# Doppler Frequency Estimation in Tensor-Based Modulation via Post-CPD Maximum Likelihood

Kassem Saied, Maxime Guillaud

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Kassem Saied, Maxime Guillaud. Doppler Frequency Estimation in Tensor-Based Modulation via Post-CPD Maximum Likelihood. WCNC 2026 - IEEE Wireless Communications and Networking Conference, Apr 2026, Kuala Lumpur, Malaysia. hal-05470265

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# Doppler Frequency Estimation in Tensor-Based Modulation via Post-CPD Maximum Likelihood

Kassem SAIED, Maxime GUILLAUD

Inria, INSA Lyon, CITI Laboratory, UR3720, F-69621 Villeurbanne, France,
email: {kassem.saied@insa-lyon.fr, maxime.guillaud@inria.fr}

**Abstract**—Tensor-Based Modulation (TBM) is a novel wave-
form design enabling efficient unsourced massive random access
(UMAC) for future wireless networks. By leveraging tensor
algebra and Canonical Polyadic Decomposition (CPD), TBM
allows to perform blind user separation at the receiver. However,
the impact of Doppler frequency shifts on TBM has not yet
been addressed. This paper shows that Doppler effects remain
encoded mode-wise within the TBM factors, and hence do not
disrupt the blind user separation approach. Furthermore, the
Doppler shifts associated to each user can be accurately esti-
mated through a post-CPD per-user Maximum-Likelihood (ML)
estimator. Simulation results show that the proposed approach
only requires a limited number pilot symbols while achieving
accurate Doppler recovery at low SNRs, thereby restoring the
decoding performance to a level nearly identical to the ideal zero-
Doppler case. This makes the proposed approach well suited for
short-packet URLLC and IoT scenarios.

***Index Terms***—Doppler effect, ML estimator, TBM, UMAC, URLLC, IoT.

I. INTRODUCTION

Massive machine-type communications (mMTC) and ultra-
reliable low-latency communications (URLLC) are central to
6G wireless networks, driven by applications such as the
Internet of Things (IoT), industrial automation, and vehicular
networks. These scenarios require supporting a very large
number of devices transmitting short packets sporadically,
without prior coordination with the base station. This has
motivated the development of unsourced massive random
access (UMAC) schemes, in which all users employ a common
codebook and the receiver jointly decodes the set of active
transmissions [1], [2].

The problem of efficient access in sporadic short-packet
transmission has triggered renewed interest in waveform and
coding design. Classical solutions developed for mobile broad-
band, relying on closed-loop mechanisms for interference miti-
gation through synchronization, resource allocation and accu-
rate channel state acquisition, are not suitable for machine-
to-machine (M2M) communication with millions of low-rate
nodes [3]. In such contexts, central coordination becomes
prohibitive and every frame must carry sufficient reference
information for detection and parameter estimation, including
time-of-arrival and Doppler frequency [4].

Early ALOHA-based random access schemes [5] were
widely adopted in both terrestrial and satellite IoT networks,
but they suffer from poor spectral efficiency due to collisions
at high contention rates. Several improvements such as Con-
tention Resolution Diversity Slotted ALOHA (CRDSA) [6]
and Coded Slotted ALOHA (CSA) [7] aimed to mitigate this
issue, while compressed sensing-based methods [8] and the
UMAC paradigm [1] reframed the problem from a coding-
theoretic perspective. Several waveforms have been proposed
to tackle the challenges of sporadic short-packet transmission.
For instance, the Quasi-Cyclic Small Packet (QCSP) frame
[9] combines non-binary coding with CCSK modulation for
self-synchronization at low Signal-to-Noise Ratio (SNR) [10],
[11].

