HW1, ECE301, Fall 2008, Prof. Boutin

Question

Compute the power and energy of the signal

$$ x(t)=cos(t) $$ .

We will find the energy in one cycle of the cosine waveform.

$E=\int_0^{2\pi}{|cos(t)|^2dt}$

$=\frac{1}{2}\int_0^{2\pi}(1+cos(2t))dt$

$=\frac{1}{2}(t+\frac{1}{2}sin(2t))|_{t=0}^{t=2\pi}$

$=\frac{1}{2}(2\pi+0-0-0)$

$=\pi$

We will find the average power in one cycle of the cosine waveform.

$E=\frac{1}{2\pi-0}\int_0^{2\pi}{|cos(t)|^2dt}$

$=\frac{1}{2\pi-0}\frac{1}{2}\int_0^{2\pi}(1+cos(2t))dt$

$=\frac{1}{4\pi}(t+\frac{1}{2}sin(2t))|_{t=0}^{t=2\pi}$

$=\frac{1}{4\pi}(2\pi+0-0-0)$

$=\frac{1}{2}$