⚡ electrical engineering

Wavenumber-domain signal processing for holographic MIMO: Foundations, methods, and future directions

This article presents a comprehensive overview of the wavenumber-domain signal processing framework for holographic MIMO systems, detailing its theoretical foundations, key technologies such as channel modeling and multiplexing, and future research directions to address the limitations of classical methods in subwavelength apertures.

Original authors: Zijian Zhang, Linglong Dai

Published 2026-03-02

📖 5 min read🧠 Deep dive

CC0 1.0

Original authors: Zijian Zhang, Linglong Dai

Original paper dedicated to the public domain under CC0 1.0 (http://creativecommons.org/publicdomain/zero/1.0/). ✨ 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 send a message across a crowded room using a flashlight.

The Old Way (Traditional MIMO):
In the past, we used a flashlight with a few distinct, separate beams (like a multi-beam laser pointer). We aimed these beams at specific angles to reach different people. This worked fine when the people were far away and the room was huge. The beams were like discrete "lanes" on a highway. If you tried to use this same flashlight for someone standing right next to you, the beams would be too wide and blurry, missing the target or hitting the wrong person. This is what current 5G networks do: they use a grid of antennas spaced far apart, assuming everyone is "far away."

The New Way (Holographic MIMO):
Now, imagine replacing that flashlight with a giant, glowing, flexible sheet of light (a holographic surface). This sheet is covered in millions of tiny, invisible switches (sub-wavelength elements). Instead of shooting a few beams, this sheet can shape the light into any pattern you want. It can focus a tiny, sharp dot of light on someone's nose from across the room, or create a wide, soft glow for a group nearby. It can even bend the light to go around corners.

The Problem:
The problem is that the old math we used to control the flashlight (called the DFT or "Discrete Fourier Transform") is like a ruler with only inch marks. It's great for measuring long distances, but it's terrible for measuring tiny details or things that are very close. If you try to use inch-mark measurements to describe the curve of a tiny bubble, you get a jagged, inaccurate line. In wireless terms, this causes "energy leakage"—your signal spills over and gets messy, especially when the receiver is close or the antenna sheet is huge.

The Solution (Wavenumber-Domain Processing):
This paper introduces a new way of thinking called the Wavenumber Domain.

Think of the old method (DFT) as trying to describe a complex painting by only using a few large, blocky Lego bricks. You can get the general shape, but the details are lost.

The Wavenumber Domain is like having a box of infinite, tiny, smooth clay particles. Instead of forcing the signal into a rigid grid of angles, this method breaks the signal down into a continuous flow of "waves" (like ripples in a pond).

The Analogy: Imagine the air in the room is filled with invisible sound waves. The old method only listens to specific notes on a piano (Discrete). The new method listens to the entire spectrum of sound, including the tiny, continuous hums between the notes.
Why it matters: Because the holographic sheet is so dense, the "ripples" of the signal are continuous, not stepped. The Wavenumber method understands that these ripples change shape depending on how far away you are (spherical waves). It doesn't care if the receiver is 1 meter away or 100 meters away; it just describes the shape of the wave perfectly.

What Can We Do With This?
The paper explains how using this "continuous wave" thinking unlocks superpowers for 6G:

Multiplexing (Sending More Data):
- Old Way: Like sending letters in separate, fixed mailboxes.
- New Way: Like sending letters on different "frequencies" of a continuous river. You can pack way more data into the same space because you aren't limited by the size of the mailboxes. You can send a unique "wave pattern" to every single person in the room simultaneously without them interfering with each other.
Channel Estimation (Finding the Path):
- Old Way: Trying to guess the path by checking a few fixed spots.
- New Way: You send out a "probe" that sweeps the entire continuous river. Because the math is perfect for this shape, you can find the exact path of the signal without the "blur" or "leakage" that confused the old methods. It's like using a high-resolution MRI instead of a blurry X-ray.
Waveform Design (Shaping the Beam):
- Old Way: You have a fixed set of shapes (like a stencil). If the user moves, the shape doesn't fit.
- New Way: You can mold the signal like clay. If a user moves closer, the signal automatically reshapes itself to stay focused on them, keeping the connection strong and fast.

