Signal energy

The signal energy of $x(t)$ is

$E_x={\int }_{-\infty }^{\infty }|x(t){|}^{2}dt.$

The square ensures the integrand is always nonnegative — a signal that wiggles between positive and negative doesn’t have its energy cancel. The absolute value handles complex-valued signals.

The terminology is borrowed from physics: for a voltage signal $v(t)$ across a 1 Ω resistor, instantaneous power is $v^2/R=v^2$, and the integral $\int v^{2}dt$ is the energy delivered in joules. For abstract signals we just call this the “signal energy” and don’t worry about units.

Worked example: triangular pulse

Find the signal energy of $x(t)=3(1-|t/4|)$ for $|t|<4$, zero otherwise — a triangle of height 3 centered at the origin, going to zero at $t=\pm 4$.

$E_x={\int }_{-4}^{4}9(1-|t|/4{)}^{2}dt.$

The integrand is even (squaring a symmetric function), so

$E_x=18{\int }_0^4(1-t/4{)}^{2}dt=18{\int }_0^4\left(1+\frac{t^2}{16}-\frac{t}{2}\right)dt=18{\left[t+\frac{t^3}{48}-\frac{t^2}{4}\right]}_0^4=18\cdot \frac{4}{3}=24.$

So $E_x=24$ joules-if-you-like.

Worked example: stepped rectangles

Find the energy of

$x(t)=[2\mathrm{rect}(t/2)-4\mathrm{rect}((t+1)/4)]u(t+2).$

The step $u(t+2)$ is zero for $t<-2$ and 1 for $t>-2$. So the signal is zero before $t=-2$ and the bracketed expression afterward. The bracket has two rectangles: $2\mathrm{rect}(t/2)$ is height 2 on $(-1,1)$, and $4\mathrm{rect}((t+1)/4)$ is height 4 on $(-3,1)$.

Squaring:

$[2\mathrm{rect}(t/2)-4\mathrm{rect}((t+1)/4){]}^2=4{\mathrm{rect}}^2(t/2)+16{\mathrm{rect}}^2((t+1)/4)-16\mathrm{rect}(t/2)\mathrm{rect}((t+1)/4).$

Each ${\mathrm{rect}}^2=\mathrm{rect}$ (values are only 0 or 1). The cross-term product is 1 only on the intersection $(-1,1)\cap (-3,1)=(-1,1)$. Integrating each piece on $(-2,\infty )$:

$\int 4\mathrm{rect}(t/2)dt=4\cdot 2=8,\int 16\mathrm{rect}((t+1)/4)dt=16\cdot 3=48,\int 16\mathrm{rect}(t/2)\mathrm{rect}((t+1)/4)dt=16\cdot 2=32.$

(The second integral goes from $-2$ not $-3$ because of the $u(t+2)$ factor.) So $E_x=8+48-32=24$.

Notice that no hard integrals were involved — the work was bookkeeping the intervals and heights. This is the typical shape of an energy calculation.

Energy vs power

Some signals — sinusoids, periodic signals in general — have infinite energy. For those we use average signal power instead. The signals split cleanly into energy signals (finite, nonzero energy) and power signals (finite, nonzero average power), with neither type belonging to the other.

In the frequency domain, total energy can also be computed by Parseval’s theorem:

$E_x={\int }_{-\infty }^{\infty }|X(f){|}^{2}df,$

with $X(f)$ the Fourier transform of $x$. The two integrals must give the same number — useful as a sanity check.