Variational Soft Symbol Detection (VSSD)

• Variational Soft Symbol Detection (VSSD) is a variational inference framework that replaces exact APP detection with optimization of a tractable surrogate posterior.
• It applies to multiple channel models—including MAI, ISI, MIMO, and underwater channels—using Gaussian, discrete, and DDF-aided variants for soft symbol and LLR estimation.
• The method integrates turbo processing and EM extensions to efficiently manage complexity and convergence, yielding improved performance in iterative decoding systems.

Variational Soft Symbol Detection (VSSD) denotes a class of variational-inference-based soft detectors for linear Gaussian communication channels in which exact a posteriori probability (APP) detection is replaced by optimization over a tractable approximate posterior. In the foundational variational framework for soft-in-soft-out detection in interference channels, the method is developed for multiple-access interference (MAI), inter-symbol interference (ISI), and multiple-input multiple-output (MIMO) channels, with a concrete emphasis on turbo multiuser detection (0809.0032). Later work uses the same variational logic under the explicit VSSD label for affine frequency division multiplexing (AFDM) and for delay-scale-spread underwater acoustic channels, where it produces soft symbol probabilities and bit-level log-likelihood ratios (LLRs) with low-complexity iterative updates (Zheng et al., 5 Jul 2025, Halder et al., 29 Aug 2025).

1. Channel models and detection objective

The canonical VSSD setting is a linear observation model in which an observed vector is generated by a channel matrix, a symbol vector, and additive Gaussian noise. In the 2008 formulation, the received real vector satisfies

$r = S A b + n,$

where $S$ is an $N\times K$ known channel or spreading matrix, $A=\mathrm{diag}(A_1,\dots,A_K)$ is a diagonal amplitude matrix, $b\in\mathcal X^K$ is the transmitted symbol vector, and $n\sim \mathcal N(0,\sigma_n^2 I)$ (0809.0032). After matched filtering or Cholesky-whitening, the same model is written in compact form as

$$y = H b + w,\qquad w\sim \mathcal N(0,\sigma_n^2 I).$$

The stated channel interpretations include chip-space signatures in CDMA, convolutional/ISI mixing, and MIMO mixing matrices (0809.0032).

In coded systems, the symbol prior $p(b)$ is supplied by decoder-side information, typically factorized across bit levels via LLRs. The detector’s target is the set of soft marginals $p(b_k\mid r)$, but exact APP evaluation is exponentially complex in the number of users or symbols $K$ (0809.0032). VSSD addresses precisely this complexity bottleneck: it preserves soft-input/soft-output operation while avoiding exhaustive marginalization.

Later VSSD formulations retain the same structural objective in complex-valued models. For AFDM, the effective demodulated model is

$S$0

with $S$1 drawn from a $S$2-ary QAM alphabet under a factorized uniform prior (Zheng et al., 5 Jul 2025). In delay-scale-spread channels, the receiver operates on

$S$3

with $S$4 and i.i.d. discrete-uniform symbol priors (Halder et al., 29 Aug 2025). Across these settings, the common problem is approximate marginal inference for discrete symbols in a dense linear Gaussian observation model.

2. Variational free energy, KL minimization, and ELBO formulations

The foundational variational construction introduces a tractable approximate posterior $S$5 and selects it by minimizing the Kullback–Leibler divergence to the true posterior,

$S$6

Equivalently, one minimizes the variational free energy

$S$7

or, using $S$8,

$S$9

Up to an additive constant, this free energy is the KL divergence to the exact posterior, so restricting $N\times K$0 to a simpler family converts APP detection into a tractable optimization problem (0809.0032).

The later AFDM and delay-scale-spread formulations write the same idea in evidence-lower-bound language. In AFDM, the detector minimizes $N\times K$1 or, equivalently, maximizes

$N\times K$2

under a mean-field factorization $N\times K$3 (Zheng et al., 5 Jul 2025). The DS-spread formulation likewise adopts a fully factorized approximate posterior

$N\times K$4

and maximizes

$N\times K$5

Because the symbol prior is uniform in that setting, only the likelihood term affects the per-symbol coordinate update (Halder et al., 29 Aug 2025).

