🔬 applied physics

Analog Computing with Hybrid Couplers and Phase Shifters

This paper establishes the theoretical conditions and systematic design methods for implementing linear microwave networks using only hybrid couplers and phase shifters to perform analog matrix-vector computations, specifically validating the approach with a hardware prototype that successfully computes a 4×4 discrete Fourier transform.

Original authors: Matteo Nerini, Xuekang Liu, Bruno Clerckx

Published 2026-03-27

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Matteo Nerini, Xuekang Liu, Bruno Clerckx

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 solve a massive math problem, like sorting a million songs by genre or translating a book instantly. Today, we use digital computers for this. Think of a digital computer as a super-fast, super-precise accountant. It takes the data, breaks it down into tiny bits (0s and 1s), processes them one by one in a line, and writes down the answer. It's incredibly accurate, but it has a bottleneck: it has to move the data from memory to the processor and back, over and over again. This takes time and energy, especially for huge, real-time tasks like 6G networks or advanced radar.

Now, imagine a different kind of machine: an analog computer. Instead of an accountant, think of it as a giant, magical water pipe system.

In this paper, the researchers built a new kind of "water pipe" system using microwaves (the same kind of waves that carry your Wi-Fi and cell signals). Here is the simple breakdown of what they did:

1. The Magic Pipes: Hybrid Couplers and Phase Shifters

To build their machine, they used only two simple, cheap building blocks:

Hybrid Couplers: Imagine a T-junction in a pipe. If you pour water in one side, it splits perfectly into two streams. If you pour from the other side, it mixes them. In the microwave world, these devices split or combine signals, but with a special twist: they can change the "timing" (phase) of the waves as they mix.
Phase Shifters: Imagine a pipe that is slightly longer or shorter. If you send a wave through a longer pipe, it arrives a tiny bit later than a wave through a short pipe. This device just delays the signal by a specific amount.

By connecting these two simple pieces together in a complex maze, the researchers created a "microwave computer."

2. The "Instant" Calculation

Here is the magic part: The math happens as the signal travels.

In a digital computer, to multiply a matrix (a grid of numbers) by a vector (a list of numbers), the processor has to do millions of tiny addition and multiplication steps. It's like walking through a maze, stopping at every turn to do a calculation.

In this Microwave Linear Analog Computer (MiLAC), you simply pour your input signals into one end of the pipe maze. As the waves travel through the T-junctions (couplers) and long pipes (phase shifters), they naturally interfere with each other. They add up, cancel out, and shift timing automatically.

By the time the waves reach the other end of the maze, the answer is already there! The output signals are the result of the calculation.

Speed: It happens at the speed of light. There is no "processing time" other than the time it takes for light to travel through the circuit.
Energy: It uses almost no extra energy because the "calculation" is just physics happening naturally.

3. What Can This Machine Do?

The researchers asked: "What kind of math can we do with just these two simple pipes?"

They proved that you can build a machine to perform three very famous and useful mathematical transformations:

The DFT (Discrete Fourier Transform): This is the "superpower" of signal processing. It takes a messy signal (like a song with all its notes mixed together) and separates it into its individual frequencies (the bass, the treble, the vocals). It's how your phone knows what voice commands you are saying.
The Hadamard Transform: A mathematical tool used for compressing data and error correction (keeping your data safe from noise).
The Haar Transform: A tool used for image compression and wavelet analysis (like how JPEGs work).

The paper provides a "recipe" (a systematic design method) to build these machines for any size, as long as the size is a power of two (like 4, 8, 16, 32 inputs).

4. The Real-World Test

Theory is great, but does it work in real life?
The team built a physical prototype on a circuit board (like a motherboard, but for microwaves). They built a 4x4 DFT machine.

The Problem: Real pipes aren't perfect. The metal isn't 100% smooth, and the connectors add a tiny bit of delay. This is like having a pipe with a slight leak or a bend that slows the water down.
The Fix: They developed a "calibration" method. Think of it like tuning a guitar. They measured the errors and then applied a mathematical "twist" to the input or output to cancel out the imperfections.
The Result: After tuning, the machine worked almost perfectly. The output matched the theoretical math with very high accuracy.

The Big Picture: Why Does This Matter?

We are running out of speed in digital computers. We can't make transistors much smaller, and moving data around is becoming too slow and energy-hungry for future technologies like 6G and autonomous driving.

This paper shows a path forward: Let the physics do the math.

Instead of asking a digital brain to calculate a matrix multiplication, we can just build a physical pipe system that is the multiplication. It's like asking a river to sort rocks by size as it flows, rather than picking them up one by one with a robot arm.

In short: This research proves we can build ultra-fast, low-energy "microwave brains" using simple, cheap components to handle the massive data processing needs of our future, all by letting waves do the heavy lifting.

1. Problem Statement

Modern digital computing faces significant bottlenecks in the post-Moore's Law era, particularly regarding the von Neumann bottleneck (latency and energy costs of data movement between memory and processors) and the power consumption of high-speed Analog-to-Digital Converters (ADCs) required for ultra-wideband signals. While analog computing offers a pathway to ultra-fast, low-latency processing by performing computations directly on physical carriers (e.g., electromagnetic waves), there is a lack of systematic frameworks for designing microwave networks that can perform specific, useful linear transformations using only fundamental, low-cost components.

