Even and Odd Signals

An even signal, whether in continuous-time or discrete-time, is defined by its symmetry with its time-reversed version. Mathematically, this is represented as

for continuous-time signals and

for discrete-time signals. Even signals exhibit symmetry around the vertical axis, meaning the signal for negative time values mirrors that for positive time values.

In contrast, a signal is termed odd if it does not match its time-reversed counterpart, represented by

for continuous-time signals and

for discrete-time signals. Odd signals are characterized by their antisymmetrical nature about the vertical axis.

Any continuous-time signal can be expressed as a combination of even and odd components. This decomposition is given by:

​Here, the first term on the right side is an even function, while the second term is an odd function.

Complex signals can also be decomposed into even and odd components using conjugate symmetries. The product of an even function and an odd function results in an odd function. Furthermore, multiplying two functions of the same type—either both even or both odd—yields an even function.

Additionally, the algebraic operations of addition and subtraction follow specific rules: adding or subtracting two even functions results in an even function, and adding or subtracting two odd functions produces an odd function. These properties are fundamental in signal analysis, enabling the decomposition and simplification of complex signals into manageable components.