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Computation-Limited Signals: A Channel Capacity Regime Constrained by Computational Complexity

Saulo Queiroz, João P. Vilela, Edmundo Monteiro

TL;DR

The Spectro-Computational (SC) analysis is presented, a novel mathematical framework designed to enhance classic concepts of information theory – such as data rate, spectral efficiency, and capacity – to accommodate the computational complexity overhead of signal processing.

Abstract

In this letter, we introduce the computational-limited (comp-limited) signals, a communication capacity regime in which the signal time computational complexity overhead is the key constraint -- rather than power or bandwidth -- to the overall communication capacity. We present the Spectro-Computational (SC) analysis, a novel mathematical framework that enhances classic concepts of information theory -- such as throughput, spectral efficiency and capacity -- to account for the signal processing computational complexity overhead. We consider a specific Shannon regime under which capacity is expected to get arbitrarily large as channel resources grow. Under that regime, we identify the conditions under which the time complexity overhead causes capacity to decrease rather than increasing, thereby creating the case for the comp-limited regime. We also provide examples of the SC analysis and show the OFDM waveform is comp-limited unless the lower-bound computational complexity of the $N$-point DFT problem verifies as $Ω(N)$, which remains an open challenge.

Computation-Limited Signals: A Channel Capacity Regime Constrained by Computational Complexity

TL;DR

The Spectro-Computational (SC) analysis is presented, a novel mathematical framework designed to enhance classic concepts of information theory – such as data rate, spectral efficiency, and capacity – to accommodate the computational complexity overhead of signal processing.

Abstract

In this letter, we introduce the computational-limited (comp-limited) signals, a communication capacity regime in which the signal time computational complexity overhead is the key constraint -- rather than power or bandwidth -- to the overall communication capacity. We present the Spectro-Computational (SC) analysis, a novel mathematical framework that enhances classic concepts of information theory -- such as throughput, spectral efficiency and capacity -- to account for the signal processing computational complexity overhead. We consider a specific Shannon regime under which capacity is expected to get arbitrarily large as channel resources grow. Under that regime, we identify the conditions under which the time complexity overhead causes capacity to decrease rather than increasing, thereby creating the case for the comp-limited regime. We also provide examples of the SC analysis and show the OFDM waveform is comp-limited unless the lower-bound computational complexity of the -point DFT problem verifies as , which remains an open challenge.
Paper Structure (23 sections, 2 theorems, 12 equations, 2 figures)

This paper contains 23 sections, 2 theorems, 12 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Background and Rationale
  3. Asymptotic Notation
  4. A Case for a Time Complexity-Constrained Signal Regime
  5. Shannon Capacity Regimes
  6. Channel Resources and Time Complexity
  7. What If Channel Resources Grow Arbitrarily?
  8. Fundamentals of Spectro-Computational Complexity
  9. System Model and Assumptions
  10. Computational Time and Algorithmic Throughput
  11. Spectro-Computational Throughput
  12. Spectro-Computational Efficiency
  13. Computation-Limited Signals
  14. SC Algorithmic Capacity
  15. The Comp-Limited Signal Regime
  16. ...and 8 more sections

Key Result

Lemma 1

Under the fixed SNR regime of the Shannon capacity (discussed in Section subsubsec:snrregime), the SC throughput (Eq. eq:sc) nullifies as $W\to\infty$ unless $B(W)=\Omega(T(W))$.

Figures (2)

  • Figure 1: Communication system model of the SC analysis (receiver-side omitted).
  • Figure 2: FFT baseband processor comparison: 512-point (right-hand side) vs. 64-point (left-hand side). The 512-point processor produces a $8\times$ faster signal at the penalty of consuming larger computational resources. We show that this gain nearly halves if the signal data rate also accounts for runtime and both processors are provided the same computational resources.

Theorems & Definitions (5)

  • Lemma 1: Condition of Scalability
  • proof
  • Definition 1: Comp-Limited Signal Regime
  • Theorem 1: Comp-Limited OFDM Signal
  • proof