Signal power energy exercise CT ECE301S18 sin - Rhea

Question

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

$ x(t)= \sin (2 \pi t) $

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

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Answer 1=

$ \begin{align} E_{\infty}&=\lim_{T\rightarrow \infty}\int_{-T}^T |\sin(2 \pi t)|^2 dt \quad \\ &= \lim_{T\rightarrow \infty}\int_{-T}^T |(cos(2t) + j*sin(2t))|^2 dt \quad \\ &= \lim_{T\rightarrow \infty}\int_{-T}^T {\sqrt{(cos(2t))^2 + (sin(2t))^2}}^2 dt \quad \\ & = \lim_{T\rightarrow \infty}\int_{-T}^T 1 dt \quad \\ &= \lim_{T\rightarrow \infty} t \Big| ^T _{-T} \quad \\ &=\infty. \quad \end{align} $

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

$ \begin{align} P_{\infty}&=\lim_{T\rightarrow \infty} {1 \over {2T}} \int_{-T}^T |e^{(2jt)}|^2 dt \quad \\ &= \lim_{T\rightarrow \infty} {1 \over {2T}} \int_{-T}^T 1 dt \quad \\ & = \lim_{T\rightarrow \infty} {1 \over {2T}} t \Big| ^T _{-T} \quad \\ & = \lim_{T\rightarrow \infty} {1 \over {2T}} T - {1 \over {2T}} (-T) \quad \\ & = \lim_{T\rightarrow \infty} {1 \over {2}} + {1 \over {2}} \quad \\ &= 1 \end{align} $

So $ P_{\infty} = 1 $.

Answer 2

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