Is FFT Fast Enough for Beyond-5G Communications?

In this paper, we study the impact of computational complexity on the throughput limits of the {\color{black}fast Fourier transform (FFT)} algorithm for {\color{black}orthogonal frequency division multiplexing(OFDM)} waveforms. Based on the spectro-computational {\color{\corcorrecao}complexity} (SC) analysis, {\color{\corcorrecao} we verify that the complexity of an $N$-point FFT grows faster than the number of bits in the OFDM symbol.} Thus, we show that FFT nullifies the OFDM throughput on $N$ unless the $N$-point discrete Fourier transform (DFT) problem verifies as $\Omega(N)$, which remains a "fascinating" open question in theoretical computer science. Also, because FFT demands $N$ to be a power of two $2^i$ ($i>0$), the spectrum widening leads to an exponential complexity on $i$, i.e. $O(2^ii)$. To overcome these limitations, {\color{\corcorrecao} we consider the alternative frequency-time transform formulation of vector OFDM (V-OFDM), in which an $N$-point FFT is replaced by $N/L$ ($L$$>$$0$) smaller {\color{\corcorrecao}$L$-point} FFTs to mitigate the cyclic prefix overhead of OFDM. Building on that, we replace FFT by the straightforward DFT algorithm to release the V-OFDM parameters from growing as powers of two and to benefit from flexible numerology (e.g., $L=3$, $N=156$). Besides, by setting $L$ to $\Theta(1)$, the resulting solution can run linearly on $N$ (rather than exponentially on $i$) while sustaining a non null throughput as $N$ grows. }