In practical applications, the signal we deal with is usually a finite duration one. Continuous prolate spheroidal wave functions (PSWFs) were proposed by Slepian and are useful for analyzing the characters of the finite duration continuous Fourier transform. Based on the PSWF, the finite fractional Fourier transform was developed. In this paper, for digital signal processing application, we derive discrete-to-discrete prolate spheroidal wave functions. Then, we define the finite duration discrete fractional Fourier transform (fi-DFRFT) based on it. We can use the fi-DFRFT for filter design, multiplexing, modulation, encryption, and optical system simulation. The fi-DFRFT has the advantage of less complexity and is useful for deal with the noise that is chirp-like and finite duration.
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