Even Signal & Odd Signal

Even signal is a signal which satisfies the relation f(t) = f(−t). Even signal is symmetric about the y-axis. Example: cost.

f(t) = cost

f(−t) = cos(−t) = cost

Hence, f(t) = f(−t)

Odd signal is a signal which satisfies the relation f(t) = −f(−t). Odd signal is symmetric about the origin. Example: sinωt.

f(t) = sint

f(−t) = sin(−t) = −sint

Hence, f(t) = −f(−t)

Properties

The time derivative of even signal gives an odd signal and the time derivative of even signal gives an even signal.

3. The even part of a signal f(t) is given as

4. The odd part of a signal f(t) is given as

Therefore, f(t) = fe(t) + fo(t)

5. f(t) + f(−t) → It always represents an even signal.

Proof: Let x(t) = f(t) + f(−t)

x(−t) = f(−t) + f(t)

Therefore, x(t) = x(−t)

Hence x(t) is an even signal.

6. f(t) − f(−t) → It always represents an odd signal.

Proof: Let x(t) = f(t) − f(−t)

x(−t) = f(−t) − f(t)

x(−t) = −[−f(−t) + f(t)] = −x(−t)

Therefore, x(t) = −x(−t)

Hence x(t) is an odd signal.

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