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# Belief Propagation Decoding of Tensor-Based
Modulation for Unsourced Random Access

Sweta Suresh, Maxime Guillaud

► To cite this version:

Sweta Suresh, Maxime Guillaud. Belief Propagation Decoding of Tensor-Based Modulation for Unsourced Random Access. IZS 2026 - International Zurich Seminar on Information and Communication, Feb 2026, Zurich, Switzerland. hal-05451488

HAL Id: hal-05451488

https://hal.science/hal-05451488v1

Submitted on 10 Jan 2026

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Distributed under a Creative Commons CC BY 4.0 - Attribution - International License
# Belief Propagation Decoding of Tensor-Based Modulation for Unsourced Random Access

Sweta Suresh, Maxime Guillaud

Inria, INSA Lyon, CITI Laboratory UR3720, Villeurbanne, France

Email: {sweta.suresh, maxime.guillaud}@inria.fr

**Abstract**—We target the problem of random access over a non-coherent flat-fading single-input multiple-output (SIMO) channel with short packets. Specifically, we consider an unsourced joint coding and modulation approach based on the idea of tensor-based modulation (TBM) introduced by Decurninge et al., where the component codebooks are restricted to phase-shift keying (PSK) symbols. At the receiver, we leverage the fact that PSK symbols are on the unit circle to relax the problem to the continuous domain; we propose a low-complexity message passing belief propagation (BP) multi-user decoding algorithm, to jointly estimate the information bearing symbols and the channel coefficients. The proposed method allows efficient inter-user interference cancellation without prior knowledge of the identity of the active user nor of their channels. We establish its convergence behaviour and derive its complexity and further demonstrate the effectiveness of the algorithm for multiple antenna receiver system in comparison with meta-converse lower bound.

## I. INTRODUCTION

The design of efficient waveforms, error correction codes and receivers for massive random access with short packet transmission over non-coherent fading channels is a fundamental challenge in modern wireless communications. The traditional multi-user receiver architectures based on activity detection, channel estimation, and coherent detection which are the state-of-the-art for large blocklengths typically do not scale well when the number of active users is large.

The unsourced random access paradigm [1], [2] introduced an original perspective on this problem: it only mandates that the transmitted messages be recovered, but not that the receiver associates them with a specific transmitter; hence all users use the same codebook, conversely to the classical (sourced) approaches which typically involve user-specific pilot sequences. This assumption allows for elegant theoretical performance analysis [1]; furthermore it suggests a new class of codebook design and receiver architectures and has led to several practical schemes [3]–[6] (in practice, the user identity can be encoded within the data payload, hence the unsourced assumption only affects the physical-layer design). The fundamental limits of unsourced massive random access over fading channels have been analyzed in [7].

In this work, we introduce a variation of the tensor-based modulation approach from [6], where the sequence transmitted by each user is obtained from multi-linear spreading of information-bearing symbols. While [6] relies on tensor algebraic arguments for blind multi-user separation and encodes information using generic vector codebooks, in this work we further constrain the component vectors of the tensors to be

made up of unit-norm information-bearing symbols. The key contributions of this paper are as follows.

* We leverage the constant modulus property to introduce a continuous relaxation of the problem where the information symbols are assumed by the decoder to be continuous variables lying on the unit circle, allowing the use of directional statistics to characterize the posterior distributions of the information and channel state variables.

* We derive a message-passing BP algorithm to compute an approximation of the maximum a posteriori (MAP) solution with reasonable complexity thanks to the use of parametric distributions with conjugate priors.

* We present experimental convergence and error rate results, and benchmark the method against the finite block-length meta-converse bound [8].

* We analyze the complexity of the proposed BP algorithm, and show that it is significantly lower than maximum-likelihood (ML) decoding and scalable with both modulation order and number of users.

**Notations:** Scalar, vectors and matrices are denoted as lowercase, bold lower-case, bold upper-case variables respectively, e.g. $s, v, M$. $\otimes$ denote a Kronecker product. The conjugate, transpose and Hermitian operators are written as $s^*, v^T, M^H$. $vec\{\cdot\}$ denotes the vectorization operation. $\angle$ and $|\cdot|$ respectively denote the angle and absolute value operators for arguments in $\mathbb{C}$.

