Inverse Fourier Transform of δ(ω - ω₀) is ______.

Question

Question

This question was previously asked in

DFCCIL Executive S&T 2018 Official Paper

Answer 1

Concept:

Time-shifting property of Fourier transform:

If X(ω) is the Fourier transform of x(t),

---(1)

Duality property of Fourier transform:

If X(ω) is the Fourier transform of x(t),

Calculation:

Fourier Transform of δ(t) = 1

Now by definition of inverse Fourier transform

Now using time-shifting property from equation (1)

Hence option (3) is the correct answer.

Important Points

Unit Impulse function:

A continuous-time unit impulse function δ(t), also called a Dirac delta function is defined as:

δ (t) = ∞ , t = 0

= 0, otherwise

The unit impulse function is represented by an arrow with the strength of ‘1’ which represents its area.

Properties of Delta function:

Scaling Property:

Multiplication Property:

X(t).δ(t – t0) = x(t0)δ(t – t0)

Sampling Property:

Important Expressions:

X(t).δ(t) = x(0)δ(t)