This paper investigates semi-blind channel estimation for massive multiple-input multiple-output (MIMO) systems. To this end, we first estimate a subspace based on all received symbols (pilot and payload) to provide additional information for subsequent channel estimation. This additional information enhances minimum mean square error (MMSE) channel estimation. Two variants of the linear MMSE (LMMSE) estimator are formulated, where the first one solves the estimation within the subspace, and the second one uses a subspace projection as a preprocessing step. Theoretical derivations show that the latter method achieves superior mean square error performance for uncorrelated Rayleigh fading. Further, we provide asymptotical insights on how the proposed MMSE-based channel estimation strategy outperforms the unbiased Cramer-Rao bound. Subsequently, we introduce parameterizations of these semi-blind LMMSE estimators based on two different conditional Gaussian latent models, i.e., the Gaussian mixture model and the variational autoencoder. Both models learn the propagation environment’s underlying channel distribution based on training data and serve as generative priors for our semi-blind channel estimation. Extensive simulations on real-world measurement data and spatial channel models show that the proposed methods achieve superior performance compared to state-of-the-art semi-blind channel estimators in terms of MSE.

This paper investigates using side information in minimum mean square error (MMSE) estimation. We propose a direction-of-arrival (DoA)-aided two-stage channel estimation technique that utilizes information about the dominant direction of the channel. To this end, the decomposition of the MMSE channel estimation into two orthogonal subspaces is formulated. After estimating the channel along the dominant direction, we utilize a Gaussian mixture model to estimate the conditionally Gaussian distributed random vector, which represents the multipath propagation. The proposed two-stage estimator allows pre-computing the respective estimation filters, tremendously reducing the computational complexity. Numerical simulations depict the superior performance of our proposed two-stage estimation approach compared to state-of-the-art methods.

In this work, we propose a Gaussian mixture model (GMM)-based pilot design scheme for downlink (DL) channel estimation in single- and multi-user multiple-input multiple-output (MIMO) frequency division duplex (FDD) systems. In an initial offline phase, the GMM captures prior information on the channel statistics through training, which is then utilized for pilot design. In the single-user case, the GMM is utilized to construct a codebook of pilot matrices and, once shared with the mobile terminal (MT), can be employed to determine a feedback index at the MT. This index selects a pilot matrix from the constructed codebook, eliminating the need for online pilot optimization. We further establish a sum conditional mutual information (CMI)-based pilot optimization framework for multi-user MIMO (MU-MIMO) systems. Based on the established framework, we utilize the GMM for pilot matrix design in MU-MIMO systems. The analytic representation of the GMM enables the adaptation to any signal-to-noise ratio (SNR) level and pilot configuration without re-training. Additionally, an adaption to any number of MTs is facilitated. Extensive simulations demonstrate the superior performance of the proposed pilot design scheme compared to state-of-the-art approaches. The performance gains can be exploited, e.g., to deploy systems with fewer pilots.

In this work, we propose to utilize a variational autoencoder (VAE) for channel estimation (CE) in underdetermined (UD) systems. The basis of the method forms a recently proposed concept in which a VAE is trained on channel state information (CSI) data and used to parameterize an approximation to the mean squared error (MSE)-optimal estimator. The contributions in this work extend the existing framework from fully-determined (FD) to UD systems, which are of high practical relevance. Particularly noteworthy is the extension of the estimator variant, which does not require perfect CSI during its offline training phase. This is a significant advantage compared to most other deep learning (DL)-based CE methods, where perfect CSI during the training phase is a crucial prerequisite. Numerical simulations for hybrid and wideband systems demonstrate the excellent performance of the proposed methods compared to related estimators.

In this work, we propose two methods that utilize data symbols in addition to pilot symbols for improved channel estimation quality in a multi-user system, so-called semi-blind channel estimation. To this end, a subspace is estimated based on all received symbols and utilized to improve the estimation quality of a Gaussian mixture model-based channel estimator, which solely uses pilot symbols for channel estimation. Both of the proposed approaches allow for parallelization. Even the precomputation of estimation filters, which is beneficial in terms of computational complexity, is enabled by one of the proposed methods. Numerical simulations for real channel measurement data available to us show that the proposed methods outperform the studied state-of-the-art channel estimators.

