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FTN Signaling: Spectral Efficiency from BPSK to 16-QAM

Emre Cerci, Adem Cicek, Melda Yuksel, Gokhan M. Guvensen, Enver Cavus, Yaser Dalveren, Halim Yanikomeroglu

arXiv:2610.12066v1cs.ITeess.SPmath.IT

TL;DR

The paper asks whether FTN acceleration can improve spectral efficiency across constellations without the measured optimum being a receiver limitation. It evaluates finite-alphabet symbol-wise AIR with a common channel-shortening MMSE detector and finds that the best acceleration becomes less aggressive as constellation order increases.

  • Problem

    The study asks whether a short-memory detector is sufficient for 16-QAM so its reported optimum reflects the information rate rather than a receiver limitation.

    Source: B. 16-QAM Complexity and ν-Convergence · p. 2

  • Method

    The paper computes finite-alphabet symbol-wise AIR for BPSK, QPSK, and 16-QAM with a common MMSE channel-shortening detector, using an independent BCJR benchmark to validate BPSK rates.

    Source: §I · p. 1

  • Results

    SE gains over Nyquist are 33% for BPSK, 14% for QPSK, and 4.5% for 16-QAM, with τ⋆ shifting from 0.65 to 0.75 to 0.90.

    Source: §I · p. 1, A. Optimum SE for Different… · p. 3

  • Takeaways & Limitations

    Useful FTN acceleration should be selected jointly with constellation order, operating SNR, and receiver complexity.

    Source: E. Operating-Point Dependence · p. 4

Abstract

from arXiv · show

Faster-than-Nyquist (FTN) signaling can improve spectral efficiency by transmitting symbols closer together than the Nyquist limit, at the cost of additional inter-symbol interference (ISI). In this letter, we study the finite-alphabet symbol-wise achievable information rate (AIR) of FTN signaling for BPSK, QPSK, and 16-QAM under a transmit-power constraint and a common minimum mean square error (MMSE) channel-shortening (CS) detection framework, cross-validated for BPSK against an independent reduced-state Ungerboeck Bahl-Cocke-Jelinek-Raviv (BCJR) benchmark. Our results show a clear trend. Lower-order constellations can benefit from more aggressive time acceleration, while higher-order constellations become less tolerant of acceleration and achieve their best spectral efficiency closer to the Nyquist limit. At a reference signal-to-noise ratio (SNR) of 6 dB the optimum acceleration factor shifts from $\tau^\star\approx0.65$ for BPSK to 0.75 for QPSK and 0.90 for 16-QAM. We further show that this trend is preserved in coded systems and remains robust to the detector memory. The impact of FTN operation on the instantaneous-to-average power ratio (IAPR) is also examined. Overall, the results highlight how constellation order should be considered when selecting the operating point for FTN signaling.

I. INTRODUCTION

FTN packs symbols more densely than Nyquist signaling, trading increased ISI for potential spectral-efficiency gains. The paper compares finite-alphabet rates across constellation orders under a common power constraint and detection framework.

II. SYSTEM MODEL

  • FTN transmits one symbol every τT seconds, packing 1/τ times more symbols into the same bandwidth at the cost of ISI; AIR in bit/symbol is expressed as spectral efficiency in bit/s/Hz.The system model holds total transmit power constant while sweeping τ, keeping the comparison fair.Source: §I · p. 1, §II · p. 2

I. INTRODUCTION

  • Prior finite-alphabet FTN rate studies often focus on a particular modulation format or operating point, leaving how constellation order shifts the SE-maximizing acceleration factor less broadly examined.The paper interprets the differences through the constellations’ minimum Euclidean distances.Source: §I · p. 1
  • The comparison computes symbol-wise AIR for BPSK, QPSK, and 16-QAM using fixed power normalization and a common M-ary CS-MMSE detector.A reduced-state Ungerboeck BCJR detector is used only to validate the BPSK CS-MMSE rate.Source: §I · p. 1
  • All three constellations benefit from FTN acceleration, but the benefit shrinks and the optimal acceleration moves closer to Nyquist as constellation order increases.The study also examines whether this trend persists with larger detector memory and in coded BICM systems, and evaluates the IAPR cost.Source: §I · p. 1

