Channel Estimation based on Gaussian Mixture Models with Structured Covariances

Abstract

We present new fundamental results for the mean square error (MSE)-optimal conditional mean estimator (CME) in one-bit quantized systems for a Gaussian mixture model (GMM) distributed signal of interest, possibly corrupted by additive white Gaussian noise (AWGN). We first derive novel closed-form analytic expressions for the Bussgang estimator, the well-known linear minimum mean square error (MMSE) estimator in quantized systems. Afterward, closed-form analytic expressions for the CME in special cases are presented, revealing that the optimal estimator is linear in the one-bit quantized observation, opposite to higher resolution cases. Through a comparison to the recently studied Gaussian case, we establish a novel MSE inequality and show that that the signal of interest is correlated with the auxiliary quantization noise. We extend our analysis to multiple observation scenarios, examining the MSE-optimal transmit sequence and conducting an asymptotic analysis, yielding analytic expressions for the MSE and its limit. These contributions have broad impact for the analysis and design of various signal processing applications.

Learning a Gaussian Mixture Model From Imperfect Training Data for Robust Channel Estimation

“…By lowercase bold symbols, we refer to the vectorized expressions of (1), e.g., h = vec(H), which we use interchangeably. We consider E[h] = 0 and E[∥h∥ 2 2 ] = 𝑁 rx 𝑁 tx , which can be ensured by pre-processing, allowing to define the SNR as 1/𝜂 2 .…”

“…Assuming knowledge of the observation's SNR, the LS estimate is normalized as Ĥinit = (1 + 𝜂 2 ) − 1 2 ĤLS since the deployed DM is variance-preserving, cf. (2). Afterward, the observation is transformed into the angular domain via Ĥang = fft( Ĥinit ).…”

“…We additionally compare the proposed DM to the GMMbased estimator from [1] with either full covariance matrices with 𝐾 = 128 components ("GMM") or the Kronecker version thereof with (𝐾 rx , 𝐾 tx ) = (16, 8) components ("GMM Kron"), also yielding a total of 𝐾 = 128 components, cf. [17]. We also evaluate the score-based channel estimator from [9] based on a convolutional RefineNet architecture with 𝐷 = 6 residual blocks with 𝐿 = 7 layers with 𝑊 = 32 channels after the first layer and a maximum channel size of 𝐶 max = 128, and a kernel size of 𝑘 = 3 in each dimension.…”

“…This success is built on the great importance of inferring knowledge of the unknown and generally complex channel distribution of, e.g., a base station (BS) environment through a representative dataset. Consequently, the development of advanced channel estimation methodologies has ensued, primarily relying on state-of-the-art generative models such as Gaussian mixture models (GMMs) [1], mixture of factor analyzers (MFAs) [2], generative adversarial networks (GANs) [3], or variational autoencoders (VAEs) [4].…”

Semi-Blind Joint Channel and Symbol Estimation in IRS-Assisted Multiuser MIMO Networks

“…Sequential channel estimation methods are proposed in [15]- [17], where the BS-UE and different direct or IRS-assisted channels are estimated sequentially. The work [18] considers a one-bit IRS for the direction of arrival (DoA) estimation by solving an atomic-norm-based sparse recovery problem, while [19] and [20] propose neural network-based channel estimation methods for IRS-assisted networks.…”