  Adaptive Fourier decomposition (AFD, precisely 1-D AFD or Core-AFD) was
originated for the goal of positive frequency representations of signals. It
achieved the goal and at the same time offered fast decompositions of signals.
There then arose several types of AFDs. AFD merged with the greedy algorithm
idea, and in particular, motivated the so-called pre-orthogonal greedy
algorithm (Pre-OGA) that was proven to be the most efficient greedy algorithm.
The cost of the advantages of the AFD type decompositions is, however, the high
computational complexity due to the involvement of maximal selections of the
dictionary parameters. The present paper offers one formulation of the 1-D AFD
algorithm by building the FFT algorithm into it. Accordingly, the algorithm
complexity is reduced, from the original $\mathcal{O}(M N^2)$ to $\mathcal{O}(M
N\log_2 N)$, where $N$ denotes the number of the discretization points on the
unit circle and $M$ denotes the number of points in $[0,1)$. This greatly
enhances the applicability of AFD. Experiments are carried out to show the high
efficiency of the proposed algorithm.
