Define symmetric and antisymmetric signal

In digital signal processing and mathematical signal analysis, symmetric and antisymmetric signals are defined based on their behavior when reflected around a central point, typically the origin. Here’s a detailed explanation:

Symmetric Signal

A symmetric signal, also known as an even signal, is one that remains unchanged when reflected about the vertical axis. Mathematically, a signal x[n]x[n]x[n] is symmetric if it satisfies the following condition:

x[n]=x[−n]x[n] = x[-n]x[n]=x[−n]

This means that the signal values at positive and negative indices are the same. In other words, the signal looks the same on both sides of the y-axis (time axis in discrete signals).

Example of a Symmetric Signal

Consider the signal x[n]={2,3,4,3,2}x[n] = \{2, 3, 4, 3, 2\}x[n]={2,3,4,3,2}.

This signal is symmetric because:

• x[−2]=x[2]=2x[-2] = x[2] = 2x[−2]=x[2]=2
• x[−1]=x[1]=3x[-1] = x[1] = 3x[−1]=x[1]=3
• x[0]=4x[0] = 4x[0]=4

Antisymmetric Signal

An antisymmetric signal, also known as an odd signal, is one that changes sign when reflected about the vertical axis. Mathematically, a signal x[n]x[n]x[n] is antisymmetric if it satisfies the following condition:

x[n]=−x[−n]x[n] = -x[-n]x[n]=−x[−n]

This means that the signal values at positive and negative indices are equal in magnitude but opposite in sign. In other words, the signal is inverted on one side of the y-axis.

Example of an Antisymmetric Signal

Consider the signal x[n]={−2,−3,0,3,2}x[n] = \{-2, -3, 0, 3, 2\}x[n]={−2,−3,0,3,2}.

This signal is antisymmetric because:

• x[−2]=−x[2]=−2x[-2] = -x[2] = -2x[−2]=−x[2]=−2
• x[−1]=−x[1]=−3x[-1] = -x[1] = -3x[−1]=−x[1]=−3
• x[0]=0x[0] = 0x[0]=0

Graphical Representation

1. Symmetric Signal:

   • A plot of a symmetric signal would show a mirror image about the y-axis.

2. Antisymmetric Signal:

   • A plot of an antisymmetric signal would show a mirror image about the y-axis with inverted (negative) values.

Mathematical Properties

1. Symmetric (Even) Signal:

   • x[n]=x[−n]x[n] = x[-n]x[n]=x[−n]
   • Example: cos?(n)\cos(n)cos(n) is an even function.

2. Antisymmetric (Odd) Signal:

   • x[n]=−x[−n]x[n] = -x[-n]x[n]=−x[−n]
   • Example: sin?(n)\sin(n)sin(n) is an odd function.

Decomposition of Signals

Any signal x[n]x[n]x[n] can be decomposed into its symmetric (even) and antisymmetric (odd) components:

x[n]=xe[n]+xo[n]x[n] = x_e[n] + x_o[n]x[n]=xe?[n]+xo?[n]

where xe[n]x_e[n]xe?[n] is the even component and xo[n]x_o[n]xo?[n] is the odd component, defined as:

xe[n]=x[n]+x[−n]2x_e[n] = \frac{x[n] + x[-n]}{2}xe?[n]=2x[n]+x[−n]?

xo[n]=x[n]−x[−n]2x_o[n] = \frac{x[n] - x[-n]}{2}xo?[n]=2x[n]−x[−n]?

Summary

• Symmetric Signal (Even): x[n]=x[−n]x[n] = x[-n]x[n]=x[−n]
• Antisymmetric Signal (Odd): x[n]=−x[−n]x[n] = -x[-n]x[n]=−x[−n]

Understanding symmetric and antisymmetric signals is crucial in signal processing, as it helps in analyzing and decomposing signals, simplifying computations, and understanding the behavior of systems and filters.