Conjugate symmetric Signal & Conjugate anti-symmetric Signal

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

Example-1

f(t) = ejt

f(−t) = ej(−t)

f*(−t) = e(−j)(−t) = ejt = f(t)

Hence, f(t) = f*(−t)

Example-2

f(t) = eot

f(−t) = eo(−t)

f*(−t) = e(−j)ωo(−t) = eo= f(t)

Hence, f(t) = f*(−t)

As we know that f(t) = eo= cosωot + j sinωot

Real part, fr(t) = cosωot

Imaginary part, fi(t) = sinωot

fr(−t) = cosωo(−t) = cosωot = fr(t)

fi(−t) = sinωo(−t) = −sinωot = −fi(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 conjugate 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) = jejt

f(−t) = jej(−t)

f*(−t) = (−j)e(−j)(−t) = −jejt = −f(t)

Hence, f(t) = −f*(−t)

Example-2

f(t) = jeot

f(−t) = jeo(−t)

f*(−t) = (−j)e(−j)ωo(−t) = −jeo= −f(t)

Hence, f(t) = −f*(−t)

As we know that f(t) = jeo= j[cosωot + j sinωot]

Real part, fr(t) = −sinωot

Imaginary part, fi(t) = cosωot

fr(−t) = −sinωo(−t) = sinωot = −fr(t)

fi(−t) = cosωo(−t) = cosωot = fi(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 conjugate anti-symmetric signal.

Properties

1. The conjugate symmetric part of a signal f(t) is given as

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

Therefore, f(t) = fec(t) + foc(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.

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