Time shifting property of CTFT

Statement: If $x(t) \leftrightarrow X(j\omega)$, then a time-shifted signal has the transform

$$x(t - t_0) \;\longleftrightarrow\; e^{-j\omega t_0}\,X(j\omega)$$

A delay of $t_0$ seconds leaves the magnitude spectrum unchanged and adds a linear phase $-\omega t_0$.

Proof: By definition,

$$\mathcal{F}\{x(t-t_0)\} = \int_{-\infty}^{\infty} x(t-t_0)\,e^{-j\omega t}\,dt$$

Put $\tau = t - t_0$, so $t = \tau + t_0$ and $dt = d\tau$; the limits stay $-\infty$ to $\infty$:

$$\begin{aligned} \mathcal{F}\{x(t-t_0)\} &= \int_{-\infty}^{\infty} x(\tau)\,e^{-j\omega(\tau+t_0)}\,d\tau \\ &= e^{-j\omega t_0}\int_{-\infty}^{\infty} x(\tau)\,e^{-j\omega\tau}\,d\tau \\ &= e^{-j\omega t_0}\,X(j\omega) \end{aligned}$$

Hence proved. Similarly, an advance gives $x(t+t_0) \leftrightarrow e^{j\omega t_0}X(j\omega)$.

Interpretation:

• $|e^{-j\omega t_0}X(j\omega)| = |X(j\omega)|$: shifting does not change which frequencies are present or their strengths.
• $\angle = \angle X(j\omega) - \omega t_0$: each frequency component is delayed by the same time $t_0$, which needs a phase shift proportional to frequency (linear phase). This is why distortionless systems need linear phase.

Example: $e^{-a|t|} \leftrightarrow \frac{2a}{a^2+\omega^2}$, so $e^{-a|t-2|} \leftrightarrow \frac{2a}{a^2+\omega^2}e^{-j2\omega}$. Likewise $\delta(t-t_0) \leftrightarrow e^{-j\omega t_0}$.