It is asked to evaluate the energy and power of the signal
$$x(t)=10\cos(100t+30°)-5\sin(220t-50°)$$
Since it is periodic, I need to find
$$\int _{-\infty}^\infty |x(t)|^2 dt \text{ and } \frac{1}{T}\int_{-T}^T |x(t)|^2 dt$$
Where \$T\$ is the period of \$x(t)\$ (which is \$18\$). By Parseval's theorem, we know the energy is conserved when we do a Fourier transform and I was trying to use it (but I couldn't). What is the best way of evaluating those integrals?
Best Answer
For any periodic signal you get
$$\int_{-\infty}^{\infty}|x(t)|^2dt=\infty$$
i.e. the integral does not converge, and, consequently, the energy is infinite. The power is finite and can be computed from the following formula (which differs from yours by a factor of \$\frac12\$):
$$\overline{x^2(t)}=\frac{1}{2T}\int_{-T}^T|x(t)|^2dt$$
Note that \$\cos^2 x=\frac12 (1+\cos(2x))\$ and \$\sin^2 x=\frac12 (1-\cos(2x))\$. So after integrating over one period, the terms with double frequency and also the cross-term cancel out. So you finally get for the power
$$\overline{x^2(t)}=10^2\cdot \frac12 + 5^2\cdot\frac12=62.5$$
Note that the actual value of \$T\$ is irrelevant. This is a good thing because the value you got is wrong.