Analysis and Approximation of a Spatially Wideband Antenna Array Factor

This work investigates the spatially wideband (SWB) antenna array factor (AF) of uniform linear arrays. First, the SWB AF approximation is derived for narrowband (NB) signals and a large aperture with element spacing satisfying the Nyquist criterion. The derivation accounts for different spatial and spectral windows. Next, an approximation of the SWB AF for a wideband (WB) signal is developed under uniform spatial and spectral weighting. The analysis shows that for fully populated arrays, increasing the bandwidth effectively suppresses the AF sidelobes. Finally, a universal SWB AF approximation is introduced, which is based on recognizing the SWB AF as a spatially variant convolution. In this formulation, the SWB AF is expressed as a convolution of the spatially narrowband (SNB) AF and the SWB kernel, providing insight into how the bandwidth and the spectral weighting affect the resulting SWB AF. The proposed approximation is shown to be accurate for a wide range of bandwidths and element spacings, including sparse arrays. In particular, for sparse arrays, the bandwidth enables suppression of grating-lobe amplitudes by spreading their energy over a wider angular range. An approximation of the grating lobe envelope as a function of the bandwidth-aperture product is provided.