Theory

This section provides detailed explanations of the mathematical theory and algorithms used in PyRth.

Contents:

• Overview of Network Identification by Deconvolution
  • Introduction
  • Step- and impulse-response
  • Logarithmic time axis
  • Time-constant spectrum
  • From spectrum to RC network
  • Inverse route starting from \(Z(s)\)
  • Thermal Structure Functions
• Differentiation
  • Why differentiation is hard
  • Optimal regression filtering
• Fourier Deconvolution
  • Problem description
  • Frequency-domain windows
• Bayesian Deconvolution
• LASSO Deconvolution
  • Mathematical Formulation
  • Comments
• Structure Functions
  • Transmission-Line Basics
  • Uniform RC Line
  • Non-Uniform RC Line ⇔ Structure Function
• From Spectrum to Foster Network
• Polynomial Long-Division
• J-fraction Route to the Cauer Ladder
  • Starting point: rational impedance
  • J-fraction representation
  • From J– to S-fraction
  • Algorithm outline
• Khatwani’s Method
  • Prerequisites
  • Markov parameters
  • Polynomial inverse by Newton doubling
  • Triangular calculation table
• Sobhy’s Method
  • Initialisation
  • Iterative filling rule
  • Extracting H and h
• de Boor–Golub Algorithm
  • Two equivalent forms of the impedance
  • Recurrence relation
  • Recovering the Cauer coefficients
  • Algorithm summary
• Lanczos Method
  • Foster Network as Differential-Equation
  • Lanczos in a Nutshell
  • Algorithm
  • Mapping to Cauer Elements
• Optimization Techniques
  • Why Optimise?
  • Three-Stage Pipeline
  • Structure Simplification Objective
  • Impedance-Fitting Objective
  • Assessing Accuracy
  • Practical Tips