Pilot Allocation and Receiver Design for Cell-Free Massive MIMO ISAC Systems

TL;DR

This paper presents a novel pilot allocation framework and a Gaussian belief propagation receiver design for cell-free massive MIMO ISAC systems, enhancing both communication and sensing performance with reduced complexity.

Contribution

It introduces a manifold optimization-based pilot allocation and a GaBP-based receiver, improving system sum rate, sensing capabilities, and computational efficiency in CF-mMIMO ISAC systems.

Findings

Achieves communication performance comparable to state-of-the-art algorithms.

Provides superior sensing capabilities due to unimodular pilot design.

Reduces computational complexity with the GaBP-based receiver.

Abstract

This paper tackles two key challenges in cell-freemassive multiple input multiple output (CF-mMIMO) systems:efficient pilot allocation and practical receiver design. To thisend, we introduce a novel pilot allocation framework leveragingmanifold optimization to maximize the system sum rate, wherepilot sequences are designed as nearly orthogonal sequences. Theproposed pilot design enforces unimodularity constraints in thefrequency domain, ensuring pilots are suitable for both communi-cation and sensing tasks. Additionally, a gaussian belief propaga-tion (GaBP)-based receiver is introduced, providing near-optimaldetection performance with substantially reduced computationalcomplexity. Simulation results demonstrate that the proposedpilot allocation method achieves communication performancecomparable to state-of-the-art (SotA) algorithms, while deliveringsuperior sensing capabilities due to its unimodular pilot design.The GaBP-based receiver achieves robust performance andlower complexity compared to conventional approaches. Thesecontributions advance the practical deployment of CF-mMIMOfor integrated sensing and communications (ISAC).

Equations

$$g_{\ell ,k}={\beta }_{\ell ,k}^{1/2}{h}_{\ell ,k},\text{vskip}-2.15277pt$$

$${\mathbf{y}}_{\ell }^{\text{p}}=\sqrt{{\rho }_p}\sum _{k=1}^K{g}_{\ell ,k}{\mathbf{f}}_k^{\mathrm{H}}+{\mathbf{n}}_{\ell }\in {\mathbb{C}}^{\tau \times 1},\text{vskip}-4.30554pt$$

$${\check{y}}_{\ell ,k}={\mathbf{f}}_k^{\mathrm{H}}{\mathbf{y}}_{\ell }^{\text{p}}=\tau \sqrt{{\rho }_p}{g}_{\ell ,k}+\sqrt{{\rho }_p}\sum _{k^{\prime }\ne k}^K{g}_{\ell ,k^{\prime }}{\mathbf{f}}_k^{\mathrm{H}}{\mathbf{f}}_{k^{\prime }}+{\mathbf{f}}_k^{\mathrm{H}}{\mathbf{n}}_{\ell }.\text{vskip}-4.30554pt$$

$${\hat{g}}_{\ell ,k}=\frac{\mathbb{E}\left[g_{\ell ,k}{\check{y}}_{\ell ,k}\right]}{\mathbb{E}\left[|{\check{y}}_{\ell ,k}{|}^2\right]}=c_{\ell ,k}{\check{y}}_{\ell ,k},\text{vskip}-2.15277pt$$

$$c_{\ell ,k}=\frac{\tau \sqrt{{\rho }_p}{\beta }_{\ell ,k}}{{\rho }_p{\sum }_{k^{\prime }=1}^K{\beta }_{\ell ,k^{\prime }}|{\mathbf{f}}_k^{\mathrm{H}}{\mathbf{f}}_{k^{\prime }}{|}^2+\tau },\text{vskip}-2.15277pt$$

$${\gamma }_{\ell ,k}=\mathbb{E}\left[|{\hat{g}}_{\ell ,k}{|}^2\right]=\frac{{\tau }^2{\rho }_p{\beta }_{\ell ,k}^2}{{\rho }_p{\sum }_{k^{\prime }=1}^K{\beta }_{\ell ,k^{\prime }}|{\mathbf{f}}_k^{\mathrm{H}}{\mathbf{f}}_{k^{\prime }}{|}^2+\tau }.\text{vskip}-2.15277pt$$

