In this letter, we develop a continuous fluid antenna (FA) framework for uplink channel estimation in cell-free massive multiple-input and multiple-output (CF-mMIMO) systems. By modeling the wireless channel as a spatially correlated Gaussian random field, channel estimation is formulated as a Gaussian process (GP) regression problem with motion-constrained spatial sampling. Closed-form expressions for the linear minimum mean squared error (LMMSE) estimator and the corresponding estimation error are derived. A fundamental comparison with discrete port-based architectures is established under identical position constraints, showing that continuous FA sampling achieves equal or lower estimation error for any finite pilot budget, with strict improvement for non-degenerate spatial correlation models. Numerical results validate the analysis and show the performance gains of continuous FA sampling over discrete baselines.
using a single radio frequency (RF) chain. Since wireless channels exhibit significant spatial variations over distances on the order of the carrier wavelength, even small antenna displacements can result in different channel realizations [6].
Most existing FAS-assisted studies discretize the fluid motion into a finite set of antenna ports and optimize port selection during data transmission or uplink training [7], [8]. While such discrete port-based architectures provide performance gains over fixed-position antennas (FPA), they suffer from two fundamental limitations. First, they approximate the inherently continuous motion of the fluid antenna using a finite set of candidate positions, leading to information loss. Second, they require frequent port switching, which may introduce latency, energy consumption, and hardware constraints, particularly when switching occurs at symbol-level time scales.
Recent works have proposed continuous fluid antenna (FA) models to analyze fading statistics, outage probability, and level-crossing behavior in single-link scenarios [9], [10]. However, a unified and rigorous treatment of continuous FA motion in CF-mMIMO systems, particularly from the perspective of uplink channel estimation, remains largely unexplored. In particular, it is unclear to what extent discrete port selection fundamentally limits channel estimation accuracy compared to continuous FA sampling under realistic position constraints.
Addressing the limitations of discrete port-based FAS, this paper develops a continuous FA framework for uplink channel estimation in CF-mMIMO systems, as shown in Fig. 1. Each AP is equipped with a single FA whose radiating element changes the position continuously during the training phase, enabling spatial channel sampling without discretization. By modeling the wireless channel as a spatially correlated complex Gaussian random field, uplink channel estimation is formulated as a Gaussian process (GP) regression problem with motion-constrained sampling. We derive closed-form expressions for the linear minimum mean-squared error (LMMSE) estimator and the corresponding estimation error, explicitly revealing the role of antenna motion and spatial correlation. A fundamental performance comparison with discrete portbased architectures is then established. It is shown that, for any finite pilot budget, continuous FA sampling achieves an estimation error no larger than that of any discrete scheme with a finite number of ports, with strict improvement for non-degenerate spatial correlation models. Numerical results validate the analysis and illustrate the resulting gains under practical system parameters.
As shown in Fig. 1, we consider the uplink of a CF-mMIMO system composed of L geographically distributed APs jointly serving K single-antenna user equipments (UEs) on the same time-frequency resource. We assume the APs are connected to a central processing unit (CPU) through fronthaul links and cooperate in channel estimation and data detection. Also, time-division duplexing (TDD) operation is assumed, such that uplink and downlink channels are reciprocal. Each AP is equipped with a single FA, whose radiating element can continuously change the position along a one-dimensional segment of finite length ℓ, normalized by the carrier wavelength. The FA is connected to a single RF chain, so that at any given time instant only one antenna position is active at each AP. The FA motion is controlled locally at each AP and is assumed to be synchronized with the uplink pilot transmission phase.
Following the user-centric CF-mMIMO paradigm [3], each user k ∈ {1, . . . , K} is served only by a subset of APs denoted by L k ⊆ {1, . . . , L}, which is determined based on large-scale fading coefficients. Conversely, each AP l serves a subset of users denoted by K l . This association is assumed to remain fixed over the considered coherence block. Each UE transmits pilot symbols for channel estimation followed by uplink data symbols. Additionally, perfect synchronization among UEs and APs is assumed, and all signals are narrowband such that frequency-flat fading applies.
Let x l ∈ [0, ℓ] denote the instantaneous position of the fluid antenna at AP l, measured along the antenna’s movement axis and normalized by the carrier wavelength. The uplink channel coefficient between UE k and AP l when the FA is located at position x l is denoted by h k,l (x l ) ∈ C.
We model the small-scale fading component of the uplink channel as a spatially continuous complex Gaussian random field indexed by the antenna position, in contrast to discrete parametric sparse-array models. Specifically, for each UE-AP pair (k, l), the channel h k,l (x) is modeled as a zero-mean, circularly symmetric, spatially correlated complex Gaussian random field 1 satisfying E[h k,l (x)] = 0, and
where β k,l > 0 is the large-scale fading coefficient accounting for path-loss and shadowing between UE k and AP l, and κ(x, x ′ ) is a no
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