_0 n\}$ is periodic if and only if $ \frac{w_0}{2\pi}$ is a rational number.
Above observations also hold for complex exponential signal $ \{x[n]\}=\{e^{jw_0 n}\}$
 
 
 
 
 
 
 
 
 
 
 
 
 
 
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