Representing Antenna Fields with Plane-Wave Expansions

A common representation of antenna fields is in terms of plane wave expansions. In AntennaFieldRepresentations.jl, plane-wave expansions are represented by a struct PlaneWaveExpansion{P, S, C} which is a subtype of AntennaFieldRepresentation{P, C}. The type parameters have the following meaning

A PlaneWaveExpansion of Radiated type corresponds to a far field pattern $\bm{F}(\vartheta, \varphi)$ and a PlaneWaveExpansion of Incident type corresponds to a plane wave spectrum $\bm{P}(\vartheta, \varphi)$ as described in more detail in the theory section.

Constructors for a PlaneWaveExpansion

To generate a PlaneWaveExpansion, use the following constructor:

PlaneWaveExpansion(P::PropagationType, samplingstrategy::S, Eθ::Matrix{C}, Eϕ::Matrix{C}, wavenumber::Number) where{S <: SphereSamplingStrategy, C}

The input arguments for the constructor are

• P <: PropagationType : Can be Radiated, Absorbed, or Incident
• samplingstrategy::SphereSamplingStrategy: Struct defining the sampling of the $k$-space unit sphere. Can be RegularθRegularϕSampling or GaussLegendreθRegularϕSampling.
• Eθ::Matrix{C}: θ-component amplitudes of the plane waves. Dimensions must match the samplingstrategy.
• Eϕ::Matrix{C}: ϕ-component amplitudes of the plane waves. Dimensions must match the samplingstrategy.
• wavenumber : Wavenumber $\omega = 2\pi \, f$

Stored Samples of a PlaneWaveRepresentation

In a PlaneWaveExpansion, an antenna far field pattern $\bm{F}(\vartheta, \varphi)$ or a plane wave spectrum $\bm{P}(\vartheta, \varphi)$ is representated by a discrete set of samples in the $\vartheta, \varphi$-domain ^[1] .

The samples should be chosen such that

• integration over the sphere is fast and accurate
• local interpolation is sufficently accurate
• the conversion $\bm{F}(\hat{\bm{k}}) \rightarrow \bm{F}(-\hat{\bm{k}})$ has small computational cost (ideally should not rely on interpolation)
• redundancy in the representation is kept minimal

By default, any far-field pattern or plane-wave spectrum with a corresponding spherical mode order of $L$ is sampled according to a GaussLegendreθRegularϕSampling with

\[N_\vartheta=L+1\]

and

\[N_\varphi=2L+2 \,.\]

Although the $\varphi$-sampling is slightly redundant ($2L+1$ samples would suffice) the even number of $\varphi$-samples ensures that for every sampling point there is another sampling point exactly in the oppsoite direction. This has obvious benefits for the conversion $\bm{F}(\hat{\bm{k}}) \rightarrow \bm{F}(-\hat{\bm{k}})$ as well as for pattern interpolation, when the interpolation scheme requires samples at either side of the coordinate poles at $\vartheta\in \{0,\pi\}$.

Methods Special to PlaneWaveExpansions

In addition to the methods defined in the interface of AntennaFieldRepresentation, a PlaneWaveExpansion supports the following methods:

References

This should not lead to any ambiguities because the relation between $\hat{\bm{k}}$ and the tuple $\vartheta, \varphi$ is one-to one, as each encodings uniquely define the same point on the unit sphere.