Joint Noise Sensing & Decoding

• Joint Noise Sensing and Decoding is a communication paradigm where latent noise and transmitted signals are jointly inferred, resulting in improved decoding accuracy.
• It employs factor-graph message passing, Bayesian inference, and structured approximations to simultaneously estimate noise characteristics and decode messages.
• The approach enhances performance in domains such as phase-noise channels, impulsive-noise OFDM, and quantum error correction while balancing computational complexity and error-rate gains.

Joint noise sensing and decoding denotes receiver and end-to-end communication architectures in which latent disturbance variables are inferred together with transmitted information, rather than estimated in a front-end stage and then treated as fixed during decoding. In the literature, the latent quantity may be a Wiener phase trajectory, impulsive-noise amplitudes and hidden states, a correlated noise realization on another channel, a quasiparticle-density field in a quantum device, a channel/state sequence, or a packet-presence hypothesis; the common feature is that code constraints, modulation structure, pilots, feedback, and dynamical priors contribute jointly to both state inference and message recovery (Shayovitz et al., 2012, Nassar et al., 2013, Nayak et al., 18 Mar 2026, Li et al., 19 Jan 2025). Some later systems extend the idea to implicit noise-aware reconstruction, in which decoding and denoising are unified without a separate explicit noise estimator (Liang et al., 11 May 2025, Chen et al., 2024).

1. Scope and historical development

A coding-theoretic precursor appears in the analysis of Raptor codes over the BIAWGNC, where joint decoding of the LT and precode components is treated as a more efficient alternative to tandem decoding, and tandem decoding is identified as a subcase of the general model [0701103]. In that early setting, the emphasis is joint decoding of coupled code components under channel noise, rather than explicit latent-noise estimation, but it establishes an important structural theme: decoding performance can improve when constituent inference modules exchange soft information instead of operating in a strict cascade [0701103].

Subsequent work broadened the topic from component-wise code coupling to explicit latent-state inference. In LDPC-coded satellite links with strong phase noise, the receiver jointly infers coded symbols and the full phase-noise trajectory on a factor graph, motivated by the failure of simple PLL-based tracking under sparse pilots and broad, multimodal phase posteriors (Shayovitz et al., 2012). OFDM receivers in impulsive noise then pushed the idea further by jointly estimating channel taps, impulsive noise samples and states, transmitted symbols, coded bits, and information bits in a single Bayesian model solved approximately by loopy belief propagation, GAMP, forward-backward inference, and SISO decoding (Nassar et al., 2013). Closely related phase-noise receivers for coded ISI channels and MIMO systems use BP/MF/EP hybrids, smoother-detector architectures, multivariate Tikhonov approximations, or variational Bayes so that oscillator phase processes and coded data are inferred together rather than sequentially (Wang et al., 2016, Krishnan et al., 2013).

More recent work makes the scope broader still. In “Noise Recycling,” decoding one orthogonal channel is used to estimate a realized noise sample path that is then recycled to assist decoding on another channel with correlated noise, yielding a modular alternative to monolithic joint decoders (Cohen et al., 2020). In asynchronous short-packet reception over the BI-AWGN channel, joint detection and decoding uses payload structure for packet detection and can approach synchronous performance rapidly (Obermüller et al., 2024). In quantum error correction, syndrome trajectories are used not only to infer Pauli faults but also to estimate an underlying quasiparticle-density field that governs correlated physical error rates (Nayak et al., 18 Mar 2026). In information-theoretic JSMC, receivers jointly decode messages and reconstruct state sequences through simultaneous decoding of messages and compressed state descriptions (Li et al., 19 Jan 2025).