II. SYSTEM MODEL

  • The symbol-wise rate is a mismatched-decoding lower bound when the detector posterior differs from the true posterior, and equals mutual information when they match.The model uses the detector-computed posterior q for the transmitted symbol.Source: §II · p. 2

III. ESTIMATION METHODOLOGY

The section introduces the estimation methodology by outlining its common detector, the rationale for 16-QAM memory, and the rate-validation checks.

III. ESTIMATION METHODOLOGY

  • The methodology covers a shared detector, justification of its short 16-QAM memory, and validation checks for reported rates.Source: §III · p. 2

A. Unified M-ary CS-MMSE Detector

A common CS-MMSE detector shortens the FTN response to a central-tap target and applies a causal BCJR, with complexity traded against model mismatch.

A. Unified M-ary CS-MMSE Detector

  • The MMSE prefilter retains the 2ν+1 dominant central taps; remaining taps and filtered noise are modeled as additional white Gaussian noise of matched variance.The shortened response is applied as a causal channel, yielding an exact Forney-type BCJR over its full 2ν-symbol memory with M^(2ν) trellis states.Source: A. Unified M-ary CS-MMSE Detector · p. 2
  • An Ungerboeck-type shortening receiver uses M^ν states for the same target span, trading a small, quantifiable model mismatch for an exponentially smaller trellis.Source: A. Unified M-ary CS-MMSE Detector · p. 2
  • The reported working memories are ν = 3 for QPSK and ν = 1 for 16-QAM.The adequacy of ν = 1 for 16-QAM is established in the following subsection.Source: A. Unified M-ary CS-MMSE Detector · p. 2

B. 16-QAM Complexity and ν-Convergence

The section tests whether the ν = 1 detector is adequate for 16-QAM, using a captured-energy criterion and a full ν = 2 comparison.

B. 16-QAM Complexity and ν-Convergence

  • The ν = 1 detector is used for the main sweep, with its adequacy assessed to distinguish the rate optimum from a receiver limitation.The section notes that reduced-state hard pruning can discard the true path in deep ISI and produce invalid negative auxiliary rates.Source: B. 16-QAM Complexity and ν-Convergence · p. 2
  • Ecap ≥ 0.99 is the self-consistency criterion; for β = 0.3, it holds for τ ≥ 0.85, defining the reliable window.Outside that window, 16-QAM results are reported as looser lower bounds; the criterion is not proof of MAP optimality.Source: B. 16-QAM Complexity and ν-Convergence · p. 2
  • A full ν = 2 detector with 65,536 states is run for τ ∈ [0.80, 1.00] as a reference for the ν = 1 results.The vectorized CS-MMSE BCJR is validated against the reference detector to within 10^-9 bit at ν ≤ 1.Source: B. 16-QAM Complexity and ν-Convergence · p. 2

C. Validation

The validation checks the AIR estimates against known or near-optimal references and reports uncertainty with confidence intervals.

C. Validation

  • At τ = 1, all three Monte Carlo AIR estimates match analytical AWGN mutual information within 0.005 bit, and all simulated rates remain below Cg.For BPSK, the CS-MMSE rate matches the validated near-MAP BCJR benchmark within confidence intervals at every τ.Source: C. Validation · p. 2
  • The BPSK reference uses an Ungerboeck-metric BCJR; at τ = 0.6, Mred = 128 reproduces the unpruned 4096-state BCJR AIR.The CS-MMSE BPSK rate matches this benchmark within confidence intervals at every τ.Source: C. Validation · p. 2
  • Fig. 1 plots SE against τ for three constellations, marking optima with stars and 95% confidence intervals with error bars.Each point is averaged over independent frames, and the per-symbol AIR term is averaged as-is rather than clamped to zero.Source: Figure 1 · p. 3, C. Validation · p. 3

IV. RESULTS AND DISCUSSION

At 6 dB, all three constellations peak below the Nyquist limit, but the SE gain declines and the optimum moves closer to τ = 1 as constellation order rises.