$${\text{R}}_k={\log }_2\left(1+{\text{SINR}}_k\right),\text{vskip}-2.15277pt$$

$${\text{SINR}}_k=\frac{{\rho }_p{\left({\sum }_{\ell =1}^L{\gamma }_{\ell k}\right)}^2}{{\rho }_p{\sum }_{\begin{array}{c}{k}^{\prime }\ne k\end{array}}^K{\left({\sum }_{\ell =1}^L\frac{{\gamma }_{\ell k}{\beta }_{\ell k^{\prime }}}{{\beta }_{\ell k}}\right)}^2|{\bar{\mathbf{f}}}_k^H{\bar{\mathbf{f}}}_{k^{\prime }}{|}^2+{\rho }_p{\sum }_{k^{\prime }=1}^K{\sum }_{\ell =1}^L{\gamma }_{\ell k}{\beta }_{\ell k^{\prime }}+{\sum }_{\ell =1}^L{\gamma }_{\ell k}}$$

$$\underset{\bar{\mathbf{F}}\in {\mathbb{C}}^{\tau \times K}}{\max }$$

$$|[\mathcal{F}\{\bar{\mathbf{F}}\}{]}_{(i,k)}|=1,\forall i\in \{1,\dots ,\tau \},k\in \{1,\dots ,K\},$$

$$\underset{\mathbf{X}\in \mathcal{M}}{\min }f(\mathbf{X}),$$

$${\mathcal{S}}^{d-1}=\left\{\mathbf{x}\in {\mathbb{R}}^d:{\mathbf{x}}^{\top }\mathbf{x}=1\right\},$$

$$T_{\mathbf{x}}{\mathcal{S}}^{d-1}=\left\{\mathbf{z}\in {\mathbb{R}}^d:{\mathbf{x}}^{\top }\mathbf{z}=0\right\},$$

$$R_{\mathbf{x}}(\mathbf{z})=\frac{\mathbf{x}+\mathbf{z}}{|\mathbf{x}+\mathbf{z}|}.$$

$$\mathcal{C}(\tau ,K)=\left\{\mathbf{X}\in {\mathbb{C}}^{\tau \times K}:|\mathbf{X}|={\mathbb{J}}_{\tau \times K}\right\},$$

$$⟨\mathbf{u},\mathbf{v}{⟩}_{\mathbf{X}}=\mathrm{Re}\left(\mathrm{Tr}\left({\mathbf{u}}^{\mathrm{H}}\mathbf{v}\right)\right),$$

$$\mathcal{T}(\mathbf{u},\mathbf{v})=\mathbf{u}-\Re ({\mathbf{u}}^{\mathrm{H}}\otimes \mathbf{v})\otimes \mathbf{v}.$$

$$\frac{\partial {\text{R}}_k}{\partial {\mathbf{f}}_k}=\frac{1}{\ln (2)}\frac{1}{1+{\text{SINR}}_k}\frac{\partial {\text{SINR}}_k}{\partial {\mathbf{f}}_k}.$$

$$\frac{\partial {\text{SINR}}_k}{\partial {\mathbf{f}}_k}=\frac{1}{\text{D}}\frac{\partial \text{N}}{\partial {\mathbf{f}}_k}-\frac{\text{N}}{{\text{D}}^2}\frac{\partial \text{D}}{\partial {\mathbf{f}}_k}.$$

$$\text{N}={\rho }_p{\left(\sum _{\ell =1}^L{\gamma }_{\ell k}\right)}^2,$$

$$\frac{\partial \text{N}}{\partial {\mathbf{f}}_k}=2{\rho }_p\left(\sum _{\ell =1}^L{\gamma }_{\ell k}\right)\sum _{\ell =1}^L\frac{\partial {\gamma }_{\ell k}}{\partial {\mathbf{f}}_k},$$

$$\frac{\partial {\gamma }_{\ell k}}{\partial {\mathbf{f}}_k}=-2\frac{{\gamma }_{\ell k}^2}{{\beta }_{\ell ,k}}\sum _{k^{\prime }=1}^K{\beta }_{\ell ,k^{\prime }}({\mathbf{f}}_k^{\mathrm{H}}{\mathbf{f}}_{k^{\prime }}){\mathbf{f}}_{k^{\prime }}.$$

$${\text{D}}_1={\rho }_p\sum _{\begin{array}{c}{k}^{\prime }\ne k\end{array}}^K{\left(\sum _{\ell =1}^L\frac{{\gamma }_{\ell k}{\beta }_{\ell k^{\prime }}}{{\beta }_{\ell k}}\right)}^2|{\mathbf{f}}_k^{\mathrm{H}}{\mathbf{f}}_{k^{\prime }}{|}^2,$$

$${\text{D}}_2={\rho }_p\sum _{k^{\prime }=1}^K\sum _{\ell =1}^L{\gamma }_{\ell k}{\beta }_{\ell k^{\prime }},{\text{D}}_3=\sum _{\ell =1}^L{\gamma }_{\ell k}.$$

