Question

Let x(t) be any causal signal with Fourier transform X(ω) = R(ω) + jI(ω). Show that given either R(w) or I (ω), we can compute the other using the formulas:

I(ω) = - 1/π ∫[infinity] to ₋[infinity](R(ω'))/(ω - ω') dω'

Answer 1

To compute either R(ω) or I(ω), you can use the given formula: I(ω) = - 1/π ∫[infinity] to −∞(R(ω'))/(ω - ω') dω'. Start by substituting X(ω) = R(ω) + jI(ω) into the Fourier transform definition. Split the integral into real and imaginary parts, equating R(ω) to the real part and I(ω) to the imaginary part of X(ω).

Step-by-step explanation:

To show that given either R(ω) or I(ω), we can compute the other using the given formulas:

I(ω) = - 1/π ∫[infinity] to −∞(R(ω'))/(ω - ω') dω'.

We can start by substituting X(ω) = R(ω) + jI(ω) into the definition of the Fourier transform:

X(ω) = ∫[infinity] to −∞ x(t)e-jωt dt

Split the integral into two parts:

X(ω) = ∫[infinity] to −∞ R(ω)e-jωt dt + j∫[infinity] to −∞ I(ω)e-jωt dt

Equating the real and imaginary parts:

R(ω) = Re[X(ω)] = 1/π ∫[infinity] to −∞ R(ω')/(ω - ω') dω'

I(ω) = Im[X(ω)] = -1/π ∫[infinity] to −∞ R(ω')/(ω - ω') dω'