## II. SYSTEM MODEL

We consider the uplink transmission from $K_a$ single-antenna transmitters to a single receiver equipped with $N$ antennas. Each user transmits $B$ information bits over $T$ channel uses. The signal transmitted by the $k^{th}$ user over the $T$ channel uses is denoted by $s^{(k)} \in \mathbb{C}^T$, where $k \in \{1, 2, \dots, K_a\}$. A block-fading quasi-static channel model is assumed, with the channel vector for user $k$ represented as $\mathbf{h}^{(k)} \in \mathbb{C}^N$. Each element of $\mathbf{h}^{(k)}$, denoted as $h_n^{(k)}$, where $n \in \{1, 2, \dots, N\}$, follows a complex Gaussian distribution with zero mean and variance $\sigma_h^2$, leading to Rayleigh distributed magnitude and uniform channel phase. The vectorized form of the received signal is expressed as

$$y = \sum_{k=1}^{K_a} s^{(k)} \otimes \mathbf{h}^{(k)} + \mathbf{w}, \quad (1)$$

where $y \in \mathbb{C}^{TN}$ and $\mathbf{w}$ is the complex additive white Gaussian noise (AWGN) noise with zero mean and $\sigma^2$ variance.
### III. MODULATION

#### A. Tensor-based Modulation

The TBM modulation scheme from [6], is adopted in this paper. In this framework, the transmitted signal from $k^{th}$ user is modeled as a tensor of order $d$, with mode dimensions $T_1, T_2, \dots, T_d$, such that $\prod_{i=1}^d T_i = T$. This tensor can be vectorized into a signal of dimension $T$. We assume $T_1, T_2, \dots, T_d \ge 2$ and constrain $\mathbf{s}^{(k)}$ to be the vectorized form of the tensor product of $d$ vectors $\mathbf{x}_1^{(k)} \cdots \mathbf{x}_d^{(k)}$, yielding a rank-1 tensor for each user. In this work, we focus on the case where the $x_{i,j}^{(k)}$ are unit-modulus, i.e. they are restricted to the unit circle; hence the proposed approach is a special case of TBM as proposed in [6]. Specifically, the signal component in the $i^{th}$ mode is represented as $\mathbf{x}_i^{(k)} = [1, x_{i,2}^{(k)}, x_{i,3}^{(k)}, \dots, x_{i,j}^{(k)}, \dots, x_{i,T_i}^{(k)}] \in \mathbb{C}^{T_i}$, where $i \in \{1, 2, \dots, d\}, j \in \{1, 2, \dots, T_i\}$; the first symbol is a reference (pilot) symbol and the remaining $T_i - 1$ symbols are information bearing. The sequence transmitted by user $k$ is expressed as

$$ \mathbf{s}^{(k)} = \mathbf{x}_1^{(k)} \otimes \mathbf{x}_2^{(k)} \otimes \cdots \otimes \mathbf{x}_d^{(k)} \in \mathbb{C}^{\prod_{i=1}^d T_i} = \mathbb{C}^T. \quad (2) $$

#### B. Constellation and Prior Distribution

The information-bearing symbols in $\mathbf{x}_i^{(k)}$ are modulated using the $M$-ary phase shift keying (M-PSK) modulation, denoted by $C_i$. The constellation points are equally spaced on the unit circle, with the corresponding discrete probability distribution function (pdf) given by

$$ f_{X_{i,j}^{(k)}}(x_{i,j}^{(k)}) = \sum_{x \in C_i} \frac{1}{M} \delta(x_{i,j}^{(k)} - x), \quad (3) $$

where $X_{i,j}^{(k)}$ denotes the $j^{th}$ information symbol of the $i^{th}$ mode of the $k^{th}$ user, and $\delta(x)$ is the Dirac delta function. Hence, each user encodes $B = \log_2 M \sum_i (T_i - 1)$ bits of information into their transmitted sequence.

### IV. RECEIVER DESIGN: BELIEF PROPAGATION MODEL FOR TBM

In line with the unsourced random access approach introduced in [1], we aim to design a receiver which decodes the messages of all $K_a$ active users, without associating each message to the a specific user. The receiver simultaneously receives signals from all $K_a$ transmitters as described by eq. (1). To jointly estimate channel and the transmit information from each user, in this paper, we propose a belief propagation (BP)-based receiver framework. This section presents, in detail, the proposed BP model tailored for TBM-based modulation with the transmit symbols constrained to be on the complex unit circle. We assume that the channel distribution is known at the receiver. We begin by defining the factor graph representation.

