Energy or power: x(t) = A cos2πft

Energy and power signals

• Energy signal: a signal with finite, non-zero total energy, $0 < E < \infty$, where $E = \int_{-\infty}^{\infty} |x(t)|^2\,dt$. Its average power is zero. Example: a single rectangular pulse, $e^{-at}u(t)$.
• Power signal: a signal with finite, non-zero average power, $0 < P < \infty$, where $P = \lim_{T\to\infty}\frac{1}{T}\int_{-T/2}^{T/2} |x(t)|^2\,dt$. Its energy is infinite. Example: periodic signals such as sinusoids, the unit step.

A signal cannot be both; some signals (e.g. $e^{t}$, a ramp $t\,u(t)$) are neither.

Is $x(t) = A\cos 2\pi f t$ an energy or power signal?

Let $f = f_0 = 1/T_0$.

Energy:

$$\begin{aligned} E &= \lim_{T\to\infty}\int_{-T/2}^{T/2} A^2\cos^2(2\pi f_0 t)\,dt \\ &= \lim_{T\to\infty}\int_{-T/2}^{T/2} \frac{A^2}{2}\left[1+\cos(4\pi f_0 t)\right]dt \\ &= \lim_{T\to\infty}\left[\frac{A^2 T}{2} + \frac{A^2\sin(2\pi f_0 T)}{4\pi f_0}\right] = \infty \end{aligned}$$

The second term stays bounded while $A^2T/2$ grows without limit, so the energy is infinite. It is not an energy signal.

Power:

$$\begin{aligned} P &= \lim_{T\to\infty}\frac{1}{T}\int_{-T/2}^{T/2} A^2\cos^2(2\pi f_0 t)\,dt \\ &= \lim_{T\to\infty}\frac{1}{T}\left[\frac{A^2 T}{2} + \frac{A^2\sin(2\pi f_0 T)}{4\pi f_0}\right] \\ &= \frac{A^2}{2} + \lim_{T\to\infty}\frac{A^2\sin(2\pi f_0 T)}{4\pi f_0 T} = \frac{A^2}{2} \end{aligned}$$

(Same result by averaging over one period $T_0$, since the signal is periodic.)

Since $0 < P = A^2/2 < \infty$ and $E = \infty$:

Answer: $x(t) = A\cos 2\pi f t$ is a power signal with average power $P = \dfrac{A^2}{2}$ (normalised to a 1 $\Omega$ load); its energy is infinite.