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− | <math>E_\infty = \int_{-\infty}^\infty |x(t)|^2\dt</math> | + | [[Category:problem solving]] |
+ | [[Category:ECE301]] | ||
+ | [[Category:ECE]] | ||
+ | |||
+ | =Example of computation of Signal energy and Signal Power = | ||
+ | A [[Signals_and_systems_practice_problems_list|practice problem on "Signals and Systems"]] | ||
+ | ---- | ||
+ | <math>x(t) = \sqrt(t)</math> | ||
+ | |||
+ | <math>x_1(t) = \cos(t) + \jmath\sin(t)</math> | ||
+ | |||
+ | ---- | ||
+ | |||
+ | <math>E_\infty = \int_{-\infty}^\infty |x(t)|^2\,dt</math> | ||
+ | |||
+ | <math>=\int_{-\infty}^\infty |\sqrt(t)|^2\,dt</math> | ||
+ | <math>=\int_0^\infty t\,dt</math> | ||
+ | <math>=.5*t^2|_0^\infty</math> | ||
+ | <math>=.5(\infty^2 - 0^2)</math> | ||
+ | <math>E_\infty = \infty</math> | ||
+ | |||
+ | |||
+ | <math>P_\infty = lim_{T \to \infty} \ 1/(2T) \int_{-T}^T |x(t)|^2\,dt</math> | ||
+ | |||
+ | <math>= lim_{T \to \infty} \ 1/(2T) \int_{-T}^T |\sqrt(t)|^2\,dt</math> | ||
+ | <math>= lim_{T \to \infty} \ 1/(2T) .5*t^2|_0^T</math> | ||
+ | <math>= lim_{T \to \infty} \ 1/(2T) * .5(T^2 - 0^2)</math> | ||
+ | <math>= lim_{T \to \infty} \ 1/(2T) * .5T^2</math> | ||
+ | <math>= lim_{T \to \infty} \ 1/(4T)*T^2</math> | ||
+ | <math>= lim_{T \to \infty} T/4</math> | ||
+ | <math>P_\infty = \infty</math> | ||
+ | |||
+ | |||
+ | |||
+ | <math>|x_1(t)| = \sqrt{\cos^2(t)+\sin^2(t)}=1</math> | ||
+ | |||
+ | ---- | ||
+ | |||
+ | |||
+ | <math>E_\infty = \int_{-\infty}^\infty |x_1(t)|^2\,dt</math> | ||
+ | |||
+ | <math>= \int_{-\infty}^\infty |1|^2 \,dt</math> | ||
+ | <math>= t|_{-\infty}^\infty</math> | ||
+ | <math>E_\infty = \infty</math> | ||
+ | |||
+ | |||
+ | <math>P_\infty = lim_{T \to \infty} \ 1/(2T) \int_{-T}^T |x_1(t)|^2\,dt</math> | ||
+ | |||
+ | <math>= lim_{T \to \infty} \ 1/(2T) \int_{-T}^T |1|^2 \,dt</math> | ||
+ | <math>= lim_{T \to \infty} \ 1/(2T) * t|_{-T}^T</math> | ||
+ | <math>= lim_{T \to \infty} \ 1/(2T) * (T- (-T))</math> | ||
+ | <math>= lim_{T \to \infty} \ 1/(2T) * (2T)</math> | ||
+ | <math>= lim_{T \to \infty} \ 1</math> | ||
+ | <math>P_\infty = 1</math> | ||
+ | ---- | ||
+ | [[Signals_and_systems_practice_problems_list|Back to Practice Problems on Signals and Systems]] |
Latest revision as of 10:09, 16 September 2013
Example of computation of Signal energy and Signal Power
A practice problem on "Signals and Systems"
$ x(t) = \sqrt(t) $
$ x_1(t) = \cos(t) + \jmath\sin(t) $
$ E_\infty = \int_{-\infty}^\infty |x(t)|^2\,dt $
$ =\int_{-\infty}^\infty |\sqrt(t)|^2\,dt $ $ =\int_0^\infty t\,dt $ $ =.5*t^2|_0^\infty $ $ =.5(\infty^2 - 0^2) $
$ E_\infty = \infty $
$ P_\infty = lim_{T \to \infty} \ 1/(2T) \int_{-T}^T |x(t)|^2\,dt $
$ = lim_{T \to \infty} \ 1/(2T) \int_{-T}^T |\sqrt(t)|^2\,dt $ $ = lim_{T \to \infty} \ 1/(2T) .5*t^2|_0^T $ $ = lim_{T \to \infty} \ 1/(2T) * .5(T^2 - 0^2) $ $ = lim_{T \to \infty} \ 1/(2T) * .5T^2 $ $ = lim_{T \to \infty} \ 1/(4T)*T^2 $ $ = lim_{T \to \infty} T/4 $
$ P_\infty = \infty $
$ |x_1(t)| = \sqrt{\cos^2(t)+\sin^2(t)}=1 $
$ E_\infty = \int_{-\infty}^\infty |x_1(t)|^2\,dt $
$ = \int_{-\infty}^\infty |1|^2 \,dt $ $ = t|_{-\infty}^\infty $
$ E_\infty = \infty $
$ P_\infty = lim_{T \to \infty} \ 1/(2T) \int_{-T}^T |x_1(t)|^2\,dt $
$ = lim_{T \to \infty} \ 1/(2T) \int_{-T}^T |1|^2 \,dt $ $ = lim_{T \to \infty} \ 1/(2T) * t|_{-T}^T $ $ = lim_{T \to \infty} \ 1/(2T) * (T- (-T)) $ $ = lim_{T \to \infty} \ 1/(2T) * (2T) $ $ = lim_{T \to \infty} \ 1 $
$ P_\infty = 1 $