In this paper, we prove a compressive sensing guarantee for restricted measurement domains in spherical near-field to far-field transformations for antenna metrology. We do so by first defining Slepian functions on a measurement sub-domain $R$ of the rotation group $\sot$, the full domain of the linear inverse problem associated with spherical near-field to far-field transformations. Then, we transform the inverse problem from the measurement basis, the bounded orthonormal system of band-limited Wigner $D$-functions on $\sot$, to the Slepian functions in a way that limits increases to signal sparsity. Contrasting methods using Wigner $D$-functions that require measurements on all of $\sot$, we show that the orthogonality structure of the Slepian functions only requires measurements on the sub-domain $R$, which is select-able. Due to the particulars of this approach and the inherent presence of Slepian functions with low concentrations on $R$, our approach gives the highest accuracy when the signal under study is well concentrated on $R$. We provide numerical examples of our method in comparison with other classical and compressive sensing approaches. In terms of reconstruction quality, we find that our method outperforms the other compressive sensing approaches we test and is at least as good as classical approaches but with a significant reduction in the number of measurements.
翻译:在本文中, 我们证明这是限制测量域的压缩感应保证, 球性近场至远场天线计量仪转换为Floor- field $D- 功能。 我们这样做的方式是首先界定Slepian函数, 用于旋转组的测量分域 $\sot$, 这是与球性近场至远场变换相关的线性反问题的全部领域。 然后, 我们将反向问题从测量基础, 带带限制的Wigner $D- 功能系统, 以美元为单位, 至Slepian 函数, 从而限制对信号偏移的增加。 使用Wigner $D$- 函数的对比方法, 需要测量全部 $\sot$, 我们显示, Slepian 函数的反向结构, 只需要对可选择的子值 $美元进行测量。 由于这种方法的特殊性, 以及Slepian 函数以美元为低浓度的内在存在, 我们的方法, 我们的方法以最精确的精确性的方法 。