Tensor-Based Modulation (TBM) [12] is another approach
tailored to UMAC problem; it is based on multi-linear spread-
ing and leverages tensor algebra through the unique decom-
position properties of low-rank tensors. In TBM, each user
encodes its data as a rank-one tensor, enabling blind user
separation at the receiver via Canonical Polyadic Decompo-
sition (CPD). TBM is suitable for multi-antenna block fad-
ing channels [12], [13], achieving efficient unsourced access
without explicit channel estimation. It has been shown in
[14] that TBM is robust to timing offsets when associated
to an OFDM waveform. Despite these advances, the impact
of Doppler frequency offsets, which can significantly degrade
TBM decoding performance, remains an open problem and
has not yet been addressed.

In this paper, we address this problem. We analytically show
that, under Doppler distortion, each user's contribution remains
a distinct rank-1 component in the tensorized received model,
ensuring successful user separation in the presence of random
Doppler shifts. We propose a post-CPD maximum likelihood
(ML) Doppler estimator that can achieve high accuracy with
a limited amount of pilot symbols, making it particularly
suitable for short-packet UMAC scenarios such as IoT and
URLLC. Simulation results with QAM modulation and polar
codes confirm the effectiveness of the approach, demonstrating
that accurate Doppler estimation and correction restore Frame-
Error Rate (FER) performance close to the ideal zero-Doppler
case, even at low SNRs.

The rest of this article is organized as follows. Section II
introduces the system model and the Doppler-impaired TBM
signal formulation. Section III introduces the proposed post-

<footnote>This work has been partially supported by the ANR under grant WARM-M2M (ANR-24-CE25-2514) and under the "France 2030" program under grant NF-PERSEUS (ANR-22-PEFT-0004).</footnote>
CPD ML Doppler estimator and its formulation. Section IV presents simulation results and performance analysis. Finally, Section V concludes the article and outlines possible directions for future research.

## II. TBM SYSTEM MODEL AND DOPPLER CHARACTERIZATION

Let us consider a UMAC system, where $K_a$ single-antenna users transmit simultaneously to a receiver with $N$ antennas, such as a satellite or a base station. We assume that the $K_a$ users are active for the entire duration of the considered block; due to the adopted unsourced transmission assumption, the total number of potential users in the system plays no role in the analysis [1]. We assume a time-synchronized channel, while each user undergoes an independent Doppler frequency shift $f_k$, and thus all users transmit simultaneously and synchronously over a block of $T$ channel uses. The objective of this section is to model the TBM demodulation process and to highlight how the Doppler effect manifests itself within the tensor structure of the received signal.

<figure><img src="image_2.png" /><figcaption>Fig. 1. Overall Communication Principle per one user.</figcaption></figure>

### A. Transmitter side

As illustrated in Fig. 1, each active user encodes a block of $B_{inf}$ information bits using a Forward Error Correction (FEC) code, resulting in $B_{enc}$ coded bits. According to the TBM approach, the sequence $s_k$ of symbols transmitted over the channel by the $k$-th user is expressed as a Kronecker (or tensor) product of $D$ vectors. Specifically,

$$s_k = \mathbf{x}_{0,k} \otimes \mathbf{x}_{1,k} \otimes \cdots \otimes \mathbf{x}_{D-1,k}, \quad (1)$$

where each $\mathbf{x}_{i,k} \in \mathbb{C}^{T_i}$ represents the modulation vector of mode $i$, $\otimes$ denotes the Kronecker product, and the overall block length satisfies $T = \prod_{i=0}^{D-1} T_i$. Within tensor mode $i$, vector $\mathbf{x}_{i,k}$ embeds one or several reference (pilot) symbols, as well as information-bearing complex scalar symbols from a standard modulation; without loss of generality, we will henceforth assume a QAM constellation with $q$ states. The mode vectors are assumed to be normalized as $\|\mathbf{x}_{i,k}\|^2 = T_i$, leading to a constant transmitted energy per user $\|s_k\|^2 = T$.

