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The design of an antipodal exponentially corrugated tapered-slot antenna to be used in a microwave system for brain stroke detection is presented. The antenna is designed to operate across the band from 1 GHz to 4 GHz that is widely used for head imaging. Tapered corrugations are used to reduce the size of the antenna and to improve its directivity especially at the lower end of the band. The antenna has dimensions of (95 mm × 120 mm) using the high dielectric constant substrate Rogers R03010 (thickness × 1.28 mm, and dielectric constant×10.6). The results indicate an antenna with a 10-dB return loss bandwidth and better than 10 dB front-to-back ratio across the band from 1.1 GHz to 4 GHz.