Fast Algorithms in Spherical Measurement Setups

Very efficient post-processing algorithms exist for measurements of the $S_{12}$-parameter between a radiating probe antenna and a rotating receiving antenna under test (AUT). The AUT is rotated around the coordinate origin by the angles $\vartheta$ and $\varphi$. The probe antenna remains at a fixed position in the positive $z$-diretion but may be rotated around the $z$-axis by the angle $\chi$ to realize different polarizations of the incident field at the AUT. The measurement setup is depicted in the figure below.

Fast Evaluation of $S_{12}$-Parameters in Spherical Measurement Setups

The $S_{12}$-parameter can be calculated by (cf. [1, eq. (4.40)])

\[S_{12}(\vartheta, \varphi, \chi)=\sum \limits_{s=1}^2 \sum \limits_{\ell=1}^L \sum \limits_{m=-\ell}^\ell \sum \limits_{\mu=-\ell}^\ell \beta_{s\ell m}^{\mathrm{aut}}\,\mathrm{e}^{-\mathrm{j} m \varphi}\, \mathrm{d}_{m,\mu}^{\ell}(\vartheta)\, \mathrm{e}^{-\mathrm{j} \mu \chi}\,\alpha_{s\ell \mu}^{(1),\mathrm{pro}}\,.\]

Here. $\alpha_{s\ell \mu}^{(1),\mathrm{pro}}$ are the spherical mode coefficients of the incident probe antenna field (normalized to a unit excitation of the probe), $\beta_{s\ell m}^{\mathrm{aut}}$ are the spherical receive coefficients of the AUT and $\mathrm{d}_{m,\mu}^{\ell}(\vartheta)$ is the so-called Wigner-d-matrix which has a finite Fourier-series expansion

\[\mathrm{d}_{m,\mu}^{\ell}(\vartheta)= \mathrm{j}^{m-\mu} \sum \limits_{m_\vartheta=-\ell}^\ell \, \Delta_{m_\vartheta,\mu}^\ell\, \Delta_{m_\vartheta,m}^\ell\, \mathrm{e}^{-\mathrm{j} m_\vartheta \, \vartheta}\, ,\]

where the deltas

\[\Delta_{m_\vartheta,m}^\ell= \mathrm{d}_{m,\mu}^{\ell}(\mathrm{\pi}/2)\]

correspond to the calue of the Wigner-d-matrix at $\vartheta=\mathrm{\pi/2}$.

Substituting the series expansion into the original equation, one obtains

\[S_{12}(\vartheta, \varphi, \chi)= \sum \limits_{\mu=-L}^L \mathrm{e}^{-\mathrm{j} \mu \chi} \underbrace{ \sum \limits_{m=-L}^L \mathrm{e}^{-\mathrm{j} m \varphi}\, \underbrace{ \sum \limits_{m_\vartheta=-L}^L \mathrm{e}^{-\mathrm{j} m_\vartheta \, \vartheta}\, \underbrace{ \mathrm{j}^{m-\mu} \sum \limits_{\ell=\max(|m|,|m_\vartheta|,1)}^L \Delta_{m_\vartheta,\mu}^\ell\, \Delta_{m_\vartheta,m}^\ell\, \underbrace{ \sum \limits_{s=1}^2 \alpha_{s\ell \mu}^{(1),\mathrm{pro}}\, \beta_{s\ell m}^{\mathrm{aut}} }_{u_{\mu,m,\ell}} }_{\tilde{v}_{\mu,m,m_\vartheta}} }_{v_{\mu,m}(\vartheta)} }_{w_\mu(\vartheta,\varphi)}\,.\]

The form of the above equation gives rise to an efficient algorithm to evaluate the $S_{12}$-parameter on a spherical measurement surface by successively calculating $\alpha_{s\ell \mu}^{(1),\mathrm{pro}}\, \beta_{s\ell m}^{\mathrm{aut}} \rightarrow u_{\mu,m,\ell} \rightarrow \tilde{v}_{\mu,m,m_\vartheta} \rightarrow v_{\mu,m}(\vartheta) \rightarrow w_\mu(\vartheta,\varphi)$. The occurring Fourier series are evaluated using a fast Fourier transform giving rise to the resulting data available at a regular spherical sampling grid.