A. Optimum SE for Different Constellations

  • SE gains over Nyquist are 33%, 14%, and 4.5% for BPSK, QPSK, and 16-QAM, while τ⋆ shifts from 0.65 to 0.75 to 0.90.The QPSK optimum is a flat plateau over τ⋆ ∈ [0.70, 0.80], with a nominal peak at 0.75; Table I summarizes the optimum and gain for each constellation.Source: A. Optimum SE for Different… · p. 3, Table I · p. 3
  • Only ν = 3 results are reported for QPSK because ν = 2 under-reports the rate by up to 0.015 bit near the optimum, while leaving τ⋆ unchanged.The ν = 2 results are omitted.Source: A. Optimum SE for Different… · p. 3
  • For 16-QAM, the τ = 1 SE is 1.69 bps/Hz, and its peak within τ ≥ 0.85 is statistically significant over that baseline.The τ = 1 value is 2.20/(1 + β) = 1.69 bps/Hz; points below τ = 0.85 are looser lower bounds.Source: A. Optimum SE for Different… · p. 3
  • The full ν = 2 detector reproduces 16-QAM ν = 1 SE within confidence intervals (|∆R| ≤ 0.003 bit) and retains τ⋆ = 0.90.The section concludes that this optimum is information-limited rather than detector-limited.Source: A. Optimum SE for Different… · p. 3
  • Fig. 2 shows QPSK and 16-QAM SE versus τ at 0 dB, with open squares marking 16-QAM points where Ecap < 0.99.Source: Figure 2 · p. 3

B. Minimum-Distance View of Constellation-Dependent τ

Under the power constraint, denser constellations lose symbol-wise AIR more steeply as acceleration increases, shifting their optimal acceleration closer to the Nyquist limit. The section also finds no statistically significant low-SNR QPSK-over-16-QAM region and reports that the order-dependent trend persists for coded BICM metrics.

B. Minimum-Distance View of Constellation-Dependent τ

  • Under the power constraint, the optimum τ shifts toward larger values as constellation order grows: τ⋆=0.65 for BPSK and τ⋆=0.90 for 16-QAM.The 1/τ packing gain balances AIR loss; smaller dmin makes QPSK and 16-QAM more sensitive than BPSK to the same SNR reduction.Source: B. Minimum-Distance View of… · p. 3

C. Can a Lower-Order Constellation Overtake a Higher One?

  • At 0 dB, the QPSK and 16-QAM SE curves show no statistically significant QPSK-over-16-QAM region; their 6-dB values at τ=0.50 are inseparable within confidence intervals.16-QAM points with Ecap < 0.99 are looser lower bounds, so they do not support a constellation-ranking conclusion.Source: C. Can a Lower-Order Constellation… · pp. 3–4

D. Bit-wise GMI for Coded BICM

  • For coded BICM, QPSK's bit-wise GMI and symbol-wise AIR coincide, while 16-QAM's GMI penalty is 1.1% at τ=1 and 1.7% at its optimum, preserving τ⋆=0.90.The comparison is shown for QPSK (top) and 16-QAM (bottom) at ρ1=6 dB.Source: D. Bit-wise GMI for Coded BICM · p. 4, Figure 3 · p. 4

E. Operating-Point Dependence

The SE-maximizing acceleration depends on operating SNR as well as constellation and detector constraints. At higher SNR, reported acceleration optima move lower, while reliability-gated values can be lower bounds rather than nominal optima.