$$\frac{\partial {\text{D}}_1}{\partial {\mathbf{f}}_k}$$

$$-4{\rho }_p\sum _{\begin{array}{c}{k}^{\prime }\ne k\end{array}}^K\left(\sum _{\ell =1}^L\sum _{k^{\prime }=1}^K\frac{{\gamma }_{\ell k}^2}{{\beta }_{\ell k}^3}{\beta }_{\ell ,k^{\prime }}^2({\mathbf{f}}_k^{\mathrm{H}}{\mathbf{f}}_{k^{\prime }}){\mathbf{f}}_{k^{\prime }}\right)|{\mathbf{f}}_k^{\mathrm{H}}{\mathbf{f}}_{k^{\prime }}{|}^2,$$

$$\frac{\partial {\text{D}}_2}{\partial {\mathbf{f}}_k}=-2{\rho }_p\sum _{k^{\prime }=1}^K\sum _{\ell =1}^L\frac{{\gamma }_{\ell k}^2}{{\beta }_{\ell k}^2}\left(\sum _{k^{\prime }=1}^K{\beta }_{\ell k^{\prime }}({\mathbf{f}}_k^{\mathrm{H}}{\mathbf{f}}_{k^{\prime }}){\mathbf{f}}_{k^{\prime }}\right),$$

$$\frac{\partial {\text{D}}_3}{\partial {\mathbf{f}}_k}=-2\sum _{\ell =1}^L\frac{{\gamma }_{\ell k}^2}{{\beta }_{\ell k}^2}\left(\sum _{k^{\prime }=1}^K{\beta }_{\ell k^{\prime }}({\mathbf{f}}_k^{\mathrm{H}}{\mathbf{f}}_{k^{\prime }}){\mathbf{f}}_{k^{\prime }}\right).$$

$$R_{\text{net}}=B\frac{1-\tau /T}{2}\sum _{k=1}^K{\text{R}}_k,$$

$${\beta }_{\ell ,k}={\text{PL}}_{\ell ,k}\cdot 10^{\frac{{\sigma }_{\mathrm{sh}}{z}_{\ell ,k}}{10}},$$

$$r_k\triangleq {\mathbf{x}}^H{\mathbf{J}}_k\mathbf{x},k=0,1,\dots ,N-1,$$

$${\mathbf{J}}_k\triangleq \left[\begin{array}{cc}0& {\mathbf{I}}_{N-k}\\ 0& 0\end{array}\right].$$

Full text

I Introduction

Large variations in data rates are intrinsic to cellular networks and persist even with advanced access point (AP) technology such as multiple input multiple output (MIMO) [2, 3]. Cell-free massive MIMO systems, where several APs jointly serve users, was introduced to mitigate these variations, suppress inter-cell interference, and remove cell boundaries, thus improving service uniformity [4, 5, 6, 7]. These systems typically operate in time-division duplex (TDD) mode, relying on uplink pilots for channel state information (CSI) acquisition.

However, due to limited coherence time, pilot symbols must be reused, resulting in pilot contamination that degrades CSI quality and limits performance [4]. Optimal pilot assignment is therefore crucial in CF-mMIMO, but the non-deterministic polynomial-time (NP)-hard nature of the problem motivates low-complexity heuristic solutions. Existing approaches include random and greedy allocation schemes [4], spatial-separation and clustering-based methods [8, 9], and graph-theoretic formulations [10, 11], which primarily aim at reducing pilot-induced interference. More recent works instead optimize pilot assignment to maximize the system sum rate, using techniques such as Tabu search and Hungarian-based iterative algorithms, achieving near-optimal performance at manageable complexity [12, 13].

With the advent of integrated sensing and communications (ISAC), pilot signals are required to support both communication and sensing tasks, such as ranging and radar parameter estimation [14, 15, 16], making unimodular (unit-circle) pilot designs yielding favorable autocorrelation properties for sensing [17, 18] ideal to CF-mMIMO ISAC systems.

