Given a discrete 1D signal $f(n)$ over the support $-N/2 \leq n \leq N/2$, where $N$ is even, and given an arbitrary scalar value $\alpha$, the definition of 1-D fractional Fourier transform (FrFT) (also known as Chirp-Z transform, see Bailey-1991) is given by,
\begin{eqnarray} F^{\alpha}(k) = \sum\limits_{n=-N/2}^{N/2} f(n) e^{-i\frac{2\pi k\alpha n}{N+1}} , -N/2 \leq k \leq N/2 \end{eqnarray}
When $\alpha = 1$, the 1-D FrFT reduces to 1-D normal FFT.
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Consider figure above, we know that there is a fast way to compute both 1D FFT and 1D FrFT, but given a 1D data (shown in red) can we also compute the in-between discrete Fourier points (shown in blue) in a fast way ?
The fractional FFT can be used multiple times to gather exactly the points desired, but it seems rather inefficient , since no one factor will yield the desired spacing (shown in blue). With 1 selection of $\alpha$ we can gather 2 points (see the data in black).
I know there is a non-uniformly spaced FFT , but it requires some user parameters and its an approximation, but for this special arrangement is there something in the literature that is or can be adapted to be algebraically exact and fast ?
