# Is FFT Fast Enough for Beyond-5G Communications?

In this work, we consider the complexity and throughput limits of the Fast Fourier Transform (FFT) algorithm having in mind the unprecedented number of points (subcarriers) N expected in future waveforms. Based on the spectro-computational analysis, we verify that the FFT complexity to process an N-subcarrier symbol grows faster than the number of bits in the symbol. Thus, the useful throughput of FFT nullifies as N grows. Also, because FFT demands N to be a power of two 2^i (for some i>0), the spectrum widening causes the FFT complexity to grow exponentially on i, i.e. O(2^ii). To overcome these limitations, we propose the Parameterized DFT (PDFT) algorithm, which builds on the parameterized complexity technique and the classic O(N^2) DFT algorithm to replace an N-point DFT into N/n (n>0) smaller n-point DFTs. By setting n=Θ(1), we get a O(N) algorithm whose resulting waveform matches OFDM in its vectorized form (i.e., Vector OFDM) but with the N=2^i constraint relaxed. Besides, we also show that PDFT becomes multiplierless for n=2, requiring only Θ(N) complex sums. We believe our results constitute a relevant step towards the practical deployment of future extremely wide multicarrier signals.

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