Asymptotically Optimal Local Receiver in Uplink CF-mMIMO: A Functional-Variational Analysis

In cell-free massive multiple-input multiple-output (CF-mMIMO) systems, the canonical uplink local receiver is the local minimum mean square error (LMMSE) receiver with large-scale fading decoding (LSFD) at the central processing unit (CPU). The LSFD coefficients are derived under the use-and-then-forget (UatF) lower bound of the ergodic rate, and computing these coefficients introduces additional fronthaul overhead and computational complexity at the CPU. This paper investigates local receiver design directly from the true ergodic-rate objective under perfect local channel state information (CSI). By introducing an expectation-based constraint and leveraging large-system random matrix theory, we develop a functional-variational approach that yields the asymptotically optimal quasi-LMMSE (Q-LMMSE) receiver in closed form. A key insight is that the Q-LMMSE receiver shares the same direction as the conventional LMMSE receiver, differing only by an instantaneous CSI-dependent scalar, and thus incurs the same per-access point (AP) complexity. More importantly, this scalar varies across APs and implicitly provides adaptive weighting for the direct summation at the CPU, thereby completely eliminating the need for statistical LSFD coefficients and the associated CPU-side computational overhead. Numerical results demonstrate that the proposed Q-LMMSE receiver consistently outperforms the LMMSE-LSFD benchmark in terms of the ergodic rate, achieving approximately a {5\%} gain when the number of antennas per AP is low, while operating with strictly lower system-level complexity.