Practice Question on "Signals and Systems"
Topic: Signal Energy and Power
Contents
Question
Compute the energy $ E_\infty $ and the power $ P_\infty $ of the following discrete-time signal
$ x[n] = \left\{ \begin{array}{ll} \left(\frac{1}{1+j}\right)^n & \text{ if } n>=0,\\ 0 & \text{otherwise}. \end{array} \right. $
Answer 1
$ \begin{align} E_{\infty}&=\sum_{n=0}^N |\left(\frac{1}{1+j}\right)^n|^2 \\ &= \sum_{n=0}^N (\left(\frac{1}{1+j}\right)^n * \left(\frac{1}{1-j}\right)^n) \\ &= \sum_{n=0}^N \left(\frac{1}{(1+j)(1-j)}\right)^n \\ &= \sum_{n=0}^N (\frac{1}{2})^n \\ &= \frac{1}{1-\frac{1}{2}} \\ &= 2 \\ \end{align} $
So $ E_{\infty} = 2 $
$ \begin{align} P_{\infty}&=\lim_{N\rightarrow \infty}{1 \over {2N+1}}\sum_{n=0}^N |\left(\frac{1}{1+j}\right)^n|^2 \\ \text{Similar to math above, the expression can be derived towards}\\ &= \lim_{N\rightarrow \infty}{1 \over {2N+1}}\sum_{n=0}^N (\frac{1}{2})^n \\ &= \lim_{N\rightarrow \infty}{1 \over {2N+1}} \frac{1(1-(\frac{1}{2})^{N+1})}{1-\frac{1}{2}} \\ &= \lim_{N\rightarrow \infty}{1 \over {2N+1}}2 (1-(\frac{1}{2})^{N+1}) \\ &= \lim_{N\rightarrow \infty}{1 \over {2N+1}} (2-\frac{1}{2^N}) \\ &= \lim_{N\rightarrow \infty} \left(\frac{2-\frac{1}{2^N}}{2N+1} \right) \\ &= \frac{2}{\infty}\\ &= 0 \\ \end{align} $
So $ P_{\infty} = 0 $.
Answer 2
write it here.
Answer 3
write it here.