Radiating Hertzian Dipole Probe and Receiving AUT

If we want to calculate the transmission between a dipole probe and the AUT, we need to know the spherical wave expansion of the probe field at the AUT location. The incident field at the coordinate origin of a Hertzian dipole, oriented in $x$- direction with a given dipole moment $I\ell$ at a given location $\bm{r}= r\, \bm{e}_z$ on the $z$-axis, is characterized by the spherical wave coefficients

\[\alpha_{1,\ell,1}^{(1)}=\dfrac{-\mathrm{j}}{4} \,k\, \sqrt{Z_{\mathrm{F}}}\, I\ell \, \sqrt{\dfrac{2\ell+1}{\pi}}\, h_\ell^{(2)}(kr)\]

\[\alpha_{1,\ell,-1}^{(1)}=\dfrac{-\mathrm{j}}{4} \,k\, \sqrt{Z_{\mathrm{F}}}\, I\ell \, \sqrt{\dfrac{2\ell+1}{\pi}}\, h_\ell^{(2)}(kr)\]

\[\alpha_{2,\ell,1}^{(1)}=\dfrac{1}{4} \,k\, \sqrt{Z_{\mathrm{F}}}\, I\ell \, \sqrt{\dfrac{2\ell+1}{\pi}}\, \dfrac{1}{kr} \, \dfrac{\mathrm{d}}{\mathrm{d} kr} \left\{kr\,h_\ell^{(2)}(kr)\right\}\]

\[\alpha_{2,\ell,-1}^{(1)}=\dfrac{-1}{4} \,k\, \sqrt{Z_{\mathrm{F}}}\, I\ell \, \sqrt{\dfrac{2\ell+1}{\pi}}\, \dfrac{1}{kr} \, \dfrac{\mathrm{d}}{\mathrm{d} kr} \left\{kr\,h_\ell^{(2)}(kr)\right\}\,,\]

where $h_\ell^{(2)}(kr)$ are the spherical Hankel functions of second kind. All incident field coefficients with $|m|\neq 1$ are equal to zero.

Regularly Sampled Far Field of AUT

The far field pattern of the AUT can be calculated by determining the received signal of a plane wave illumination traveling into the desired propagation directions. To determine the correctly normalized far-field pattern (with unit V), we utilize the spherical expansion of a $x$-polarized plane wave traveling into negativ $z$-direction with the coefficients

\[\alpha_{1,\ell,1}^{(1)}=- \mathrm{j}^{\ell}\, \sqrt{\dfrac{2\ell+1}{\pi}}\, \dfrac{\sqrt{Z_{\mathrm{F}}}}{2}\]

\[\alpha_{2,\ell,1}^{(1)}=- \mathrm{j}^{\ell}\, \sqrt{\dfrac{2\ell+1}{\pi}}\, \dfrac{\sqrt{Z_{\mathrm{F}}}}{2}\]

\[\alpha_{1,\ell,-1}^{(1)}=- \mathrm{j}^{\ell}\, \sqrt{\dfrac{2\ell+1}{\pi}}\, \dfrac{\sqrt{Z_{\mathrm{F}}}}{2}\]

\[\alpha_{2,\ell,-1}^{(1)}= \mathrm{j}^{\ell}\, \sqrt{\dfrac{2\ell+1}{\pi}}\, \dfrac{\sqrt{Z_{\mathrm{F}}}}{2}\, .\]

All incident field coefficients with $|m|\neq 1$ are equal to zero. Using the reciprocity relation $\beta_{s\ell m}^{\mathrm{aut}}=(-1)^{m}\, \alpha_{s,\ell,- m}^{(4),\mathrm{aut}}$, we can efficiently calculate the desired far fields for regularly sampled $\vartheta$ and $\varphi$.