#### A. Factor Graph

Combining the channel model and the TBM scheme from eqs. (1) and (2) yields

$$ \mathbf{y} = \sum_{k=1}^{K_a} \mathbf{x}_1^{(k)} \otimes \mathbf{x}_2^{(k)} \otimes \cdots \otimes \mathbf{x}_d^{(k)} \otimes \mathbf{h}^{(k)} + \mathbf{w}. \quad (4) $$

<figure><img src="image_3.png" /><figcaption>Fig. 1: Factor Graph</figcaption></figure>

In this work, we tackle the problem of estimating the posterior marginals of the information-bearing symbols $x_i^{(k)}$ in order to obtain a maximum a posteriori (MAP) estimator of the transmitted information given the received signal $\mathbf{y}$, based on the observation model from eq. (4). The vector equality in (4) describes a system of $TN$ multi-linear equations, which can be represented as the factor graph in Fig. 1, where the square nodes represent the function nodes, while the circular nodes represent the variable nodes. An edge indicates the dependency among variables. The second row (from the top) contains variable nodes of the information symbols and channel coefficients $(\mathbf{x}_i^{(k)}, \mathbf{h}^{(k)})$. The first row contains function nodes representing the prior distributions of these variables. The fourth row corresponds to variable nodes representing observed values ($\mathbf{y}$) and the third row contains function nodes that represent the likelihood distributions. The factor graph structure is determined by the tensor configuration at the transmitter, which depends on $d, T_i, T, N, K_a$.

1) **Directional Statistics and von Mises Distribution:** The proposed decoder is built upon the idea of relaxing the discrete problem (based on the M-PSK constellations) into a continuous problem, where the $x_i^{(k)}$ are merely assumed to lie on the unit circle. This relaxation allows to resort to directional statistics to summarize the belief about these variables using a parametric density function defined on the unit circle; specifically, we use the von Mises distribution, which we denote by $vM(\mu, \kappa)$, defined by its probability density function (p.d.f.)

$$ f_{x|\mu,\kappa}(x) = \frac{1}{2\pi I_0(\kappa)} \exp\{\mathbb{R}\{\kappa\mu^* x\}\} \quad (5) $$

where $\mu \in \mathbb{C}$ s.t. $|\mu| = 1$ and $\kappa$ are respectively the mean and concentration parameters [9], $I_0$ denotes the modified Bessel function of $0^{th}$ order and $\mathbb{R}\{\cdot\}$ extracts the real part, and $x \in \mathbb{C}$ is on the complex unit circle. Note that this relaxed assumption can potentially reduce receiver complexity and memory usage for large PSK modulations, as only $(\mu, \kappa)$ need to be stored, instead of $M$ beliefs corresponding to the entire M-PSK constellation.

2) **Approximation of Prior Probability:** The true prior distribution of information bearing symbols in $\mathbf{x}_i^{(k)}$ follows eq. (3), i.e., a discrete uniform p.d.f. over the constellation
points. At the receiver, this assumption may be relaxed into a continuous problem by assuming a uniform prior over the unit circle, with p.d.f. $f_{\text{unif}}(x) = \frac{1}{2\pi}$. However, it is desirable to further approximate the prior using a von Mises distribution with small concentration parameter. As will be shown below, this ensures that the beliefs obtained in subsequent iterations also belong to the same parametric family (conjugate prior). Furthermore, note that the considered observation model in (4) and hence the resulting MAP estimation problem is symmetric by permutation of the user indices; this stems directly from the unsourced nature of the considered scheme, in which the objective is to estimate all transmitted messages without attributing them to one user. Assigning distinct random means $\mu_{x_{i,j}^{(k)}}$ to the priors of the symbols allows to break the symmetry between the users, which prevents the algorithm from getting stuck in a local extremum. The corresponding information symbol prior is given by

$$f_{X_{i,j}^{(k)}}(x) = \text{vM}(\mu_{x_{i,j}^{(k)}}, \kappa_0). \quad (6)$$

The Kullback-Leibler distance between $\text{vM}(\mu, \kappa)$ and $f_{\text{unif}}$ can be derived using [10] as

$$D_{\text{KL}}(f_{\text{unif}}|\text{vM}(\mu, \kappa)) = \log I_0(\kappa), \quad (7)$$

which shows that $\text{vM}(\mu, \kappa)$ accurately approximates the uniform distribution over the unit circle for small $\kappa$ (our numerical implementation uses $\kappa_0 = 10^{-4}$, which corresponds to a KL distance of $2.5 \cdot 10^{-9}$). For pilot symbols, the prior is a Dirac impulse at $x = 1$ since these are fixed. The prior assumed for the channel coefficients $\mathbf{h}^{(k)}$ corresponds to the true model (see Sec. II), given by

$$f_{H_n^{(k)}}(h) = \frac{1}{\pi \sigma_h^2} \exp \left\{ \frac{-|h|^2}{\sigma_h^2} \right\}. \quad (8)$$

3) Dependency Sets: Next, we define two sets to capture the dependencies of $m^{\text{th}}$ observation $y(m)$ on information symbols ($x_{i,j}^{(k)}$) and channel coefficients ($h_n^{(k)}$), with $m \in \{1, \dots, TN\}$:

* $\psi_x^{(k)}(m)$: the set of information symbols from user $k$ contributing to $y(m)$,

* $\psi_h^{(k)}(m)$: the set of channel coefficients from user $k$ contributing to $y(m)$.

