Purpose of Even-Odd Signal Decomposition in Oppenheim Signal Processing Theory

Signals can be in nature or we artificially produce it. In nature no signals are going to be in even or odd. so other option we have is to produce.So my next question is why [B]we want to produce such signal(even and odd signal) ?[/B]

content form my signal book by oppenheim
[QUOTE]any signal can be broken into sum of even and odd signals[/QUOTE]

[B]why we need to do even-odd decomposition for the signal.what is the use in that ?[/B]

Thank you in advance

Hi Bhuvanesh,

I have here articles containing discussion about Even and Odd signals.

1. Signals and Systems
2. Signals and Systems 2

Hope it helps,

We say that a continuous signal x(t) is even if x(t)=x(−t) for all t. Similarly, x(t) is odd if x(t)=−x(−t) for all t.
Note that if x(t) is odd, x(0)=0.
Some common even signals you will be familiar with are x(t)=cos(t) and x(t)=t2. Some common odd signals you will be familiar with are x(t)=sin(t) and x(t)=t3.
There are exactly analogous definitions for discrete signals.
x[n] is even if x[n]=x[−n] for all n. x[n] is odd if x[n]=−x[−n] for all n.
Many signals, of course, are neither even nor odd. x(t)=Aebt, the exponential, is neither even nor odd. Neither are most time-shifted functions.
However, we are going to make use of the interesting fact that any signal x[t] that is defined for all t, can be expressed as the sum of an even signal and an odd signal.
This feels like quite a strong result. Suppose
x(t)=xE(t)+xO(t), where xE(t) and xO(t) are even and odd signals respectively.
Then x(−t)=xE(−t)+xO(−t).
But xE(−t)=xE(t), and xO(−t)=−xO(t).
So from that we can see that
x(−t)=xE(t)−xO(t).
Adding that to the equation we started with, we get:
x(t)+x(−t)=2xE(t)+xO(t)−xO(t)=2xE(t).
xE(t)=x(t)+x(−t)2
If instead we subtract the two equations, we get:
x(t)−x(−t)=2xO(t)
xO(t)=x(t)−x(−t)2
We're not quite done, because we started by assuming these signals exist. But we know that x(t) is defined for all t, so the odd and even components are well-defined, and it's clear that
x(t)+x(−t)2 is always even and
x(t)−x(−t)2 is always odd.

An even signal is any signal f such that f(t)=f(−t). Even signals can be easily spotted as they are symmetric around the vertical axis. An odd signal, on the other hand, is a signal f such that f(t)=−f(−t).
Using the definitions of even and odd signals, we can show that any signal can be written as a combination of an even and odd signal. That is, every signal has an odd-even decomposition. To demonstrate this, we have to look no further than a single equation.
f(t)=12(f(t)+f(−t))+12(f(t)−f(−t))
By multiplying and adding this expression out, it can be shown to be true. Also, it can be shown that f(t)+f(−t) fulfills the requirement of an even function, while f(t)−f(−t) fulfills the requirement of an odd function.

An even signal is any signal f such that f(t)=f(−t). Even signals can be easily spotted as they are symmetric around the vertical axis. An odd signal, on the other hand, is a signal f such that f(t)=−f(−t).
Using the definitions of even and odd signals, we can show that any signal can be written as a combination of an even and odd signal. To demonstrate this, we have to look no further than a single equation.
f(t)=12(f(t)+f(−t))+12(f(t)−f(−t))
By multiplying and adding this expression out, it can be shown to be true.