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Low-Complexity Sequential Detection Framework for Single-Channel Co-Frequency Signal Separation

Heng Wang, Kexian Gong, Peng Sun, Wei Wang, Hua Jiang

arXiv:2609.03986v1eess.SP

TL;DR

Single-channel co-frequency signal separation is hindered by the high complexity of benchmark methods. The paper formulates separation as sequential trellis search through SDS and SO-SDS, achieving substantial computational savings while preserving separation performance or limiting LLR degradation.

  • Problem

    Benchmark SCCFS separation algorithms can have prohibitive complexity, with PSP-based complexity growing exponentially with channel length.

  • Method

    A sequential-detection trellis framework uses SDS for hard decisions and SO-SDS with state completeness verification for bit-level LLR extraction.

  • Results

    Under 8PSK, SDS has BER comparable to HPSP and TVD while using 0.05% and 1.62% of their runtimes; SO-SDS reduces runtime by about 60% for BPSK and 92% for QPSK versus SO-PSP.

  • Takeaways & Limitations

    The proposed algorithms provide a favorable trade-off between separation performance and computational complexity, with increasing complexity advantages at higher modulation orders.

Abstract

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Practical separation of single-channel co-frequency signals (SCCFSs) is hindered by the prohibitive computational complexity of benchmark algorithms. To address this issue, we propose a low-complexity separation framework based on sequential detection (SD), in which signal separation is cast as a sequential path search over a trellis. To support both hard-decision detection and log-likelihood ratio (LLR) extraction, we develop two algorithms within this framework: the SD-based separation (SDS) algorithm and its soft-output variant (SO-SDS). Furthermore, SDS employs a windowing strategy combined with dynamic pruning to concentrate computational resources on high-probability paths, thereby enabling efficient detection of transmitted symbol sequences. Building upon SDS, SO-SDS further incorporates a state completeness verification mechanism (SCVM) to estimate bit LLRs, thus facilitating subsequent soft decoding. Numerical results show that, compared to benchmark algorithms, SDS achieves significant complexity reduction without degrading separation performance, while SO-SDS offers notable computational savings with only modest LLR accuracy loss. Notably, the computational complexity advantage of the proposed algorithms over benchmark algorithms grows substantially with increasing modulation order.

I. INTRODUCTION

Single-channel co-frequency signal separation is important for overlapping communication signals but remains computationally difficult. The paper introduces sequential-detection methods that reduce search complexity while supporting both hard decisions and soft information extraction.

  • Motivation: Existing PSP-based separation offers high accuracy but its ML-driven complexity grows exponentially with channel length, limiting practical feasibility.SCSS methods jointly recover all sub-signals, while PSP avoids relying on power disparities among them.
  • Proposed framework: The proposed framework casts SCCFS separation as sequential path search over a trellis, using variable-length path metrics instead of exhaustive candidate-path enumeration.It supports the SDS hard-decision algorithm and SO-SDS soft-output extension.
  • SDS: SDS iteratively extends the highest-metric candidate path, dynamically prunes low-probability paths, and uses a sliding window to reduce traceback depth, computation, and memory.These mechanisms concentrate the search on high-probability regions and promote rapid convergence.
  • SO-SDS: SO-SDS adds state completeness verification so competing paths for every possible symbol value can provide bit-level LLRs for subsequent channel decoding.LLRs are computed from path-metric differences between candidate paths associated with different symbol values.
  • System model: The system model represents the received SCCFS as a noisy superposition of two pulse-shaped, closely spaced co-frequency sub-signals with known channel parameters.Each terminal encodes, interleaves, modulates, and pulse-shapes its sequence before superposition and matched filtering.

III. SEPARATION FRAMEWORK AND ALGORITHMS

The framework converts SCCFS separation into sequential trellis path search, using a path metric adapted to variable-length candidates. Truncation and a modulation-dependent bias make metric computation practical while preserving discrimination between correct and incorrect paths.

