| Line 30: | Line 30: | ||
&= \sum_{n=0}^N \left(\frac{1}{(1+j)(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 \\ | &= \sum_{n=0}^N (\frac{1}{2})^n \\ | ||
| − | &=\ | + | &= \frac{1}{1-\frac{1}{2}} \\ |
| + | &= 2 \\ | ||
\end{align}</math> | \end{align}</math> | ||
| − | So <math>E_{\infty} = | + | So <math>E_{\infty} = 2</math> |
| − | + | ||
| − | + | ||
<math>\begin{align} | <math>\begin{align} | ||
Revision as of 09:48, 22 January 2018
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=-N}^N |j|^2 \\ &= \lim_{N\rightarrow \infty}{1 \over {2N+1}}\sum_{n=-N}^N {(\sqrt{jj^*})}^2 \\ &= \lim_{N\rightarrow \infty}{1 \over {2N+1}}\sum_{n=-N}^N {(\sqrt{-j^2})}^2 \\ &= \lim_{N\rightarrow \infty}{1 \over {2N+1}}\sum_{n=-N}^N 1 \\ &= \lim_{N\rightarrow \infty}{1 \over {2N+1}}\sum_{n=0}^{2N} 1 \\ &= \lim_{N\rightarrow \infty}{2N+1 \over {2N+1}} \\ &= \lim_{N\rightarrow \infty}{1}\\ &= 1 \\ \end{align} $
So $ P_{\infty} = 1 $.
--Rgieseck 21:35, 12 January 2011
Answer 2
write it here.
Answer 3
write it here.
