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Signal Energy expended from <math>t_1\!</math> to <math>t_2\!</math> for CT functions is given by the formula <math>E = \int_{t_1}^{t_2} \! |x(t)|^2\ dt</math>
 
Signal Energy expended from <math>t_1\!</math> to <math>t_2\!</math> for CT functions is given by the formula <math>E = \int_{t_1}^{t_2} \! |x(t)|^2\ dt</math>
  
The total signal energy for a signal can be found by taking the limits for the integral <math>t_1\!</math> and <math>t_2\!</math> as <math>-inf\!</math> and <math>inf\!</math> respectively
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The total signal energy for a signal can be found by the formula <math>E = \int_{-\infty}^{\infty} \! |x(t)|^2\ dt</math>
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For DT signals, the total energy is given by the formula <math>E_{\infty} = \sum^{\infty}_{n=-\infty} |x[n]|^2 \!</math>
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 +
Example:
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<math>x(t) = e^{j (\pi t)}\!</math>
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 +
<math>E = \int_{-\infty}^{\infty} \! |e^{j (\pi t)}|^2\ dt</math>
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 +
<math>E = \int_{-\infty}^{\infty} \sqrt(\cos^2(\pi t)+\sin^2(\pi t))^2 dt\!</math>
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<math>E = \int_{-\infty}^{\infty} 1 dt \!</math>
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<math>E = x|_{-\infty}^{\infty}\!</math>
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<math>E = \infty</math>
  
 
== Signal Power ==
 
== Signal Power ==
 
For CT functions, the power of a signal from <math>t_1\!</math> to <math>t_2\!</math> is given by the function <math>P_{avg}=\frac{1}{t_2-t_1} \int_{t_1}^{t_2} |x(t)|^2\ dt \!</math>
 
For CT functions, the power of a signal from <math>t_1\!</math> to <math>t_2\!</math> is given by the function <math>P_{avg}=\frac{1}{t_2-t_1} \int_{t_1}^{t_2} |x(t)|^2\ dt \!</math>
  
The total signal power is given by the function <math>P_{inf}=\lim_{t->inf}  \frac{1}{2t} \int_{-t}^{t} |x(t)|^2\ dt \!</math>
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The total signal power is given by the function <math>P_{\infty}=\lim_{t \to \infty}  \frac{1}{2t} \int_{-t}^{t} |x(t)|^2\ dt \!</math>
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Total signal power for DT signals is given by the formula <math>P_{\infty} = \lim_{N \to \infty} \frac{1}{2N+1} \sum^{N}_{n=-N} |x[n]|^2\!</math>
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 +
Example:
 +
 
 +
<math>x(t) = e^{j (\pi t)}\!</math>
 +
 
 +
<math>P_{\infty}=\lim_{t \to \infty}  \frac{1}{2t} \int_{-t}^{t} \! |e^{j (\pi t)}|^2\ dt</math>
 +
 
 +
<math>P_{\infty}=\lim_{t \to \infty}  \frac{1}{2t} \int_{-t}^{t} \sqrt(\cos^2(\pi t)+\sin^2(\pi t))^2 dt</math>
 +
 
 +
<math>P_{\infty}=\lim_{t \to \infty}  \frac{1}{2t} \int_{-t}^{t} \! 1 dt</math>
 +
 
 +
<math>P_{\infty}=\lim_{t \to \infty}  \frac{1}{2t} 2t</math>
  
<math>\sum^{N}_{n=-N}</math>
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<math>P_{\infty} = 1\!</math>

Latest revision as of 06:34, 5 September 2008

Signal Energy

Signal Energy expended from $ t_1\! $ to $ t_2\! $ for CT functions is given by the formula $ E = \int_{t_1}^{t_2} \! |x(t)|^2\ dt $

The total signal energy for a signal can be found by the formula $ E = \int_{-\infty}^{\infty} \! |x(t)|^2\ dt $

For DT signals, the total energy is given by the formula $ E_{\infty} = \sum^{\infty}_{n=-\infty} |x[n]|^2 \! $

Example:

$ x(t) = e^{j (\pi t)}\! $

$ E = \int_{-\infty}^{\infty} \! |e^{j (\pi t)}|^2\ dt $

$ E = \int_{-\infty}^{\infty} \sqrt(\cos^2(\pi t)+\sin^2(\pi t))^2 dt\! $

$ E = \int_{-\infty}^{\infty} 1 dt \! $

$ E = x|_{-\infty}^{\infty}\! $

$ E = \infty $

Signal Power

For CT functions, the power of a signal from $ t_1\! $ to $ t_2\! $ is given by the function $ P_{avg}=\frac{1}{t_2-t_1} \int_{t_1}^{t_2} |x(t)|^2\ dt \! $

The total signal power is given by the function $ P_{\infty}=\lim_{t \to \infty} \frac{1}{2t} \int_{-t}^{t} |x(t)|^2\ dt \! $

Total signal power for DT signals is given by the formula $ P_{\infty} = \lim_{N \to \infty} \frac{1}{2N+1} \sum^{N}_{n=-N} |x[n]|^2\! $

Example:

$ x(t) = e^{j (\pi t)}\! $

$ P_{\infty}=\lim_{t \to \infty} \frac{1}{2t} \int_{-t}^{t} \! |e^{j (\pi t)}|^2\ dt $

$ P_{\infty}=\lim_{t \to \infty} \frac{1}{2t} \int_{-t}^{t} \sqrt(\cos^2(\pi t)+\sin^2(\pi t))^2 dt $

$ P_{\infty}=\lim_{t \to \infty} \frac{1}{2t} \int_{-t}^{t} \! 1 dt $

$ P_{\infty}=\lim_{t \to \infty} \frac{1}{2t} 2t $

$ P_{\infty} = 1\! $

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