This common variational structure is central to the identity of VSSD. Exact Bayesian detection is replaced not by an ad hoc linearization alone, but by optimization of a surrogate objective—variational free energy or ELBO—whose stationary points define approximate soft posteriors. The 2008 framework explicitly states that this viewpoint provides unified and rigorous justifications for detectors proposed on radically different grounds and facilitates convenient joint detection and decoding when error-control codes are incorporated (0809.0032).

3. Principal detector families and update mechanisms

Within the original framework, two principal approximating families are emphasized: Gaussian VSSD and discrete mean-field VSSD. A DDF-aided discrete variant modifies initialization in the first iteration. Their essential structure is summarized below (0809.0032).

Variant	Approximate posterior	Stated per-iteration cost
Gaussian VSSD	$N\times K$6	One $N\times K$7 solve, $N\times K$8, or iterative $N\times K$9 with structure
Discrete VSSD	$A=\mathrm{diag}(A_1,\dots,A_K)$0 with binary means $A=\mathrm{diag}(A_1,\dots,A_K)$1	$A=\mathrm{diag}(A_1,\dots,A_K)$2 per flooding sweep
DDF-aided discrete VSSD	First iteration uses triangularized channel filter $A=\mathrm{diag}(A_1,\dots,A_K)$3	DDF initialization adds $A=\mathrm{diag}(A_1,\dots,A_K)$4

For Gaussian VSSD, the prior and likelihood are approximated as Gaussian:

$A=\mathrm{diag}(A_1,\dots,A_K)$5

Minimizing the free energy yields closed-form updates

$A=\mathrm{diag}(A_1,\dots,A_K)$6

and

$A=\mathrm{diag}(A_1,\dots,A_K)$7

Here $A=\mathrm{diag}(A_1,\dots,A_K)$8 is the prior mean vector and $A=\mathrm{diag}(A_1,\dots,A_K)$9 comes from the decoder. The detector then forms an output LLR from the Gaussian marginal $b\in\mathcal X^K$0 via

$b\in\mathcal X^K$1

The framework states that this implementation requires one $b\in\mathcal X^K$2 linear solve per iteration, for example via Cholesky (0809.0032).

For discrete VSSD, the approximate posterior is factorized into independent binary marginals

$b\in\mathcal X^K$3

Substituting this family into the free energy and minimizing by coordinate descent yields mean-field updates. The stated recursion updates the posterior mean through

$b\in\mathcal X^K$4

with $b\in\mathcal X^K$5 formed from the prior LLR and interference-cancellation terms involving the channel column $b\in\mathcal X^K$6 and the off-diagonal correlation matrix $b\in\mathcal X^K$7 (0809.0032). Under a flooding schedule, all $b\in\mathcal X^K$8 users are updated in parallel from the previous sweep; under sequential scheduling, the updates are applied one at a time in an SIC-style manner with immediate interference cancellation.

The DDF-aided discrete variant modifies only the first pass: it replaces $b\in\mathcal X^K$9 by a triangularized channel filter $n\sim \mathcal N(0,\sigma_n^2 I)$0 to mimic DDF cancellation and obtain a better initialization, after which ordinary flooding or sequential discrete VSSD resumes (0809.0032). The same source states that this variational framework unifies known SISO detectors—including decorrelator, MMSE, SIC, Wang–Poor, and DDF—as special cases of variational free-energy minimization.

4. Turbo processing, parameter estimation, and extension beyond BPSK

VSSD is explicitly formulated as a soft-in/soft-out block and is therefore naturally embedded in a turbo loop. In the multiuser setting, each outer iteration consists of a detection step and a decoding step. The detector uses current prior means and variances from the decoder to run one or more VSSD sweeps and outputs extrinsic LLRs $n\sim \mathcal N(0,\sigma_n^2 I)$1. These are passed to APP decoders such as BCJR, turbo, or LDPC decoders, which return updated extrinsic LLRs $n\sim \mathcal N(0,\sigma_n^2 I)$2. The next VSSD pass then uses

$n\sim \mathcal N(0,\sigma_n^2 I)$3

with prior means updated, for example, by $n\sim \mathcal N(0,\sigma_n^2 I)$4 (0809.0032).

