as the last post shown the formula of discrete Fourier transform:

a_n = \sum_{i=0}^m { x_i e^{i (n-1) 2\pi (i-1) /m}}

where (m+1) is the number of data-point, n is from 1 to (m+1). if n is bigger then m+1, it is a_n is start to repeat. for example, n = m+1+r

a_{m+1+r} = a_{r}

Therefore,

a_{m+1-r} = a_{-r}

Thus, we can rearrange the a_n from -(m+1)/2  to (m+1)/2. This, set the center be zero-frequency. for example, if m+1 = 1000, then we can have n from 1 to 1000, or -499 to 0 and 1 to 500. since n=1 mean frequency = 0, we can shift n by 1 unit and get n from -500 to 499.

this it very useful if out signal is actually from 2 signals, 1 is raw signal and the other is a reference signal. the signal we get is by subtracting these 2. thus, when the raw signal is very close to the reference signal, the frequency is very small and the peak of Fourier spectrum will go to the edge and hardly recognized. However, by shift the spectrum and set the a_0 at the center. we can see How the raw signal is different from the reference signal.

 

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