The Future Challenges:
The authors admit this is still new and tricky.

The Hardware: Building a sheet with millions of tiny switches is hard and expensive.
The Math: Calculating with "infinite" waves is computationally heavy.
The Reality: Real rooms have walls and people that scatter signals. The new math needs to be robust enough to handle a messy real world, not just a perfect theory.

In Summary:
This paper argues that to build the super-fast, ultra-connected 6G networks of the future, we need to stop thinking of antennas as a grid of dots and start thinking of them as a continuous, flowing sheet of energy. By using Wavenumber-Domain Signal Processing, we can finally control this energy with the precision of a hologram, unlocking speeds and capacities that were previously impossible. It's the difference between painting with a thick brush and painting with a laser.

1. Problem Statement

The emergence of Holographic Multiple-Input Multiple-Output (H-MIMO) systems, which utilize spatially quasi-continuous electromagnetic (EM) apertures with subwavelength antenna spacing, presents a fundamental challenge to conventional wireless signal processing.

Limitations of Classical Models: Traditional MIMO systems rely on half-wavelength ( $\lambda/2$ ) spacing and far-field assumptions, utilizing the Discrete Fourier Transform (DFT) to decompose signals into angular basis vectors.
The Mismatch: In H-MIMO, the antenna spacing is much smaller than $\lambda/2$ $λ /2$ , and the array aperture is often electromagnetically large, placing many links in the near-field (radiating near-field) regime.
- Basis Mismatch: DFT bases assume plane waves at discrete angles, failing to capture the spherical wave propagation and angle-distance coupling inherent in near-field H-MIMO.
- Energy Leakage: The classical DFT representation suffers from severe "energy leakage" and loss of sparsity because it cannot accurately model the continuous spatial correlation and evanescent waves present in subwavelength apertures.
The Need: A new signal processing framework is required that is physically consistent with Maxwell's equations, antenna-spacing independent, and capable of handling both near-field and far-field regimes without the artifacts of discrete grid sampling.

2. Methodology: Wavenumber-Domain Framework

The paper proposes a Wavenumber-Domain Signal Processing framework as the natural solution.

Core Concept: Instead of discretizing space into antenna elements, the EM field on the holographic surface is modeled as a superposition of continuous plane-wave modes (Fourier harmonics) indexed by their wavenumbers ( $k_x, k_y$ ).
Physical Basis: This approach leverages spatial Fourier plane-wave decomposition. It treats the H-MIMO channel as a continuous linear mapping governed by Maxwell's equations.
Key Advantages:
- Continuity: It provides a continuous spectrum of wavenumbers rather than a fixed discrete grid, eliminating basis mismatch.
- Invariance: The representation is independent of antenna spacing and distance, naturally capturing spherical wave propagation and subwavelength spatial correlation.
- Sparsity: H-MIMO channels exhibit "clustered sparsity" in the wavenumber domain, where only specific ranges of wavenumbers carry significant power, enabling efficient processing.

3. Key Contributions and Technical Methods

The paper categorizes existing research and methodologies into four main areas of wavenumber-domain signal processing:

A. Wavenumber-Division Multiplexing (WDM)

Concept: Analogous to OFDM in the frequency domain, WDM treats the spatial aperture as a domain for orthogonal plane-wave modes.
Mechanism: Transmit currents are decomposed into orthogonal spatial sinusoids (wavenumbers). Each mode carries an independent data stream.
Performance: When the receiver aperture is sufficiently large, WDM can achieve the same sum-rate as optimal water-filling MIMO precoding but with simpler implementation (activating specific spatial patterns).

