The ability of tapered leaky-wave antennas (LWAs) to synthesize directive radiation diagrams with suppression of the radiation in prescribed angular regions is studied in this work. The synthesis is based on an iterative Fourier technique that has been adapted for the case of LWAs, taking into account that the far field is related to the antenna illumination through an inverse Fourier transform. Results show that the simultaneous tapering of both phase and leakage constants of the leaky mode allow to obtain wide and deep radiation nulls while maintaining the high-gain scanning pattern of LWAs. The limitations of this unconventional tapering technique are also reported, and a practical design is validated with simulations.