The framework identifies three scheduling regimes for turbo VSSD. In flooding, all users are detected in parallel and then decoded in parallel. In sequential scheduling, users are detected and decoded one by one in SIC fashion. In a hybrid schedule, the detector runs sequentially across users while storing extrinsics, and the decoders then operate in parallel (0809.0032). These are not separate inference principles; they are operational arrangements for exchanging soft information between variational detection and channel decoding.

A second extension concerns uncertain channel parameters. When amplitudes $n\sim \mathcal N(0,\sigma_n^2 I)$5 or noise variance $n\sim \mathcal N(0,\sigma_n^2 I)$6 are not accurately known, the detector wraps a parameter update around variational data detection through variational EM. The joint objective is

$n\sim \mathcal N(0,\sigma_n^2 I)$7

with $n\sim \mathcal N(0,\sigma_n^2 I)$8. The E-step fixes $n\sim \mathcal N(0,\sigma_n^2 I)$9 and minimizes the free energy with respect to $$y = H b + w,\qquad w\sim \mathcal N(0,\sigma_n^2 I).$$0 by running VSSD. The M-step fixes $$y = H b + w,\qquad w\sim \mathcal N(0,\sigma_n^2 I).$$1 and minimizes with respect to the parameters, including the stated update

$$y = H b + w,\qquad w\sim \mathcal N(0,\sigma_n^2 I).$$2

The source further notes that closed-form updates exist when $$y = H b + w,\qquad w\sim \mathcal N(0,\sigma_n^2 I).$$3 is Gaussian and $$y = H b + w,\qquad w\sim \mathcal N(0,\sigma_n^2 I).$$4 is conjugate Gaussian/Gamma (0809.0032).

The same paper also extends BPSK-based SISO detection schemes to arbitrary square QAM constellations. Each symbol $$y = H b + w,\qquad w\sim \mathcal N(0,\sigma_n^2 I).$$5 is represented in $$y = H b + w,\qquad w\sim \mathcal N(0,\sigma_n^2 I).$$6 bits with Gray mapping; the variational approximation is then factorized across bit levels,

$$y = H b + w,\qquad w\sim \mathcal N(0,\sigma_n^2 I).$$7

By writing the QAM symbol as a weighted sum of its bit indicators and re-deriving mean-field updates on bit LLRs, the method yields “bit-level equalization and soft detection” (BLESD), with stated complexity $$y = H b + w,\qquad w\sim \mathcal N(0,\sigma_n^2 I).$$8 per iteration (0809.0032). This directly addresses a common misconception that the original variational framework is restricted to BPSK or real-valued detection; the source explicitly states a rigorous extension to arbitrary square QAM.

5. Complexity, convergence properties, and performance characterizations

The complexity statements attached to VSSD are variant-specific rather than uniform. In the original framework, Gaussian VSSD requires one $$y = H b + w,\qquad w\sim \mathcal N(0,\sigma_n^2 I).$$9 matrix solve per iteration, giving $p(b)$0 complexity, or $p(b)$1 with conjugate-gradient and structured $p(b)$2. Discrete VSSD requires $p(b)$3 per full flooding sweep, since each user update is $p(b)$4. DDF initialization adds $p(b)$5 for triangular-filter multiplications, and the EM extension adds $p(b)$6 per outer iteration for parameter updates (0809.0032).

Convergence guarantees are likewise differentiated. The 2008 formulation states that the free energy is non-increasing under each coordinate or variational update. For Gaussian VSSD, the free energy is convex in $p(b)$7, so there is a unique global minimizer. For discrete mean-field VSSD, the free energy $p(b)$8 is nonconvex; the source therefore describes convergence to a good local minimum as the typical outcome, and notes that SIC-style scheduling or DDF-aided initialization plus multiple sweeps usually improves the result. It further states that linear convergence can be shown under mild conditions (0809.0032). This is an important clarification: VSSD is a controlled approximation to APP detection, not an exact replacement.

The original performance examples emphasize turbo multiuser detection. In a 4-user CDMA system with cross-correlation $p(b)$9, turbo VSSD with Gaussian flooding is reported to reach the near-single-user bound in 4–5 iterations and to outperform hybrid Wang–Poor and decorrelator-based SISO methods. Discrete VSSD with DDF-aided initialization is reported to match Gaussian VSSD up to moderate loads while using lower complexity. Under imperfect knowledge of channel amplitude or noise variance, the EM-extended versions are stated to recover most of the lost SNR within 3–6 EM iterations (0809.0032).

