This is an interactive plot I made illustrating the effect of time delay in the frequency domain.

On the left, I plot the time domain signal, which consists of a simple impulse that has been delayed by some number of samples. You can use the slider on the bottom to adjust the applied time delay.

On the right, I plot the fast Fourier Transform (FFT) of the delayed impulse. We can observe that delay essentially creates a phase rotation to the frequency domain samples. Each sample delay in time adds one phase spiral in frequency.

Once we go past 16 samples of delay (there are 32 samples in total), the number of spirals in the frequency domain starts decreasing. This continues until we have 31-sample delay, which has only a single phase spiral in frequency domain.

If we compare the FFT for delay=1 and delay=31, we see that they both have a single phase spiral, but with opposite orientations. This is because FFT, which is basically a discrete Fourier transform (DFT), assumes that the signal is periodic (in frequency and in time). So a delay of 31 samples is equivalent to an advance of 1 sample (or a delay of -1 sample).