Question

Derive the inverse Fourier transform for each of the following spectra:
(a) X(jω)= δ(ω−ω₀)

Answer 1

To derive the inverse Fourier, transform of X(jω)= δ(ω−ω₀), substitute ω₀ for t₀ in the relationship between the Fourier transform pair and apply the inverse Fourier transform.

Step-by-step explanation:

To derive the inverse Fourier, transform of X(jω)= δ(ω−ω₀), we can use the relationship between the Fourier transform pair:

F{δ(t-t₀)} = e^(-jωt₀)

Substituting ω₀ for t₀ in the equation, we get:

F{δ(t-ω₀)} = e^(-jωω₀).

So, the inverse Fourier transform of X(jω)= δ(ω−ω₀) is given by:

f(t) = F^(-1){X(jω)} = e^(jωω₀).