The aggregate dependency sets across all users are defined as:

* information symbol dependency set:

$$\Psi_x(m) = \{\psi_x^{(1)}(m), \dots, \psi_x^{(K_a)}(m)\},$$

* channel dependency set:

$$\Psi_h(m) = \{\psi_h^{(1)}(m), \dots, \psi_h^{(K_a)}(m)\}.$$

4) Likelihood Function and Function Nodes Definition: The likelihood functions are now given by

$$f_{Y(m)|\Psi_x(m), \Psi_h(m)} \left( y | \Psi_x(m), \Psi_h(m) \right) = \frac{1}{\pi \sigma^2} \exp \left\{ -\frac{1}{\sigma^2} \left| y(m) - \sum_k \prod_i (\psi_x^{(k)}(m) \psi_h^{(k)}(m)) \right|^2 \right\}, \quad (9)$$

where the product is among all dependent variables at $m$. For example, the received symbol at index $TN$ can be written as

$$y(TN) = \sum_{k=1}^{K_a} \left( \prod_{i=1}^{d} x_{i,T_i}^{(k)} \right) \otimes h_N^{(k)} + w(TN), \quad (10)$$

where $w(TN)$ denotes the additive noise. In this case, $\psi_x(m)^{(k)} = \{x_{i,T_i}^{(k)}\}_{i=1}^d$ and $\psi_h^{(k)}(m) = \{h_N^{(k)}\}$. Finally, the function nodes in the factor graph are summarised as

$$f_{X,n_x} = f_{X_{i,j}^{(k)}} = \text{vM}(\mu_{x_{i,j}^{(k)}}, \kappa_0)$$

$$f_{H,n_h} = f_{H_n^{(k)}} = \mathcal{N}(0, \sigma_h^2)$$

$$f_m = f_{Y(m)|\Psi_x(m), \Psi_h(m)} = \mathcal{N} \left( \prod_i (\psi_x^{(k)}(m) \psi_h^{(k)}(m)), \sigma^2 \right), \quad (11)$$

where $n_x \in \{1, \dots, N_x\}$ (with $N_x$ denoting the total number of information symbols) and $n_h \in \{1, \dots, NK_a\}$ (total number of channel coefficients). The messages are iteratively passed along every edge of the factor graph to estimate the channel coefficients and information symbols. The detailed procedure is described in the subsequent section.

## B. Message Passing

Let $\lambda_{v \to f}^{[t]}$, $\lambda_{f \to v}^{[t]}$ denote the message passed from a variable node (v)/function node (f) to a function node (f)/variable node (v) at iteration t, respectively.

1) Iteration 0: In the 0th iteration, known as the initialization step of belief propagation, the messages passed correspond to the prior density of the respective nodes. These messages are denoted as

$$\begin{align}
\lambda_{y_m \to f_m}^{[0]} &= \mathcal{N} \left( \prod_i (\psi_x^{(k)}(m) \psi_h^{(k)}(m)), \sigma^2 \right) \notag \\
\lambda_{X_{i,j}^{(k)} \to f_m}^{[0]} &= \text{vM} \left( \mu_{x_{i,j}^{(k)}}, \kappa_0 \right) \tag{12} \\
\lambda_{H_n^{(k)} \to f_m}^{[0]} &= \mathcal{N}(0, \sigma_h^2), \notag
\end{align}$$

where $\lambda_{y_m \to f_m}^{[0]}$, $\lambda_{X_{i,j}^{(k)} \to f_m}^{[0]}$, $\lambda_{H_n^{(k)} \to f_m}^{[0]}$ denote the messages passed from the received signal $y(m)$, the transmit signal $x_{i,j}^{(k)}$, channel coefficient $h_n^{(k)}$, respectively to the associated likelihood function node $f_m$ at 0th iteration.

