Question

Determine the odd and even parts of a signal example

Answer 1

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Table of Contents

Definition

Examples

Properties of Even and Odd Signals

Examples

Definition

A signal x(t) is said to be,

Even if,

(1)

x(t)=x(−t)

Odd if,

(2)

x(t)=−x(−t)

(3)

x(t)=−x(−t)

The following figures illustrate clearly,

ch1.1.6.1.jpg

Any signal x(t) can be written as the sum of an even signal and odd signal.

(4)

x(t)=xe(t)+xo(t)

where, xe(t) is the even part and xo(t) is the odd part.

(5)

x(t)=12[x(t)+x(−t)]+12[x(t)−x(−t)]

So,

(6)

xe(t)=12[x(t)+x(−t)]⇒xe(−t)=12[x(−t)+x(t)]

Therefore,

(7)

xe(t)=xe(−t)⇒xe(t) is even

And,

(8)

xo(t)=12[x(t)−x(−t)]⇒xo(−t)=12[x(−t)−x(t)]

Therefore,

(9)

xo(t)=−xo(−t)⇒xo(t) is odd

Examples

Example 1

Example 2

Properties of Even and Odd Signals

*Addition/Subtraction:

Even Signal ± Even Signal =Even signal_

Odd Signal ± Odd Signal =Odd signal

Even Signal ± Odd Signal = We can't say anything

*Multiplication:

Even * Even = Even

Odd * Odd = Even

Even * Odd= Odd

*Integrals:

If x(t) is odd then ∫A−Ax(t)dt=0

Example: ∫1−1sin3(t)dt=0

If x(t) is even then ∫A−Ax(t)dt=2∫A0x(t)dt

*Conjugate Symmetry:

Suppose x(t) is a complex signal ⇒ x(t)=a(t)+jb(t)=r(t)ejθ(t)

Even Signal is Conjugate Symmetric Signal if-

x(t)=x∗(−t)

and Conjugate Anti-Symmetric if-

x(t)=−x∗(−t)⇒x∗(t)=−x(−t)⇒−x(t)=−x∗(−t)

If x(t) is real →X(jω) is conjugate symmetric.

If x(t) is real and even →X(jω) is real.

If x(t) is complex, conjugate symmetric →X(jω) is real.

Examples

Example 1

Example 2

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Answer 2

Hey mate,

Answer is refer to the attachment

Hope it helps you

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