Even/odd decomposition of a signal

• Even signal: symmetric about the vertical axis, $x(-t) = x(t)$ (DT: $x[-n] = x[n]$). Examples: $\cos\omega t$, $t^2$, $e^{-|t|}$.
• Odd signal: antisymmetric about the origin, $x(-t) = -x(t)$. An odd signal is always zero at $t = 0$. Examples: $\sin\omega t$, $t$, $t^3$.

Even/odd decomposition

Let any signal be written as the sum of an even part and an odd part:

$$x(t) = x_e(t) + x_o(t) \qquad (1)$$

where $x_e(-t) = x_e(t)$ and $x_o(-t) = -x_o(t)$. Replace $t$ by $-t$:

$$x(-t) = x_e(-t) + x_o(-t) = x_e(t) - x_o(t) \qquad (2)$$

Adding (1) and (2), and subtracting (2) from (1):

$$\begin{aligned} x_e(t) &= \frac{1}{2}\left[x(t) + x(-t)\right] \\ x_o(t) &= \frac{1}{2}\left[x(t) - x(-t)\right] \end{aligned}$$

These parts always exist for any $x(t)$, so every signal can be decomposed into even and odd components. The same holds for DT signals: $x_e[n] = \frac{1}{2}(x[n]+x[-n])$, $x_o[n] = \frac{1}{2}(x[n]-x[-n])$.

Example

$x(t) = e^{t}$:

$$x_e(t) = \frac{e^{t}+e^{-t}}{2} = \cosh t, \qquad x_o(t) = \frac{e^{t}-e^{-t}}{2} = \sinh t$$

Check: $\cosh t + \sinh t = e^{t}$.

Example 2: $x(t) = u(t)$ gives $x_e(t) = \frac{1}{2}$ for all $t \neq 0$ and $x_o(t) = \frac{1}{2}\,\text{sgn}(t)$.

Useful properties: even × even = even, odd × odd = even, even × odd = odd; the integral of an odd signal over $[-T, T]$ is zero.