Energy, Power, Periodicity, and Signal Symmetries

Energy and Power Signals

A signal can be classified by whether it carries finite energy, finite average power, or neither.

Energy

For a discrete-time signal:

\[E = \sum_{n=-\infty}^{\infty}\vert x[n]\vert^2\]

For a continuous-time signal:

\[E = \int_{-\infty}^{\infty}\vert x(t)\vert^2\,dt\]

Energy is measured in joules (or volts-squared-seconds). It is the total “content” of the signal over all time.

Average Power

\[P = \lim_{N\to\infty}\frac{1}{2N+1}\sum_{n=-N}^{N}\vert x[n]\vert^2\] \[P = \lim_{T\to\infty}\frac{1}{2T}\int_{-T}^{T}\vert x(t)\vert^2\,dt\]

Power is energy per unit time, measured in watts (or volts-squared).

The Three Classes

Class	Energy $E$	Power $P$	Examples
Energy signal	$0 < E < \infty$	$P=0$	Finite-duration pulses, $a^n u[n]$ for $\vert a\vert<1$, windowed signals
Power signal	$E=\infty$	$0 < P < \infty$	$u[n]$, $\cos(\omega_0 n)$, periodic sequences, random noise
Neither	$E=\infty$	$P=\infty$	$x[n]=n\,u[n]$, growing exponentials

A signal cannot be both an energy signal and a power signal: a finite-energy signal spread over infinite time has zero average power.

Worked Examples

1. Finite pulse. $x[n]=1$ for $0\le n\le 4$, zero otherwise.

\[E = \sum_{n=0}^{4}1^2 = 5 < \infty, \qquad P = 0\]

Energy signal.

2. Unit step. $x[n]=u[n]$.

\[E = \sum_{n=0}^{\infty}1 = \infty\] \[P = \lim_{N\to\infty}\frac{N+1}{2N+1} = \frac{1}{2}\]

Power signal with average power $1/2$.

3. Decaying exponential. $x[n]=a^n u[n]$ with $\vert a\vert<1$.

\[E = \sum_{n=0}^{\infty}a^{2n} = \frac{1}{1-a^2} < \infty\]

Energy signal. Its average power over infinite time is zero.

4. Sinusoid. $x[n]=A\cos(\omega_0 n)$ with $\omega_0 = 2\pi m/N$.

\[P = \frac{A^2}{2}\]

Power signal with $E=\infty$. This is why sinusoids have a power spectral density rather than an energy spectral density.

5. Ramp. $x[n]=n\,u[n]$.

\[E = \sum n^2 = \infty, \qquad P = \lim_{N\to\infty}\frac{1}{2N+1}\sum_{n=0}^{N}n^2 = \infty\]

Neither.

Exam rule of thumb: finite-duration or decaying → energy signal. Infinite-duration with constant amplitude → power signal. Growing without bound → neither.

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Periodic Signals

A discrete-time signal is periodic with period $N$ if

\[x[n+N]=x[n] \quad \forall n\]

for some positive integer $N$. The fundamental period $N_0$ is the smallest such $N$.

A continuous-time signal is periodic with period $T$ if $x(t+T)=x(t)$; the fundamental period is the smallest positive $T$.

Periodicity of Discrete Sinusoids

A discrete sinusoid $A\cos(\omega_0 n+\phi)$ is periodic only if

\[\frac{\omega_0}{2\pi}=\frac{m}{N}\]

is rational. Then $N$ is the fundamental period.

Signal	$\omega_0/2\pi$	Periodic?
$\cos(\pi n/4)$	$1/8$	Yes, $N=8$
$\cos(0.1\pi n)$	$1/20$	Yes, $N=20$
$\cos(0.3n)$	irrational	No
$\cos(n)$	irrational	No

This is a discrete-only trap. Every continuous sinusoid is periodic; a discrete sinusoid sampled from one may not be.

Sum of Periodic Signals

If $x_1[n]$ has period $N_1$ and $x_2[n]$ has period $N_2$, their sum is periodic only when $N_1/N_2$ is rational, with period

\[N = \operatorname{lcm}(N_1,N_2)\]

Example. $x[n]=\cos(\pi n/4)+\cos(\pi n/6)$ has $N_1=8$, $N_2=12$, so the sum repeats every $\operatorname{lcm}(8,12)=24$ samples.

