lecture series

“A Linear System Theory Approach to Wave Propagation”

The premise of this lecture series is that the traditional physicist’s approach to teaching electromagnetic theory – which entails scalar and vector potentials, the method of separation of variables, spherical coordinates, and the frequent use of special functions – obscures some of its most important implications. Instead, we treat Maxwell’s equations as descriptive of a linear space/time-invariant system, amenable to tools of linear system theory that are familiar to signal processing engineers and communication theorists: convolutions and space/time Fourier transforms, performed in space/time Cartesian coordinates.

Rather than starting with electromagnetic wave propagation, instead, we begin with acoustic wave propagation. There are good reasons for this. First, the acoustic field is a scalar field, whereas the electromagnetic field is a vector field. Hence, the notation for the acoustic field is less cluttered than for the electromagnetic field. Second, the physics of the acoustic field is simpler than that of the electromagnetic field, and more intuitive. Third, with the exception of polarization, all of the wave propagation phenomenology associated with electromagnetics is manifested in acoustics.

The required background is linear system theory, comfort with convolutions and Fourier transforms, and familiarity with functions of complex variables.

This lecture series will benefit both signal processing researchers and communication theorists. It will equip them to understand advanced topics, such as wave equation migration filtering, wireless power transfer via evanescent wave coupling, super-directive antenna arrays, large intelligent surfaces, and holographic MIMO.

Outline

Lecture 1

• Systems of antennas and circuit theory
  • Impedance matrix
  • Real and reactive power
  • Fundamentals of wireless power transfer
  • Wireless power transfer efficiency

Lecture 2

• Introduction to acoustics
  • pressure/strain relation
  • distributed source
  • Newton’s second law
  • 3D wave equation for pressure, resulting from a distributed source
• 1D wave propagation
  • 1D plane-waves
  • Helmholtz equation with distributed source
  • solution of Helmholtz equation for the pressure field via Green’s function

Lecture 3

• 1D wave propagation (continued)
  • solution of Helmholtz equation is spatial frequency domain
    • review of Cauchy integral & residue theorems
    • inverse spatial Fourier transform via residues
  • source of limited extent: pressure field inside the source vs. outside the source
  • computation of power
  • self-impedance of source
  • mutual impedance between two sources

Lecture 4

• Solution of 3D Helmholtz equation for pressure in spatial Fourier domain
  • residues, and the plane-wave representation of pressure field
  • ordinary plane waves and evanescent plane waves
• Plane-wave representation of spherical wave
• Greene’s function representation for pressure field

Lecture 5

• Pressure field: inside distributed source vs. outside
• Degrees of freedom associated with an array of sources
• Real power and conservation of energy

Lecture 6

• Equivalent ways of computing power
  • Direct integration over distributed source
  • Power carried by propagating plane-waves
  • Far-field power

Lecture 7

• Self-impedance of distributed source
• Mutual impedance between two distributed sources
• Physically realistic source models

Lecture 8

• Review of Maxwell’s equations with distributed electric/magnetic current sources
• Direct solution of Maxwell’s equations in space/time Fourier domain for E and H fields
• Residues, and the plane-wave representation of the electromagnetic field

Lecture 9

• Power: real and reactive
• Poynting vector

Lecture 10

• Electromagnetic sources
• Self-impedance and mutual impedance