The core question addressed is: What class of linear transformations can be computed in the analog domain using exclusively hybrid couplers and phase shifters? Furthermore, can these components be systematically arranged to compute practical signal processing transforms like the Discrete Fourier Transform (DFT)?

2. Methodology

The authors propose a theoretical framework and experimental validation for Microwave Linear Analog Computers (MiLACs) constructed solely from two fundamental components:

Hybrid Couplers: 4-port lossless, reciprocal networks that split/combine signals with specific phase relationships (in-phase and quadrature).
Phase Shifters: 2-port networks that introduce a controllable phase delay.

Theoretical Approach:

System Modeling: The network is modeled as a multiport microwave system characterized by a scattering matrix ( $S$ ). The input-output relationship is linear ($v = Wu$), where $W$ is the transmission scattering matrix ( $S_{21}$ ).
Decomposition Theorem: The authors derive a necessary and sufficient condition for a matrix $W$ $W$ to be implementable. They prove that any such matrix must be decomposable into a product of alternating permutation matrices ( $P$ $P$ ) and block-diagonal matrices ( $D$ $D$ ).
- The block-diagonal matrices consist of $2 \times 2$ blocks (realizable by hybrid couplers with specific phase shifts) and $1 \times 1$ blocks (realizable by phase shifters).
- The structure is: $W = P_L D_L \dots P_1 D_1 P_0$ .
Recursive Design: Leveraging the recursive nature of the Fast Fourier Transform (FFT) and similar algorithms, the authors demonstrate that the DFT, Hadamard, and Haar transforms satisfy this decomposition condition. They provide systematic design procedures to construct the physical network (series and parallel interconnections of couplers and shifters) for any transform size $N = 2^L$ .

Experimental Approach:

Prototype Fabrication: A hardware prototype was designed and fabricated using microstrip technology on a dielectric substrate to implement a $4 \times 4$ DFT.
Calibration: A systematic calibration procedure was developed to mitigate fabrication errors and connector imperfections. This included:
1. Phase-based calibration: Removing global phase offsets (e.g., from SMA connectors).
2. Vector-based calibration: Using a diagonal input-output calibration matrix to correct per-port amplitude and phase imbalances via a regularized non-linear least-squares optimization.

3. Key Contributions

Characterization of Computable Matrices: The paper establishes a rigorous mathematical condition (Theorem 1) defining the exact class of linear transformations computable by networks of hybrid couplers and phase shifters.
Systematic Design for DFT, Hadamard, and Haar Transforms:
- DFT: Proved that the DFT matrix can be decomposed according to the necessary condition. A systematic method is provided to build $N \times N$ DFT networks for any power-of-two size. The circuit complexity scales as $O(N \log N)$ , requiring $N/2 \log_2 N$ hybrid couplers.
- Hadamard Transform: Demonstrated that the Hadamard matrix also fits the decomposition criteria, with similar circuit complexity to the DFT.
- Haar Transform: Showed that the Haar wavelet transform is also realizable, but with a simpler circuit structure requiring only $N-1$ hybrid couplers.
Hardware Validation: Successful fabrication and measurement of a $4 \times 4$ DFT microwave circuit.
Calibration Framework: Development of a vector-based calibration method that significantly reduces measurement error, proving that analog imperfections can be compensated without digital re-computation of the core logic.

4. Results

Theoretical Validation: The decomposition proofs confirmed that DFT, Hadamard, and Haar transforms are theoretically achievable using the proposed components.
Experimental Performance:
- Raw Measurements: Initial measurements showed significant phase discrepancies (approx. $-43^\circ$ ) due to connectors and amplitude errors, resulting in a high Normalized Mean Squared Error (NMSE) of ~57%.
- Phase Calibration: Removing the global phase offset reduced the NMSE to 1.456%, while the cosine distance (measuring structural fidelity) remained extremely low (0.066%), indicating the transformation structure was preserved.
- Vector Calibration: Applying the full diagonal calibration further reduced the NMSE to 0.091% and cosine distance to 0.037%.
- Conclusion: The measured results showed high correlation with theoretical expectations, confirming successful analog-domain DFT computation.

5. Significance

Low-Latency, High-Bandwidth Computing: This work demonstrates a practical pathway to perform matrix-vector multiplications at the speed of light ( $O(1)$ complexity) without digital sampling, bypassing the ADC bottleneck.
Hardware Efficiency: By using only hybrid couplers and phase shifters (standard microstrip components), the approach offers a low-cost, passive, and energy-efficient alternative to digital processors for specific linear algebra tasks.
Scalability: The systematic design rules allow for the construction of larger networks (e.g., $8 \times 8$ , $16 \times 16$ ) following the same principles, making it a scalable solution for future 6G communications and radar systems requiring real-time wideband signal processing.
Bridging Theory and Practice: The successful calibration and hardware validation bridge the gap between theoretical microwave analog computing and practical implementation, showing that fabrication tolerances can be managed effectively.

In summary, this paper provides a foundational framework for Microwave Linear Analog Computers (MiLACs), proving that fundamental microwave components can be systematically arranged to perform complex, essential signal processing transforms like the DFT with high accuracy and negligible latency.