Numerical Integration Study for Spherical Near-Field to Far-Field Transformations

This paper presents a numerical integration study for spherical near field (NF) to far field (FF) transformations. The trapezoidal and Simpson 1/3 numerical integration methods are employed in the NF to FF transformations algorithm. The performance of both numerical integrations is investigated. In this study, a seven patch array antenna is employed. Simulated FF results are performed by three dimensional (3D) simulation program (CST Microwave Studio) and used as the reference. The collected NF sampling data is divided into two cases; 1) over-sampled NF angles of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">θ = φ = 2∘</tex> and 2) under-sampled NF angles of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">θ = φ = 12∘</tex> . The investigation reveals that, in the case of under-sampled near field (NF) angles of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">θ = φ = 12∘</tex> , the trapezoidal demonstrates lower errors of the FF radiation pattern in comparison to those of Simpson 1/3. Particularly, the trapezoidal is suitable for the under-sampled NF measurement case.