E. Operating-Point Dependence

  • As reference SNR rises from 0 to 24 dB, τ⋆ moves to smaller values; the SE at τ=1 and τ=τ⋆ both approach their k/(1+β) ceilings, with the FTN gain widening their gap.At high SNR, finite-alphabet AIR approaches k=log2 M bits per symbol, so the 1/τ symbol-rate gain dominates over a wider τ range.Source: E. Operating-Point Dependence · p. 4, Figure 4 · p. 4
  • At ρ1≥12 dB, the reliability gate confines τ to at least 0.80, yielding a 0.96 bps/Hz BPSK SE bound, whereas at 6 dB it admits τ⋆=0.65.The high-SNR tabulated optima are lower bounds on useful acceleration depth, not measured optima.Source: E. Operating-Point Dependence · p. 4
  • At 6 dB, a separate reliability-gated ν=2 QPSK sweep gives τ⋆=0.85 with a 10.0% gain, versus τ⋆=0.75 with a 14.1% gain in the ν=3 Fig. 1 sweep.The gated sweep is a conservative lower bound rather than the nominal optimum.Source: E. Operating-Point Dependence · p. 4
  • At 6 dB, the reliability-gated 16-QAM entry is 3.6% versus 4.5% in the other sweep, with τ⋆=0.90 in both; the difference reflects a smaller Monte Carlo budget.The reliability gate does not bind at this optimum.Source: E. Operating-Point Dependence · p. 4

F. IAPR Cost of Operating at the Optima

Operating at the SE optima raises IAPR relative to Nyquist signaling, with the size of the penalty varying by constellation. The section compares IAPR CCDFs at Nyquist and at each constellation's optimum.

F. IAPR Cost of Operating at the Optima

  • Fig. 5 plots the IAPR CCDF against threshold γ for each constellation at τ=1 and at its SE optimum; acceleration densifies pulse overlap and makes the waveform peakier.The plotted IAPR is Pr(IAPR > γ), with IAPR defined as |s(t)|^2/E[|s(t)|^2].Source: F. IAPR Cost of Operating at the Optima · p. 4, Figure 5 · p. 5
  • At the 10^-3 outage level, acceleration raises IAPR by 1.3 dB for BPSK (4.3 to 5.6 dB), versus 0.6 dB for QPSK (3.9 to 4.5 dB) and 16-QAM (5.3 to 5.9 dB).The respective optima are τ⋆=0.65 for BPSK and milder optima for QPSK and 16-QAM.Source: F. IAPR Cost of Operating at the Optima · p. 5, Figure 5 · p. 5
  • At common τ=0.80, IAPR is ordered QPSK < BPSK < 16-QAM, separating the constellation effect from the acceleration effect.The normalized IAPR is invariant to power scaling, so these curves do not depend on the power convention.Source: F. IAPR Cost of Operating at the Optima · p. 5

V. CONCLUSION

The conclusion is that all considered constellations can gain SE through FTN acceleration, but the optimal acceleration becomes less aggressive as constellation order increases. It also reports that this trend is not a detector-memory artifact and persists for coded BICM, with moderate IAPR increase.

V. CONCLUSION

  • FTN acceleration improves SE for BPSK, QPSK, and 16-QAM, while the optimal acceleration becomes less aggressive as constellation order increases.The order-dependent trend is understood through denser constellations' smaller minimum Euclidean distances under constant-power normalization and their interaction with ISI.Source: §V · p. 5
  • A larger-memory 16-QAM detector gives essentially the same rates and optimum as the short-memory detector, so the reported optimum is not a detector artifact.The trend also persists at lower SNRs.Source: §V · p. 5
  • Coded-BICM bit-wise GMI leads to the same conclusions with only a small 16-QAM penalty, while IAPR increases moderately at FTN operating points.The conclusion identifies coded-system performance and more powerful deep-FTN detectors for higher-order constellations as future work.Source: §V · p. 5
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