|M.Sc Student||Margolis Evgeny|
|Subject||Reconstruction of Periodic Bandlimited Signals from|
|Department||Department of Electrical Engineering||Supervisor||Professor Yonina Eldar|
Digital signal processing theory relies on sampling a continuous-time signal to obtain a discrete-time representation of the signal. In many practical situations the data is sampled nonuniformly and 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 and appropriate way to deal with reconstruction from finitely many samples.
In this work, two closed-form algorithms for reconstructing a periodic bandlimited signal from its nonuniform samples are developed. The advantages and disadvantages of each method are analyzed. Specifically, it is shown that in the first algorithm the set of reconstruction functions constitute a basis, and it provides consistent reconstruction of the signal. The second approach, in which the reconstruction functions constitute a frame, is shown to be more stable in noisy environments.
We also focus on the practical aspects of the reconstruction process, such as numerical stability and efficient implementation. Not surprisingly, we show that uniform sampling results in the most stable reconstruction algorithm. We then consider two special distributions of sampling points: Uniform and recurrent nonuniform, and show that for these cases the reconstruction formula as well as the stability analysis of the algorithm are simplified significantly. In particular, we introduce a continuous-time and discrete-time filterbank implementation of the reconstruction from recurrent nonuniform samples, which leads to efficient interpolation and reconstruction methods.
We then develop methods for reconstruction of two dimensional function f(r,q) given in polar coordinates, which is 2pi-periodic in q. As an application of these algorithms, we apply them to reconstruction of medical images from their frequency domain samples, which are usually taken in polar coordinates.