This work extends two previously proposed semi-blind channel estimators to a more general multi-user multiple-input-multiple-output (MU-MIMO) system. These estimators utilize data symbols in addition to pilot symbols to enhance the channel estimation quality. Based on all received signals, a subspace is calculated, which enhances the Gaussian mixture model based channel estimator. To estimate this subspace, we consider the inherent additional degrees of freedom in terms of precoding in MU-MIMO systems. Numerical simulations for different scenarios show that the extended methods outperform the studied state-of-the-art channel estimators.

Discrete Fourier transform (DFT) codebook-based solutions are well-established for limited feedback schemes in frequency division duplex (FDD) systems. In recent years, data-aided solutions have been shown to achieve higher performance, enabled by the adaptivity of the feedback scheme to the propagation environment of the base station (BS) cell. In particular, a versatile limited feedback scheme utilizing Gaussian mixture models (GMMs) was recently introduced. The scheme supports multi-user communications, exhibits low complexity, supports parallelization, and offers significant flexibility concerning various system parameters. Conceptually, a GMM captures environment knowledge and is subsequently transferred to the mobile terminals (MTs) for online inference of feedback information. Afterward, the BS designs precoders using either directional information or a generative modeling-based approach. A major shortcoming of recent works is that the assessed system performance is only evaluated through synthetic simulation data that is generally unable to fully characterize the features of real-world environments. It raises the question of how the GMM-based feedback scheme performs on real-world measurement data, especially compared to the well-established DFT-based solution. Our experiments reveal that the GMM-based feedback scheme tremendously improves the system performance measured in terms of sum-rate, allowing to deploy systems with fewer pilots or feedback bits.

We propose a versatile feedback scheme for both single- and multi-user multiple-input multiple-output (MIMO) frequency division duplex (FDD) systems. Particularly, we propose utilizing a Gaussian mixture model (GMM) with a reduced number of parameters for codebook construction, feedback encoding, and precoder design. The GMM is fitted offline at the base station (BS) to uplink training samples to approximate the channel distribution of all possible mobile terminals (MTs) within the BS cell. Subsequently, a codebook is constructed, with each element based on one GMM component. Extracting directional information from the codebook or exploiting the GMM’s sample generation ability facilitates joint precoder design for a multi-user MIMO system using state-of-the-art precoding algorithms. After offloading the GMM to the MTs, they can easily determine their feedback by selecting the index of the GMM component with the highest responsibility for their received pilot signal. This strategy exhibits low complexity and supports parallelization. Simulations demonstrate that the proposed approach outperforms conventional methods, which either estimate the channel and utilize a Lloyd codebook or use a deep neural network to determine the feedback in terms of spectral efficiency or sum-rate. The performance gains can be exploited to deploy systems with fewer pilots or feedback bits.

This paper investigates a channel estimator based on Gaussian mixture models (GMMs). We fit a GMM to given channel samples to obtain an analytic probability density function (PDF) which approximates the true channel PDF. Then, a conditional mean estimator (CME) corresponding to this approximating PDF is computed in closed form and used as an approximation of the optimal CME based on the true channel PDF. This optimal estimator cannot be calculated analytically because the true channel PDF is generally not available. To motivate the GMM-based estimator, we show that it converges to the optimal CME as the number of GMM components is increased. In numerical experiments, a reasonable number of GMM components already shows promising estimation results.

We study the problem of predicting channel coefficients from one-bit quantized observations in an environment of a moving user who sends pilots to a base station. To start with, we propose a prediction algorithm which consists of two stages. The first stage aims at reconstructing the high-resolution (pre-quantization) receive signal. The second stage then predicts channel coefficients from this reconstructed signal. A drawback of this algorithm is that certain second moments of the channel statistics are required. In case of high-resolution (no quantization) observations, a recently introduced neural network based approach was able to predict channels even without the use of second order statistics. A low-SNR formulation of the proposed two stage algorithm motivates us to employ the neural network based method also in the case of one-bit quantization. Numerical simulations demonstrate the validity of this approach. We observe that the obtained channel predictor can compete with the algorithm that makes use of the second order statistics.