If X(ω) = δ(ω - ω₀) then x(t) is

Concept:

Fourier Transform:

It is used for frequency analysis of any Bounded Input and Bounded Output (BIBO) signal.

Fourier Transform for any function x(t) is given by

Inverse Fourier Transform for any function X(ω) is given by

Frequency shifting

If X(ω) has the inverse Fourier transform as x(t).

Calculation:

δ (ω) is impulse function that exists only at t = 0

so its inverse Fourier transform will be

Given signal is δ(ω – ω0)

Inverse Fourier transform is

Impulse function exists at ω0.

We know that the area of the impulse is 1.

Important Points

The delta function is a normalized impulse, that is, sample number zero has a value of one, while all other samples have a value of zero.

For this reason, the delta function is frequently called the unit impulse. 

Fourier Transform of δ (t) is 1.