Motivated by these observations, we propose a manifold-optimization scheme to design optimized unimodular pilot sequences for ISAC. The problem is reformulated as an unconstrained optimization task on the unit-modulus manifold, enabling the use of efficient tangent-space-based algorithms that exploit the geometry of the feasible set and scale well to large systems [19, 20]. Unlike conventional pilot allocation methods that assign predefined orthogonal sequences, the proposed approach jointly designs all pilot symbols under unimodularity constraints, effectively mitigating pilot contamination while preserving favorable sensing properties. This design naturally supports ISAC operation, as recently advocated in [21, 22], by enabling the same pilots to be used for reliable channel estimation and high-resolution sensing, without exploiting data symbols for sensing purposes. Simulation results show that the proposed algorithm consistently outperforms benchmark pilot allocation schemes, achieving higher sum rates and enhanced sensing capability, while maintaining communication performance comparable to state-of-the-art (SotA) methods. Overall, this work demonstrates that manifold-based unimodular pilot design is a promising and flexible tool for joint communication and sensing in CF-mMIMO systems.

II System Model

Consider a CF-mMIMO system with $L$ AP s serving $K$ user equipment (UE), in the same time-frequency resource block. Each UE and AP is equipped with a single antenna, and the channel between the $\ell$-th AP and the $k$-th UE is given by

[TABLE]

where $\beta_{\ell,k}\in\mathbb{R}$ is the large scale fading coefficient and $h_{\ell,k}\sim\mathcal{CN}(0,1)$ are independent and identically distributed (i.i.d) complex Gaussian random variables representing the small scale fading coefficients.

II-A Pilot Training Phase

During the pilot training phase, each UE s transmits a pilot signal of length $\tau$ to the AP s. We assume a predefined matrix $\mathbf{F}\in\mathbb{C}^{\tau\times\tau}$ that contains the orthogonal pilot sequences assigned to the UE s. Due to the limited duration of the wireless channel’s coherence interval, the number of available orthogonal pilot sequences is much smaller than the number of UE s (i.e. $K>\tau$), leading to pilot reuse.

More precisely, the $k$-th UE transmits a pilot choosen as the $k$-th column of the matrix $\mathbf{F}$ and is denoted as $\mathbf{f}_{k}\in\mathbb{C}^{\tau\times 1}$ that satisfies $\frac{1}{\tau}\|\mathbf{f}_{k}\|^{2}=1$ and $\mathbf{f}_{k}^{H}\mathbf{f}_{k^{\prime}}=0$ for $k\neq k^{\prime}$. The received pilot signal at the $\ell$-th AP is given by

[TABLE]

where $\rho_{p}$ is the pilot power and $\mathbf{n}_{\ell}\sim\mathcal{CN}(\mathbf{0},\mathbf{I}_{\tau})$ is a vector of circularly symmetric complex Gaussian noise at the $\ell$-th AP.

The $\ell$-th AP performs pilot-based channel estimating based on the received signal

[TABLE]

Following previous works e.g. [8, 12], the channel coefficient between the $\ell$-th AP and the $k$-th UE is estimated using the minimum mean square error (MMSE) estimator, i.e.

[TABLE]

where $c_{\ell,k}$ is the MMSE coefficient given by

[TABLE]

and the power of the estimated channel coefficient is

[TABLE]

The achievable uplink rate for the $k$-th user is given by

[TABLE]

where the SINR for the $k$-th user is as in equation (10).

II-B Sensing Constraint

Under the ISAC paradigm, pilot signals used for channel estimation in wireless systems can be repurposed for sensing applications, such as radar parameter estimation (RPE), without requiring additional dedicated resources [23]. To that end, we propose to design pilot sequences that serve a dual purpose: enabling accurate channel estimation for communication and providing favorable properties for sensing tasks. This dual use of pilots for both communication and sensing has not been previously addressed in the context of pilot design for CF-mMIMO, and thus represents a novel contribution of this work. As shown in [24], perfect detection of the RPE is achieved when the elements of the radar signal are each unimodular complex numbers (i.e., $|\bm{r}|=1,\forall n$) in the frequency domain. This property avoids the infamous masking effect, where echoes from strong targets can obscure those from weak targets, making it difficult to detect and estimate weaker signals.

III Manifold Optimization for Pilot Allocation

III-A Problem Formulation

The pilot allocation problem in cell-free massive MIMO systems is fundamentally challenging due to the limited number of available orthogonal pilot sequences and the need to serve a large number of users. Traditional pilot assignment strategies, such as random or greedy algorithms, often focus on minimizing pilot contamination or maximizing spatial separation, but they do not directly optimize the overall system throughput. Furthermore, the emergence of ISAC applications introduces additional requirements on the pilot design, such as unimodularity in the frequency domain for sensing tasks.