  • Framework: The proposed framework formulates SCCFS separation as sequential path search over a trellis, with SDS for hard decisions and SO-SDS for soft-output separation.Its path metric is tailored to variable-length candidate paths and measures their proximity to the transmitted symbol sequences.
  • Path metric: The path with the highest path metric contains symbol sequences closest to the true transmitted sequences.
  • Metric approximation: Truncating pulse-response vectors avoids impractical length-2L computations by retaining only their core entries.The approximation relies on the sharp central lobe and fast-decaying sidelobes of the pulse response.
  • Metric approximation: The proposed constant approximation reduces the relevant complexity from O(M 2L′) to O(1) while preserving the path-discrimination bias.The bias favors correct paths and suppresses spurious extensions.
  • Metric approximation: Experiments identify λ = 1.5 as a good balance between algorithmic performance and computational complexity.In theory, λ > 1 guarantees a positive expected bias for correct candidate paths, but truncation perturbations motivate a slightly-above-one practical setting.

B. The SDS Algorithm

SDS performs hard symbol-sequence detection by repeatedly extending the highest-metric candidate, pruning weak paths, and limiting traceback through a sliding window. This concentrates computation on promising paths while reducing memory overhead.

  • SDS procedure: SDS iteratively expands the highest-metric candidate, prunes low-metric candidates, and uses time-domain windowing to reduce computational overhead.The window restricts detection memory to recent symbols.
  • Initialization: SDS initializes windowed candidate paths to cover all M 2 symbol pairs before sequential path extension.Each initialized path fixes its final symbol pair while other symbols are drawn independently over the constellation.
  • Path search: At each iteration, SDS extends the current best path, evaluates branch metrics, sorts candidates, and updates the stack’s best and worst paths.
  • Decision output: When a window reaches its maximum length, SDS outputs the oldest symbol estimate and removes that entry before continuing the search.
  • Scope: SDS is designed for hard symbol decisions and is followed by SO-SDS for scenarios requiring soft information for channel decoding.

C. The SO-SDS Algorithm

SO-SDS extends SDS to estimate bit-level LLRs for soft decoding. It uses state completeness verification to ensure every symbol hypothesis is represented, then approximates LLRs from path-metric differences over explored paths.

  • SO-SDS overview: SO-SDS integrates state completeness verification into SDS and computes bit-level LLRs for soft decoding.The method stores path metrics by symbol position and constellation value.
  • LLR estimation: SO-SDS estimates symbol-pair LLRs from path-metric differences between hypotheses and a reference all-zero symbol pair.The required path metrics correspond to candidates sharing the same decision position and competing symbol hypotheses.
  • State completeness verification: SCVM explicitly extends a candidate whenever a possible symbol value is absent, ensuring every constellation value is covered at each symbol position.
  • LLR estimation: Because exhaustive length-N path enumeration is infeasible, SO-SDS approximates global maximum metrics using explored length-n paths dominated by recent observations.
  • LLR estimation: The algorithm converts symbol-hypothesis metrics into bit LLRs by aggregating symbols mapped to one and zero at each bit position.

D. Complexity Analysis

SDS and SO-SDS have complexity independent of channel-memory length because they explore only M 2 branches per iteration. Benchmark methods instead search exponentially many M 2L′ branches, making the proposed methods increasingly attractive for large L′.

  • Computational complexity: SDS and SO-SDS explore only M 2 branches per iteration, whereas benchmark algorithms search over M 2L′ branches.SDS and SO-SDS require pQM 2 branch-metric evaluations; HPSP and SO-PSP require pM 2L′, while TVD requires 2pM 2L′.
  • Computational complexity: The complexity of SDS and SO-SDS is independent of L′, contrasting with the exponential growth of benchmark methods.This makes the proposed methods suitable for high-fidelity channel models with large L′.
  • Memory usage: TVD uses twice the memory operations of HPSP, while SO-TVD adds negligible memory overhead because of its simpler LLR computation.
  • Memory usage: SO-SDS requires more memory operations than SDS because of its additional stack updates.

IV. NUMERICAL RESULTS AND ANALYSIS

The numerical evaluation defines the compared detectors and metrics, then reports average BER after the appropriate detection or decoding stage.

  • Hard-decision detectors are evaluated by BER after demodulation, whereas SO-PSP, SO-TVD, and SO-SDS are evaluated after channel decoding.
  • MBER is defined as the average BER of the two separated signals, z0 and z1.

A. Experimental Conditions

The experiments use repeated coded-signal simulations under fixed hardware, modulation, noise, and algorithm-parameter settings.

