⚡ electrical engineering

Variational Bayes Estimation for Affine-Precoded Superimposed Pilots in Partially Connected Dual-Wideband Tera-Hertz MU-MIMO Systems

This paper proposes two affine precoding-based system models for partially connected dual-wideband terahertz MU-MIMO systems and develops a variational Bayesian inference algorithm to achieve robust, high-precision channel estimation using superimposed pilots while jointly estimating channel coefficients and learning sparsity structures.

Original authors: Abhisha Garg, Suraj Srivastava, Aditya K. Jagannatham

Published 2026-06-08

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CC BY 4.0

Original authors: Abhisha Garg, Suraj Srivastava, Aditya K. Jagannatham

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are trying to have a group conversation in a massive, echoing cathedral (the Terahertz communication system). The room is so big and the sound so high-pitched that the echoes behave strangely: some sounds bounce off the walls at different angles depending on their pitch, and the sound gets absorbed by the air itself. This is what engineers call the "dual-wideband effect"—a messy mix of time delays and frequency shifts that makes it incredibly hard to hear anyone clearly.

To fix this, the people in the room (the users) need to know exactly how the sound travels to the listener (the Base Station). This knowledge is called Channel State Information (CSI).

The Problem: The "Pilot" Bottleneck

Usually, to figure out the acoustics of the room, everyone stops talking and shouts a specific test word (a pilot) so the listener can measure the echo. But in a high-speed data world, stopping to shout test words wastes valuable time and slows down the actual conversation (data transmission).

The Solution: Instead of stopping, the speakers whisper their test words while they are talking. This is called Superimposed Pilots (SIP). It's like trying to hear a specific ringtone while someone is singing a song; you have to separate the ringtone from the singing to understand the room's acoustics.

The Two Strategies

The paper proposes two different ways to organize this "whispering while singing" scenario for a group of people:

The "Group Huddle" Approach (CP-JCE):
- Everyone uses the same secret code (precoding) to hide their test words.
- The listener tries to figure out the acoustics for the entire group at once.
- Analogy: Imagine the whole choir singing a chord together, and the conductor tries to figure out how the whole choir sounds in the room in one go.
- Pros: It's faster to set up because you only need two "codes" for the whole group.
- Cons: It's harder to untangle the mess, leading to slightly more confusion (error) when the group gets too big.
The "Soloist" Approach (USP-DCE):
- Every person gets their own unique secret code.
- The listener separates the group into individuals and figures out the acoustics for one person at a time.
- Analogy: Imagine each choir member has a unique instrument. The conductor listens to the violinist, then the flutist, then the drummer, figuring out the room's echo for each one individually.
- Pros: It's much more accurate because the listener isn't confused by overlapping signals.
- Cons: It takes more time and effort to generate and manage all the unique codes.

The Magic Tool: Variational Bayes

How does the listener actually separate the whispers from the singing in such a noisy, complex room? The paper uses a mathematical "super-brain" called Variational Bayes Estimation.

Think of this like a detective solving a mystery:

The Clues: The listener hears the mixed-up signal (singing + test words).
The Guess: The detective makes an educated guess about the room's acoustics (the channel).
The Refinement: The detective doesn't just guess once. They use a special method to constantly update their guess, learning from the "noise" and the "sparsity" (the fact that in these high-tech rooms, most of the sound paths are actually empty, and only a few are real).
The Result: The detective gets a highly accurate map of the room, even without knowing the exact noise level beforehand.

What the Paper Found

The researchers tested these two strategies in a simulated high-tech environment (using 64 antennas and terahertz frequencies). Here is the verdict:

Accuracy: The "Soloist" (USP-DCE) approach was more accurate. It figured out the room's acoustics better because it avoided the confusion of mixing everyone's signals together.
Speed & Complexity: The "Group Huddle" (CP-JCE) approach was computationally lighter (easier for the computer to process) because it handled everyone as one big block, but it made slightly more mistakes.
The Trade-off:
- If you have a very fast-moving environment (like a car driving by), the Group Huddle is better because it sets up faster.
- If you need the highest possible clarity and the environment is stable, the Soloist approach wins.

The Bottom Line

This paper is about finding the best way to listen to a group of people in a noisy, high-tech room without stopping the conversation. By using a clever mathematical detective (Variational Bayes) and two different organizational strategies (Group vs. Solo), they showed how to get crystal-clear communication even when the physics of the room is trying to mess things up.

Technical Summary: Variational Bayes Estimation for Affine-Precoded Superimposed Pilots in Partially Connected Dual-Wideband Tera-Hertz MU-MIMO Systems

Problem Statement
This work addresses the challenge of accurate Channel State Information (CSI) acquisition in Terahertz (THz) Multi-User Multiple Input Multiple Output (MU-MIMO) systems operating under a "dual-wideband" effect. This phenomenon arises from the coexistence of frequency-wideband effects (traditional frequency selectivity) and spatial-wideband effects (differential delays across large antenna arrays), leading to beam-split effects where Angle-of-Arrival (AoA) and Angle-of-Departure (AoD) vary across subcarriers.

