In signals and systems, signals can be classified based on their symmetry properties into even signals and odd signals. This classification is important in Fourier series, Fourier transform, and signal analysis,
Table of Contents
Even Function
A signal \(x(t)\) is called an even function if it satisfies the condition
\[
x(t) = x(-t)
\]
This means the signal is symmetric about the vertical axis \((t = 0)\).
Properties of Even Signals
• Mirror symmetry about the y-axis
• The left side of the signal is identical to the right side
• The integral of an even function over symmetric limits becomes
\[
\int_{-a}^{a} x(t)\,dt = 2\int_{0}^{a} x(t)\,dt
\]
Examples of Even Signals
\(x(t) = t^2\)
\(x(t) = \cos(\omega t)\)
\(x(t) = |t|\)
Odd Function
A signal \(x(t)\) is called an odd function if it satisfies the condition
\[
x(t) = -x(-t)
\]
This means the signal is antisymmetric about the origin.
Properties of Odd Signals
• Rotational symmetry about the origin
• The signal changes sign when time is reversed
• The integral of an odd function over symmetric limits is
\[
\int_{-a}^{a} x(t)\,dt = 0
\]
Examples of Odd Signals
\(x(t) = t\)
\(x(t) = \sin(\omega t)\)
Decomposition of a Signal
Any signal \(x(t)\) can be expressed as the sum of an even component and an odd component.
Even component:
\[
x_e(t) = \frac{x(t) + x(-t)}{2}
\]
Odd component:
\[
x_o(t) = \frac{x(t) – x(-t)}{2}
\]
Thus,
\[
x(t) = x_e(t) + x_o(t)
\]
This decomposition is widely used in Fourier analysis and signal processing.
Graphical Interpretation
Even Signal
• Symmetrical about the vertical axis
• Folding the graph along \(t=0\) gives identical halves
Odd Signal
• Symmetrical about the origin
• Rotating the graph 180° around the origin gives the same signal
Importance in Signals and Systems
Even and odd signals are important because:
• They simplify Fourier series calculations
• They help analyze system symmetry properties
• Many signals in engineering can be decomposed into even and odd parts
For example:
• Even signals produce cosine terms in Fourier series
• Odd signals produce sine terms
Comparison of Even and Odd Functions
| Feature | Even Function | Odd Function |
|---|---|---|
| Mathematical condition | \(x(t)=x(-t)\) | \(x(t)=-x(-t)\) |
| Symmetry | About vertical axis | About origin |
| Example | \(\cos(\omega t)\) | \(\sin(\omega t)\) |
| Integral over symmetric limits | Non-zero | Zero |