HW1, ECE301, Fall 2008, Prof. Boutin

Question

Compute the energy and the power of the function

$f(t)=\cos \left( t-2 \right).$

A time shift should not effect the energy or power of periodic function over one period (0 to 2 $\pi$ in this case).

I used this as the original function.

$$ u = (t-2) $$

$E=\int_{-2}^{2\pi-2}{|cos(u)|^2du}$

$E=\frac{1}{2}\int_{-2}^{2\pi-2}(1+cos(2(u)))du$

$E=\frac{1}{2}((u+\frac{1}{2}sin(2(u)))|_{u=-2}^{u=2\pi-2}$

$E=\frac{1}{2}(2\pi-2 + .378 -(-2 - .378))$

$E=\pi$

$P=\frac{1}{2\pi-0}\int_{-2}^{2\pi-2}{|cos(u)|^2du}$

$P=\frac{1}{2\pi-0} *{\frac{1}{2}}\int_{-2}^{2\pi-2}(1+cos(2u))du$

$P=\frac{1}{4\pi}((u)+\frac{1}{2}sin(2u))|_{u=-2}^{u=2\pi-2}$

$P=\frac{1}{4\pi}(2\pi-2+.378-(-2+.378))$

$P=\frac{1}{2}$