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Analog Computing with Hybrid Couplers and Phase Shifters

Matteo Nerini, Xuekang Liu, Bruno Clerckx

Abstract

Analog computing with microwave signals can enable exceptionally fast computations, potentially surpassing the limits of conventional digital computing. For example, by letting some input signals propagate through a linear microwave network and reading the corresponding output signals, we can instantly compute a matrix-vector product without any digital operations. In this paper, we investigate the computational capabilities of linear microwave networks made exclusively of two low-cost and fundamental components: hybrid couplers and phase shifters, which are both implementable in microstrip. We derive a sufficient and necessary condition characterizing the class of linear transformations that can be computed in the analog domain using these two components. Within this class, we identify three transformations of particular relevance to signal processing, namely the discrete Fourier transform (DFT), the Hadamard transform, and the Haar transform. For each of these, we provide a systematic design method to construct networks of hybrid couplers and phase shifters capable of computing the transformation for any size power of two. To validate our theoretical results, a hardware prototype was designed and fabricated, integrating hybrid couplers and phase shifters to implement the $4\times4$ DFT. A systematic calibration procedure was subsequently developed to characterize the prototype and compensate for fabrication errors. Measured results from the prototype demonstrate successful DFT computation in the analog domain, showing high correlation with theoretical expectations. By realizing an analog computer through standard microwave components, this work demonstrates a practical pathway toward low-latency, real-time analog signal processing.

Analog Computing with Hybrid Couplers and Phase Shifters

Abstract

Analog computing with microwave signals can enable exceptionally fast computations, potentially surpassing the limits of conventional digital computing. For example, by letting some input signals propagate through a linear microwave network and reading the corresponding output signals, we can instantly compute a matrix-vector product without any digital operations. In this paper, we investigate the computational capabilities of linear microwave networks made exclusively of two low-cost and fundamental components: hybrid couplers and phase shifters, which are both implementable in microstrip. We derive a sufficient and necessary condition characterizing the class of linear transformations that can be computed in the analog domain using these two components. Within this class, we identify three transformations of particular relevance to signal processing, namely the discrete Fourier transform (DFT), the Hadamard transform, and the Haar transform. For each of these, we provide a systematic design method to construct networks of hybrid couplers and phase shifters capable of computing the transformation for any size power of two. To validate our theoretical results, a hardware prototype was designed and fabricated, integrating hybrid couplers and phase shifters to implement the DFT. A systematic calibration procedure was subsequently developed to characterize the prototype and compensate for fabrication errors. Measured results from the prototype demonstrate successful DFT computation in the analog domain, showing high correlation with theoretical expectations. By realizing an analog computer through standard microwave components, this work demonstrates a practical pathway toward low-latency, real-time analog signal processing.

Paper Structure

This paper contains 21 sections, 12 theorems, 112 equations, 18 figures, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. System Model
  3. Main Result
  4. Proof of the Main Result
  5. Proof of Sufficiency
  6. Proof of Necessity
  7. Analog Computation of the DFT
  8. Examples of Networks Computing the DFT
  9. Analog Computation of the Hadamard Transform
  10. Analog Computation of the Haar Transform
  11. Experimental Verification
  12. Performance Evaluation Without Calibration
  13. Performance Evaluation With Phase-Based Calibration
  14. Performance Evaluation With Vector-Based Calibration
  15. Related Work
  16. ...and 6 more sections

Key Result

Proposition 1

(Series of networks) Consider a series system consisting of a first $2N$-port network with scattering matrix $\mathbf{Q}\in\mathbb{C}^{2N\times 2N}$ and a second $2N$-port network with scattering matrix $\mathbf{R}\in\mathbb{C}^{2N\times 2N}$, partitioned as where $\mathbf{Q}_{XY},\mathbf{R}_{XY}\in\mathbb{C}^{N\times N}$, for $X,Y\in\{1,2\}$. In this series, the last $N$ ports of the first netwo

Figures (18)

  • Figure 1: High-level representation of a microwave linear analog computer.
  • Figure 2: Model of a microwave linear analog computer.
  • Figure 3: Symbols of (a) a hybrid coupler, (b) a phase shifter, (c) an interconnection, (d) a permutation network, and (e) implementation of the permutation network having permutation matrix $\mathbf{P}$ given by \ref{['eq:Pexample']}.
  • Figure 4: Series of two networks having scattering matrices $\mathbf{Q}$ and $\mathbf{R}$.
  • Figure 5: Parallel of two networks having scattering matrices $\mathbf{Q}$ and $\mathbf{R}$.
  • ...and 13 more figures

Theorems & Definitions (33)

  • Definition 1
  • Definition 2
  • Definition 3
  • Definition 4
  • Definition 5
  • Proposition 1
  • proof
  • Proposition 2
  • proof
  • Definition 6
  • ...and 23 more