Finite representation of finite energy signals

In this thesis, we study how to encode finite energy signals by finitely many bits. Since such an encoding is bound to be lossy, there is an inevitable reconstruction error in the recovery of the original signal. We also analyze this reconstruction error. In our work, we not only verify the intuition that finiteness of the energy for a signal implies finite degree of freedom, but also optimize the reconstruction parameters to get the minimum possible reconstruction error by using a given number of bits and to achieve a given reconstruction error by using minimum number of bits. This optimization leads to a number of bits vs reconstruction error curve consisting of the best achievable points, which reminds us the rate distortion curve in information theory. However, the rate distortion theorem are not concerned with sampling, whereas we need to take sampling into consideration in order to reduce the finite energy signal we deal with to finitely many variables to be quantized. Therefore, we first propose a finite sample representation scheme and question the optimality of it. Then, after representing the signal of interest by finite number of samples at the expense of a certain error, we discuss several quantization methods for these finitely many samples and compare their performances.