Description
Shannon proves in his 1949 paper "Communication in the Presence of Noise", that sampling at an optimal rate gives us the ability to completely determine a signal without loss of information. By sampling, we can more effectively store, analyze and transfer data. However, this only applies to band-limited, periodic signals. Are we able to completely determine a signal which is inhomogeneous or not band-limited? In this paper we will review details of sampling and discuss approaches for extending Shannon's sampling theory to a broader range of signal types. By expanding this theory, we can then apply this to multiple real world signals for more accurate information, such as weather patterns, precipitation, cloud coverage, etc. By using Empirical Orthogonal Functions (EOF's), the decomposition of a specified data set using regression analysis to estimate the relationship between variables, we are able to capture functions/signals which are inhomogeneous. Then, using common fields for the EOF's, we are more easily able to validate our sampling and reconstruction, and in turn apply this same theory to the inhomogeneous function. There are still open areas in this research, some of these topics include aliasing, further error analysis, and potentially different types of reconstruction techniques which may be better suited for EOF's. As for now, we can show zero error for a homogeneous function using EOF's and a regression reconstruction technique for inhomogeneous fields.
