Signal Representation, Reconstruction, and Projection

Regardless of its informational content, in this tutorial we consider that a signal is an element of an inner product space with norm induced by the inner product, . Moreover, we assume that all the signals of interest belong to some finite dimensional subspace of spanned by the set . Hence, a signal can be expressed by a finite linear superposition where the coefficients , are in .

We call measurement or sampling to the process of transforming a signal into a number. Hence a measure or sample is a functional. Because we restrict considerations to linear measures the associated functional is linear and can be expressed as for some We refer the vector to as measurement vector.

Considering measurements , each of which is obtained by a measurement vector , we have a numerical representation of as given by Now we want to answer the question as to whether it is possible to reconstruct from these measurements. More precisely, we wish to find the requirements we need to impose upon the measurement vectors , so as to use the concomitant measures , as coefficients for the signal reconstruction, i.e., we wish to have (1)

By denoting (2)

where the operation indicates that acts by taking inner products, (1) is written as As will be discussed next, the above equation tells us that the measurement vectors , should be such that the operator is a projector onto .

Subsections