Let $\mathcal{P}_i$ denote the set of pilot indices in mode $i$, with cardinality $|\mathcal{P}_i|$. The number of information-bearing symbols in mode $i$ is thus $T_i - |\mathcal{P}_i|$, and as a consequence $B_{enc}$ is linked

to the total number of data-carrying QAM symbols across all modes as

$$B_{enc} = q \sum_{i=0}^{D-1} (T_i - |\mathcal{P}_i|). \quad (2)$$

The encoded information is distributed across the $D$ tensor modes, so that each user's transmitted vector results from the joint contribution of all modes through the multi-linear spreading operation described in eq. (1). It was shown in [12] that this structure can be exploited by the receiver to perform blind user separation. In this work, the pilot symbols are later exploited at the receiver side for phase and Doppler estimation after the CPD.

### B. Channel and Doppler Model

We consider a block-fading uplink channel between the $K_a$ active users and $N$-antenna receiver in the absence of Doppler effects. The baseband received signal, stacked across all receive antennas and time samples, is represented in vectorized form as

$$\mathbf{y} = \sum_{k=1}^{K_a} \mathbf{s}_k \otimes \mathbf{h}_k + \mathbf{w}, \quad (3)$$

where $\mathbf{h}_k \in \mathbb{C}^N$ is the complex channel vector of user $k$, and $\mathbf{w}$ is a zero-mean additive white Gaussian noise (AWGN) vector with independent entries of variance $\sigma^2$. We define the instantaneous SNR for user $k$ as the received signal power divided by the noise power both summed over one block of $T$ channel accesses and over the receive antennas. Taking into account the constraints $\|\mathbf{x}_{i,k}\|^2 = T_i$, this yields SNR$_k = \frac{\|\mathbf{h}_k\|^2}{N\sigma^2}$. Note that we can also define the average SNR by replacing the term $\|\mathbf{h}_k^2\|$ by its expectation over the fading process.

**Doppler modeling:** Each user experiences an independent Doppler frequency shift $f_k$ due to relative motion. This Doppler effect introduces a linearly varying phase across the transmitted symbols, modeled as a multiplicative diagonal operator:

$$\mathbf{D}(f_k) = \mathrm{diag}(e^{j2\pi f_k T_s n})_{n=0}^{T-1}, \quad (4)$$

where $T_s$ is the symbol period and $n$ denotes the sample index where $n \in [0, T-1]$. The Doppler-distorted codeword becomes $\mathbf{s}_k \rightarrow \mathbf{D}(f_k)\mathbf{s}_k$, and the received vector in (3) becomes

$$\mathbf{y} = \sum_{k=1}^{K_a} (\mathbf{D}(f_k) \mathbf{s}_k) \otimes \mathbf{h}_k + \mathbf{w}. \quad (5)$$

This model explicitly expresses the Doppler phase ramp as a linear function.

### C. TBM Structure Preservation under Doppler

In the TBM framework, each transmitted vector $\mathbf{s}_k$ in (1) is the vectorized version of a $D$-mode tensor of dimensions
$(T_0, T_1, \dots, T_{D-1})$. The relationship between the multidimensional tensor indices $(m_0, m_1, \dots, m_{D-1})$ and the corresponding flattened time-domain index $n$ follows the standard multilinear-to-vector mapping [15, Sec. 2.4], expressed as

$$n = \sum_{i=0}^{D-1} w_i m_i, \quad w_0 = 1, \quad w_i = \prod_{j=0}^{i-1} T_j. \qquad (6)$$

Here, $w_i$ denotes the stride of mode $i$, which determines how indices advance when the tensor is flattened into its one-dimensional vector representation. Using (6), the Doppler phase term of user $k$ at sample $n$ can then be written as

$$e^{j2\pi f_k T_n n} = \prod_{i=0}^{D-1} e^{j2\pi f_k T_n w_i m_i}. \qquad (7)$$

