Definition of Power Signal
A power signal is a type of signal that has a finite, non-zero average power over time. In contrast to energy signals, which are finite in energy but may have infinite power, power signals typically repeat periodically. They are characterized by their ability to maintain a constant power level.
Power of the Signal x(t) = A cos(ω₀t)
To calculate the power of the signal \( x(t) = A \cos(ω₀t) \), we follow these steps:
- Average Power Formula:
The average power \( P \) of a continuous signal is given by:
\[
P = \frac{1}{T} \int_{0}^{T} |x(t)|^2 dt
\]
where \( T \) is the period of the signal.
- Power Calculation:
For \( x(t) = A \cos(ω₀t) \):
\[
|x(t)|^2 = A^2 \cos^2(ω₀t)
\]
Using the trigonometric identity:
\[
\cos^2(θ) = \frac{1 + \cos(2θ)}{2}
\]
we can rewrite:
\[
|x(t)|^2 = A^2 \cdot \frac{1 + \cos(2ω₀t)}{2}
\]
- Average Power Calculation:
Now, substituting into the power formula:
\[
P = \frac{1}{T} \int_{0}^{T} A^2 \cdot \frac{1 + \cos(2ω₀t)}{2} dt
\]
Over one period \( T = \frac{2π}{ω₀} \):
\[
P = \frac{A^2}{2} \left( \frac{T}{T} + 0 \right) = \frac{A^2}{2}
\]
Conclusion
The average power of the signal \( x(t) = A \cos(ω₀t) \) is:
\[
P = \frac{A^2}{2}
\]
This indicates that the power signal maintains a consistent average power over time, characteristic of periodic signals.