Even and odd signals in CT and DT

Even signal

A signal that is symmetric about the vertical axis (time origin).

• CT: $x(-t) = x(t)$ for all $t$. Examples: $\cos\omega t$, $t^2$, $e^{-|t|}$, rectangular pulse centred at 0.
• DT: $x[-n] = x[n]$ for all $n$. Examples: $\cos(\pi n/4)$, $\delta[n]$, $\{1, 2, \underline{3}, 2, 1\}$.

Odd signal

A signal that is antisymmetric about the origin.

• CT: $x(-t) = -x(t)$ for all $t$. Examples: $\sin\omega t$, $t$, $t^3$, $\text{sgn}(t)$.
• DT: $x[-n] = -x[n]$ for all $n$. Examples: $\sin(\pi n/4)$, $n$, $\{-2, -1, \underline{0}, 1, 2\}$.

An odd signal must be zero at the origin: $x(0) = -x(0) \Rightarrow x(0) = 0$.

 Even: {1, 2, 3, 2, 1}
        o
     o  |  o
  o  |  |  |  o
--+--+--+--+--+-> n
 -2 -1  0  1  2

 Odd: {-2, -1, 0, 1, 2}
              o
           o  |
--+--+--o--+--+-> n
  |  o
  o
 -2 -1  0  1  2

Decomposition: any signal can be written as $x_e + x_o$ with

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

Properties: even × even = even, odd × odd = even, even × odd = odd; $\sum_{n=-N}^{N}x_o[n] = 0$.