Conventional pilot-based estimation methods incur significant bandwidth overhead, reducing data rates. While Superimposed Pilots (SIP) embed low-power pilots within data to mitigate this, they introduce interference and estimation complexity, particularly in THz channels characterized by high dimensionality, inherent sparsity, and severe model uncertainty due to molecular absorption and reflection losses. Existing methods often struggle with structural errors, convergence issues, or the inability to effectively model the sparse structure of THz channels under these complex conditions.

Methodology
The authors propose a system model for a partially connected THz MU-MIMO uplink using Orthogonal Frequency Division Multiplexing (OFDM). The Base Station (BS) utilizes a partially connected architecture with multiple subarrays, while User Equipment (UE) employs a fully-connected architecture. The THz channel is rigorously modeled to include line-of-sight (LoS), non-line-of-sight (NLoS) components, molecular absorption, and free-space losses.

To address the estimation problem, the paper introduces two distinct system models utilizing affine precoding with superimposed pilots:

Common Precoding with Joint Channel Estimation (CP-JCE): A single set of precoding matrices is used for all users, enabling joint estimation of the combined channel.
User-Specific Precoding with Decoupled Channel Estimation (USP-DCE): Unique precoding matrices are assigned to each user, allowing for the decoupling of received signals and individual channel estimation.

Both models are formulated as structured sparse channel estimation problems. The authors develop a Variational Bayesian Inference (VBI) algorithm to solve these problems. Unlike standard Bayesian approaches that rely solely on priors, this framework integrates information from the likelihood function to refine the posterior distribution. Key aspects of the methodology include:

Sparse Modeling: The channel is represented in an extended virtual domain using a sparsifying dictionary derived from array response vectors that account for spatial-wideband effects.
Prior Distributions: A Gaussian-Gamma prior is adopted to promote sparsity, where the hyperparameters governing sparsity are treated as random variables with Gamma priors.
Noise Modeling: The noise precision is modeled with a Gamma prior to adapt to actual noise conditions.
Inference: A mean-field approximation is employed to derive closed-form updates for the posterior distributions of the channel coefficients, hyperparameters, and noise precision, avoiding intractable high-dimensional integrals.

Key Contributions
According to the authors, the primary contributions of this work are:

Novel System Models: This is the first work to feature MU affine precoded SIP techniques specifically for partially connected THz MIMO systems impacted by the dual-wideband effect.
Dual-Model Framework: The proposal of two distinct architectures (CP-JCE and USP-DCE) to analyze trade-offs between joint and decoupled estimation strategies.
Robust Estimation Algorithm: The development of a VBI-based algorithm that jointly estimates channel coefficients and learns the underlying sparsity structure through hyperparameter inference. This approach is designed to be robust under severe model uncertainty without relying solely on prior assumptions.
Comprehensive Trade-off Analysis: A detailed comparison of the two proposed models regarding computational complexity, estimation accuracy (NMSE), and coherence time requirements.

Simulation Results
The performance was evaluated using a practical scenario with $U=3$ users, a 64-antenna BS, and a carrier frequency of 0.671 THz. Key findings include:

Performance vs. LS and Existing Methods: The proposed VBI approach significantly outperforms Least Squares (LS) and other state-of-the-art sparse estimation techniques (cited as [9] and [10]). The LS method fails to incorporate sparsity priors, while the other cited algorithms suffer from structural/convergence errors or suboptimal local minima.
Accuracy: The proposed method achieves a Normalized Mean Square Error (NMSE) that closely approaches the "Genie-aided" benchmark (which assumes perfect channel knowledge) at higher Signal-to-Noise Ratios (SNR).
Convergence: The USP-DCE model converges faster (9 iterations to reach NMSE of 0.1 at 5 dB SNR) compared to the CP-JCE model (19 iterations).
Trade-offs:
- Complexity: USP-DCE offers lower computational complexity ( $O(UG^3_T G^3_R)$ ) compared to CP-JCE ( $O(U^3 G^3_T G^3_R)$ ), making it more scalable as the number of users increases.
- Accuracy: USP-DCE demonstrates superior NMSE performance compared to CP-JCE, as the latter handles simultaneous multi-user estimation which introduces higher interference and complexity.
- Coherence Time: CP-JCE requires fewer precoding matrices (2 total) compared to USP-DCE ( $2 \times U$ ), resulting in a shorter required coherence time, making CP-JCE potentially more suitable for rapidly changing channels despite its higher complexity.

Significance
The paper claims significance by providing the first rigorous framework for affine precoded SIP in partially connected dual-wideband THz MU-MIMO systems. It demonstrates that Variational Bayes inference can effectively handle the high dimensionality and sparsity of THz channels while managing the trade-offs between estimation accuracy and computational complexity. The work offers a viable path toward robust channel estimation that maintains high data rates by eliminating dedicated pilot overhead, a critical requirement for next-generation THz communication systems.

The Problem: The "Pilot" Bottleneck

The Two Strategies

The Magic Tool: Variational Bayes

What the Paper Found

The Bottom Line

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