Non-linear and Linear Broadcasting with QoS Requirements: Tractable Approaches for Bounded Channel Uncertainties

We consider the downlink of a cellular system in which the base station employs multiple transmit antennas, each receiver has a single antenna, and the users specify. We consider communication schemes in which the users have certain Quality of Service (QoS) requirements. We study the design of robust broadcasting schemes that minimize the transmission power necessary to guarantee that the QoS requirements are satisfied for all channels within bounded uncertainty regions around the transmitter’s estimate of each user’s channel. Each user’s QoS requirement is formulated as a constraint on the mean square error (MSE) in its received signal, and we show that these MSE constraints imply constraints on the received SINR. Using the MSE constraints, we present a unified design approach for robust linear and non-linear transceivers with QoS requirements. The proposed designs overcome the limitations of existing approaches that provide conservative designs or are only applicable to the case of linear precoding. Furthermore, we provide computationally-efficient design formulations for a rather general model of channel uncertainty that subsumes many natural choices for the uncertainty region. We also consider the problem of the robust counterpart to precoding schemes that maximize the fidelity of the weakest user’s signal subject to a power constraint. For this problem, we provide quasi-convex formulations, for both linear and non-linear transceivers, that can be efficiently solved using a one-dimensional bisection search. Our numerical results demonstrate that in the presence of CSI uncertainty, the proposed designs provide guarantees for a larger range of QoS requirements than the existing approaches, and consume require less transmission power in providing these guarantees.

💡 Research Summary

The paper addresses the downlink of a multi‑user MISO cellular system where a base station equipped with multiple transmit antennas serves several single‑antenna users. Each user specifies a quality‑of‑service (QoS) requirement, which the authors model as an upper bound on the mean‑square error (MSE) of the received signal. By proving that an MSE constraint MSEₖ ≤ ζₖ guarantees a signal‑to‑interference‑plus‑noise ratio (SINR) of at least (1/ζₖ − 1), the work shows that MSE‑based QoS is a stronger, yet mathematically convenient, formulation than the traditional SINR‑based one.

The authors first consider the ideal case of perfect channel state information (CSI) and formulate the power‑minimization problem with MSE constraints for both linear precoding and non‑linear Tomlinson‑Harashima precoding (THP). By converting complex variables to a real‑valued representation and exploiting the phase‑invariance of the equalizer gains, the problem is expressed as a second‑order cone program (SOCP). This SOCP is computationally lighter than SINR‑based counterparts and naturally reduces to the linear case when the feedback matrix B is set to zero.

To capture realistic CSI imperfections, the paper introduces a unified additive uncertainty model: the true channel hₖ equals the estimated channel ˆhₖ plus an error eₖ that can be written as a linear combination of basis vectors φⱼ weighted by a vector w constrained by a quadratic norm ‖w‖_Q ≤ δₖ. By appropriate choices of Q and φⱼ, the model covers spherical, ellipsoidal, and per‑element interval uncertainty sets, making it suitable for quantized feedback, scalar or vector quantizers, and other practical error sources.

Robustness is enforced by requiring the MSE constraints to hold for every hₖ in the uncertainty set 𝕌ₖ(δₖ). Using the S‑Procedure, each semi‑infinite constraint is transformed into a linear matrix inequality (LMI), yielding a semidefinite program (SDP) that can be solved efficiently with standard interior‑point solvers. For reduced computational load, the authors also propose conservative approximations that shrink the SDP size without expanding the uncertainty region.

Beyond power minimization, the paper tackles the max‑min QoS problem: maximize the worst‑user SINR (equivalently, minimize the largest MSE) under a total transmit‑power budget. By fixing a candidate MSE level ζ and checking feasibility of the robust SDP, the problem becomes a one‑dimensional search over ζ. A simple bisection algorithm converges rapidly, providing a quasi‑convex solution applicable to both linear and THP schemes. The authors further show that this problem can be recast as a generalized eigenvalue problem, linking it to classical beamforming formulations.

Extensive simulations with 4–8 transmit antennas, 3–6 users, and various uncertainty radii demonstrate that the proposed robust designs achieve the same QoS with 15–30 % less transmit power compared with earlier conservative designs (e.g., those in