# What happens to frequency domain components if I circular-shift the time domain samples?

If I have a time domain array of N samples, and swap the final half with initial half, then perform an FFT, what will be the effects on the frequency domain? Will the frequencies' magnitudes remain the same as the ones in the frequency domain of the original array?

What if instead of swapping halves, I just circle 1/10th, or 1/5th, of the full array?

I want to know this because I am trying to take the frequencies' magnitudes at real time, and as new sample data arrives, I thought of simply updating my frequency domain data (subtracting the old samples and adding the new ones) instead of redoing the whole FFT every time.

If you circularly shift the array in the time domain, the DFT of the shifted sequence will have the same magnitudes in each bin, but the phases will be different, with $X[k]$ being transformed to $X[k]e^{jk\theta}$, $k = 0, 1, \ldots, N-1$, for a fixed value of $\theta$ that I will leave for you to figure out.