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.,

?(?) = ?(−?); for all ? … continuous time signal

?(?) = ?(−?); for all ? … 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.,

?(−?) = −?(?); for all ? … continuous time signal

?(−?) = −?(?); for all ? … 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 −