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This paper addresses the theory, design, and applications of surface acoustic wave (SAW) Fourier-transform processors. These analog processors are shown to perform several sophisticated real-time signal-processing functions at wide bandwidth (tens of megahertz) making them attractive for use in radar, sonar, and communication equipments. Theoretical results show how specific arrangements of physically realizable SAW chirp filters permit Fourier transformation of both baseband and IF input signals. The operation of SAW Fourier-transform processors, with separate real- and imaginary-baseband inputs, highlights their use as wide-band "FFT-type" processors. The application of individual SAW-based processors to spectrum analysis, network analysis, beamforming, and frequency-hopped wave-form synthesis is discussed and demonstrated. Finally, the use of combinations of two or more SAW Fourier-transform processors for cepstrum analysis and programmable correlation is described.