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It is known that one can use Shannon's theory to sample a bandlimited signal. In this paper, we introduce how to use prolate spheroidal wave functions (PSWFs) to sample a time-limited and nearly band-limited signal. PSWFs have the property of optimal energy concentration. Thus we can apply it to sampling theory to reduce the aliasing error of the recovered signal. We derive a theory that can estimate the upper bound of the error. With it, we can determine, to achieve certain accuracy, how many samples we should acquire. Moreover, we combine the proposed sampling theory with the fractional Fourier transform (FRFT). We also find an important theory, i.e., to achieve a certain degree of accuracy, the number of sampling points required for a signal is proportional to the 'area' of its time-frequency distribution.