Antenna Region

In general there are many ways to define and characterize an antenna. In chan-nel modeling we normally put the interface between the antenna region and the propagation region at some radius from the center of the antenna excitation port, i.e., the point where the transition between transmission line and open-space propagation occur. The corresponding region that is circumventing all the structures that is fixed to the antenna (or to the excitation port) is inserted into the channel as one unit, which may be referred to as the actual antenna itself, or alternatively as a super-antenna. This means that the antenna region may include, e.g., the mobile phone with casing in a cellular system, or a lap-top computer in a WLANsystem. If it can be assumed that all surrounding structures outside the antenna region is in the far-field, the antenna can be rep-resented by the polarized complex electric far-field found by measurements or electromagnetic theory, i.e., by aP2Dmodel. The far-field of a certain antenna is a deterministic property, but could be turned into a statistical representa-tion that include small-scale uncertainties by adding a stochastic component

as proposed in [85].

An antenna far-field may also be stored and handled in different ways.

There are different pros and cons for the choice of method:

Closed-form expression is of course the most convenient model for simula-tions of antennas but are only available in practice for simple structures like idealized dipoles, etc.

Sampled field data is normally what would come out of antenna range mea-surements or EMCAD tools. A disadvantage is the discrete frequency and angular samples which require interpolation to fit with a directional propagation channel.

Basis function representation is an alternative mathematical representa-tion of full sphere field data often derived from either a closed-form rep-resentation, or sampled full sphere field data with sufficient sampling density. The field is represented by a linear combination of a set of basis functions (preferably orthonormal) with different properties depending on application.

The basis function representation is commonly used in the included papers of this thesis. The far-field data originates in most cases from measured full sphere data and once the transform or modal expansion is made in combi-nation with the measurements, the file size becomes compact since only the complex basis function weights are stored. Furthermore, with the closed-form basis functions we avoid numerical complex interpolation in the angular do-main. Another useful feature of basis function representation is the physical interpretation of the basis functions where higher order modes mainly repre-sent radiating parts of objects with a larger size or at a larger distance from the actual antenna (if centered at the origin of the coordinate system of the measurement range). This means that it is possible to identify error sources in measured data that originate from objects outside of the actual antenna, e.g., from the anechoic chamber positioning system, etc.

One set of basis functions that is commonly used is the vector spherical harmonics or spherical vector modes (SVM) as described in [3]. This repre-sentation gives compact data storage and nice physical truncation properties.

Another very useful representation of especially far field data is the angular spectral domain (ASD) or 2-D Fourier decomposition [20]. To do this we need to use the trick of extending the spherical vector coordinate system to map a torus geometry such that the field (over-)representation gets periodic in both θ and φ over (−π : π). Now a 2-D discrete Fourier transform (DFT) give the angular spectral domain decomposition (referred to as the EADF in [20, 51]).

With periodic (equidistant) field data samples in (θ, φ) we may utilize the ef-ficient fast Fourier transform (FFT) algorithm. The transition between ASD andSVM is easily derived making it possible to switch very fast between the representations [37, (A1.35a)]. Thus, we can utilize the advantages of minimal storage and physical truncation methods of the SVM but switch to ASD to perform fast field calculations.

For an antenna region radius of a few wavelengths theSVMexpansion is a very efficient antenna representation since the number of required modes will be moderate. There is a rule-of-thumb that the maximum longitudinal and azimuthal mode numbers (l, m) has to be larger than approximately 2π times the electrical antenna radius r/λ plus 10 [37, (2.31)], i.e., if n = lmax= mmax

then

n & kr + 10 = 2πr

λ+ 10 (5.1)

Since the mode numbers are defined so that l = 1, 2, . . . n and m = −mmax, −mmax+ 1, . . . , mmax− 1, mmaxthe total number of modes nmodes required becomes

n_modes= 2(l²_max+ 2 ∗ l_max) (5.2) where the first 2 account for the TE plus the TM modes, and mmax = lmax. Thus, for a large antenna radius of many wavelengths the number of modes may become too large to be an efficient representation, compared to direct sampled field data. This may also be the case if the antenna radius expands into the scattering region.

Another very important issue when modeling multiple antenna systems is to keep a common phase reference point of all the antenna elements comprising the array. This is referred to as the steering vector or antenna location vector (3.1). It is possible to use theSVMmethod to find the effective radiation center of each individual antenna (pattern) by an iterative procedure that minimizes the maximum number of modes required for a certain level of field accuracy and has been found to produce reliable results [32]. This method has been used to verify the phase reference of antenna representations after being decomposed, transformed and truncated from original measured field data. Applied, e.g., on the four-antenna test terminal that was used in the experiments described in Paper II-V, see Paper II Section 3.1, gives a strong indication that the main radiating parts of the antenna manifold are located at the positions of the planar inverted-F antenna (PIFA) elements as seen in Figure 5.1. The figure shows a model of the handset with the four PIFA elements marked by blue boxes. The origins of the sub-axes of each antenna element in the graph mark the effective radiation centers found by the previously described method. Thus, it is clear that the radiation indeed stems primarily from the locations of the PIFAs and not from induced currents on the connecting cables, which is a

Figure 5.1: Geometry of a four-antenna handset (PDA) with superim-posed 3-D measured antenna patterns with the origins at the effective radiation centers.

potential problem especially for small terminal antenna evaluations. A way to avoid this potential problem would be to use the technique with optical feed cables and RF-opto converters [67, 104].