## 2 NR Representation of the Probe Voltage on the Spiral  
An NR representation of the voltage collected by a small non-directive probe on a suitable spiral, which wraps a sphere of radius \(d\) in the NF region of an offset-mountedlong AUT, is developed in this Section. Figure 1 showcases the geometry of the problem at hand. Let us consider two different spherical coordinate systems \(S(r,\vartheta,\varphi)\), whose origin \(O\) corresponds to the center of the scanning spherical surface \(M\) (onset coordinate system), and \(S^{\prime}(r^{\prime},\vartheta^{\prime},\varphi^{\prime})\) (offset coordinate system), whose origin \(O^{\prime}\) coincides with the center of the offset-mounted AUT and is obtained by shifting, by a distance \(d_{s}\), \(O\) on the \(z^{\prime}\)-axis, which coincides with the \(z\)-axis. It is worthwhile to highlight that \(S(r,\vartheta,\varphi)\) specifies the generic observation point \(P(d,\vartheta,\varphi)\) on the sphere, while \(S^{\prime}(r^{\prime},\vartheta^{\prime},\varphi^{\prime})\) is introduced for developing a suitable NR sampling representation of the probe voltage. Furthermore, since the sampling locations are determined in the reference system \(S^{\prime}\), they must be properly expressed in the reference system of interest, namely, \(S\). The following expressions provide the relationship between the respective sets of coordinates:  
\[\varphi=\varphi^{\prime};\ \vartheta=\tan^{-1}\left[r^{\prime}\sin(\vartheta^{ \prime})/\left(r^{\prime}\cos(\vartheta^{\prime})+d_{s}\right)\right] \tag{1}\]  
\[r^{\prime}=\sqrt{d^{2}-\left(d_{s}\sin(\vartheta^{\prime})\right)^{2}}-d_{s} \cos(\vartheta^{\prime}) \tag{2}\]  
\[\vartheta^{\prime}=\tan^{-1}[d\sin(\vartheta)/(d\cos(\vartheta)-d_{s})] \tag{3}\]  
As stated in [33], the voltage detected by a non-directive probe has the same effective spatial bandwidth of the AUT field. Thus, it is a spatially quasi-bandlimited function and can be effectively represented by a bandlimited function. In particular, to obtain an NR representation of the voltage \(V\) on the sphere, an optimal parameter, say \(\eta\), must be defined in order to describe any observation curve on the sphere. Furthermore, an optimal phase factor \(\mathrm{e}^{\mathrm{i}\psi(\eta)}\) must be extracted from the voltage expression, thus obtaining the so-called \"reduced voltage\":  
\[\widetilde{V}(\eta)=V(\eta)\mathrm{e}^{\mathrm{i}\psi(\eta)}, \tag{4}\]  
where \(V(\eta)\) is the voltage \(V_{p}\) or \(V_{r}\) is detected by the probe or by the rotated probe. When \(\widetilde{V}(\eta)\) is approximated with a bandlimited function, the approximation error, which exhibits step-like behavior, becomes negligible when the bandwidth exceeds a critical value \(W_{\eta}\). Therefore, the error can be controlled by choosing the bandwidth of the approximating function equal to \(\chi^{\prime}W_{\eta}\), with \(\chi^{\prime}\) being an enlargement bandwidth factor generally chosen slightly larger than one when dealing with electrically large AUTs. Figure 1: Characterization of an offset-mounted elongated AUT through a spherical spiral NF measurement system. The number of samples required to effectively represent the voltage depends on the area of the surface \(\Sigma\) modeling the AUT. Therefore, the considered offset-mounted antenna can be modeled by a rotational surface that, in order to best fit the elongated AUT geometry, could be a prolate spheroid of minor and major semi-axes equal to \(b^{\prime}\) and \(a^{\prime}\) (see Figure 2a), or a so-called rounded cylinder (see Figure 2b), i.e., a cylinder of radius \(a_{c}\) and height \(h_{c}\) ended into two hemispheres. The prolate spheroid represents a more appropriate choice when the AUT is irregularly shaped and has a maximum in the middle of its transverse section. Otherwise, the rounded cylinder represents a better solution.