Hence, the Doppler operator acting on the vectorized signal admits the Kronecker decomposition

$$\mathbf{D}(f_k) = \bigotimes_{i=0}^{D-1} \mathbf{D}_i(f_k), \quad \text{where} \\ \mathbf{D}_i(f_k) = \text{diag}(e^{j2\pi f_k T_n w_i m})^{T_i - 1}_{m=0}. \qquad (8)$$

Applying this operator to $\mathbf{s}_k$ and using repeatedly the identity $(\mathbf{A} \otimes \mathbf{B})(\mathbf{C} \otimes \mathbf{D}) = (\mathbf{A}\mathbf{C}) \otimes (\mathbf{B}\mathbf{D})$ for any matrices $\mathbf{A}, \mathbf{B}, \mathbf{C}$ and $\mathbf{D}$ yields

$$\mathbf{D}(f_k) \mathbf{s}_k = \left[ \bigotimes_{i=0}^{D-1} \mathbf{D}_i(f_k) \right] \left[ \bigotimes_{i=0}^{D-1} \mathbf{x}_{i,k} \right] = \left[ \bigotimes_{i=0}^{D-1} \mathbf{D}_i(f_k) \mathbf{x}_{i,k} \right]. \qquad (9)$$

This shows that the diagonal operator $\mathbf{D}(f_k)$ defined in (5) can thus be equivalently expressed as a Kronecker product of per-mode diagonal factors $\mathbf{D}_i(f_k)$, each introducing a deterministic linear phase ramp across its respective mode. Consequently, the received codewords in (5) can be written as

$$\mathbf{y} = \sum_{k=1}^{K_a} \left( \bigotimes_{i=0}^{D-1} \mathbf{D}_i(f_k) \mathbf{x}_{i,k} \right) \otimes \mathbf{h}_k + \mathbf{w}, \qquad (10)$$

which confirms that even under Doppler distortion, each user is associated with a rank-1 component of the received (tensorized) signal, hence users remain separable through the CPD, although each mode vector $\mathbf{x}_{i,k}$ is multiplied element-wise by a deterministic linear phase factor.

### D. CPD-based user separation

It was proposed in [12] to use the CPD to decompose signals of the type of eq. (3) into its rank-1 components. Low-rank tensors enjoy favorable unique decomposition properties, and it can be shown that the decomposition is unique under mild conditions in the noise-free case, up to a given rank which depends on the tensor dimensions. Since each rank-1 component represents the signal from one user, this approach allows for user separation (and estimation of the channel as a by-product). When noise is present, a CPD-like low-rank tensor approximation has been shown to perform well.

Considering (10) instead of (3), it is clear that this approach remains applicable to the case including Doppler effect, since

the signal part is again the sum of $K_a$ rank-1 components, with each component associated with one user, albeit affected by the Doppler according to (9). We will analyze this approach under the angle of Doppler effect mitigation.

After CPD of the reshaped received array, the estimated mode vectors associated with user $k$ satisfy

$$\hat{\mathbf{x}}_{i,k} \approx \mathbf{D}_i(f_k) \mathbf{x}_{i,k} \alpha_{i,k} + \mathbf{v}_{i,k}, \qquad (11)$$

where $\alpha_{i,k}$ is the complex scaling ambiguity inherent to CPD. The decomposition is unique up to complex scaling ambiguities $\alpha_{i,k}$, which satisfy the constraint $\prod_i \alpha_{i,k} = 1$. The Doppler frequency offset therefore, appears as a **linear phase progression** across each recovered mode, perturbed by the residual noise term $\mathbf{v}_{i,k}$. The term $\mathbf{v}_{i,k}$ represents the *residual CPD noise*, which aggregates decomposition inaccuracies and background AWGN. Its entries are well modeled as zero-mean, complex Gaussian variables with variance approximately $\sigma_{\mathbf{v}_{i,k}}^2 \approx 1/\text{SNR}_{\text{eq},i,k}$, where $\text{SNR}_{\text{eq},i,k}$ is the effective per mode post-CPD SNR. It was shown in [13] that the equivalent per mode post-CPD SNR can be expressed as

$$\text{SNR}_{\text{eq},i,k} = \frac{(TN - T_i(K_a - 1))}{T_i - 1} \text{SNR}_k. \qquad (12)$$

This formula quantifies the effective per-mode SNR enhancement (denoising) achieved through CPD-based multiuser separation.

