Question

Prove the time-shifting property of the Fourier transform
x(t) ⇔ X( ω),then x(t-t₀)⇔X(ω)e⁻ʲωᵗ₀.

Answer 1

To prove the time-shifting property of the Fourier transform, substitution is used to show that the Fourier transform of x(t-t0) is X(ω)e−jωt0.

Step-by-step explanation:

The question asks to prove the time-shifting property of the Fourier transform. The Fourier transform of a function x(t) is defined as X(ω), and we are to show that the Fourier transform of x(t-t0) is X(ω)e−jωt0. Here's the step-by-step proof:

By the definition of the Fourier transform:

X(ω) = ∫ x(t)e−jωtdt

To find the Fourier transform of x(t-t0):

Let u = t - t0, then du = dt.

Now the transform becomes:

X(ω)e−jωt0 = ∫ x(u)e−jω(u + t0)du

Expanding the exponent:

X(ω)e−jωt0 = ∫ x(u)e−jωue−jωt0du

Since e−jωt0 is a constant with respect to u, it can be taken outside the integral:

X(ω)e−jωt0 = e−jωt0∫ x(u)e−jωudu

This completes the proof, showing that x(t-t0) indeed transforms to X(ω)e−jωt0.