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Digital processing techniques are based on representing a continuous-time signal by a discrete set of samples. This paper treats the problem of reconstructing a periodic bandlimited signal from a finite number of its nonuniform samples. In practical applications, only a finite number of values are given. Extending the samples periodically across the boundaries, and assuming that the underlying continuous time signal is bandlimited, provides a simple way to deal with reconstruction from finitely many samples. Two algorithms for reconstructing a periodic bandlimited signal from an even and an odd number of nonuniform samples are developed. In the first, the reconstruction functions constitute a basis while in the second, they form a frame so that there are more samples than needed for perfect reconstruction. The advantages and disadvantages of each method are analyzed. Specifically, it is shown that the first algorithm provides consistent reconstruction of the signal while the second is shown to be more stable in noisy environments. Next, we use the theory of finite dimensional frames to characterize the stability of our algorithms. We then consider two special distributions of sampling points: uniform and recurrent nonuniform, and show that for these cases, the reconstruction formulas as well as the stability analysis are simplified significantly. The advantage of our methods over conventional approaches is demonstrated by numerical experiments.