### III. POST-CPD MAXIMUM-LIKELIHOOD DOPPLER ESTIMATOR

#### A. Principle of Post-CPD Doppler Estimation

After the CPD of the received TBM tensor, each recovered mode vector for a given user exhibits a deterministic linear phase ramp induced by the Doppler shift, as shown in (11). While one could, in principle, estimate the Doppler frequency separately from each mode, this approach neglects the intrinsic phase coupling among modes implied by the Kronecker structure of the tensorized signal. Moreover, per-mode estimation suffers from fundamental resolution and identifiability limitations, as each mode contains a reduced number of pilot samples and distinct stride factors $w_i$ that scale the effective frequency grid differently. In contrast, we propose a *joint vectorized ML formulation* which coherently combines all pilot observations into a single correlation domain, thereby exploiting the full Kronecker coupling across modes.

Let us consider again the sets $\mathcal{P}_i$ of indices of the pilot symbols in modes $i = 0 \dots D-1$, as defined in Sec. II-A. Their Cartesian product $\mathcal{P}_\times = \mathcal{P}_0 \times \dots \times \mathcal{P}_{D-1}$ defines the multidimensional pilot coordinates within the tensor. Using the stride-based mapping from eq. (6) gives the corresponding flattened pilot indices $\mathcal{P} = \{n \mid (m_0, \dots, m_{D-1}) \in \mathcal{P}_\times\}$, which identify the pilot positions in the vectorized codeword. Figure 2 illustrates pilot placement for the two-mode case, where per-mode pilots form a two-dimensional lattice whose flattened indices map to pilot positions in the one-dimensional TBM vector.
<figure><img src="image_3.png" /><figcaption>Fig. 2. From 2-D per-mode pilots to 1-D flattened positions in TBM.</figcaption></figure>

Let $\mathbf{s}_k^{(P)} = [s_k[n_p]]_{n_p \in \mathcal{P}}$ denote the restriction of $\mathbf{s}_k$ to the elements containing pilot symbols of user $k$.

## B. Observation Model and Maximum-Likelihood Formulation

The joint vectorized observations used for Doppler estimation originate from the CPD mode-wise reconstructions in (11). Let

$$ \mathbf{s}'_k = \hat{\mathbf{x}}_{0,k} \otimes \hat{\mathbf{x}}_{1,k} \otimes \cdots \otimes \hat{\mathbf{x}}_{D-1,k} \quad (13) $$

denote the estimated rank-1 tensor component associated with user $k$, and $\mathbf{s}_k^{(P)}$ its restriction to the pilot positions. This vector represents a noisy observation of the true transmitted pilot subset $\mathbf{s}_k^{(P)}$, whose elements correspond to the known reference pilots visible in Fig. 2.

Here, we will assume the following simplified observation model for the symbols at the pilot positions:

$$ \mathbf{s}'_{k}^{(P)} = \mathbf{s}_{k}^{(P)} \odot e^{j2\pi f_k T_s \mathbf{n}_P} + \boldsymbol{\eta}_{k}^{(P)}, \quad (14) $$

where $\mathbf{n}_P = [n_p]_{n_p \in \mathcal{P}}$ and $\boldsymbol{\eta}_k^{(P)}$ denotes an equivalent complex Gaussian noise vector with assumed independent and identically distributed entries of variance $\sigma_{\eta,k}^2$. Based on eq. (13), its variance is approximated as the aggregated contribution of all modes, $\sigma_{\eta,k}^2 \approx \sum_{i=0}^{D-1} \frac{1}{\text{SNR}_{\text{eq},i,k}}$. Consequently,

$$ \text{SNR}' = \frac{1}{\sigma_{\eta,k}^2} = \left( \sum_{i=0}^{D-1} \frac{T_i - 1}{TN - T_i (K_a - 1)} \right)^{-1} \text{SNR}_k, \quad (15) $$

provides a practical link between the channel SNR and the equivalent effective variance used in the likelihood following formulation.

