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This paper addresses the spatial processing of signals collected by a linear array of sensors that feeds a filter-and-sum, data-independent beamformer. When the frequency band spanned by the signals to be processed is extremely wide, a given array can be shorter than the wavelength (at the lowest frequencies) and, at the same time, too scarcely populated for a correct sampling of the wavefield (at the highest frequencies). Superdirectivity and aperiodic sparse layouts are possible solutions to these two problems. However, these two solutions have never been considered jointly to achieve a broadband beam pattern with a desired profile. Through a mixed stochastic and analytic optimization, a method is proposed herein that synthesizes a sparse array layout and the tap coefficients of the beamformer filters to provide a broadband beam pattern that is superdirective and robust against the fluctuation of the sensors' characteristics, which is free from any grating lobes and possesses a controlled side-lobe level. Different types of beam patterns, from the frequency-invariant pattern to the maximum-directivity pattern, can be obtained and the synthesized solutions retain their validity for any steering direction inside a given interval. The functioning of the method is proven by considering a microphone array with four different design targets and by discussing the performance and the robustness of the synthesized solutions.