Coded FTN Results Overview

• The paper shows that advanced detection methods, including CNN and SDR detectors, achieve near-optimal BER performance and 30–50% throughput gains compared to Nyquist systems.
• Coded FTN experiments quantify spectral efficiency improvements and computational savings by comparing BER/BLER curves and complexity scaling under varied modulation and coding schemes.
• Analytical bounds confirm that integrating LDPC and polar codes with FTN signaling enhances coding rates and error reduction, making it viable for URLLC and IoT applications.

Faster-than-Nyquist (FTN) signaling represents a class of transmission schemes in which symbols are intentionally packed faster than dictated by the Nyquist limit, producing controlled intersymbol interference (ISI) to increase signaling density. In the context of coded FTN systems, evaluation of bit error rate (BER), block error rate (BLER), achievable spectral efficiency, and computational complexity are central to assessing practical viability for next-generation communications. Coded FTN results—interpreted here as channel-coded FTN transmission experiments and associated theoretical bounds—demonstrate how modern detection algorithms, especially those based on deep learning and innovative sequence estimation methods, enable near-Nyquist reliability at higher throughput and reduced computational cost.

1. FTN System Model and Performance Metrics

Coded FTN studies consistently utilize linear modulation with acceleration factor $\tau < 1$ (Nyquist interval $T$, symbol interval $\tau T$), where the signal model takes the form $s(t) = \sum_k a_k\,g(t - k\tau T)$, with $g(t)$ a unit-energy, typically root-raised-cosine pulse. Introduction of coding schemes such as LDPC and polar codes allows measurement of coded performance via BER and BLER under additive white Gaussian noise (AWGN), fading channels, and ISI levels determined by $\tau$. Spectral efficiency scales as $\eta = R/\tau$ for code rate $R$, indicating substantial throughput gains as $\tau$ decreases.

Key metrics reported in simulation and theory include:

• BER/BLER curves versus SNR for various FTN compressions and code rates.
• Achievable spectral efficiency at fixed BER, often compared to coded Nyquist or MAP limits.
• Computational cost per symbol (multiplications, additions, algorithmic complexity exponents).
• Sensitivity to modulation order, ISI length, and channel conditions.

Numerical tables, such as Table A in (Tokluoglu et al., 18 Aug 2025), exemplify these results by quantifying cost savings and performance gaps across modulations and FTN compressions.

2. Detection Architectures for Coded FTN

Detection in coded FTN systems faces the challenge of severe ISI induced by time packing ($\tau < 1$). Several detector architectures are prominent:

• Structured CNN detectors (Tokluoglu et al., 18 Aug 2025): Use triplet-aware fixed kernel layers to learn ISI patterns at specific lags, with hierarchical filter allocation (more filters for strong ISI taps).
• Deep Learning Assisted Sum-Product Algorithm (DL-SPDA) (Liu et al., 2020): A small CNN node augments the conventional factor graph detector, correcting residual ISI and message correlations during turbo equalization.
• Symbol-by-symbol sequence estimation (SSSgbKSE) (Cerci et al., 2021, Caglan et al., 2020): Performs recursive local ISI cancellation with go-back re-estimation, enabling complexity scaling linear in ISI length.
• Semidefinite relaxation (SDR) detectors with LLR lookup tables (Cicek et al., 2020): Casts FTN detection as a lifted convex problem, and produces soft outputs with LUT-aided LLR conversion.
• CNN-based equalizers with skip connections (Filippo et al., 20 Jan 2025): Enable direct mapping from complex ISI-corrupted blocks to LLR vectors suitable for LDPC decoding.

In coded scenarios, outputs from these detectors are processed as soft inputs for the channel decoder (LDPC, polar), with several studies employing belief-propagation for LDPC or successive cancellation for polar codes.

3. Coded FTN Simulation and Experimental Results

Coded FTN experiments reveal critical insights into performance versus complexity trade-offs:

• Near-optimal BER: CNN detectors with LDPC decoding attain BER and BLER performance comparable to BCJR+LDPC, particularly for $\tau \geq 0.7$ (gap $\leq 0.05$–$0.2$ dB at BER=$10^{-4}$) (Tokluoglu et al., 18 Aug 2025, Filippo et al., 20 Jan 2025).
• Spectral efficiency gains: Simulation and theoretical models consistently observe throughput enhancements proportional to $1/\tau$ (e.g., $25\%$ at $\tau=0.8$, $50\%$ at $\tau=0.67$) (Li et al., 29 Dec 2025, Zhang et al., 2024).
• Computational cost reductions: Structured CNNs and SDR detectors offer linear or polynomial complexity scaling, versus exponential in ISI length for M-BCJR or MAP detectors, with up to $84\%$ LUT-area reduction reported for CNNs under QPSK ($\tau=0.9$), and similar scaling in polar-coded SDR-based FTN (Tokluoglu et al., 18 Aug 2025, Cicek et al., 2020).
• Robustness to modulation and fading: Deep learning detectors maintain competitive BER for 16/64-QAM, and exhibit resilience in quasi-static Rayleigh channels (Tokluoglu et al., 18 Aug 2025). NB-LDPC and polar codes are particularly effective for short packets and URLLC (Cerci et al., 2021, Caglan et al., 2020).
• Error probability reduction at fixed coding rate: The finite blocklength normal approximation confirms that lowering $\tau$ not only boosts maximal coding rate but also reduces BLER for a given $R$ and packet length (Kim, 2023).