Under this i.i.d. Gaussian assumption, the probability density function (pdf) of each observed pilot sample, conditioned on the corresponding transmitted symbol and a trial Doppler frequency $f$, is given by

$$ p(s'_{k}[n_p] | s_k[n_p], f) = \frac{1}{\pi \sigma_{\eta,k}^2} \exp\left(-\frac{|s'_{k}[n_p] - s_k[n_p]e^{j2\pi f T_s n_p}|^2}{\sigma_{\eta,k}^2}\right). \quad (16) $$

Since the noise samples are independent, the joint likelihood of the observed pilot vector, conditioned on the known transmitted pilots and a trial Doppler $f$, is obtained as the product of these individual densities, expressed as $p(\mathbf{s}'_{k}^{(P)}|\mathbf{s}_{k}^{(P)}, f)$

$$ \begin{aligned} & \propto \prod_{n_p \in \mathcal{P}} \exp\left(-\frac{|s'_{k}[n_p] - s_k[n_p]e^{j2\pi f T_s n_p}|^2}{\sigma_{\eta,k}^2}\right) \\ & \propto \exp\left(-\frac{1}{\sigma_{\eta,k}^2} \sum_{n_p \in \mathcal{P}} |s'_{k}[n_p] - s_k[n_p]e^{j2\pi f T_s n_p}|^2\right). \end{aligned} \quad (17) $$

Maximizing the likelihood (17) with respect to $f$ is equivalent to minimizing the quadratic error term in the exponent. Expanding the norm and discarding $f$-independent constants leads to the maximum-likelihood Doppler estimator

$$ \hat{f}_{d_k} = \arg \max_f \Re \left\{ \sum_{n_p \in \mathcal{P}} s'_{k}[n_p] (s_k[n_p])^* e^{-j2\pi f T_s n_p} \right\}. \quad (18) $$

where $\Re\{\cdot\}$ denotes the real part.

*Proof:* From (17), maximizing the likelihood over $f$ is equivalent to

$$ \hat{f}_k = \arg \min_f \| \mathbf{s}'_{k}^{(P)} - \mathbf{s}_{k}^{(P)} \odot e^{j2\pi f T_s \mathbf{n}_P} \|^2. $$

Let $\mathbf{y} = \mathbf{s}'_{k}^{(P)}$ and $\mathbf{a}(f) = \mathbf{s}_{k}^{(P)} \odot e^{j2\pi f T_s \mathbf{n}_P}$. Expanding the quadratic term gives

$$ \|\mathbf{y} - \mathbf{a}(f)\|^2 = \mathbf{y}^H \mathbf{y} + \mathbf{a}^H(f) \mathbf{a}(f) - 2 \Re\{\mathbf{y} \mathbf{a}^H(f)\}. $$

Since $\mathbf{y}^H \mathbf{y}$ and $\mathbf{a}^H(f) \mathbf{a}(f) = \|\mathbf{s}'_{k}^{(P)}\|^2$ do not depend on $f$, minimizing the error is equivalent to

$$ \hat{f}_k = \arg \max_f \Re\{\mathbf{y} \mathbf{a}^H(f)\} $$

which corresponds to (18).

This one-dimensional search identifies the Doppler frequency that best compensates for the phase rotation across the observed pilots. It is performed over a predefined frequency range using a coarse-to-fine grid, optionally refined through parabolic or Newton interpolation [16]. Each evaluation requires $O(|\mathcal{P}|)$ operations, where $|\mathcal{P}| = \prod_i |\mathcal{P}_i|$.

