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Compute the energy $ E_\infty $ and the power $ P_\infty $ of the following discrete-time signal

x[n] = j

What properties of the complex magnitude can you use to check your answer?


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Answer 1

$ \begin{align} E_{\infty}&=\lim_{N\rightarrow \infty}\sum_{n=-N}^N |j|^2 \\ &= \lim_{N\rightarrow \infty}\sum_{n=-N}^N {(\sqrt{jj^*})}^2 \\ &= \lim_{N\rightarrow \infty}\sum_{n=-N}^N {(\sqrt{-j^2})}^2 \\ &= \lim_{N\rightarrow \infty}\sum_{n=-N}^N 1 \\ &=\infty. \\ \end{align} $


So $ E_{\infty} = \infty $.

$ \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 \over {2N+1}} \\ &= \lim_{N\rightarrow \infty}{2 \over {2}} \\ &= 1 \\ \end{align} $


So $ P_{\infty} = 1 $. 
Instructors comment: Good job! The answer is correct and the justification is very clear. Now can someone compute the power? --Mboutin 19:31, 13 January 2011 (UTC)

--Rgieseck 21:35, 12 January 2011

Answer 2

write it here.

Answer 3

write it here.


Back to ECE301 Spring 2011 Prof. Boutin

Alumni Liaison

Ph.D. 2007, working on developing cool imaging technologies for digital cameras, camera phones, and video surveillance cameras.

Buyue Zhang