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Constrained Capacity for Faster-than-Nyquist Signaling in Frequency-Selective Fading Channels

Zichao Zhang, Melda Yuksel, Gokhan M. Guvensen, Halim Yanikomeroglu

arXiv:2609.06639v1cs.ITeess.SPmath.IT

TL;DR

Finite-alphabet constrained capacity for FTN over frequency-selective fading channels remains limited. This paper uses DFT precoding and CP/CS to analyze parallel eigenchannels, and formulates a mismatched AIR for finite blocks without guards. FTN outperforms Nyquist in the reported comparisons, while mismatched AIR approaches constrained capacity as block length grows.

  • Problem

    Constrained-capacity analysis of FTN over frequency-selective fading channels remains limited for practical finite-alphabet inputs.

    Source: §I · p. 1

  • Method

    The paper analyzes DFT-precoded FTN over frequency-selective fading using CP/CS eigenchannels and formulates mismatched AIR for finite blocks without guards.

    Source: §I · p. 1, §IV · p. 4

  • Results

    FTN outperforms the Nyquist RRC benchmark, while mismatched AIR is lower at short block lengths and approaches constrained capacity as N increases.

    Source: §V · p. 5

  • Takeaways & Limitations

    For 16QAM, DFT-precoded FTN achieves spectral-efficiency gains over Nyquist before reaching the finite-alphabet saturation limit.

    Source: §VI · p. 6

Abstract

from arXiv · show

In this paper, we investigate the constrained capacity of discrete Fourier transform (DFT)-precoded faster-than-Nyquist (FTN) signaling over frequency-selective channels with finite-alphabet inputs. With a cyclic prefix (CP) and cyclic suffix (CS), the FTN and multipath induced intersymbol interference (ISI) is decomposed into parallel eigenchannels, whose gains are jointly determined by the folded FTN spectrum and the channel frequency response. Based on this decomposition, we derive the constrained capacity for finite-alphabet constellations and formulate a mismatched decoding achievable information rate for DFT-precoded FTN signaling without CP/CS, quantifying the finite-block rate loss caused by imperfect diagonalization. We find that even under mismatched decoding, FTN significantly improves upon Nyquist transmission.

I. INTRODUCTION

FTN can improve spectral efficiency within the same bandwidth, but intentional ISI complicates detection. This paper addresses limited constrained-capacity analysis for practical finite-alphabet inputs by analyzing DFT-precoded FTN over frequency-selective fading channels, with and without CP/CS.

I. INTRODUCTION

  • FTN transmits symbols at a compressed interval δT within the same bandwidth, increasing symbol rate while intentionally introducing ISI that complicates detection.Here T is the Nyquist symbol interval and δ ∈ (0, 1] is the acceleration factor.Source: §I · p. 1
  • DFT precoding is attractive because CP and CS let the DFT matrix diagonalize the FTN ISI matrix into parallel channels.Practical constellation symbols are loaded in the frequency domain and precoded into the time domain.Source: §I · p. 1
  • Finite-alphabet constrained-capacity analysis for FTN remains limited, despite practical systems using discrete modulations such as BPSK, QPSK, and QAM.Earlier work derived bounds for finite-alphabet time-domain FTN symbols and studied binary FTN as δ → 0.Source: §I · p. 1
  • With CP and CS, DFT precoding diagonalizes the FTN ISI matrix; this work extends the analysis to finite-alphabet inputs over frequency-selective fading channels.The paper also considers omitted CP/CS, where finite-block precoding leaves residual inter-eigenchannel interference and motivates a mismatched AIR analysis.Source: §I · pp. 1–2

II. SYSTEM MODEL

The system uses DFT-precoded finite-alphabet symbols, pulse shaping, and a frequency-selective tapped-delay channel. With suitable CP and CS, the combined FTN and multipath ISI becomes circulant and is diagonalized into parallel eigenchannels whose gains combine the folded FTN spectrum and channel response.

II. SYSTEM MODEL

  • The data are iid finite-alphabet symbols with unit energy and independent across symbol indices; the frequency-domain vector is DFT-precoded under uniform power allocation.The text gives QPSK and 16QAM as examples.Source: §II · p. 2
  • The pulse-shaped signal uses interval δT, with a root-raised cosine pulse in the analysis; the authors state that other pulse shapes are also applicable.The channel is frequency-selective fading with AWGN and J taps, where tap j arrives at delay jδT.Source: §II · p. 2
  • CP length L + J and CS length L make the banded combined FTN and multipath channel matrix cyclic.These lengths account for pulse memory L and the J-tap delayed channel matrices.Source: §II · p. 2
  • Applying the IDFT at the receiver diagonalizes the cyclic channel, yielding parallel eigenchannels with gains λ_kH_k.The λ_k values sample the folded FTN spectrum, while H_k represents the frequency-selective channel response.Source: §II · pp. 2–3
  • The folded spectrum is periodic with period 1, and its support approximately determines the fraction of nonzero eigenvalues.When N is sufficiently large, the transformed noise covariance is diagonal and the noise samples are uncorrelated.Source: §II · p. 3

1) Power constraint:

For sufficiently large blocks, CP and CS overhead is treated as negligible in the average transmit-power constraint. The channel can then be approximated by parallel DFT-domain subchannels.

1) Power constraint:

  • For sufficiently large N, CP and CS overhead is negligible for the average transmit-power constraint.The constraint is stated under a physical transmit-power limit P_TX.Source: 1) Power constraint · p. 3
  • With D†GD ≈ D†G_cD = Λ for large enough N, the frequency-selective FTN channel is represented by parallel scalar subchannels.The approximation enables the subsequent per-eigenchannel input-output relationship.Source: 1) Power constraint · p. 3

III. THE CONSTRAINED CAPACITY DERIVATION

The derivation identifies active eigenchannels and sums their finite-alphabet information to obtain constrained capacity and spectral efficiency for DFT-precoded FS-FTN.

