HW1, ECE301, Fall 2008, Prof. Boutin

Question

Compute the power and energy of the signal

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

Energy

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 $

Energy

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} $