The sampling theorem associated with the fractional Fourier transform can be looked as the convolution of the sinc kernel with infinite sequence of signal points and chirp signal modulations. But in most practical applications we only have finite number of samples, which makes a perfect reconstruction of the original signal impossible. To solve this problem, we obtain a new formula for perfect reconstruction of discrete chirp-periodic signal points based on the fractional Fourier transform in this paper. The method is equivalent to trigonometrically interpolation by fractional Fourier series expansion and can be looked as a generalization of the classical results.