B. Wavenumber-Domain Channel Estimation

Challenge: DFT-based estimation fails due to energy leakage from angle mismatch.
Solution: Utilizing compressed sensing techniques tailored for the continuous wavenumber domain.
Methodology:
- Designing probing waveforms (e.g., multi-sinusoidal spatial pilots) that target specific wavenumber bands.
- Representing the channel as an integral over wavenumbers using a finite Fourier basis.
- Algorithm: The paper highlights the Graph-Cut Swap Expansion (GCSE) algorithm, which iteratively selects the most significant wavenumber modes. Simulations show this outperforms DFT-sampled angular-domain methods by eliminating energy leakage.

C. Wavenumber-Domain Waveform Design

Single-User: Replacing DFT codebooks with wavenumber-domain codebooks (steering vectors for plane waves on a finer grid).
- Result: Wavenumber beams maintain constant rate over distance, whereas DFT beams degrade in the near-field due to physical mismatch (high-angle DFT beams have no physical counterpart in the near field).
Multi-User: Introduction of Pattern-Division Multiplexing (PDM).
- Mechanism: Optimizes current-density distribution functions (patterns) by transforming the design problem into the projection of these patterns onto finite Fourier bases.
- Result: Creates nearly mutually orthogonal patterns that balance beam directionality and inter-user interference, maximizing sum-rate.

D. Integrated Sensing and Communications (ISAC)

Application: The unified wavenumber framework allows simultaneous allocation of spatial modes for data transmission and radar sensing.
Wideband ISAC: Addresses "beam squint" in wideband near-field scenarios. It proposes joint wavenumber-frequency waveform designs where frequency-dependent wavenumber beams direct energy to users while illuminating sensing targets, enabling high-resolution distance-angle estimation.

E. Hardware-Constrained Designs

Practicality: Addresses limitations like limited RF chains and phase-only control (e.g., in Reconfigurable Intelligent Surfaces - RIS).
Approach: Projects desired wavenumber waveforms onto feasible sets defined by hardware constraints (e.g., hybrid analog-digital precoding). It suggests using RIS to implement analog wavenumber transforms with low power consumption.

4. Results and Findings

Performance Gains: H-MIMO systems using wavenumber-domain processing (specifically CAPAs - Continuous Aperture Arrays) demonstrate superior array gain and multiplexing gain compared to Spatially Discrete Phased Arrays (SPDAs).
Robustness: Wavenumber-based estimators and beamformers are robust against near-field effects and do not suffer from the performance degradation seen in DFT-based systems as users move closer to the array.
Sparsity: The channel gain in the wavenumber domain is shown to be clustered-sparse, validating the use of compressed sensing for efficient channel estimation.

5. Significance and Future Directions

Theoretical Shift: The paper establishes the wavenumber domain not merely as a subset of DFT, but as a physically grounded basis essential for accurate modeling of next-generation EM systems. It bridges the gap between classical information theory and Electromagnetic Information Theory (EIT).
6G Standardization: As 6G moves toward near-field communications and extremely large-scale apertures, the wavenumber domain offers a unified framework for defining channel representations, codebooks, and feedback mechanisms.
Open Challenges Identified:
1. Capacity Theory: Developing a comprehensive information-theoretic framework that integrates EM properties (like evanescent modes) into capacity limits.
2. Tri-Hybrid Architectures: Combining analog EM processing (lens/RIS) with hybrid MIMO to scale spatial dimensions without increasing RF complexity.
3. Mutual Coupling (MC): Moving beyond treating MC as a nuisance to exploiting it for enhanced spatial degrees of freedom.
4. Robustness: Designing waveforms robust to hardware imperfections and environmental scattering.
5. Network Integration: Balancing computational complexity with performance gains for scalable network deployment.

Conclusion:
This paper provides a foundational overview of how wavenumber-domain signal processing resolves the limitations of classical MIMO in the context of H-MIMO. By shifting from discrete angular grids to continuous plane-wave decomposition, it enables precise control over EM fields, unlocking new spatial degrees of freedom and paving the way for high-efficiency 6G communications.