| Summary: | In this paper, we study the impact of computational complexity on the throughput limits of the fast Fourier transform (FFT) algorithm for orthogonal frequency division multiplexing (OFDM) waveforms. Based on the spectro-computational complexity (SC) analysis, we verify that the complexity of an <inline-formula> <tex-math notation="LaTeX">$N$ </tex-math></inline-formula>-point FFT grows faster than the number of bits in the OFDM symbol. Thus, we show that FFT nullifies the OFDM throughput on <inline-formula> <tex-math notation="LaTeX">$N$ </tex-math></inline-formula> unless the <inline-formula> <tex-math notation="LaTeX">$N$ </tex-math></inline-formula>-point discrete Fourier transform (DFT) problem verifies as <inline-formula> <tex-math notation="LaTeX">$\Omega (N)$ </tex-math></inline-formula>, which remains a “fascinating” open question in theoretical computer science. Also, because FFT demands <inline-formula> <tex-math notation="LaTeX">$N$ </tex-math></inline-formula> to be a power of two <inline-formula> <tex-math notation="LaTeX">$2^{i}$ </tex-math></inline-formula> (<inline-formula> <tex-math notation="LaTeX">$i>0$ </tex-math></inline-formula>), the spectrum widening leads to an exponential complexity on <inline-formula> <tex-math notation="LaTeX">$i$ </tex-math></inline-formula>, i.e. <inline-formula> <tex-math notation="LaTeX">$O(2^{i}i)$ </tex-math></inline-formula>. To overcome these limitations, we consider the alternative frequency-time transform formulation of vector OFDM (V-OFDM), in which an <inline-formula> <tex-math notation="LaTeX">$N$ </tex-math></inline-formula>-point FFT is replaced by <inline-formula> <tex-math notation="LaTeX">$N/L$ </tex-math></inline-formula> (<inline-formula> <tex-math notation="LaTeX">$L > 0$ </tex-math></inline-formula>) smaller <inline-formula> <tex-math notation="LaTeX">$L$ </tex-math></inline-formula>-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., <inline-formula> <tex-math notation="LaTeX">$L=3$ </tex-math></inline-formula>, <inline-formula> <tex-math notation="LaTeX">$N=156$ </tex-math></inline-formula>). Besides, by setting <inline-formula> <tex-math notation="LaTeX">$L$ </tex-math></inline-formula> to <inline-formula> <tex-math notation="LaTeX">$\Theta (1)$ </tex-math></inline-formula>, the resulting solution can run linearly on <inline-formula> <tex-math notation="LaTeX">$N$ </tex-math></inline-formula> (rather than exponentially on <inline-formula> <tex-math notation="LaTeX">$i$ </tex-math></inline-formula>) while sustaining a non null throughput as <inline-formula> <tex-math notation="LaTeX">$N$ </tex-math></inline-formula> grows.
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