  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 $n=\Theta(1)$, 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 $\Theta(N)$ complex sums. We believe
our results constitute a relevant step towards the practical deployment of
future extremely wide multicarrier signals.
