| Line 16: | Line 16: | ||
&=\int_{-\infty}^\infty \frac{1+\cos(10t)}{2} dt \\ | &=\int_{-\infty}^\infty \frac{1+\cos(10t)}{2} dt \\ | ||
&=\int_{-\infty}^\infty \frac{1}{2} + {\frac{1}{2}}\cos(10t) dt \\ | &=\int_{-\infty}^\infty \frac{1}{2} + {\frac{1}{2}}\cos(10t) dt \\ | ||
| − | &= (t + {\frac{1}{10}}\sin(10t) \Big| ^T _{-T} | + | &= (t + {\frac{1}{10}}\sin(10t)) \Big| ^T _{-T}\\ |
| + | &=\infty\\ | ||
| + | |||
\end{align} | \end{align} | ||
</math> | </math> | ||
| + | |||
| + | <math class="inline">E_{\infty} = \infty </math>. | ||
| + | |||
| + | |||
| + | |||
| + | <math class="inline">P_{\infty} = \frac{1}{2} </math>. | ||
Revision as of 18:43, 1 December 2018
Problem
Compute the energy and the power of the CT sinusoidal signal below:
$ x(t)= \cos (5t) $
Solution
$ \begin{align} \left|\cos(5t)\right|^{2} = \cos^2(5t) \\ \cos^2(5t) = \frac{1+\cos(10t)}{2} \end{align} $
$ \begin{align} E_{\infty} &=\int_{-\infty}^\infty \cos^2(5t) dt \\ &=\int_{-\infty}^\infty \frac{1+\cos(10t)}{2} dt \\ &=\int_{-\infty}^\infty \frac{1}{2} + {\frac{1}{2}}\cos(10t) dt \\ &= (t + {\frac{1}{10}}\sin(10t)) \Big| ^T _{-T}\\ &=\infty\\ \end{align} $
$ E_{\infty} = \infty $.
$ P_{\infty} = \frac{1}{2} $.