Motivated by these considerations, we seek a pilot allocation strategy that not only mitigates pilot contamination but also maximizes the sum rate of the system, while satisfying constraints relevant for both communication and sensing. This leads to a highly non-convex optimization problem, where the pilot combining matrix must be designed to optimize the achievable rates under nonlinear constraints.

To address this, we formulate pilot allocation as a manifold optimization problem.

[TABLE]

where $\mathcal{F}\{\cdot\}$ denotes the Fourier transform operation, and the unimodularity constraint ensures that each element of the pilot matrix has unit modulus in the frequency domain, which as shown in the previous section, is crucial for perfect autocorrelation properties and avoiding the masking effect in sensing applications.

This formulation captures the essence of the pilot allocation challenge in cell-free massive MIMO systems with ISAC constraints: it is a non-convex optimization problem with nonlinear constraints, making it intractable for conventional methods.

III-B Formal Definitions

Manifold optimization is a powerful tool for solving optimization problems on manifolds, which are spaces that are smooth and possibly not linear. Our goal is to project our problem as an unconstrained optimization problem on a manifold

[TABLE]

such that simple optimization algorithms can be applied.

The smoothness can be modelled by the unit Sphere in $\mathbb{R}^{d}$, which is defined as

[TABLE]

The definition above can be used to capture the idea that $\mathcal{S}^{d-1}$ can be locally approximated by a linear space. This is called the tangent space of the manifold and is defined as

[TABLE]

When utilizing the most fundamental methods to adress problem 12 (i.e the gradient descent method), it must be noted that $\mathcal{S}^{d-1}$ is not a linear space and the gradient descent method cannot be directly applied.

To address this issue, the notion that smooth manifolds can be locally approximated by linear spaces is used. Moreover the retraction operator is presented as a way to project the results of the optimization steps back to the manifold

[TABLE]

In case of our problem the pilot matrix to be designed $\bar{\mathbf{F}}$ is part of the set of all $\tau\times K$ matrices with the entries on the unit circle. The search space for $\bar{\mathbf{F}}$ is a product of several circles, which is an embedded submanifold of $\mathbb{C}^{\tau\times K}$ and can be formally defined as

[TABLE]

This manifold is endowed with a Riemannian metric, which is defined as the inner product on the tangent space

[TABLE]

where $\mathbf{u},\mathbf{v}\in T_{\mathbf{X}}\mathcal{C}(\tau,K)$ are tangent vectors at the point $\mathbf{X}$ on the manifold.

However, as it was mentioned above, the gradient of the objective function is not defined on the manifold. To address this issue, we can use the Euclidean gradient of the objective function, and then project it onto the tangent space of the manifold at the point $\mathbf{X}$, using:

[TABLE]

III-C Conjugate Gradient Ascent algorithm

In the following subsection, we describe the Conjugate Gradient Ascent algorithm used to find the optimal pilot combining matrix $\bar{\mathbf{F}}$ on the complex circle manifold. This algorithm iteratively updates the pilot matrix to maximize the sum rate objective, while ensuring the unimodularity constraint is satisfied at each step. The procedure is summarized in Algorithm 1.

The algorithm starts by initializing $\bar{\mathbf{F}}^{(0)}$ as a random matrix on the manifold $\mathcal{C}(\tau,K)$. At each iteration, the Riemannian gradient of the objective function is computed and projected onto the tangent space of the manifold.

The step size $\alpha^{(i)}$ is determined via Armijo line search to ensure sufficient increase in the objective function [25]. The pilot matrix is then updated using the retraction operator, which projects the new iterate back onto the manifold. The process repeats until the change in the objective function falls below a predefined threshold $\epsilon$, or the maximum number of iterations is reached.

To compute the gradient of the achievable rate with respect to the pilot sequence $\mathbf{f}_{k}$, we apply the chain rule

[TABLE]

Let $\text{SINR}_{k}=\frac{\text{N}}{\text{D}}$, where N and D denote the numerator and denominator, respectively. The gradient is then

[TABLE]

III-C1 Gradient of the Numerator

[TABLE]

[TABLE]

where

[TABLE]

III-C2 Gradient of the Denominator

The denominator consists of three terms $\text{D}=\text{D}_{1}+\text{D}_{2}+\text{D}_{3}$, where

[TABLE]

The gradients of each term are

[TABLE]

[TABLE]

[TABLE]

By combining the above results, the overall gradient $\frac{\partial\text{R}_{k}}{\partial\mathbf{f}_{k}}$ can be efficiently computed and used in the manifold optimization algorithm 1.