2. Probabilistic and information-theoretic foundations

Most explicit formulations introduce a latent variable that governs the observation law. In strong phase-noise channels, a standard model is

$r_k \approx c_k e^{j\theta_k} + n_k,\qquad \theta_k = \theta_{k-1} + \Delta_k,\qquad \Delta_k \sim \textsl{N}(0,\sigma_\Delta^2),$

so decoding requires inference over both coded symbols and the continuous Wiener phase trajectory (Shayovitz et al., 2012). In impulsive-noise OFDM, each tone obeys

$Y_k[q] = H_k[q]S_k[q] + N_k[q],$

while the time-domain impulsive component is modeled by a Bernoulli-Gaussian-mixture prior with optional Markov state dynamics; the receiver therefore infers channel coefficients, impulsive amplitudes, hidden impulse states, symbols, coded bits, and information bits together (Nassar et al., 2013). In radiation-induced quantum noise, the latent field evolves as

$X_t = A_t X_{t-1} + \gamma C_t,$

and syndrome data are used to infer both error mechanisms and the spatiotemporal quasiparticle field $X_{1:T}$ that drives the error probabilities (Nayak et al., 18 Mar 2026).

From an information-theoretic viewpoint, the same idea appears as a state-estimation constraint coupled to message decoding. For point-to-point JSMC, the capacity-distortion function is

$$C(D)=\max_{(X,V,h)\in\mathcal P_D} I(X;Y)-I(V;S_T\mid X,Y),$$

which makes the tradeoff explicit: the achievable communication rate is reduced by the rate needed to describe the state information relevant to reconstruction at distortion $D$ (Li et al., 19 Jan 2025). A related but distinct formulation appears in joint communication and channel discrimination, where the same input distribution $P_X$ governs both the communication rate and the sensing error exponents: $$R \le I(P_X,P_{Z|X}),\qquad E_0 \le D(W_s\|W\mid P_X),\qquad E_1 \le D(W_s\|V\mid P_X),$$ with the sensor discriminating between two possible sensing channels while the receiver decodes the message (Wu et al., 2022).

A recurring consequence is that the decoder is no longer merely a terminal stage consuming fixed likelihoods. It becomes part of the sensing loop. In explicit-state models, decoder priors sharpen the latent-state posterior, and the refined posterior in turn changes symbol likelihoods or channel priors (Shayovitz et al., 2012, Nassar et al., 2013, Wang et al., 2016, Krishnan et al., 2013). In simultaneous-decoding JSMC, compressed state descriptions can improve both state estimation and message recovery because they are decoded jointly with the messages rather than after them (Li et al., 19 Jan 2025).

3. Principal algorithmic architectures

One major family is factor-graph message passing on coupled code and state graphs. For strong phase noise in LDPC-coded systems, forward and backward SPA recursions over the phase chain are combined with decoder-to-symbol priors and symbol-to-decoder extrinsic information; because the phase posterior becomes multimodal under sparse pilots, the messages are represented as mixtures of Tikhonov components rather than a single mode (Shayovitz et al., 2012). The central complexity problem is that the mixture order grows as $NM$, so the paper formulates mixture reduction as KL minimization and uses a sequential clustering procedure with CMVM and a KL threshold $\mu$, while allowing adaptive mixture size rather than a fixed reduced order (Shayovitz et al., 2012).

A second family uses structured approximations for dense linear mixing and nonlinear factors. In impulsive-noise OFDM, channel-GAMP and noise-GAMP operate on the DFT-induced linear subgraphs, forward-backward tracks Markov impulse states, and a SISO decoder feeds soft priors back into both channel and noise estimation (Nassar et al., 2013). In coded ISI with phase noise, BP is used on the Gaussian phase-noise chain, MF handles the nonlinear likelihood $Y_k[q] = H_k[q]S_k[q] + N_k[q],$0, and EP Gaussianizes symbol messages in the equalizer; the key approximation is a second-order Taylor expansion of the MF message rather than first-order linearization of the observation model (Wang et al., 2016). For MIMO phase noise, three distinct approximations are developed: an SPA-MAP receiver based on multivariate Tikhonov canonical distributions, a Gauss-MAP smoother-detector framework that incorporates phase posterior covariance into detection, and a VB-MAP factorization $Y_k[q] = H_k[q]S_k[q] + N_k[q],$1 (Krishnan et al., 2013).

