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Compute the Energy and Power of the signal <math>x(t)=\dfrac{2t}{t^2+5}</math> between 3 and 5 seconds. | Compute the Energy and Power of the signal <math>x(t)=\dfrac{2t}{t^2+5}</math> between 3 and 5 seconds. | ||
==Energy== | ==Energy== | ||
| + | |||
| + | <math>E=\int_{t_1}^{t_2}x(t)dt | ||
<math>E=\int_3^{5}{\dfrac{2t}{t^2+5}dt}</math> | <math>E=\int_3^{5}{\dfrac{2t}{t^2+5}dt}</math> | ||
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==Power== | ==Power== | ||
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| + | <math>P=\dfrac_{1}^{{t_2}-{t_1}}\int_{t_1}^{t_2}x(t)dt | ||
| + | |||
<math>P=\dfrac{1}{5-3}\int_3^{5}{\dfrac{2t}{t^2+5}dt}</math> | <math>P=\dfrac{1}{5-3}\int_3^{5}{\dfrac{2t}{t^2+5}dt}</math> | ||
Revision as of 14:35, 4 September 2008
Compute the Energy and Power of the signal $ x(t)=\dfrac{2t}{t^2+5} $ between 3 and 5 seconds.
Energy
$ E=\int_{t_1}^{t_2}x(t)dt <math>E=\int_3^{5}{\dfrac{2t}{t^2+5}dt} $
$ U=t^2+5 $
$ dU=2tdt $
Limits:
$ U(3)=3^2+5=9+5=14 $
$ U(5)=5^2+5=25+5=30 $
$ E=\int_{14}^{30}\dfrac{du}{U} $
$ E=\ln U |_{U=14}^{U=30} $
$ E=\ln 30 - \ln 14 $
$ E=\ln {30/14} $
Power
$ P=\dfrac_{1}^{{t_2}-{t_1}}\int_{t_1}^{t_2}x(t)dt <math>P=\dfrac{1}{5-3}\int_3^{5}{\dfrac{2t}{t^2+5}dt} $
$ U=t^2+5 $
$ dU=2tdt $
Limits:
$ U(3)=3^2+5=9+5=14 $
$ U(5)=5^2+5=25+5=30 $
$ P=\int_{14}^{30}\dfrac{du}{U} $
$ P=\ln U |_{U=14}^{U=30} $
$ P=\ln 30 - \ln 14 $
$ P=\ln {30/14} $