2) Iteration t: At iteration t, the message passed to a information symbol node $x_{i,j}^{(k)}$ from a function node $f_m$ is expressed as eq. (13) (at the bottom of the next page), where the integration is performed over all variables in the dependency sets $\Psi_x(m)$ and $\Psi_h(m)$, except for $x_{i,j}^{(k)}$ and includes the updated messages from previous iteration. Substituting the values from eqs. (6), (8) and (9) in (13) yields eq. (14), where $\hat{\sigma}_h^{2[t-1]}$, $\hat{\mu}_{h_n^{(k)}}^{[t-1]}$, $\hat{\kappa}_{x_{i,j}^{(k)}}^{[t-1]}$, $\hat{\mu}_{x_{i,j}^{(k)}}^{[t-1]}$ denote the estimates of variance and mean of channel, estimates of concentration parameter and mean of information symbols respectively at iteration $t-1$. Note that the integration is carried out over the complex unit circle for $x_{i,j}^{(k)}$ and over the entire complex plane for $h_n^{(k)}$. The integrand in eq. (14) consists of products of complex Gaussian and von Mises like functions. The solution to (14) can be approximated by a von Mises distribution using moment
matching, yielding $\lambda_{f_m \to X_{i,j}^{(k)}}^{[t]} = \text{vM}(\mu_{f_m \to X_{i,j}^{(k)}}^{[t]}, \kappa_{f_m \to X_{i,j}^{(k)}}^{[t]})$ with parameters

$$ 
\begin{aligned}
\mu_{f_m \to X_{i,j}^{(k)}}^{[t]} &= \frac{(\eta_{f_m \to X_{i,j}^{(k)}}^{[t]})^*}{|\eta_{f_m \to X_{i,j}^{(k)}}^{[t]}|} \\
\kappa_{f_m \to X_{i,j}^{(k)}}^{[t]} &= |\eta_{f_m \to X_{i,j}^{(k)}}^{[t]}|,
\end{aligned} 
\quad (15) $$

where

$$ \eta_{f_m \to X_{i,j}^{(k)}}^{[t]} = \frac{S_k^{[t-1]}(m) - \sum_{\bar{k}} \mathcal{I}_{\bar{k}}^{[t-1]}(m)}{\sigma^2 + N \hat{\sigma}_h^{2[t-1]}}, \quad (16) $$

$$ S_k^{[t-1]}(m) = 2y^*(m) \prod_{\psi_x^{(k)}(m) \setminus x_{i,j}^{(k)}} \hat{\mu}_{x_{i,j}^{(k)}}^{[t-1]} \prod_{\psi_h^{(k)}(m)} \hat{\mu}_{h_n^{(k)}}^{[t-1]}, \quad (17) $$

is the signal term, and the interference from other users is

$$ \mathcal{I}_{\bar{k}}^{[t-1]}(m) = 2 \prod_{\psi_x^{(k)}(m) \setminus x_{i,j}^{(k)}} \hat{\mu}_{x_{i,j}^{(k)}}^{[t-1]} \prod_{\psi_h^{(k)}(m)} \hat{\mu}_{h_n^{(k)}}^{[t-1]} \\ \qquad \prod_{\psi_x^{(k)}(m) \setminus x_{i,j}^{(k)}} \hat{\mu}_{x_{i,j}^{(k)}}^{*[t-1]} \prod_{\psi_h^{(k)}(m)} \hat{\mu}_{h_n^{(k)}}^{*[t-1]}, \quad (18) $$

where $\bar{k} \in \{1, \dots, K_a\} \setminus k$. Similarly, the messages to each channel coefficient node follow a Gaussian distribution, where in eq. (17) and eq. (18), the product of transmit variables is among all $\{\psi_x^{(k)}(m)\}$ and the product of channel coefficients is among $\{\psi_h^{(k)}(m) \setminus h_n^{(k)}\}$. The distribution is hence given by $(\lambda_{f_m \to h_n^{(k)}}^{[t]} = N(\mu_{f_m \to h_n^{(k)}}^{[t]}, \sigma_{f_m \to h_n^{(k)}}^{2[t]})$ where the parameters of the distribution are $\mu_{f_m \to h_n^{(k)}}^{[t]} = S_k^{[t-1]}(m) - \sum_{\bar{k}} \mathcal{I}_{\bar{k}}^{[t-1]}(m); \sigma_{f_m \to h_n^{(k)}}^{2[t]} = \sigma_h^2 + \sigma^2$.

