Gaussian Mixture Model (GMM) Estimator

This repository provides an estimator for the problem of recovering the channel vector $\mathbf{h} \in \mathbb{C}^N$ from the observation $$\mathbf{y} = \mathbf{A} \mathbf{h} + \mathbf{n}$$ where $\mathbf{A} \in \mathbb{C}^{m\times N}$ is the observation matrix and $\mathbf{n} \sim \mathcal{N}_{\mathbb{C}}(\mathbf{0}, \mathbf{\Sigma})$ is a realization of a complex Gaussian zero-mean random variable with known covariance matrix $\mathbf{\Sigma} \in \mathbb{C}^{m\times m}$. It is assumed that the channel $\mathbf{h} \sim f_{\mathbf{h}}$ is a realization of a random variable with unknown probability density function (PDF) $f_{\mathbf{h}}$.

Using the estimator is a two-step process.

Step 1: Train the estimator

Training data $\{\mathbf{h}_t\}_{t=1}^{T}$ is used to fit a GMM $$f^{(K)}(\mathbf{h}) = \sum_{k=1}^K p(k) \mathcal{N}_{\mathbb{C}}(\mathbf{h}; \mathbf{\mu}_k, \mathbf{C}_k)$$ where $K$ is the number of GMM components and $p(k)$, $\mathbf{\mu}_k$, and $\mathbf{C}_k$ are the mixing coefficient, mean vector, and covariance matrix of component $k$, respectively. After the fitting process is done, the GMM $f^{(K)}$ approximates the unknown PDF $f_{\mathbf{h}}$ of the channels.

Step 1 only needs to be done once and it can be done in a preparatory offline training phase.

Step 2: Estimate channels

Assuming that $f^{(K)}$ is a good approximation of $f_{\mathbf{h}}$, the GMM estimator is the function $g$ which minimizes the mean square error

$$\mathrm{E}\big[\| \mathbf{h} - g(\mathbf{y}) \|^2\big].$$

The GMM estimator $g$ can be found in closed form and is implemented in gmm_estimator.py. The file examples.py demonstrates how gmm_estimator.py can be used. See also the next section below.

For more details, kindly take a look at the first two references below.

Requirements

The code is written in Python. Required packages are numpy, scikit-learn, and scipy. To generate plots, matplotlib is required as well.

How to use the estimator

The file examples.py demonstrates the use of the GMM estimator in various settings. Invoking

python example.py

runs all examples. Alternatively,

python example.py --nr n

runs example $n$. The examples demonstrate the following:

1. The observation matrix $\mathbf{A} = \mathbf{I}$ is the identity matrix and full GMM covariance matrices are used.
2. The observation matrix is a selection matrix and full GMM covariance matrices are used.

References

This repository is joint work of Michael Koller and Benedikt Fesl.

Submodule

The implementation makes use of Benedikt Fesl's repository GMM_cplx which allows fitting a GMM with complex-valued quantities.

Papers

The following reference provides more details and properties of the GMM estimator.

• Koller, Fesl, Turan, Utschick, "An Asymptotically MSE-Optimal Estimator Based on Gaussian Mixture Models," IEEE Trans. Signal. Process., 2022. [IEEEXplore] [arXiv]

The estimator has been used in the following references.

• N. Turan, B. Fesl, M. Grundei, M. Koller, and W. Utschick, “Evaluation of a Gaussian Mixture Model-based Channel Estimator using Measurement Data,” in Int. Symp. Wireless Commun. Syst. (ISWCS), 2022. [IEEEXplore]
• B. Fesl, M. Joham, S. Hu, M. Koller, N. Turan, and W. Utschick, “Channel Estimation based on Gaussian Mixture Models with Structured Covariances,” in 56th Asilomar Conf. Signals, Syst., Comput., 2022, pp. 533–537. [IEEEXplore]
• B. Fesl, N. Turan, M. Joham, and W. Utschick, “Learning a Gaussian Mixture Model from Imperfect Training Data for Robust Channel Estimation,” IEEE Wireless Commun. Lett., 2023. [IEEEXplore]
• M. Koller, B. Fesl, N. Turan and W. Utschick, "An Asymptotically Optimal Approximation of the Conditional Mean Channel Estimator Based on Gaussian Mixture Models," IEEE Int. Conf. Acoust., Speech, Signal Process. (ICASSP), 2022, pp. 5268-5272. [IEEEXplore] [arXiv]
• B. Fesl, A. Faika, N. Turan, M. Joham, and W. Utschick, “Channel Estimation with Reduced Phase Allocations in RIS-Aided Systems,” in IEEE 24th Int. Workshop Signal Process. Adv. Wireless Commun. (SPAWC), 2023. [arXiv]
• N. Turan, B. Fesl, M. Koller, M. Joham, and W. Utschick, “A Versatile Low-Complexity Feedback Scheme for FDD Systems via Generative Modeling,” 2023, arXiv preprint: 2304.14373. [arXiv]
• N. Turan, B. Fesl, and W. Utschick, "Enhanced Low-Complexity FDD System Feedback with Variable Bit Lengths via Generative Modeling," in 57th Asilomar Conf. Signals, Syst., Comput., 2023. [arXiv]
• N. Turan, M. Koller, B. Fesl, S. Bazzi, W. Xu and W. Utschick, "GMM-based Codebook Construction and Feedback Encoding in FDD Systems,"in 56th Asilomar Conf. Signals, Syst., Comput., 2022, pp. 37-42. [IEEEXplore]