Which transformation in frequency domain equals a x-axis shift of a signal in time domain?

Question

I have discrete Fourier transformation results from measurements. Looking at the signal from the time domain perspective, I want to shift the signal on the $x$-axis to the left or right.

Which transformation in the frequency-domain results in a left/right shift in the time-domain?

Answer 1

If $s(t)$ is your signal in time domain, you want to make the operation $s(t+\Delta t)$, which according to the Fourier's transform properties, is equivalent to the operation : $$ S(f)e^{j2\pi f\Delta t} $$ $S(f)$ being $s(t)$ Fourier's transform.

In case of a discrete Fourier's transform, which is defined as $$ S[k] = \sum^{N-1}_{n=0}{s[n]e^{\frac{-i2\pi kn}{N}}} $$ with $s$ the signal and N the number of points of both signal and DFT.

The operation to be applied in order to induce a shift by a number c of samples to $s[n]$ and obtain $s[n+c]$ is : $$ e^{\frac{-i2\pi kc}{N}}S[k] $$

Which in matlab code would be done with :

exp(-j*2*pi*(1:(N-1))*c/N).*S_fdt;