Given that Fourier transform of x(t) is X (ω), what is the Fourier transform of x (t-t0)?

Explanation:

Let the original signal be x(t), and its Fourier transform is denoted by X(ω). If the signal is time-shifted by t₀, the new signal becomes x(t-t₀). The Fourier transform of the time-shifted signal is given by:

X_shifted(ω) = e^-jωt₀ × X(ω)

This equation signifies that the Fourier transform of a time-shifted signal is the Fourier transform of the original signal multiplied by a complex exponential term e^-jωt₀, which introduces a phase shift in the frequency domain.

Derivation:

The Fourier transform of a signal x(t) is defined as:

X(ω) = ∫_-∞^∞ x(t) e^-jωt dt

If the signal is time-shifted by t₀, the new signal becomes x(t-t₀). Substituting this into the Fourier transform definition:

X_shifted(ω) = ∫_-∞^∞ x(t-t₀) e^-jωt dt

Let τ = t - t₀, so t = τ + t₀. Changing the variable of integration:

X_shifted(ω) = ∫_-∞^∞ x(τ) e^{-jω(τ + t₀)} dτ

Separating the exponential terms:

X_shifted(ω) = ∫_-∞^∞ x(τ) e^-jωτ × e^-jωt₀ dτ

The term e^-jωt₀ is independent of τ, so it can be factored out of the integral:

X_shifted(ω) = e^-jωt₀ × ∫_-∞^∞ x(τ) e^-jωτ dτ

The integral is the Fourier transform of the original signal x(t), which is X(ω). Thus:

X_shifted(ω) = e^-jωt₀ × X(ω)

This proves that the Fourier transform of the time-shifted signal is the original Fourier transform multiplied by the complex exponential phase shift.

Correct Option Analysis:

The correct option is:

Option 1: e^-jωt₀ × X(ω)

This option accurately represents the Fourier transform of the time-shifted signal x(t-t₀). The phase shift introduced by the complex exponential term e^-jωt₀ accounts for the time shift in the frequency domain