SC · Signal basics

Even and Odd Parts of a Signal

Any signal splits uniquely into an even part with mirror symmetry and an odd part with point symmetry. Fold the signal against its own reflection to extract these two, and as a bonus learn to size a signal by energy versus power.

Recombine even and odd to rebuild the original

The gold dashed curve is the original signal x(t). The mirror-symmetric even part alone reproduces only half of it. Only adding both the even and odd pieces brings the original back exactly.

Tap to switch the piece.

Chosen piece and formula

xₑ = [x(t) + x(−t)] / 2

The even part is the signal averaged with its own reflection, so it is mirror-symmetric. It cannot carry the original’s tilt, so it matches only half.

Half

Fold the signal against its reflection

An even function has x(−t) = x(t), mirror-symmetric about the vertical axis. An odd function has x(−t) = −x(t), point-symmetric about the origin. Add any signal to its reflection x(−t) and halve to keep only the symmetric part; subtract and halve to keep only the antisymmetric part.

Observexₑ = [x(t) + x(−t)] / 2

The even part is sum with reflection, halved.

Choosexₒ = [x(t) ?] / 2

The odd part is difference with reflection, halved.

Fill inx(t) = xₑ ? xₒ

Even plus odd is the original.

On your ownE = ?

Total energy is the integral of |x|².

Even plus odd is always the original

Adding the two formulas gives [x(t)+x(−t)]/2 + [x(t)−x(−t)]/2 = x(t): the reflected terms cancel and the original returns exactly. So every signal is written uniquely as the sum of its even and odd parts. This split often simplifies work — for instance, an integral meeting symmetry can drop half its terms to zero.

Sizing it: energy or power

A signal’s size is measured by two yardsticks. If the total energy E = ∫|x(t)|² dt is finite, it is an energy signal, and its average power is 0; vanishing signals like pulses live here. By contrast, a signal that lasts forever, like a periodic one, has E = ∞ yet a finite average power P, so it is called a power signal. No signal can be both.

Back to the first screen

The even part alone, being mirror-symmetric, matched only half of the tilted original. Adding the odd part cancelled the reflected terms and brought the gold dashed original back exactly. Every signal splits this way, uniquely, into two independent pieces: symmetric (even) and antisymmetric (odd). And a signal’s size is read as energy if it vanishes, power if it persists. Symmetry measures shape; energy and power measure size — different yardsticks.

Any signal splits uniquely into an even part xₑ(t) = [x(t) + x(−t)]/2 and an odd part xₒ(t) = [x(t) − x(−t)]/2. Even means x(−t) = x(t) (mirror symmetry); odd means x(−t) = −x(t) (point symmetry). A signal’s size is measured by total energy E = ∫|x(t)|² dt and average power. If E is finite it is an energy signal (P = 0); if E = ∞ but P is finite it is a power signal.

On to the next units

Symmetry becomes a lever that cuts future work. In the Fourier transform, even signals map to cosines (real) and odd signals to sines (imaginary), fixing the symmetry of the spectrum. The energy–power split becomes the bridge of Parseval’s theorem, measuring the same size in time and frequency, and the foundation of the power spectrum. The next unit, properties of a system, classifies not signals but the systems that transform them — linear, time-invariant, causal.