A key advantage of the proposed post-CPD estimation strategy lies in the fact that the ML search is applied after tensor decomposition, i.e., the estimator (18) is applied independently for each user, which is typically orders of magnitude less complex than tackling the joint estimation of multiple parameters associated with interfering signals. Furthermore, this approach benefits from the denoising effect of the CPD. This yields a substantial benefit: the estimation operates under the equivalent per-mode SNR defined in (15), which is significantly higher than the raw SNR observed before decomposition.

# IV. SIMULATION RESULTS

This section evaluates the performance of the proposed post-CPD ML Doppler estimator and its impact on the TBM transmission.

## A. Simulation Setup

We consider a TBM symbol of shape [64, 50], corresponding to two tensor modes with respective lengths $T_0 = 64$ and $T_1 = 50$. Each active user encodes a block of $B_{\text{info}} = 300$ information bits using a polar code (the encoding follows the standard construction available in the open-source implementation by McBain [17]) with $R_c = 0.708$, yielding $B_{\text{enc}} = 424$ coded bits that are subsequently mapped to QAM symbols with $q = 4$ bits per symbol. For synchronization and Doppler estimation, four QAM symbols in each mode
are reserved as pilots, i.e., |Pᵢ| = 4. The symbol duration is set to Tₛ = 150 µs. The modulated sequence is then organized according to (1), where each mode vector **x**ᵢ,ₖ ∈ ℂ³ᵢ follows the normalization ||**x**ᵢ,ₖ||² = Tᵢ, ensuring a total symbol energy of ||**s**ₖ||² = T = 64 × 50 = 3200. With those parameters, each user transmits at a rate of B<sub>info</sub>/T ≈ 0.09 bits per channel use.

Pilot placement plays a crucial role in the accuracy of the estimator; however the actual reference symbol locations depend non-trivially on the per-mode pilot symbol allocation through eq. (6), making the problem difficult to analyze. Different pilot placements were tested to analyze empirically the sensitivity of the proposed ML estimator to spatial distribution. After exploring several configurations, the placements P₀ = [0, 19, 44, 63] and P₁ = [0, 14, 31, 49] were selected, as they consistently yielded the lowest estimation-error variance across the tested SNR range. In general, and consistent with observations from both simulations and prior literature, an effective guideline for such estimation problems is to distribute pilots irregularly across the entire frame to ensure good frequency resolution and minimize ambiguity effects.

The Doppler frequency of each user is uniformly drawn
from the range [-2, 2] kHz. At the receiver, CPD-based
user separation is performed, followed by post-CPD Doppler
estimation and polar decoding after frequency correction. All
results are presented as a function of the SNR per receive
antenna as defined in Section II-B.

B. Pre- and Post-CPD ML estimator

First, we evaluate the impact of the CPD-based denoising
step on the accuracy of the proposed Doppler estimation
method. We consider a single active user ($K_a = 1$) and one
receive antenna ($N = 1$), under the same TBM configura-
tion as described previously. This simplified setup enables
a clear comparison between the *pre-CPD* and *post-CPD*
ML estimators. Under these assumptions, eq. (10) becomes
$\mathbf{y} = \mathbf{D}(f_{d_1})\mathbf{s}_k\mathbf{h}_k + \mathbf{w}$ which is similar in structure to (14).
Therefore, the ML estimator derived in Sec. III-B is directly
applicable. Note that the noise variance for the pre-CPD
estimator corresponds to the channel noise level (with SNR$_k$),
while the post-CPD estimator is applied to the recovered tensor
components from (14), where the CPD has a denoising effect
and effectively reduces the equivalent per-mode noise variance
as established in (15).