Fast Reconstruction of Expansion Coefficients from $S_{12}$-Parameters in Spherical Measurement Setups

The $S_{12}$-parameter can be calculated by (cf. [1, eq. (4.40)])

\[S_{12}(\vartheta, \varphi, \chi)= \sum \limits_{\mu=\pm 1} \mathrm{e}^{-\mathrm{j} \mu \chi} \underbrace{ \sum \limits_{m=-L}^L \mathrm{e}^{-\mathrm{j} m \varphi}\, \underbrace{ \sum \limits_{\ell=\max(|m|,|m_\vartheta|,1)}^L \mathrm{d}_{\mu,m}^\ell(\vartheta)\, \underbrace{ \sum \limits_{s=1}^2 \alpha_{s\ell \mu}^{(1),\mathrm{pro}}\, \beta_{s\ell m}^{\mathrm{aut}} }_{u_{\mu,m,\ell}} }_{v_{\mu,m}(\vartheta)} }_{w_\mu(\vartheta,\varphi)}\,.\]

Here, $\mathrm{d}_{m,\mu}^{\ell}(\vartheta)$ is the so-called Wigner-d-matrix, $\alpha_{s\ell \mu}^{(1),\mathrm{pro}}$ are the spherical mode coefficients of the incident probe antenna field (normalized to a unit excitation of the probe), and $\beta_{s\ell m}^{\mathrm{aut}}$ are the spherical receive coefficients of the AUT which we aim to reconstruct from the measured data. Note, that a first-order probe is assumed here (i.e., all incident modes with $\mu\neq \pm 1$ are zero), reducing the number of summation terms in the $\mu$-summation to two.

Starting at $S_{12} (\vartheta, \varphi, \chi)$, we can successively calculate $S_{12}(\vartheta, \varphi, \chi) \rightarrow w_\mu(\vartheta,\varphi) \rightarrow v_{\mu,m}(\vartheta) \rightarrow u_{\mu,m,\ell}$ using the orthogonality relations

\[\int \limits_{0}^{2\pi} \mathrm{e}^{\, \mathrm{j} (m-m') \varphi} \, \mathrm{d} \varphi = 2\pi \, \delta_{m,m'}\]

and

\[\int \limits_{0}^{\pi} \mathrm{d}_{m,m'}^\ell(\vartheta) \, \mathrm{d}_{m,m'}^{\ell'}(\vartheta) \, \sin(\vartheta) \, \mathrm{d} \vartheta = \dfrac{2}{2\ell+1} \, \delta_{\ell,\ell'}\,\]

from where the $\beta_{s\ell m}^{\mathrm{aut}}$ can be found by solving a $2\times 2$ system of linear equations

\[ \begin{bmatrix} {u_{+1,m,\ell}}\\[0.5em] {u_{-1,m,\ell}} \end{bmatrix} = \begin{bmatrix} \alpha_{1,\ell ,+1}^{(1),\,\mathrm{pro}} & \alpha_{2,\ell ,+1}^{(1),\,\mathrm{pro}} \\[0.5em] \alpha_{1,\ell ,-1}^{(1),\,\mathrm{pro}} & \alpha_{2,\ell ,-1}^{(1),\,\mathrm{pro}} \end{bmatrix} \, \begin{bmatrix} {\beta}_{1, \ell, m }^{\mathrm{aut}}\\[0.5em] {\beta}_{2, \ell, m }^{\mathrm{aut}} \end{bmatrix}\, .\]

The individual transformation steps are as follows. Since we only have to consider $\mu$-values for $\mu = \pm 1$, we can calculate the $\rightarrow w_\mu(\vartheta,\varphi)$ from the $S_{12} (\vartheta, \varphi, \chi)$-values measured at $\chi \in \{0, \pi/2\}$ by

\[w_{+1}(\vartheta,\varphi)= \dfrac{1}{2} \left[ S_{12} (\vartheta, \varphi, \chi=0) + \mathrm{j}\, S_{12} (\vartheta, \varphi, \chi=\pi/2) \right]\]

and

\[w_{-1}(\vartheta,\varphi)= \dfrac{1}{2} \left[ S_{12} (\vartheta, \varphi, \chi=0) - \mathrm{j}\, S_{12} (\vartheta, \varphi, \chi=\pi/2) \right]\]

References

• [1] J. E. Hansen, ed., Spherical Near-Field Antenna Measurements, The Institution of Engineering and Technology, Michael Faraday House, Six Hills Way, Stevenage SG1 2AY, UK: IET, 1988.