If the ratio is irrational — e.g. $\cos(n)+\cos(\sqrt{2}n)$ — the sum is not periodic, even though both parts individually are not periodic either. A sum of two non-periodic sinusoids with commensurate frequencies can still be non-periodic.

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Even and Odd Signals

Definitions

\[\text{Even: } x[-n]=x[n]\] \[\text{Odd: } x[-n]=-x[n]\]

Graphically, an even signal is mirror-symmetric about the vertical axis; an odd signal is antisymmetric and must pass through the origin ($x[0]=0$ for a finite-valued odd sequence).

Even/Odd Decomposition

Every signal is the sum of an even part and an odd part:

\[x[n] = x_e[n]+x_o[n]\]

where

\[x_e[n]=\frac{x[n]+x[-n]}{2}, \qquad x_o[n]=\frac{x[n]-x[-n]}{2}\]

Proof. Compute $x_e[-n]$ and $x_o[-n]$; the definitions swap sign, so $x_e[-n]=x_e[n]$ and $x_o[-n]=-x_o[n]$. Adding them recovers $x[n]$. $\blacksquare$

Worked example. $x[n]={2,\ 1,\ 4,\ -1,\ 3}$ with the arrow at $n=0$ (so $x[0]=4$).

• $x[-n]$ reverses the sequence: ${3,\ -1,\ 4,\ 1,\ 2}$
• $x_e[n]=\frac{1}{2}(x[n]+x[-n])={2.5,\ 0,\ 4,\ 0,\ 2.5}$
• $x_o[n]=\frac{1}{2}(x[n]-x[-n])={-0.5,\ 1,\ 0,\ -1,\ 0.5}$

Check: $x_e+x_o={2,1,4,-1,3}$. Correct.

Cross-Checks

Property	Even	Odd
$x[-n]$	$x[n]$	$-x[n]$
$x[0]$	any value	must be 0
Product of two even	even	—
Product of two odd	even	—
Product of even and odd	—	odd
Sum of two even	even	—
Sum of two odd	odd	—

A periodic signal that is also even needs only a cosine series; a periodic signal that is odd needs only a sine series. This is the origin of the cosine/sine split in the Fourier series.

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Symmetric and Antisymmetric Signals

The 2024 exam asks for these directly. For continuous time:

\[\text{Symmetric: } x(t)=x(-t)\] \[\text{Antisymmetric: } x(t)=-x(-t)\]

They are the continuous-time names for even and odd. The same decomposition applies:

\[x(t)=x_e(t)+x_o(t), \qquad x_e(t)=\frac{x(t)+x(-t)}{2}, \qquad x_o(t)=\frac{x(t)-x(-t)}{2}\]

Example. $x(t)=e^{-2t}u(t)$.

• $x(-t)=e^{2t}u(-t)$
• $x_e(t)=\frac{1}{2}\left(e^{-2t}u(t)+e^{2t}u(-t)\right)$ — a two-sided decaying exponential, symmetric
• $x_o(t)=\frac{1}{2}\left(e^{-2t}u(t)-e^{2t}u(-t)\right)$ — antisymmetric

Note $x_e(0)=\frac{1}{2}$ while $x(0)=1$. For a signal with a jump at the origin, the even part takes the average of the two one-sided limits.

Why Symmetry Matters

Operation	Result
Fourier transform of a real even signal	Real and even
Fourier transform of a real odd signal	Imaginary and odd
Autocorrelation $r_{xx}[k]$	Always even
Convolution of even with even	Even
Energy of $x_e$ plus energy of $x_o$	Equals energy of $x$ (they are orthogonal)

The last row is a useful shortcut: for a real signal the even and odd parts are orthogonal, so

\[E_x = E_{x_e}+E_{x_o}\]

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Quick Reference

Concept	Formula	Test
Energy	$\sum \vert x[n]\vert^2$	Finite and non-zero?
Power	$\lim \frac{1}{2N+1}\sum \vert x[n]\vert^2$	Finite and non-zero?
Periodicity	$x[n+N]=x[n]$	Is $\omega_0/2\pi$ rational?
Even	$x[-n]=x[n]$	Mirror about $y$-axis
Odd	$x[-n]=-x[n]$	Mirror plus sign flip, $x[0]=0$
Decomposition	$x_e+x_o$	Always possible