## 1. Introduction

[2] Near-field antenna measurements and near-field to far-field transformations have been a subject of extensive research efforts over the years [*Larsen*, 1977; *Yaghjian and Wittmann*, 1985; *Yaghjian*, 1986; *Hansen*, 2009]. Recently, probe pattern compensated near-field to far-field transformation techniques, known as probe correction techniques, have fascinated different researchers as evidenced, for example, by *Laitinen et al.* [2005], *Schmidt et al.* [2008], *Pogorzelski* [2008], and *Laitinen et al.* [2010]. These probe correction techniques are required to accurately determine the far field from probe signals gathered in the near field of the antenna under test (AUT) [*Yaghjian*, 1986].

[3] Until recently, the first-order probe correction technique [*Wacker*, 1974; *Jensen*, 1975], described in detail by *Hansen* [1988], has been the preferred technique for spherical near-field measurements due to its computational efficiency and stability. The application of this technique requires a specific probe to be used which has only first-order azimuthal variations in its pattern. Also scanning must occur on a rectangular ϕ-θ scan grid at a fixed measurement distance. Although being computationally powerful, the use of the first-order technique has been seen problematic in a few respects. In particular, the probe requirement imposed by the first-order technique has been considered a problem [*Laitinen et al.*, 2005], but also confining to a regular ϕ-θ grid and a fixed measurement distance may be unfavorable in some cases.

[4] Several probe correction techniques have been recently presented, allowing a full probe correction for probes possessing also zeroth-order or higher than first-order variations in their azimuthal pattern. Although providing simplifications in the probe requirements, these so-called high-order probe correction techniques [*Laitinen et al.*, 2005; *Laitinen*, 2008; *Pogorzelski*, 2008] require, similar to the first-order probe correction technique, the probe signals to be available on a regular ϕ-θ scan grid. These techniques have a higher computational complexity compared to the first-order probe correction technique.

[5] Plane wave based near-field far-field transformation techniques with full probe correction inspired by the Fast Multipole Method (FMM) [*Coifman et al.*, 1993] and Multilevel Fast Multipole Method (MLFMM) [*Chew et al.*, 2001] have been recently presented by *Schmidt et al.* [2008] and *Schmidt and Eibert* [2009]. Advantageously, these techniques are applicable to arbitrary and irregular measurement contours. Moreover, the computational complexity of the multilevel plane wave based technique [*Schmidt and Eibert*, 2009] is *O*(*N* log *N*) per iteration of the employed iterative linear equation solver, which is lower than that of the first-order probe correction technique. Hereby, *N* refers to the number of measurement points. In terms of the computational complexity, the plane wave based multilevel transformation is superior to the other probe correction techniques presented so far. However, the plane wave expansion is an interpolatory representation of the AUT antenna pattern demanding a certain amount of oversampling for the interpolations to work. This does, however, not require to increase the measurement sample density and dependent on the chosen interpolations the oversampling of the internal AUT pattern representations can be very low. In contrast, the modal expansions are hierarchical orthogonal series expansions, where the individual modes can be processed independently and any sampling redundancy can be removed. Therefore, it is desirable to work with orthogonal expansions where possible.

[6] Recent work presented by *Schmidt et al.* [2010] has shown that the azimuthal Fourier mode expansion with Fast Fourier Transform preprocessing can be successfully utilized in conjunction with the single level plane wave based near-field far-field transformation [*Schmidt et al.*, 2008] to speed-up the computations for spherical near-field measurements at the cost of restriction to measurement grids equidistantly sampled in ϕ. The purpose of this work, as a continuation of the work presented by *Schmidt et al.* [2010], is to present and demonstrate the functionality of a low complexity near-field far-field transformation technique with full probe correction that is based on Fast Fourier Transform preprocessing and the multilevel plane wave based transformation algorithm. This technique is essentially similar to the work of *Schmidt et al.* [2010], but it is now employed with the multilevel transformation algorithm inspired by the MLFMM. The technique takes advantage of the orthogonality property of the azimuthal Fourier modes in the same manner as done in the first-order probe correction. Then, it exploits the low complexity plane wave based near-field transformation for the remaining computations. In this way, as compared to the work of *Schmidt and Eibert* [2009], a lower overall computation time is achieved and the computational complexity of the technique is not increased. It is, however, noted that the computational improvements are reached at the expense that arbitrary measurement contours are not applicable anymore. The exploitation of the orthogonality property of the azimuthal Fourier modes restricts the applicable measurement grids to those which are rotationally symmetric around the *z*-axis and where the probe signals are available equidistantly in ϕ from 0° to 360° − Δϕ, where Δϕ is the ϕ step. Hence, this new technique is applicable for “body of revolution” scan surfaces.

[7] Section 2 of this paper presents the theory of the technique. Section 3 describes the simulations, and section 4 describes the measurements for the validation of the technique. Conclusions are presented in section 5.