Even and Odd signals

### Definition

A signal $x(t)$ is said to be,

Even if,

(1)
\begin{align} x(t)=x(-t) \nonumber \end{align}

Odd if,

(2)
\begin{align} x(t)=-x(-t) \nonumber \end{align}
(3)
\begin{align} x(t)=-x(-t) \nonumber \end{align}

The following figures illustrate clearly,

Any signal $x(t)$ can be written as the sum of an even signal and odd signal.

(4)
\begin{align} x(t)=x_e(t)+x_o(t) \nonumber \end{align}

where, $x_e(t)$ is the even part and $x_o(t)$ is the odd part.

(5)
\begin{align} x(t)= \frac{1}{2}\big[x(t)+x(-t)\big] + \frac{1}{2}\big[x(t)-x(-t)\big]\nonumber \end{align}

So,

(6)
\begin{align} x_e(t)= \frac{1}{2}\big[x(t)+x(-t)\big] \nonumber \Rightarrow x_e(-t)= \frac{1}{2}\big[x(-t)+x(t)\big] \end{align}

Therefore,

(7)
\begin{align} x_e(t)=x_e(-t) \Rightarrow x_e(t) \text{ is even} \end{align}

And,

(8)
\begin{align} x_o(t)= \frac{1}{2}\big[x(t)-x(-t)\big] \nonumber \Rightarrow x_o(-t)= \frac{1}{2}\big[x(-t)-x(t)\big] \end{align}

Therefore,

(9)
\begin{align} x_o(t)=-x_o(-t) \Rightarrow \text{$x_o(t)$ is odd} \end{align}

### Properties of Even and Odd Signals

Even Signal $\pm$ Even Signal =Even signal_
Odd Signal $\pm$ Odd Signal =Odd signal
Even Signal $\pm$ Odd Signal = We can't say anything

*Multiplication:

Even * Even = Even
Odd * Odd = Even
Even * Odd= Odd

*Integrals:

If $x(t)$ is odd then $\int_{-A}^{A}x(t)dt=0$
Example: $\int_{-1}^{1}sin^3(t)dt=0$
If $x(t)$ is even then $\int_{-A}^{A}x(t)dt=2\int_{0}^{A}x(t)dt$

*Conjugate Symmetry:

Suppose $x(t)$ is a complex signal $\Rightarrow$ $x(t)=a(t)+jb(t) = r(t)e^{j\theta(t)}$
Even Signal is Conjugate Symmetric Signal if-
$x(t)=x^*(-t)$
and Conjugate Anti-Symmetric if-
$x(t)=-x^*(-t) \Rightarrow x^*(t)=-x(-t) \Rightarrow -x(t)=-x^*(-t)$

If $x(t)$ is real $\rightarrow X(j\omega)$ is conjugate symmetric.
If $x(t)$ is real and even $\rightarrow X(j\omega)$ is real.
If $x(t)$ is complex, conjugate symmetric $\rightarrow X(j\omega)$ is real.