A third family organizes inference around the disturbance itself. “Noise Recycling” decodes a lead orthogonal channel, forms $Y_k[q] = H_k[q]S_k[q] + N_k[q],$2, and then reduces the effective noise on channel $Y_k[q] = H_k[q]S_k[q] + N_k[q],$3 through $Y_k[q] = H_k[q]S_k[q] + N_k[q],$4, with MDST-based static order selection and confidence-based dynamic selection (Cohen et al., 2020). “Soft-input, soft-output joint detection and GRAND” goes further by treating decoding as a search over likely additive noise effects; guessed noise sequences generate candidate corrected words, Euclidean distances over those candidates produce approximate bit LLRs, and those LLRs are recycled to reorder noise guesses in subsequent turbo iterations (Sarieddeen et al., 2022). In slotted asynchronous BI-AWGN reception, DAD uses the decoder’s selected codeword in the packet-detection test, while HyPED combines preamble correlation with a payload log-likelihood based on $Y_k[q] = H_k[q]S_k[q] + N_k[q],$5 terms (Obermüller et al., 2024).

A fourth family replaces sequential recovery by simultaneous decoding. In point-to-point, broadcast, and multiple-access JSMC, messages and compressed state descriptions are decoded jointly by backward simultaneous decoding without Wyner–Ziv random binning (Li et al., 19 Jan 2025). This is especially consequential in degraded BC and MAC models, where the recovered state descriptions can assist message decoding directly, enlarging the achievable rate-distortion region relative to sequential decoding (Li et al., 19 Jan 2025).

4. Representative application domains

Carrier phase noise is the most thoroughly developed classical application. In satellite communication with $Y_k[q] = H_k[q]S_k[q] + N_k[q],$6-PSK and LDPC coding, the latent phase is modeled as a Wiener process, and multiple Tikhonov hypotheses are preserved until decoder feedback collapses the ambiguity (Shayovitz et al., 2012). In coded ISI channels, phase-noise estimation, equalization, and decoding are tightly coupled through the nonlinear observation factor and Gaussianized message passing (Wang et al., 2016). In MIMO systems with one noisy oscillator per RF chain, the effective phase on each transmit-receive pair is $Y_k[q] = H_k[q]S_k[q] + N_k[q],$7, so both uncoded MAP detection and coded soft-output detection require joint inference over symbols and a high-dimensional phase state (Krishnan et al., 2013).

Impulsive-noise OFDM is a second canonical domain. Here the sensing target is not a smooth phase trajectory but a sparse or bursty time-domain impulse process with hidden states. The receiver exploits null tones, pilots, data-tone soft information, constellation constraints, and code constraints together, rather than treating impulse suppression as a preprocessing step (Nassar et al., 2013). The state posteriors $Y_k[q] = H_k[q]S_k[q] + N_k[q],$8 provide an explicit noise-sensing output, while posterior means and variances of $Y_k[q] = H_k[q]S_k[q] + N_k[q],$9 and $X_t = A_t X_{t-1} + \gamma C_t,$0 feed back into channel estimation and demapping (Nassar et al., 2013).

Short-packet and noise-centric decoding provide a different perspective. In asynchronous BI-AWGN reception, the sensing problem is packet detection in noise, and DAD/HyPED incorporate payload structure into the detection rule instead of depending only on a preamble (Obermüller et al., 2024). In GRAND-based receivers, the unknown disturbance is represented directly as a binary or symbol-level noise effect sequence; this suggests a distinct design philosophy in which the entropy of noise, rather than the entropy of messages, determines the practical search space (Sarieddeen et al., 2022).

Quantum stabilizer decoding under radiation-induced correlated noise is a recent extension of the same principle. Standard syndrome decoding uses a fixed prior on fault mechanisms; the joint formulation instead models the quasiparticle density as a latent variable that changes local Pauli error rates, and alternates between BP-based decoding on the detector error model and latent-field estimation by a variational M-step or EKF-like pseudo-measurements (Nayak et al., 18 Mar 2026). The inferred field is also physically interpretable, providing diagnostic information for shielding, chip design, and device characterization (Nayak et al., 18 Mar 2026).

A final class of systems blurs decoding and denoising. In diffusion-based semantic communication, the transmitter intentionally creates a noisy diffusion latent, the channel contributes additional AWGN or Rayleigh fading, and the receiver performs reverse diffusion directly on the channel-corrupted latent; the method is explicitly described as implicit noise-aware generative decoding rather than explicit noise sensing (Liang et al., 11 May 2025). Transformer-based learned image compression similarly enables joint decoding and denoising from a single bitstream by decoder-side latent refinement and prompt-conditioned decoding, again without an explicit noise-state estimator (Chen et al., 2024).

