Signals and Systems – Properties of Even and Odd Signals

Even Signal

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

𝑥(𝑡) = 𝑥(−𝑡); for all 𝑡 … continuous time signal

𝑥(𝑛) = 𝑥(−𝑛); for all 𝑛 … discrete time signal

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

𝑥(−𝑡) = −𝑥(𝑡); for all 𝑡 … continuous time signal

𝑥(−𝑛) = −𝑥(𝑛); for all 𝑛 … discrete time signal

Addition and Subtraction Properties of Even and Odd Signals

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

The division of two odd signals results an even signal, i.e.,
$$\mathrm{\frac{odd\: signal}{odd \: signal}= even\: signal}$$
The division of two even signals results an even signal, i.e.,
$$\mathrm{\frac{even\: signal}{even \: signal}= even\: signal}$$
The division of an odd signal by an even signal results an odd signal, i.e.,
$$\mathrm{\frac{even\: signal}{odd \: signal}= odd\: signal}$$

The differentiation of an odd signal is an even signal, i.e.,
$$\mathrm{\frac{\mathrm{d} }{\mathrm{d} t}\left ( odd\:signal \right )= even\: signal}$$
The differentiation of an even signal is an odd signal, i.e.,
$$\mathrm{\frac{\mathrm{d} }{\mathrm{d} t}\left ( even\:signal \right )= odd\: signal}$$

The integration of an odd signal is an even signal, i.e.,
$$\mathrm{\int \left ( odd\: signal \right )= even\: signal}$$
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
$\mathrm{\int_{-\infty }^{\infty }\:x_{0}\left ( t \right )dt = 0}$	$\mathrm{\sum_{n=-\infty }^{\infty }\: x_{0}\left ( n \right )= 0}$
$\mathrm{\int_{-\infty }^{\infty }\:x_{e}\left ( t \right )dt = 2\int_{0}^{\infty }\:x_{e}\left ( t \right )dt}$	$\mathrm{\sum_{n=-\infty }^{\infty }\: x_{e}\left ( n \right )= x\left ( 0 \right )+2\sum_{n=1}^{\infty }\: x_{e}(n)}$
$\mathrm{\int_{-\infty }^{\infty }\:x_{e}\left ( t \right ).\:x_{0}\left ( t \right )dt = 0}$	$\mathrm{\sum_{n=-\infty }^{\infty }\: x_{e}\left ( n \right ).\: x_{0}\left ( n \right )= 0}$
$\mathrm{\int_{-\infty }^{\infty }\:x^{2}\left ( t \right )dt =\int_{-\infty }^{\infty }\: x_{e}^{2}\left ( t \right )dt\, +\, \int_{-\infty }^{\infty }\: x_{0}^{2}\left ( t \right )dt}$	$\mathrm{\sum_{n=-\infty }^{\infty }\:x^{2}\left ( n \right )dt = \sum_{n=-\infty }^{\infty }x_{e}^{2}(n)\, +\, \sum_{n=-\infty }^{\infty }x_{0}^{2}(n)}$