Table: Summary of BER and Complexity Results

Detector Configuration	BER Gap to MAP (at $10^{-4}$)	Complexity Reduction
CNN+LDPC ($\tau=0.7$, QPSK)	$\leq 0.2$ dB	$60\%$–$84\%$ LUT area
SSSgb1SE+NB-LDPC ($\tau=0.8$)	$-0.3$ dB (better)	$4\times$ lower OPS
SDR+Polar ($\tau=0.8$)	$0.8$ dB (single LUT SNR)	Polynomial (vs. exp.)
DL-SPDA+CC ($\tau=0.6$)	$0.35$–$0.75$ dB	$<60\%$ MAP cost

4. Analytical Bounds and Theoretical Insights

Analytical results for coded FTN derive from finite blocklength information theory:

• Normal approximation for maximum coding rate: $$R^*(n,\varepsilon;\tau) \approx C(\tau) - \sqrt{V(\tau)/n}Q^{-1}(\varepsilon) + o(1/n)$$, where $C(\tau)$ is capacity and $V(\tau)$ is dispersion (Kim, 2023, Zhang et al., 2024).
• Operating regions: For practical pulses (e.g., RRC), FTN delivers both higher channel capacity and coding rate above Nyquist for $\tau > \tau_0 = 1/(1+\beta)$. Below this threshold, capacity saturates but blocklength coding rate can still increase due to dispersion improvement.
• Meta-converse and RCU bounds: Tight theoretical bounds confirm the empirical gains observed in coded FTN, especially for short packets and stringent BLER.

The derivation and simulation of these results across SISO/MIMO, varying $n$, $R$, $\varepsilon$, and modulation order reinforce FTN's utility for high-throughput, reliable communication in low-latency, spectrum-constrained settings (Zhang et al., 2024, Kim, 2023).

5. Practical Implementation and Complexity Considerations

Coded FTN deployment hinges on achieving tractable computational demands without significant performance loss:

• Detector complexity scaling: CNN and SDR detectors scale linearly or polynomially with block length and ISI span, enabling real-time operation, with the neural network approach decoupled from modulation order size $M$ (Tokluoglu et al., 18 Aug 2025, Cicek et al., 2020).
• Coding choices: NB-LDPC and polar codes are preferred for short-packet and URLLC scenarios due to higher codeword diversity and robustness under strong ISI (Cerci et al., 2021, Caglan et al., 2020).
• Integration with frequency domain equalization and hardware acceleration: Structured receivers can further reduce complexity; however, care is needed to avoid performance loss under aggressive FTN packing ($\tau \ll 1$) (Li et al., 29 Dec 2025).
• LUT memory footprint: For SDR-based receivers, offline table construction for LLR mapping is negligible for moderate modulation orders and SNR, but may scale up for very high $M$ or fine granularity (Cicek et al., 2020).

6. Interpretations and Prospects in Coded FTN Systems

The body of coded FTN research shows that with careful matching of detector architecture, code design, and time-packing parameters, spectral efficiency increases of $30$–$50\%$ are attainable at sub-dB SNR penalty relative to coded Nyquist systems. Error probability can be significantly lowered for fixed rate and block length, and complexity reductions make FTN suitable for URLLC, IoT, and power-constrained mobile applications.

A plausible implication is that for next-generation networks (6G and beyond), coded FTN, especially when paired with advanced neural detectors and tailored FEC, can deliver practical throughput and reliability gains without prohibitive receiver complexity. Operating below the FTN threshold $\tau_0$ may unlock further performance for short-packet or high-reliability scenarios by leveraging additional signaling dimensions (Kim, 2023, Li et al., 29 Dec 2025).

Future work includes scaling to massive MIMO, joint optimization across coding, detector, and pulse design, and hardware implementation of neural FTN receivers for real-time adaptive modulation and coding (Li et al., 29 Dec 2025, Filippo et al., 20 Jan 2025).