Figure 3 shows the scatter distribution of the estimated versus true Doppler frequencies for both estimators. The results are computed over 10³ independently transmitted frames, corresponding to a regime where the FER is on the order of 10⁻⁴ (approximately at 0 dB SNR). The blue markers (pre-CPD) show a broad dispersion around the diagonal, indicating large estimation errors caused by the higher effective noise level at the receiver before decomposition. In contrast, the orange markers (post-CPD) cluster tightly along the ideal line, confirming that the Doppler shifts are accurately retrieved once the denoised tensor factors are exploited.

<figure><img src="image_4.png" /><figcaption>Fig. 3. Pre- and post-CPD ML estimators versus true Doppler frequencies over 10³ frames.</figcaption></figure>

Figure 4 illustrates the probability distribution of the Doppler estimation error for the same dataset. The post-CPD error distribution is sharply concentrated around zero, with deviations typically within ±0.1 Hz, whereas the pre-CPD distribution exhibits a much wider spread, extending over several tens and hundreds of hertz. This clear separation demonstrates that the post-CPD estimator remains unbiased under small residual CPD noise and achieves a significantly higher estimation precision.

<figure><img src="image_5.png" /><figcaption>Fig. 4. Pre- and post-CPD ML estimators: distribution of Doppler estimation error.</figcaption></figure>

The performance improvement comes from the fact that the
CPD step effectively increases the usable SNR in each mode.
As a result, reliable and fine-resolution Doppler estimation
can be achieved even with a small number of pilots and
without increasing the transmission overhead. These results
confirm the effectiveness of the post-CPD denoising process
in improving Doppler estimation accuracy and robustness,
particularly under low-SNR conditions.
### C. Pre- and Post-CPD ML estimator

In this section, we evaluate the FER performance of polar-coded TBM transmission under practical multiuser IoT and UMAC-URLLC conditions, with up to $K_a = 100$ active users and $N = 50$ receive antennas. Both Doppler-free and Doppler-impaired cases are considered, where the Doppler frequency shifts are uniformly distributed within [-2, 2] kHz. The same TBM configuration described earlier is used, with 4 pilots per mode.

Fig. 5 illustrates the FER curves with and without Doppler correction. In the uncorrected case, Doppler-induced phase drift across tensor modes causes a significant degradation in FER performance. When the proposed post-CPD ML Doppler correction is applied (white markers), the FER performance for both $N = 1$ and $N = 50$ closely follows their respective zero-Doppler baselines (filled markers). This strong alignment confirms the accuracy and robustness of the proposed estimator across all user loads ($K_a \in 1, 50, 100$), ensuring reliable per-user Doppler estimation and separation.

<figure><img src="image_6.png" /><figcaption>Fig. 5. FER performance with and without Doppler correction for polar-coded TBM transmission ([64, 50] tensor shape, 4 pilots per mode).</figcaption></figure>

After correction, the FER performance is recovered to within approximately 0.2 dB of the zero-Doppler reference. These results demonstrate that the proposed post-CPD ML estimator effectively compensates for Doppler shifts using only a few pilots, making it well suited for short-packet UMAC and IoT applications. So, the CPD stage not only provides robust multiuser separation—achieving nearly identical FER for $K_a = 100$ as in the single-user case—but also enhances the effective per-mode SNR, enabling precise Doppler estimation in regimes where conventional pre-CPD approaches typically fail.

## V. CONCLUSION

This paper presented a post-CPD maximum-likelihood Doppler estimation framework for TBM in unsourced multiple-access scenarios. By leveraging the separability and inherent denoising properties of the CPD, the proposed method accurately estimates Doppler shifts directly from the decomposed user factors using only a few pilot symbols. The usage

of the generated multidimensional pilot enables coherent estimation across tensor modes, substantially improving accuracy compared to pre-CPD or the per-mode estimators. Simulation results demonstrated that the proposed approach achieves near-zero residual Doppler error and restores polar decoding performance to within 0.2 dB of the zero-Doppler baseline, even under low-SNR conditions.

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