Fourier transform property: Time shifting

Statement: if $g(t)\leftrightarrow G(f)$, then delaying the signal by $t_0$ gives

$$g(t-t_0)\leftrightarrow G(f)\,e^{-j2\pi ft_0}$$

Proof: put $\lambda=t-t_0$ in $\int g(t-t_0)e^{-j2\pi ft}dt=\int g(\lambda)e^{-j2\pi f(\lambda+t_0)}d\lambda=e^{-j2\pi ft_0}G(f)$.

Explanation: a time shift does not change the amplitude spectrum, $\lvert G(f)e^{-j2\pi ft_0}\rvert=\lvert G(f)\rvert$; it only adds a phase $-2\pi ft_0$ that is linear in frequency. Higher frequencies get more phase shift so that all components are delayed by the same time.

Significance: this gives the distortionless transmission condition: a channel with $H(f)=Ke^{-j2\pi ft_d}$ (linear phase) only delays the signal without changing its shape.