seegongsik
SC · Signal basics

Even and Odd Parts of a Signal

Any signal splits in exactly one way into the sum of an even part, symmetric left to right, and an odd part, symmetric about the origin. Fold the signal against its own reflection to pull these two out, and as a bonus learn to measure a signal's size as energy and power.

Recombine even and odd to rebuild the original

The gold dashed curve is the original signal x(t), a bell shape tilted to the right. Switch the pieces with the tabs and you see that the even part (blue), being symmetric left to right, cannot hold this tilt, and only when the odd part (green) is added does the original come back exactly.

Tap a tab to switch the piece.
Chosen piece and formula
xₑ = [x(t) + x(−t)] / 2
The even part is the average of the signal and its reflection, so it is symmetric left to right. It cannot hold how far the original leans, so it fits only halfway.
Half

Fold the signal against its reflection

An even function satisfies x(−t) = x(t), so it is mirror-symmetric about the vertical axis, and an odd function satisfies x(−t) = −x(t), so it is point-symmetric about the origin. Add any signal to its reflection x(−t) and halve it, and only the symmetric component remains; subtract and halve, and only the antisymmetric component remains. The original in the figure is the bell shape g(t) multiplied by (1 + 0.8t), so the even part is g(t) and the odd part is 0.8t·g(t).

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 can be written in exactly one way as the sum of an even part and an odd part. This decomposition often shortens calculations, because an odd function integrated over a symmetric interval from −T to T gives 0.

Sizing it: energy or power

A signal's size is measured with two yardsticks. If the total energy E = ∫|x(t)|² dt is finite, it is an energy signal, and its average power is 0. A pulse that dies away, like the gold curve in the small figure alongside, belongs here, and its area ∫e^(−t²) dt = √π ≈ 1.77 is the energy. A periodic signal like the blue curve, on the other hand, goes on forever, so E = ∞, but its average power P is finite, and it is called a power signal. The blue curve is cos²(2.5t), so its average height of 0.5 is exactly P. A signal cannot be both, and some signals, like the ever-growing ramp t, are neither.

energy |x|² · finite areapower |x|² · persists

Back to the first screen

The even part alone was symmetric left to right, so it could fit the tilted original only halfway. Adding the odd part made the reflected terms cancel, and the gold dashed original came back exactly. Every signal splits in this one way into two independent pieces, the symmetric (even) and the antisymmetric (odd). A signal's size, meanwhile, is measured as energy if it dies away and as power if it persists, so symmetry and energy or power are different yardsticks, one for shape and one for size.

Any signal splits in exactly one way into the sum of the even part xₑ(t) = [x(t) + x(−t)]/2 and the 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 its total energy E = ∫|x(t)|² dt and its average power: if E is finite it is an energy signal (P = 0), and if E = ∞ but P is finite it is a power signal.
On to the next units

Symmetry becomes a lever that cuts down calculations from here on. In the Fourier transform, a real even signal has a real spectrum and a real odd signal a purely imaginary one, which fixes the symmetry of the spectrum. The split between energy and power leads to Parseval's theorem, which measures the same size in time and in frequency, and it underlies the power spectrum. The next unit, properties of a system, sorts not signals but the systems that change them, as linear, time-invariant and causal.