5. Performance, complexity, and empirical trade-offs

Reported gains are strongly domain-specific, and the metrics vary across PER, BER, NMSE, logical error probability, and PSNR. Even so, the empirical record is consistent with the basic thesis that coupling noise/state inference to decoding is beneficial when the latent disturbance is structured and sufficiently informative.

Domain	Representative findings	Citation
LDPC with strong phase noise	At $X_t = A_t X_{t-1} + \gamma C_t,$1 dB, where PER is around $X_t = A_t X_{t-1} + \gamma C_t,$2, the average mixture order starts at about $X_t = A_t X_{t-1} + \gamma C_t,$3 and never exceeds $X_t = A_t X_{t-1} + \gamma C_t,$4; DP requires $X_t = A_t X_{t-1} + \gamma C_t,$5 operations and $X_t = A_t X_{t-1} + \gamma C_t,$6 LUT accesses per code symbol per iteration, versus $X_t = A_t X_{t-1} + \gamma C_t,$7 and $X_t = A_t X_{t-1} + \gamma C_t,$8 in iteration 1 and $X_t = A_t X_{t-1} + \gamma C_t,$9 and $X_{1:T}$0 in iteration 2 for the proposed method	(Shayovitz et al., 2012)
OFDM in impulsive noise	JCIS outperforms conventional OFDM by about $X_{1:T}$1 dB and comes within $X_{1:T}$2 dB of the matched-filter bound; for coded 16-QAM, JCISB gains about $X_{1:T}$3 dB over JCIS	(Nassar et al., 2013)
Correlated-noise recycling	Dynamic Noise Recycling gives more than $X_{1:T}$4 dB for $X_{1:T}$5, about $X_{1:T}$6 dB for $X_{1:T}$7 at BLER $X_{1:T}$8, and about $X_{1:T}$9 dB for long LDPC at BLER $$C(D)=\max_{(X,V,h)\in\mathcal P_D} I(X;Y)-I(V;S_T\mid X,Y),$$0	(Cohen et al., 2020)
BI-AWGN joint detection and decoding	At $$C(D)=\max_{(X,V,h)\in\mathcal P_D} I(X;Y)-I(V;S_T\mid X,Y),$$1, rate is about $$C(D)=\max_{(X,V,h)\in\mathcal P_D} I(X;Y)-I(V;S_T\mid X,Y),$$2 for the genie-aided achievability and about $$C(D)=\max_{(X,V,h)\in\mathcal P_D} I(X;Y)-I(V;S_T\mid X,Y),$$3 for DAD; at $$C(D)=\max_{(X,V,h)\in\mathcal P_D} I(X;Y)-I(V;S_T\mid X,Y),$$4, DAD gains about $$C(D)=\max_{(X,V,h)\in\mathcal P_D} I(X;Y)-I(V;S_T\mid X,Y),$$5 dB over HyPED and HyPED about $$C(D)=\max_{(X,V,h)\in\mathcal P_D} I(X;Y)-I(V;S_T\mid X,Y),$$6 dB over preamble detection	(Obermüller et al., 2024)
Radiation-correlated quantum noise	For the distance-7 surface code at $$C(D)=\max_{(X,V,h)\in\mathcal P_D} I(X;Y)-I(V;S_T\mid X,Y),$$7, $$C(D)=\max_{(X,V,h)\in\mathcal P_D} I(X;Y)-I(V;S_T\mid X,Y),$$8 drops from $$C(D)=\max_{(X,V,h)\in\mathcal P_D} I(X;Y)-I(V;S_T\mid X,Y),$$9 with a uniform prior to $D$0 with offline joint estimation; for $D$1 BB qLDPC, $D$2 drops from $D$3 to $D$4	(Nayak et al., 18 Mar 2026)
Diffusion-based semantic communication	JSCNA-AD achieves PSNR $D$5 dB, up to $D$6 dB higher PSNR under harsh channel conditions, and about $D$7 inference time reduction; at Rayleigh channel, SNR $D$8 dB, ImageNet-256, it reports $D$9 dB at $P_X$0 ms	(Liang et al., 11 May 2025)
Joint decoding and denoising from one bitstream	The proposed add-on modules require $P_X$1 compute and $P_X$2 parameters relative to Base, versus $P_X$3 parameters for Fine-tuning, while denoising quality is comparable to training a separate decoder	(Chen et al., 2024)

