**Conjugate symmetric Signal** is a signal which satisfies the relation ** f(t) = f^{*}(−t). **It is also known as even conjugate signal.

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**Example-1**

f(t) = e^{jt}

f(−t) = e^{j(−t)}

f^{*}(−t) = e^{(−j)}^{(−t)} = e^{jt }= f(t)

Hence, f(t) = f^{*}(−t)

**Example-2**

f(t) = e^{jωot}

f(−t) = e^{jωo}^{(−t)}

f^{*}(−t) = e^{(−j)ωo}^{(−t)} = e^{jωot }= f(t)

Hence, f(t) = f^{*}(−t)

As we know that f(t) = e^{jωot }= cosω_{o}t + j sinω_{o}t

Real part, f_{r}(t) = cosω_{o}t

Imaginary part, f_{i}(t) = sinω_{o}t

f_{r}(−t) = cosω_{o}(−t) = cosω_{o}t = f_{r}(t)

f_{i}(−t) = sinω_{o}(−t) = −sinω_{o}t = −f_{i}(t)

From above equations, we can conclude that the **real part is always even signal** and the **imaginary part is always odd signal** for a c*onjugate symmetric signal.*

**Conjugate anti-symmetric Signal** is a signal which satisfies the relation ** f(t) = −f^{*}(−t). **It is also known as odd conjugate signal.

**Example-1**

f(t) = je^{jt}

f(−t) = je^{j(−t)}

f^{*}(−t) = (−j)e^{(−j)}^{(−t)} = −je^{jt }= −f(t)

Hence, f(t) = −f^{*}(−t)

**Example-2**

f(t) = je^{jωot}

f(−t) = je^{jωo}^{(−t)}

f^{*}(−t) = (−j)e^{(−j)ωo}^{(−t)} = −je^{jωot }= −f(t)

Hence, f(t) = −f^{*}(−t)

As we know that f(t) = je^{jωot }= j[cosω_{o}t + j sinω_{o}t]

Real part, f_{r}(t) = −sinω_{o}t

Imaginary part, f_{i}(t) = cosω_{o}t

f_{r}(−t) = −sinω_{o}(−t) = sinω_{o}t = −f_{r}(t)

f_{i}(−t) = cosω_{o}(−t) = cosω_{o}t = f_{i}(t)

From above equations, we can conclude that the **real part is always odd signal** and the **imaginary part is always even signal** for a c*onjugate anti-symmetric signal.*

## Properties

1. The c*onjugate symmetric* part of a signal f(t) is given as

2. The c*onjugate anti-symmetric* part of a signal f(t) is given as

Therefore, f(t) = f_{ec}(t) + f_{oc}(t)

*3. f(t) + f ^{*}(−t) → It always represents conjugate symmetric signal.*

**Proof:** Let x(t) = f(t) + f^{*}(−t)

x(−t) = f(−t) + f^{*}(t)

x^{*}(−t) = [f(−t) + f^{*}(t)]^{* }= f^{*}(−t) + f(t)

Therefore, x(t) = x^{*}(−t)

Hence x(t) is *conjugate symmetric* signal.

*4. f(t) − f ^{*}(−t) → It always represents conjugate anti-symmetric signal.*

**Proof:** Let x(t) = f(t) − f^{*}(−t)

x(−t) = f(−t) − f^{*}(t)

x^{*}(−t) = [f(−t) − f^{*}(t)]^{* }= f^{*}(−t) − f(t)

x^{*}(−t) = −[f(t) − f^{*}(−t)] = −x(t)

Therefore, x(t) = −x^{*}(−t)

Hence x(t) is *conjugate anti-symmetric* signal.