3) Beliefs: After deriving the messages on each edge at iteration $t$, the corresponding beliefs are computed, which is the posterior distribution of the variable. The belief of the information symbol $x_{i,j}^{(k)}$ is given as

$$ b(x_{i,j}^{(k)})^{[t]} \propto f_{X_{i,j}^{(k)}} \cdot \prod_{\tilde{m}} \lambda_{f_{\tilde{m}} \to X_{i,j}^{(k)}}^{[t]}, \quad (19) $$

where $\tilde{m}$ takes all values of $m$ that depends on $x_{i,j}^{(k)}$. Since the messages passed on edges to any information symbol node follow von Mises like distributions, the belief in eq. (19) also

follows a von Mises distribution. Substituting eqs. (6) and (15), we get

$$ b(x_{i,j}^{(k)})^{[t]} = \text{vM}\left(\hat{\mu}_{x_{i,j}^{(k)}}^{[t]}, \hat{\kappa}_{x_{i,j}^{(k)}}^{[t]}\right), \quad (20) $$

where the parameters of the distribution are $\hat{\mu}_{x_{i,j}^{(k)}}^{[t]} = \eta_{x_{i,j}^{(k)}}^{*[t]}/|\eta_{x_{i,j}^{(k)}}^{[t]}|$ and $\hat{\kappa}_{x_{i,j}^{(k)}}^{[t]} = |\eta_{x_{i,j}^{(k)}}^{[t]}|$, where

$$ \eta_{x_{i,j}^{(k)}}^{[t]} = \kappa_0 e^{-j\mu_{x_{i,j}^{(k)}}^{[t]}} + \sum_{\tilde{m}} \eta_{f_{\tilde{m}} \to X_{i,j}^{(k)}}^{[t]}. \quad (21) $$

Similarly, the belief of channel coefficients $h_n^{(k)}$ at iteration $t$ is given by

$$ b(h_n^{(k)})^{[t]} \propto f_{H_n^{(k)}} \cdot \prod_{\tilde{m}} \lambda_{f_{\tilde{m}} \to H_n^{(k)}}^{[t]}. \quad (22) $$

It follows a Gaussian distribution since each message to the channel nodes are Gaussian-like functions:

$$ b(h_n^{(k)})^{[t]} = \mathcal{N}\left(\hat{\mu}_{h_n^{(k)}}^{[t]}, \hat{\sigma}_{h_n^{(k)}}^{2[t]}\right), \quad (23) $$

where

$$ 
\begin{aligned}
\hat{\mu}_{h_n^{(k)}}^{[t]} &= \sum_{\tilde{m}} \mu_{f_{\tilde{m}} \to h_n^{(k)}}^{[t]}, \\
\hat{\sigma}_{h_n^{(k)}}^{2[t]} &= \left( \sum_{\tilde{m}} \frac{1}{\sigma_{f_m \to h_n^{(k)}}^{2[t]}} + \frac{1}{\sigma_h^2} \right)^{-1}.
\end{aligned} 
\quad (24) $$

### C. Estimation of Symbols and Channels

The beliefs eqs. (20) and (23) can be used to obtain MAP estimates of the information symbols and channel coefficients. In both cases, the maximum of the distribution occurs at the mean. Therefore, the estimates at iteration $t$ are given by

$$ \hat{x}_{i,j}^{(k)[t]} = \hat{\mu}_{x_{i,j}^{(k)}}^{[t]} \quad \text{and} \quad \hat{h}_{n}^{(k)[t]} = \hat{\mu}_{h_n^{(k)}}^{[t]}. \quad (25) $$

After convergence, the information symbol MAP estimates (which can lie anywhere on the unit circle) are rounded to the nearest PSK constellation point to yield the decoded messages.

$$ \lambda_{f_m \to X_{i,j}^{(k)}}^{[t]} = \int_{\sim X_{i,j}^{(k)}} f_m \prod_{\psi_x(m) \setminus X_{i,j}^{(k)}} \lambda_{X_{i,j}^{(k)} \to f_m}^{[t-1]} \prod_{\psi_h(m)} \lambda_{H_N^{(k)} \to f_m}^{[t-1]} \quad (13) $$

$$ 
\begin{aligned}
\lambda_{f_m \to X_{i,j}^{(k)}}^{[t]} = \int_{\sim x_{i,j}^{(k)}} & \left[ \frac{1}{\pi \sigma^2} \exp \left\{ -\frac{|y(m) - \sum_k \psi_x^{(k)}(m) \psi_h^{(k)}(m)|^2}{\sigma^2} \right\} \prod_{n,k} \left\{ \left( \frac{1}{\pi \hat{\sigma}_h^{2[t-1]}} \right) \exp \left\{ \frac{-|h - \hat{\mu}_{h_n^{(k)}}^{[t-1]}|^2}{\hat{\sigma}_h^{2[t-1]}} \right\} dh_n^{(k)} \right\} \right. \\
& \left. \prod_{x_{i,j}^{(k)} \in \Psi_x} \left\{ \left( 2\pi I_0(\hat{\kappa}_{x_{i,j}^{(k)}}^{[t-1]}) \right)^{-1} \exp \{ \mathbb{R}\{\hat{\kappa}_{x_{i,j}^{(k)}}^{[t-1]} \hat{\mu}_{x_{i,j}^{(k)}}^{*[t-1]} x_{i,j}^{(k)}\} \} dx_{i,j}^{(k)} \right\} \right]
\end{aligned} 
\quad (14) $$
## D. Parameters Update and Stopping Criteria

As shown in eqs. (15), (17) and (18), the messages depend on the parameters of variables in previous iteration. Hence, after every iteration, these must be updated using eqs. (21) and (23). The proposed algorithm runs iteratively and stops upon reaching either convergence, or a maximum number of iterations $N_{max,itr}$. The proposed algorithm is summarized in Algorithm 1.