III-D Performance Analysis

The performance of the proposed algorithms is quantitatively evaluated through simulations. The effects of shadow fading correlation and channel correlation on the achievable rate are ignored for simplicity. A total of $M$ APs and $K$ users are uniformly distributed at random within a square of size $D\times D\text{ m}^{2}$.

For all simulation scenarios, system parameters are set according to Table I, with the values $\bar{\rho}_{\mathrm{u}}$ and $\bar{\rho}_{\mathrm{p}}$ representing the transmit powers for uplink data and pilot signals, respectively; and the normalized transmit SNRs $\rho_{\mathrm{d}}^{\mathrm{cf}}$, $\rho_{\mathrm{u}}^{\mathrm{cf}}$, and $\rho_{\mathrm{p}}^{\mathrm{cf}}$ computed by dividing the respective transmit powers by the noise power. To mitigate edge effects and emulate an infinitely large network, the simulation area is wrapped at the boundaries, resulting in each square having eight neighboring regions. The net throughput per user, accounting for the overhead due to channel estimation, is evaluated as follows

[TABLE]

where $B$ is the bandwidth, $\tau$ is the number of pilot symbols, $T$ is the total number of symbols in a frame, and $\text{R}_{k}$ is the achievable rate for the $k$-th user given by (7).

The large-scale fading coefficients $\beta_{\ell,k}$ in (1) are computed using the path loss model

[TABLE]

where $\text{PL}_{\ell,k}$ is the path loss between the $\ell$-th AP and the $k$-th UE, $z_{\ell,k}\sim\mathcal{N}(0,1)$, and $\sigma_{\mathrm{sh}}$ is the standard deviation of the shadow fading in dB. To keep the simulations consistent with prior work, the path loss and other related parameters are defined as in [4].

III-D1 Communication Performance

To assess the channel estimation performance of the proposed pilot design, we adopt the net throughput per user as the primary metric, following the approach in [4].

Figure 3 illustrates the cumulative distribution function (CDF) of the uplink achievable per-user throughput for a fixed number of UEs and pilot sequences, and for 2 different numbers of APs: a moderate density of $L=40$ and a high density of $L=100$.

In Figure 3, the number of pilot sequences $\tau$ is increased while keeping the number of APs $M$ and UEs $K$ fixed. The pilot-to-UE ratio starts at $1/2$ (i.e., $\tau=K/2$) and increases up to $1$ (i.e., $\tau=K$). As the number of available pilot sequences grows, the proposed design consistently achieves higher median rates compared to conventional methods.

Figure 3 shows the median achievable per-user throughput as the number of users $K$ increases, with the number of APs $M$ and pilot sequences $\tau$ held constant. Here, $K$ ranges from $2\tau$ (moderate overload) up to $4\tau$ (heavy overload), highlighting system performance under growing user density and constrained pilot resources. The proposed pilot design consistently outperforms conventional methods, with the CDF curves shifted to the right and a notably higher minimum achievable rate, and performs only slightly worse than the tabu search.

III-D2 Sensing Performance

The autocorrelation function (ACF) of a signal is a fundamental metric for range estimation, especially in matched filtering at the receiver. Depending on the presence of a cyclic prefix (CP), the ACF can be defined as either the periodic or aperiodic self-convolution of the signal. For clarity and without loss of generality (wlg), we focus on the aperiodic case, the extension to CP-inclusive scenarios is straightforward. The aperiodic autocorrelation function (A-ACF) for a signal $\mathbf{x}\in\mathbb{C}^{N}$ is given by

[TABLE]

where $\mathbf{J}_{k}$ is the $k$-th shift matrix defined as

[TABLE]

Figure 4 illustrates the normalized sidelobe level of pilots (length $\tau=10$) designed using the proposed method, compared to conventional methods such as greedy pilot assignment and tabu search.

The proposed pilot design achieves a perfect autocorrelation property, with a single peak at the origin and zero elsewhere, which is essential for effective sensing applications. Meanwhile, the other methods exhibit a blunt ACF with significant sidelobes, indicating that they cannot effectively distinguish between different targets.

IV Conclusion

We proposed a novel, sensing enabling, pilot design algorithm for CF-mMIMO systems. The pilot design leverages manifold optimization to directly construct pilot sequences that maximize the system sum rate while enforcing unimodularity constraints for joint communication and sensing. By intelligently exploiting the concept of orthogonality, the proposed scheme reduces pilot contamination, resulting in improved overall performance. Simulation results show that the proposed pilot design achieves communication performance comparable to state-of-the-art algorithms and superior sensing capabilities due to its perfect autocorrelation properties.