Determine the values of power and energy of the following signals. Find whether signals are power, energy or neither energy nor power signals: $$ \quad x(n)=\left(\frac{1}{3}\right)^n u(n) $$

Solution:

$ \text { signal } x(n)=\left(\frac{1}{3}\right)^n u(n)\\ $

To find Energy signal of x(n):

$ \begin{aligned} E & =\sum_{n=-\infty}^{\infty}|x(n)|^2 \\\\ & =\sum_{n=-\infty}^{\infty}\left[\left(\frac{1}{3}\right)^n\right]^2 u(n) \\\\ & =\sum_{n=0}^{\infty}\left[\left(\frac{1}{3}\right)^n\right]^2 \\\\ & =\sum_{n=0}^{\infty}\left(\frac{1}{9}\right)^n \\\\ & =\frac{1}{1-9}=\frac{9}{8}\\ \end{aligned}\\ $

To find Power of x(n) :

$ P=\lim _{n \rightarrow \infty} \frac{1}{2 N+1} \sum_{n=-N}^N|x(n)|^2\\ $

$ \begin{aligned} & =\lim \frac{1}{2 N+1} \sum_{n=-N}^N\left|\left(\frac{1}{3}\right)^n\right|^2 u(n) \\\\ & =\lim \frac{1}{2 N+1} \sum_{n=0}^N\left(\frac{1}{9}\right)^n \\\\ & =\lim \frac{1}{2 N+1}\left[\frac{1-\left(\frac{1}{9}\right)^{N+1}}{1-\frac{1}{9}}\right] \\\\ & =0\\ \end{aligned}\\ $