Properties of Even and Odd Signals

Quiz

Even Signal

A signal is said to be an even signal if it is symmetrical about the vertical axis or time origin, i.e.,

$$\mathrm{x(t) \:=\: x(-t);\: \text{ for all } t \: \dotso\: \text{ continuous time signal}}$$

$$\mathrm{x(n) \:=\: x(-n);\: \text{ for all } n \: \dotso\: \text{ discrete time signal}}$$

A signal is said to be an odd signal if it is anti-symmetrical about the vertical axis, i.e.,

$$\mathrm{x(-t) \:=\: -x(t);\: \text{ for all } t \: \dotso\: \text{ continuous time signal}}$$

$$\mathrm{x(-n) \:=\: -x(n);\: \text{ for all } n \: \dotso\: \text{ discrete time signal}}$$

The addition or subtraction of two odd signals is also an odd signal, i.e.,

odd signal ± odd signal = odd signal
The addition or subtraction of two even signals is also an even signal, i.e.,

even signal ± even signal = even signal
The addition or subtraction of an odd signal and an even signal is a signal which is neither even nor odd, i.e.,

odd signal ± even signal = neither even nor odd
The addition of the dc component and even signal is an even signal, i.e.,

DC component + even signal = even signal
The addition of the dc component and odd signal is a signal that is neither even nor odd, i.e.,

DC component + odd signal = neither even nor odd

The product of two even signals is also an even signal, i.e.,

even signal × even signal = even signal
The product of two odd signals is an even signal, i.e.,

odd signal × odd signal = even signal
The product of an even signal and an odd signal (or an odd signal and an even signal) is an odd signal, i.e.,

odd signal × even signal = even signal × odd signal = odd signal

1. The division of two odd signals results an even signal, i.e.,

$$\mathrm{\frac{odd\: signal}{odd \: signal} \:=\: even\: signal}$$

2. The division of two even signals results an even signal, i.e.,

$$\mathrm{\frac{even\: signal}{even \: signal} \:=\: even\: signal}$$

3. The division of an odd signal by an even signal results an odd signal, i.e.,

$$\mathrm{\frac{even\: signal}{odd \: signal} \:=\: odd\: signal}$$

1. The differentiation of an odd signal is an even signal, i.e.,

$$\mathrm{\frac{d}{dt}\left( odd\:signal \right) \:=\: even\: signal}$$

2. The differentiation of an even signal is an odd signal, i.e.,

$$\mathrm{\frac{d}{dt}\left(even\:signal \right) \:=\: odd\: signal}$$

1. The integration of an odd signal is an even signal, i.e.,

$$\mathrm{\int \left( odd\: signal \right) \:=\: even\: signal}$$

2. The integration of an even signal is an odd signal, i.e.,

$$\mathrm{\int \left( even\: signal \right) \:=\: odd\: signal}$$

Some important expressions for the continuous-time and discrete-time even and odd signals are given in the table below −

Continuous-Time Signal	Discrete-Time Signal
$\mathrm{\int_{-\infty }^{\infty }\:x_{0}\left ( t \right )dt \:=\: 0}$	$\mathrm{\sum_{n \:=\: -\infty}^{\infty} \: x_{0}\:\left( n \right ) \:=\: 0}$
$\mathrm{\int_{-\infty }^{\infty }\:x_{e} \:\left (t \right)dt \:=\: 2\int_{0}^{\infty }\:x_{e}\left( t \right)dt}$	$\mathrm{\sum_{n \:=\: -\infty}^{\infty }\: x_{e}\left( n \right) \:=\: x\left( 0 \right) \:+\: 2\sum_{n \:=\:1}^{\infty }\: x_{e}(n)}$
$\mathrm{\int_{-\infty }^{\infty }\:x_{e}\left ( t \right ).\:x_{0}\left ( t \right )dt \:=\: 0}$	$\mathrm{\sum_{n \:=\: -\infty }^{\infty }\: x_{e}\left ( n \right ).\: x_{0}\left ( n \right ) \:=\: 0}$
$\mathrm{\int_{-\infty }^{\infty }\:x^{2}\left ( t \right )dt \:=\: \int_{-\infty }^{\infty }\: x_{e}^{2}\left ( t \right )dt\: +\: \int_{-\infty }^{\infty }\: x_{0}^{2}\left ( t \right )dt}$	$\mathrm{\sum_{n \:=\: -\infty }^{\infty }\:x^{2}\left ( n \right )dt \:=\: \sum_{n \:=\:-\infty }^{\infty }x_{e}^{2}(n)\: +\: \sum_{n \:=\: -\infty }^{\infty }x_{0}^{2}(n)}$

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