  • Experimental Conditions: Results are averaged over 1,000 Monte Carlo trials using 12,000-bit rate-1/2 convolutionally coded BPSK sequences at Es/N0 = 14 dB.
  • Experimental Conditions: The simulations run in MATLAB R2020a on a Windows 10 PC with an Intel Core i5-12400F processor and 16 GB DDR4 RAM.
  • Experimental Conditions: The reported settings use L′_0 = 1 for TVD and SO-TVD, L′_0 = 3 and L′_1 = 1 for SDS-family methods, λ = 1.5, I = 14, and D = 14M^2.

B. Simplified BM

The simplified branch-metric computation preserves useful separation behavior while substantially reducing computational overhead, with λ controlling the performance–complexity balance.

  • Simplified BM: SDS maintains MBER comparable to SDSV across noise levels while incurring much lower computational overhead in all scenarios.SDS has lower Q than SDSV at high noise but slightly higher Q at low noise.
  • Simplified BM: SDS performance decreases as λ increases initially, but λ > 1.5 increases MBER because excessively large branch metrics impair error correction.
  • Simplified BM: Fig. 4 compares SDS sensitivity to λ and SDS versus SDSV under varying noise levels.

C. Stack Size

SDS performance improves with sufficient stack and window settings, then saturates, while benchmark ML-based methods face rapidly increasing complexity as memory length grows.

  • Stack Size: When D ≥ 12M^2 and I ≥ 11, further increases provide only marginal SDS performance gains.
  • Stack Size: Increasing L′_0 improves SDS MBER until performance saturates because distant symbols contribute negligibly to the current sample.
  • Stack Size: For SDS, Q stabilizes when L′_0 ≥ 3, indicating computational complexity is insensitive to further window expansion.
  • Demodulation Performance: HPSP and TVD improve MBER with larger L′_0, but their computational complexity grows exponentially and limits large-window applicability.
  • Demodulation Performance: SDS achieves MBER comparable to HPSP and TVD while maintaining a significant computational-efficiency advantage.
  • Demodulation Performance: For 8PSK, SDS runtime is about 0.05% of HPSP runtime, and its advantage becomes more pronounced as modulation order increases.

E. Decoding Performance

The proposed algorithms maintain low computational complexity across modulation, amplitude-ratio, and noise conditions while preserving strong separation and decoding performance. SO-SDS substantially reduces runtime relative to SO-PSP with only limited LLR accuracy degradation.

  • SO-SDS reduces runtime to 39.7% and 7.8% of SO-PSP for BPSK and QPSK, respectively, with only minor LLR accuracy loss.Its LLR computation retains path-metric information from decided symbols, unlike SO-TVD’s current-branch-only estimation.
  • Increasing hr degrades z0 BER but can initially improve z1 BER before error propagation causes degradation when hr ≥ 0.7.When hr ≥ 0.5, ISIC suffers severe self-interference and markedly higher BER than other methods.
  • SDS with L′0 = 3 approaches TVD performance and exceeds HPSP with L′0 = 1, while the runtime advantage persists across hr values.At L′0 = 1, TVD outperforms HPSP and SDS in BER.
  • SO-SDS achieves lower BER than SO-TVD at Es/N0 = 4 dB while remaining slightly inferior to SO-PSP.At Es/N0 = 4.5 dB, SO-SDS slightly outperforms SO-PSP at 4 dB.
  • Under 8PSK, SDS achieves BER comparable to HPSP and TVD while requiring only 0.05% and 1.62% of their runtimes, respectively.The framework avoids exponential complexity growth as L′ increases.

APPENDIX A PROOF OF PROPOSITION 1

The appendix derives Proposition 1 by expanding indexed expressions and exploiting independence and zero-mean properties of i.i.d. transmitted symbols. Several analogous cases are omitted after establishing the representative derivation.

  • The derivation focuses on Γk because the relevant cases differ only in the range of l, with other cases obtained analogously.Adjusting the range of l yields equation (17).
  • The proof expands equation (30) using dummy indices I′ and l′ to avoid duplication with i and l.The resulting expression is then analyzed through the definitions of ϕi,I′.
  • For i.i.d. transmitted symbols from a zero-mean, unit-energy constellation, terms with I′ ≠ i or l′ ≠ l vanish in expectation.Consequently, the second and third terms in equation (31) disappear in expectation.
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