Complexity reductions are often as important as raw error-rate gains. The OFDM impulsive-noise receiver preserves $P_X$4 complexity through FFT-based GAMP, instead of the $P_X$5 complexity of the SBL baseline (Nassar et al., 2013). The BI-AWGN JDD literature derives a converse lower bound

$P_X$6

on the slot length required to satisfy false-alarm and missed-detection targets, isolating a detection-limited regime below which no joint detector-decoder can succeed (Obermüller et al., 2024). In GRAND-based receivers, worst-case complexity remains polynomial in the guess budget and number of iterations, while average complexity can grow linearly with the number of symbols because codeword hits often occur after few likely-noise guesses (Sarieddeen et al., 2022).

6. Conceptual boundaries, limitations, and open problems

A persistent conceptual boundary is the difference between explicit noise sensing and implicit noise-aware decoding. Phase-noise receivers, impulsive-noise OFDM, JSMC, and stabilizer-code QP tracking treat the disturbance as an explicit latent state and output state estimates or state posteriors (Shayovitz et al., 2012, Nassar et al., 2013, Li et al., 19 Jan 2025, Nayak et al., 18 Mar 2026). By contrast, JSCNA-AD does not separately estimate $P_X$7, instantaneous SNR, or a dedicated noise latent, and Transformer-based joint decoding/denoising from one bitstream does not explicitly estimate noise level or parameters; both are better described as implicit noise-aware reconstruction (Liang et al., 11 May 2025, Chen et al., 2024). “Noise Recycling” is also not a fully joint decoder in the classical sense: channels aid each other only through post-decoding noise estimates (Cohen et al., 2020).

Most explicit methods are also model-specialized. Phase-noise algorithms assume Wiener dynamics and often rely on Tikhonov or Gaussian approximations (Shayovitz et al., 2012, Wang et al., 2016, Krishnan et al., 2013). Impulsive-noise OFDM assumes sparse or mixture-distributed time-domain disturbances and, in the bursty case, Markov hidden states (Nassar et al., 2013). Quantum QP tracking isolates radiation-induced quasiparticle noise as the only physical noise source, so mixed-noise environments remain open (Nayak et al., 18 Mar 2026). Some methods require accurate correlation knowledge or highly reliable first-stage decoding, as in Noise Recycling (Cohen et al., 2020). Others assume known channel matrices and even channel inversion, as in diffusion-based semantic communication (Liang et al., 11 May 2025).

Approximation quality is a second recurring issue. Exact KL-optimal mixture reduction for Tikhonov mixtures is NP-hard, so strong phase-noise decoding uses heuristic clustering and CMVM (Shayovitz et al., 2012). BP/MF/EP and smoother-based receivers depend on Gaussianization or Taylor expansions of nonlinear likelihoods (Wang et al., 2016, Krishnan et al., 2013). GRAND-based soft outputs are max-log approximations over partial candidate sets, so some bit reliabilities are obtained through LLR saturation rather than explicit counter-hypothesis discovery (Sarieddeen et al., 2022). BI-AWGN DAD has strong achievability results, but a converse bound specifically tailored to DAD information rate remains open in the provided text (Obermüller et al., 2024).

Open directions are increasingly explicit in recent work. Diffusion-based semantic communication suggests conditioning reverse diffusion on channel noise level or allocating denoising steps using both semantic importance and inferred corruption severity, but this remains an open research direction rather than a present component (Liang et al., 11 May 2025). The stabilizer-code literature explicitly points toward joint inference of multiple latent noise sources, improved code/layout/schedule co-design for observability, and extension beyond QP bursts to crosstalk and control-induced correlations (Nayak et al., 18 Mar 2026). A plausible implication is that the field is converging on a common principle: whenever the disturbance is structured enough to admit a low-dimensional or compressible latent representation, and whenever code or model constraints provide informative feedback, simultaneous inference of state and data can outperform estimate-then-decode pipelines.