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Analog Computing for Signal Processing and Communications -- Part I: Computing with Microwave Networks

Matteo Nerini, Bruno Clerckx

TL;DR

This work introduces microwave linear analog computers (MiLACs) as reconfigurable, multiport microwave networks that process signals in the analog domain. By leveraging tunable admittance components, MiLACs can realize key linear-algebra operations such as the LMMSE estimator and matrix inversion with orders of magnitude lower computational complexity than digital implementations. The paper also shows how Kalman filtering can be performed using MiLAC primitives, and proposes a lossless, purely imaginary admittance design to reduce power loss. Part I thus provides a rigorous model, analysis, and practical design guidelines, while Part II discusses applications in wireless communications and beamforming in the analog domain.

Abstract

Analog computing has been recently revived due to its potential for energy-efficient and highly parallel computations. In this two-part paper, we explore analog computers that linearly process microwave signals, named microwave linear analog computers (MiLACs), and their applications in signal processing and communications. In Part I of this paper, we model a MiLAC as a multiport microwave network with tunable impedance components, enabling the execution of mathematical operations by reconfiguring the microwave network and applying input signals at its ports. We demonstrate that a MiLAC can efficiently compute the linear minimum mean square error (LMMSE) estimator and matrix inversion, with remarkably low computational complexity. Specifically, a matrix can be inverted with complexity growing with the square of its size. We also show how a MiLAC can be used jointly with digital operations to implement sophisticated algorithms such as the Kalman filter. To enhance practicability, we propose a design of MiLAC based on lossless impedance components, reducing power consumption and eliminating the need for costly active components. In Part II of this paper, we investigate the applications of MiLACs in wireless communications, highlighting their potential to enable future wireless systems by executing computations and beamforming in the analog domain.

Analog Computing for Signal Processing and Communications -- Part I: Computing with Microwave Networks

TL;DR

This work introduces microwave linear analog computers (MiLACs) as reconfigurable, multiport microwave networks that process signals in the analog domain. By leveraging tunable admittance components, MiLACs can realize key linear-algebra operations such as the LMMSE estimator and matrix inversion with orders of magnitude lower computational complexity than digital implementations. The paper also shows how Kalman filtering can be performed using MiLAC primitives, and proposes a lossless, purely imaginary admittance design to reduce power loss. Part I thus provides a rigorous model, analysis, and practical design guidelines, while Part II discusses applications in wireless communications and beamforming in the analog domain.

Abstract

Analog computing has been recently revived due to its potential for energy-efficient and highly parallel computations. In this two-part paper, we explore analog computers that linearly process microwave signals, named microwave linear analog computers (MiLACs), and their applications in signal processing and communications. In Part I of this paper, we model a MiLAC as a multiport microwave network with tunable impedance components, enabling the execution of mathematical operations by reconfiguring the microwave network and applying input signals at its ports. We demonstrate that a MiLAC can efficiently compute the linear minimum mean square error (LMMSE) estimator and matrix inversion, with remarkably low computational complexity. Specifically, a matrix can be inverted with complexity growing with the square of its size. We also show how a MiLAC can be used jointly with digital operations to implement sophisticated algorithms such as the Kalman filter. To enhance practicability, we propose a design of MiLAC based on lossless impedance components, reducing power consumption and eliminating the need for costly active components. In Part II of this paper, we investigate the applications of MiLACs in wireless communications, highlighting their potential to enable future wireless systems by executing computations and beamforming in the analog domain.

Paper Structure

This paper contains 24 sections, 2 theorems, 58 equations, 8 figures, 2 tables, 5 algorithms.

Table of Contents

  1. Introduction
  2. Model of a Microwave Linear Analog Computer
  3. Microwave Network
  4. Input Signals
  5. Output Signals
  6. Tunable Admittance Components
  7. Analysis of a Microwave Linear Analog Computer
  8. Analysis of a MiLAC with Input on Some Ports
  9. Analysis of a MiLAC with Input on All Ports
  10. Primitive Operations of a MiLAC
  11. Analog Computing for the LMMSE Estimator and Matrix Inversion
  12. LMMSE Estimator for Linear Observation Processes
  13. Computing the LMMSE Estimator with a MiLAC
  14. Computing the Covariance Matrix of the LMMSE Estimator Error with a MiLAC
  15. Inverting a Matrix with a MiLAC
  16. ...and 9 more sections

Key Result

Proposition 1

Let $\mathbf{P}\in\mathbb{C}^{P\times P}$ be an invertible matrix, and $\mathbf{P}^{-1}\in\mathbb{C}^{P\times P}$ its inverse, partitioned as where $\mathbf{A},\mathbf{A}^\prime\in\mathbb{C}^{N\times N}$, $\mathbf{B},\mathbf{B}^\prime\in\mathbb{C}^{N\times M}$, $\mathbf{C},\mathbf{C}^\prime\in\mathbb{C}^{M\times N}$, $\mathbf{D},\mathbf{D}^\prime\in\mathbb{C}^{M\times M}$, and $P=N+M$. Then, we h

Figures (8)

  • Figure 1: Representation of a $P$-port MiLAC with input on $N$ ports, where $P=N+M$.
  • Figure 2: Representation of a $4$-port MiLAC with input on $2$ ports.
  • Figure 3: Circuit to be analyzed to compute (a) the off-diagonal entry $[\mathbf{Y}]_{i,k}$ and (b) the diagonal entry $[\mathbf{Y}]_{k,k}$ as a function of the tunable admittance components.
  • Figure 4: Representation of an $N$-port MiLAC with input on all ports.
  • Figure 5: Computational complexity of computing the LMMSE estimator given an observation vector with size $Y$, for various sizes $X$ of the unknown vector.
  • ...and 3 more figures

Theorems & Definitions (10)

  • Proposition 1
  • proof
  • Remark 1
  • Remark 2
  • Proposition 2
  • proof
  • Remark 3
  • Remark 4
  • Remark 5
  • Remark 6