Question

Determine average power of the signal x(t) = cos(2πf₀t)

(where f₀ is the fundamental frequency and ‘t’ indicates continuous-time domain)
1. 1.0 W
2. 2.0 W
3. 10 W
4. 0.5 W

Answer 1

Correct Answer - Option 4 : 0.5 W

Concept:

The instantaneous power of a signal is calculated as:

\(p\left( t \right) = \smallint {s^2}\left( t \right)dt\)

The average power will be:

\({P_{avg}} = \frac{1}{T}\mathop \smallint \limits_0^T {s^2}\left( t \right)dt\)

Application:

Given signal is, x(t) = cos(2πf₀t) = cos(ω₀t)

Fundamental frequency = f₀

Time period = T = 1/f₀

Average power, \({S_{avg}} = \frac{1}{T}\mathop \smallint \limits_0^T {\cos ^2}\left( {\omega t} \right)dt\)

\( = \frac{1}{T}\mathop \smallint \limits_0^T \frac{1}{2}\left( {1 + \cos 2\omega t} \right)d\omega t\)

\( = \frac{1}{T}\left[ {\frac{1}{2}t - \frac{1}{{4\omega }}\sin 2\omega t} \right]_0^T = \frac{1}{2} = 0.5\;W\)