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Signals and Systems β 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.,
π₯(π‘) = π₯(−π‘); for all π‘ … continuous time signal
π₯(π) = π₯(−π); for all π … discrete time signal
Odd Signal
A signal is said to be an odd signal if it is anti-symmetrical about the vertical axis, i.e.,
π₯(−π‘) = −π₯(π‘); for all π‘ … continuous time signal
π₯(−π) = −π₯(π); for all π … discrete time signal
Properties of Even and Odd Signals
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
Multiplication Properties of Even and Odd Signals
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
Division Properties of Even and Odd Signals
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.,
$$\mathrm{\frac{even\: signal}{odd \: signal}= odd\: signal}$$
Differentiation Properties of Even and Odd Signals
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.,
$$\mathrm{\frac{\mathrm{d} }{\mathrm{d} t}\left ( even\:signal \right )= odd\: signal}$$
Integration Properties of Even and Odd Signals
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.,
$$\mathrm{\int \left ( even\: signal \right )= odd\: signal}$$
Important Expressions for Even and Odd Signals
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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