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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 | + | The total signal energy for a signal can be found by the formula <math>E = \int_{-\infty}^{\infty} \! |x(t)|^2\ dt</math> |
| − | For DT signals, the energy is given by the formula <math>E_{ | + | For DT signals, the total energy is given by the formula <math>E_{\infty} = \sum^{\infty}_{n=-\infty} |x[n]|^2 \!</math> |
| + | |||
| + | Example: | ||
| + | |||
| + | <math>x(t) = e^{j (\pi t)}\!</math> | ||
| + | |||
| + | <math>E = \int_{-\infty}^{\infty} \! |e^{j (\pi t)}|^2\ dt</math> | ||
| + | |||
| + | <math>E = \int_{-\infty}^{\infty} \sqrt(\cos^2(\pi t)+\sin^2(\pi t))^2 dt\!</math> | ||
| + | |||
| + | <math>E = \int_{-\infty}^{\infty} 1 dt \!</math> | ||
| + | |||
| + | <math>E = x|_{-\infty}^{\infty}\!</math> | ||
| + | |||
| + | <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_{ | + | 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> |
| + | |||
| + | 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> | ||
| + | |||
| + | 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>\ | + | <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\! $