### Algorithm 1 Belief Propagation for TBM

1: Initialize prior messages using eq. (12)

2: **for** $t = 1, 2, ..., N_{max,itr}$ **do** until convergence

3: **for** $k = 1, 2, ..., K_a$ **do**

4: **for** $i = 1, 2, ..., d$ **do**

5: **for** $j = 1, 2, ..., T_i$ **do**

6: Update messages using eq. (15)

7: Compute belief $b(x_{i,j}^{(k)})^{[t]}$ using eq. (20)

8: Compute information symbol parameters

$(\hat{\mu}_{x_{i,j}^{(k)}, \hat{k}_{x_{i,j}^{(k)}}}^{[t]}, \hat{k}_{x_{i,j}^{(k)}}^{[t]})$ using eq. (21)

9: **end for**

10: **end for**

11: Compute belief $b(h_n^{(k)})^{[t]}$ using eq. (22)

12: Compute channel parameters $(\hat{\mu}_{h_n^{(k)}, \hat{\sigma}_h^{2[t]}}^{[t]}, \hat{\sigma}_h^{2[t]})$ using eq. (23)

13: **end for**

14: **end for**

## V. SIMULATIONS RESULTS

In this section, we evaluate the performance of the algorithm from Section IV for the parameters listed in Table I. In this setup, the message transmitted by each user encodes $B = 60$ bits over $T = 200$ channel uses, yielding a spectral efficiency of 0.3 bits/channel use (bpcu) per user, and reaching an aggregate 3 bpcu for $K_a = 10$. The following subsections present key performance metrics, including the convergence behavior of the BP algorithm, message error rate, and related evaluations.

<table><caption>TABLE I: Simulation Parameters</caption><tr><td>Number of users, K<sub>a</sub></td><td>1, 5, 10</td></tr><tr><td>Number of receive antennas, N</td><td>5</td></tr><tr><td>Number of modes in TBM, d</td><td>3</td></tr><tr><td>Tensor size, (T<sub>1</sub>, T<sub>2</sub>, T<sub>3</sub>)</td><td>(8,5,5)</td></tr><tr><td>Modulation index, M</td><td>16-PSK</td></tr><tr><td>Number of channel uses, T</td><td>200</td></tr><tr><td>Number of frames transmitted</td><td>10<sup>4</sup></td></tr><tr><td>Prior concentration parameter, κ<sub>0</sub></td><td>10<sup>-4</sup></td></tr><tr><td>Variance of channel, σ<sub>h</sub><sup>2</sup></td><td>1</td></tr><tr><td>Maximum iterations for BP, N<sub>max,itr</sub></td><td>10<sup>3</sup></td></tr></table>

### A. Convergence

To examine the convergence of the proposed algorithm, a single frame is transmitted at a high signal-to-noise ratio (SNR). Figs. 2a and 2b illustrate the evolution of the estimate

and variance of the information symbols across iterations of the BP algorithm respectively, for three symbols, namely $x_{1,2}^{(1)}$, $x_{2,2}^{(1)}$, and $x_{3,2}^{(1)}$. As observed in Fig. 2a, the initial symbol estimates deviate significantly from their true values. However, with successive iterations, the estimates are refined and gradually converge towards the true values of the information symbols, which are denoted by the dotted lines. In Fig. 2b, the variance is initially large, reflecting the uncertainty in the estimates, and decreases with successive iterations, thereby enabling accurate and unambiguous symbol estimation.

<figure><img src="image_4.png" /><figcaption>Fig. 2: Performance analysis with different iterations</figcaption></figure>

### B. Distributions of Information Symbols

Fig. 3 illustrates the convergence of the posterior distribution of the information symbols conditioned on the received symbols. As shown in Fig. 3, after the first iteration, the distribution is nearly uniform, reflecting high uncertainty in the symbol estimates. With subsequent iterations, the uncertainty progressively reduces, and by iteration 14 the posterior distribution concentrates around the true symbol. This reflects the ideal transition from a uniform to an impulse-like distribution, indicating increasing certainty.

