I am new to the discrete setting of Fourier transform. Currently, I'm having trouble understanding the relation between two different definitions of the DFT, using different index sets but fixing the input vector. Say we have some signal $f^1 =(f^1_0,...,f^1_{N-1})$ and $f^2 = (f^2_{-N/2},...,f^2_{N/2-1})$, where $f^1 = f^2$. So basically, the indexing is notated differently, but we have the same signals. Is there a way to write the relation between
- $\hat{f}^1_n = \sum_{k=0}^{N-1}f^1_k exp(-2\pi i n k/N)$ for $n = 0,...,N$ and
- $\hat{f}^2_n = \sum_{k=-N/2}^{N/2-1}f^2_k exp(-2\pi i n k/N)$ for $n = -N/2,...,N/2-1$
It is clear to me that, normally, the shift of indices only outputs a shifted signal where we need to reorder the components. But as in this case we have $f^1 = f^2$ instead of one being a shifted version of another, how do we relate $\hat{f}^1$ to $\hat{f}^2$?