III. THE CONSTRAINED CAPACITY DERIVATION

  • Each eigenchannel symbol is scaled by λkHk; when λk = 0, that eigenchannel is inactive.For δ(1 + β) < 1, some eigenvalues are zero.Source: §III · pp. 3–4
  • Only eigenchannels with nonzero eigenvalues are active; zero-eigenvalue channels are effectively nulled and carry no useful information.The capacity calculation sums information over active eigenchannels and averages over transmitted time-domain symbols.Source: §III · p. 4
  • Spectral efficiency in bits/s/Hz equals the average constrained capacity over all active eigenchannels.FTN shortens the transmission duration to NδT, while only Nr frequency-domain symbols carry useful information.Source: §III · p. 4
  • SNRtx and SNRrx differ for FTN and coincide only for Nyquist signaling with δ = 1.The paper defines SNRtx from transmit power and SNRrx from symbol energy per Nyquist interval.Source: §III · p. 4

IV. MISMATCHED AIR FOR FTN SIGNALING

Without CP/CS, finite-block DFT processing leaves residual inter-eigenchannel interference, so independent eigenchannel decoding yields a mismatched AIR. As block length grows, that AIR approaches constrained capacity.

IV. MISMATCHED AIR FOR FTN SIGNALING

  • CP/CS lengths depend on both FTN pulse memory and multipath delay spread, and their guard overhead may be non-negligible for low-latency transmission.Shortened guard intervals or omitting them leaves a Toeplitz rather than circulant effective channel matrix.Source: §IV · p. 4
  • For finite block lengths, the transformed channel can remain non-diagonal, with off-diagonal entries representing residual inter-eigenchannel interference.The mismatched decoder treats eigenchannels as uncorrelated and treats that interference as noise.Source: §IV · p. 4
  • At fixed SNRtx, matched FTN curves outperform the Nyquist RRC benchmark, while the mismatched AIR approaches constrained capacity as N increases.For small N, residual inter-eigenchannel interference leaves the mismatched AIR below constrained capacity.Source: Figure 2 · p. 5
  • The mismatched AIR is a coded achievable rate: sufficiently long, well-designed codes using the same decoding metric can reliably transmit below it.Its gap from constrained capacity reflects rate loss from imperfect diagonalization and residual interference, after coding is accounted for.Source: §IV · p. 5
  • At fixed SNRrx, FTN also outperforms the Nyquist RRC benchmark.Fig. 3 compares constrained capacity and mismatched AIR across block lengths for two δ values and Nyquist sinc and RRC pulses.Source: Figure 3 · p. 5

V. SIMULATION RESULTS

For 16QAM over the frequency-selective fading channel, FTN outperforms the practical Nyquist RRC benchmark by using its excess bandwidth. The mismatched AIR falls below constrained capacity at short block lengths but approaches it as block length increases; its behavior versus acceleration depends on which SNR is fixed.

V. SIMULATION RESULTS

  • FTN outperforms the Nyquist RRC benchmark by exploiting RRC excess bandwidth under both fixed-SNRtx and fixed-SNRrx settings.Nyquist sinc is the ideal reference; the channel uses five taps with normalized average energy and an RRC pulse with β = 0.25.Source: §V · p. 5
  • At short block lengths, mismatched AIR is below constrained capacity because finite-block DFT precoding leaves residual inter-eigenchannel interference, but it approaches capacity as N increases.The mismatch loss is more pronounced at low SNR, where noise and residual interference more strongly affect the decoding metric.Source: §V · pp. 5–6
  • For fixed SNRtx, the δ = 0.3 and δ = 0.5 curves overlap because both lie in the folded-spectrum saturation region, where active-channel reduction is offset by normalization.For fixed SNRrx, those curves do not overlap: smaller δ requires larger transmit power and gives δ = 0.3 a higher rate.Source: §V · pp. 5–6
  • For fixed SNRtx, constrained capacity and mismatched AIR versus δ saturate as δ decreases, as finite-alphabet inputs and active eigenchannels limit spectral efficiency.Fig. 4 presents this comparison for 16QAM.Source: §V · p. 6, Figure 4 · p. 6
  • For fixed SNRrx, rate can continue increasing as δ decreases beyond the active-eigenchannel threshold because received symbol energy stays fixed while transmit power rises.Fig. 5 presents the fixed-SNRrx comparison for 16QAM.Source: §V · p. 6, Figure 5 · p. 6

VI. CONCLUSION

The conclusion summarizes constrained-capacity analysis for finite-alphabet DFT-precoded FTN over frequency-selective fading channels, with and without CP/CS. It reports spectral-efficiency gains over Nyquist for 16QAM, alongside finite-block and low-SNR mismatch losses.

VI. CONCLUSION

  • With CP and CS, the combined FTN and multipath ISI is decomposed into parallel eigenchannels, enabling constrained-capacity analysis for practical constellations.The analysis concerns DFT-precoded FTN over frequency-selective fading channels with finite-alphabet inputs.Source: §VI · p. 6
  • Without CP/CS, finite-block imperfect diagonalization produces residual inter-eigenchannel interference, which the paper accounts for through mismatched AIR.The mismatched AIR has a short-block loss but approaches constrained capacity as N increases; that loss is more pronounced at low SNR.Source: §VI · p. 6
  • For 16QAM, DFT-precoded FTN gains spectral efficiency over Nyquist before reaching the finite-alphabet saturation limit.The paper identifies extending the results to higher-order QAM as future work.Source: §VI · p. 6
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