<figure><img src="image_5.png" /><figcaption>Fig. 3: Posterior distributions $vM(\mu, \kappa)$ of information symbols after 1, 3 and 14 iterations, with the true value of highlighted in green.</figcaption></figure>

### C. Performance Analysis

We now present a numerical evaluation of the performance of the proposed approach. The receiver is assumed to be aware
of the number of active transmitters and always outputs exactly $K_a$ messages. We evaluate the message error rate using a definition of error event appropriate for the unsourced setting, namely the per-user probability of error (PUPE) [1]: a message is considered to be in error if it was transmitted but does not appear in the list of decoded messages. We evaluate the PUPE versus SNR using Monte-Carlo simulations. The per-user SNR is defined as

$$ \text{SNR}_{\text{per user}} = \frac{E_{\text{signal}}}{E_{\text{noise}}} = \frac{(E_{\text{avg,H}})(\|\mathbf{s}^{(k)}\|^2)}{\sigma^2 TN} = \frac{\sigma_h^2}{\sigma^2}, \quad (26) $$

where $E_{\text{signal}}$ and $E_{\text{noise}}$ denote the average signal and noise energies respectively. This performance is compared with the meta-converse lower bound from [8, Theorem 27]. Fig. 4 depicts the PUPE versus SNR for different numbers of users and number of receive antennas. As expected, the PUPE decreases with increasing SNR and increasing $N$. These results demonstrate that while the proposed modulation is suboptimal in the single-user case, it is extremenly robust to inter-user interference despite the lack of coordination between transmitters while achieving non-trivial spectral efficiency, making it suitable in massive random access scenarios.

<figure><img src="image_6.png" /><figcaption>Fig. 4: PUPE vs. SNR</figcaption></figure>

## VI. COMPLEXITY ANALYSIS

Let us start with some important general remarks. The multi-linear spreading operation in eq. (2) effectively adds redundancy at the symbol level (since $\sum_i (T_i - 1)$ information symbols are spread over $\prod_i T_i$ channel accesses) by generating multiple combinations of information symbols; hence, the proposed approach constitutes a joint coding and modulation scheme involving a non-binary code. Furthermore, Algorithm 1 performs joint demodulation, channel estimation, and multi-user decoding. Such an approach is typically impractically complex for non-trivial problem dimensions due to the iterations between binary (log-likelihood ratios) representations in the channel decoder and complex symbol representations for the equalization steps. Conversely, in the proposed approach, coding and modulation share the same representation using variables on the unit circle, allowing for considerable complexity reduction.

The stepwise complexity analysis of the proposed algorithm is derived in Table II, where $|\Psi_x|$ denotes the cardinality of the dependency set. The dominant term is $O(N^2 K_a T)$ per iteration of the proposed BP, leading to an overall complexity of $O(N^2 K_a T N_{max,itr})$. In contrast, ML-based decoding requires $O(2^{BK_a})$ and the two-step decoder introduced in [6] requires $O(TNK_a^3)$. Notably, the complexity of the proposed algorithm is independent from $M$ (the modulation index) and scales linearly with $K_a$.

<table><caption>TABLE II: Complexity Analysis per Iteration of BP</caption><tr><td>Expectation of Ψ<sub>x</sub> (in (17), (18))</td><td>TNK<sub>a</sub> + K<sub>a</sub>T|Ψ<sub>x</sub>|</td></tr><tr><td>Estimation of information symbols (25)</td><td>K<sub>a</sub>T|Ψ<sub>x</sub>|</td></tr><tr><td>Estimation of channel (25)</td><td>N<sup>2</sup>K<sub>a</sub>T</td></tr><tr><td>Update parameters (Sec. IV-D)</td><td>K<sub>a</sub>T</td></tr><tr><td>Convergence check (Sec. IV-D)</td><td>K<sub>a</sub>T</td></tr></table>

## VII. CONCLUSION

In this paper, we proposed a BP-based multi-user receiver approach for TBM, performing joint multi-user equalization, decoding, and channel estimation for short packet transmission over non-coherent quasi-static fading channels. We provided a complexity analysis and showed that it is substantially lower than for ML-based algorithms. Simulation results demonstrated the good convergence properties of the proposed algorithm and benchmarked it with the meta-converse lower bound.

## ACKNOWLEDGEMENTS

This work has been partially supported by the ANR under the "France 2030" program under grant NF-PERSEUS (ANR-22-PEFT-0004), and under grant WARM-M2M (ANR-24-CE25-2514). Numerical experiments were carried out using the Grid'5000 infrastructure.

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