Energy and Power Signals

This is a sandbox for ruminations on Energy and Power Signals

Energy Signals

• Energy signals are defined as having finite energy:

$$E=\lim_{T\rightarrow\infty}\int_{-T/2}^{T/2}|x(\tau)|^2\,d\tau<\infty$$

• All finite-duration bounded signals are energy signals
• All absolutely integrable signals such that

$$\lim_{T\rightarrow\infty}\int_{-T/2}^{T/2}|x(\tau)|\,d\tau<\infty$$

are energy signals

• Not all energy signals are absolutely integrable (for example, $$x(t)=t^{-0.8}u(t-1)$$ is an energy signal, but not an absolutely integrable signal)
• Conjecture: the integral of an energy signal is an energy signal only if the average value of the original signal is 0.
• No periodic signals are energy signals
• Energy signals have zero average power

Power Signals

• Power signals are defined as having finite average power:

$$P_{avg}=\lim_{T\rightarrow\infty}\frac{1}{T}\int_{-T/2}^{T/2}|x(\tau)|^2\,d\tau<\infty$$

• All bounded periodic signals are power signals
• The integral of a periodic power signal is a power signal only if the average value of the original signal is 0.
• Power signals have infinite energy

Singularities

• The unit step is a power signal:

$$P_{avg}=\lim_{T\rightarrow\infty}\frac{1}{T}\int_{-T/2}^{T/2}|u(\tau)|^2\,d\tau$$

$$P_{avg}=\lim_{T\rightarrow\infty}\frac{1}{T}\int_{0}^{T/2}\,d\tau$$

$$P_{avg}=\lim_{T\rightarrow\infty}\frac{1}{T}\frac{T}{2}=\frac{1}{2}$$

• The impulse function is neither an energy nor a power signal.

Simple Classifications

• Finite-duration bounded signals are energy signals
• Periodic bounded signals are power signals
• If $$x(t)$$ is an energy signal with total energy $$E_x$$, $$y(t)=K x(a(t-t_0))$$ is also an energy signal with total energy $$\frac{K^2}{|a|}E_x$$; that is a magnitude scaling scales the energy with the square of the scale, time scaling scales the energy with the inverse absolute value of the time scale, and time shift has no impact.
• If $$x(t)$$ is a power signal with average power $$P_x$$, $$y(t)=K x(a(t-t_0))$$ is also a power signal with average power $$K^2P_x$$; that is a magnitude scaling scales the power with the square of the scale, and neither time scaling nor time shift has an impact.
• If $$x(t)$$ is a periodic power signal with average power $$P_x$$, $$y(t)=x(t)~u(t-t_0)$$ is also a power signal (though not periodic) with average power $$P_x/2$$.
• The energy of a rectangular pulse of height $$H$$ and width $$W$$ is $$\begin{align*} E&=\int_0^WH^2dt\\ ~&=\left[H^2t\right]_0^W\\ ~&=H^2W \end{align*}$$
• The energy of a triangular pulse that goes from 0 to $$H$$ over time $$W$$ is $$\begin{align*} E&=\int_0^W\left(\frac{Ht}{W}\right)^2dt\\ ~&=\int_0^W\left(\frac{H^2t^2}{W^2}\right)dt\\ ~&=\left[\frac{H^2t^3}{3W^2}\right]_0^W\\ ~&=\frac{1}{3}H^2W \end{align*}$$
• The energy of a trapezoidal section that goes from $$H_1$$ at time 0 to $$H_2$$ at time $$W$$ is$$\begin{align*} E&=\int_0^W\left(H_1+\frac{(H_2-H_1)t}{W}\right)^2dt\\ ~&=\int_0^W\left(H_1^2+2H_1\frac{(H_2-H_1)t}{W}+ \frac{(H_2-H_1)^2t^2}{W^2}\right)dt\\ ~&=\left[H_1^2t+2H_1\frac{(H_2-H_1)t^2}{2W}+ \frac{(H_2-H_1)^2t^3}{3W^2}\right]_0^W\\ ~&=H_1^2W+H_1\frac{(H_2-H_1)W^2}{W}+ \frac{(H_2-H_1)^2W^3}{3W^2}\\ ~&=H_1^2W+H_1(H_2-H_1)W+ \frac{(H_2-H_1)^2W}{3}\\ ~&=\frac{3H_1^2W+3H_1H_2W-3H_1^2W+H_2^2W-2H_1H_2W+H_1^2W}{3}\\ ~&=\frac{1}{3}\left(H_1^2+H_1H_2+H_2^2\right)W \end{align*}$$

Note that if $$H_1=H_2=H$$ this gives the same value as a rectangle and if $$H_1=0, H_2=H$$ this gives the same value as a triangle.

• The energy of a signal comprised of a collection of rectangles, triangles, and trapezoids is equal to the sum of the energies of the individual rectangles, triangles, and trapezoids.