The later AFDM formulation expresses analogous convergence in ELBO language: its closed-form coordinate updates guarantee monotonic increase of the ELBO and convergence to a local optimum. In the stated QPSK, $p(b_k\mid r)$0 experiments under $p(b_k\mid r)$1 and $p(b_k\mid r)$2 multipaths, five VB iterations produce a waterfall at lower SNR and achieve more than $p(b_k\mid r)$3 gain over MPA, while the residual

$p(b_k\mid r)$4

typically falls below $p(b_k\mid r)$5 within 3–4 iterations even for $p(b_k\mid r)$6 (Zheng et al., 5 Jul 2025). These results reinforce the original point that variational optimization trades exact posterior computation for a monotone surrogate objective with practical iteration counts.

6. Later VSSD formulations and related variational detection lines

The 2025 AFDM detector presents VSSD in a fully discrete complex form. Under the mean-field law

$p(b_k\mid r)$7

each factor is a categorical distribution over the QAM alphabet with probabilities $p(b_k\mid r)$8, posterior mean $p(b_k\mid r)$9, and variance $K$0 (Zheng et al., 5 Jul 2025). The coordinate ascent updates define, for each symbol, a residual

$K$1

an effective variance

$K$2

and a scalar equivalent model

$K$3

followed by the soft-max update

$K$4

Bit-wise LLRs are then formed by summing posterior symbol probabilities over the subsets of constellation points whose labeled bit equals 1 or 0. The stated complexity is $K$5, contrasted in the paper with $K$6 for MPA (Zheng et al., 5 Jul 2025). In the reported experiments, VSSD outperforms ZF, LMMSE, and MPA across all SNRs, converges faster than MPA, and degrades only mildly as the number of multipath components increases.

The delay-scale-spread channel work of the same year places VSSD inside a broader receiver chain that also includes variational off-grid channel estimation and iterative channel-estimation/data-detection (ICED). Its detector uses the sufficient statistics

$K$7

forms for each symbol the interference-canceled residual

$K$8

and updates the factor $K$9 over each constellation point $S$00 through

$S$01

followed by soft-max normalization (Halder et al., 29 Aug 2025). The per-iteration cost is stated as approximately $S$02 after precomputing $S$03 and $S$04, in contrast with an $S$05 MMSE matrix inversion. Performance statements are waveform-dependent: replacing a 1-tap equalizer with VSSD yields a 4–5 dB gain at BER $S$06 for all waveforms, collapses the performance gap between OTFS, OCDM, and ODSS, and leaves all three outperforming OFDM by approximately 11 dB at BER $S$07. With coded transmission under estimated CSI, VSSD LLRs provide a further approximately 5.4 dB gain at BER $S$08, after which all four waveforms become essentially indistinguishable in BER; ICED adds an approximately 3 dB improvement in channel NMSE and an approximately 3 dB BER gain relative to pilot-only estimation (Halder et al., 29 Aug 2025). A plausible implication is that, in this regime, receiver inference quality can dominate nominal waveform differences.

A related but distinct line of work in MIMO detection develops an inverse-free variational Bayesian framework and unfolds it into a deep architecture called VBINet. That model approximates the posterior by $S$09, maximizes a relaxed ELBO based on a quadratic bound with diagonal $S$10, and alternates closed-form updates for the symbol distribution and the noise precision before unrolling the iteration into trainable layers (Wan et al., 2021). The resulting network outputs posterior means and variances that can be converted into bit-level LLRs, and it is reported to remain robust under noise-variance uncertainty because the $S$11 update is built into the model (Wan et al., 2021). This suggests a methodological extension of the VSSD viewpoint: variational soft detection need not terminate at hand-designed iterative receivers, but can also serve as the template for model-driven unrolled networks.

Across these strands, VSSD is best understood not as a single detector formula but as a variational design paradigm for soft symbol inference in linear channels. The defining elements are the replacement of exact APP detection by KL/ELBO optimization, the use of tractable factorized or Gaussian approximate posteriors, monotone coordinate or variational updates, and the production of soft outputs suitable for iterative decoding (0809.0032, Zheng et al., 5 Jul 2025, Halder et al., 29 Aug 2025).