  In this work, we tackle the problem of beampattern design for a transmit
architecture wherein a large reconfigurable intelligent surface (RIS) redirects
the radio frequency signals emitted by a few active antennas (sources). After
introducing a convenient signal model and discussing the impact of the signal
bandwidth, the source-RIS channel, and the system geometry on the involved
derivations, we propose to optimize the waveform emitted by each source and the
phase shifts introduced by the RIS to match a desired space-frequency
distribution of the far-field radiation pattern, which may be relevant in both
radar and communication applications. We provide a sub-optimum solution to the
considered problem under a constraint on the total power radiated by the
sources and, possibly, on the constant modulus of the involved waveforms. The
reported example demonstrates that the beampattern of an RIS-based transmit
architecture can be well shaped; in particular, for the same array size and the
same desired radiation pattern, the resulting approximation error remains
comparable to that obtained with a fully-digital MIMO array, especially when
constant-modulus waveforms are required, and smaller than that obtained with a
phased array.
