An Optimum Complete Orthonormal Basis for Signal Analysis and Design

We derive a new set of basis functions which have high energy concentration in both time and frequency. Our approach is based on transforming a signal energy maximization problem into an equivalent eigenvalue problem for a certain integral operator, and then showing that the eigenvectors of this integral operator form a complete orthonormal basis. This basis set is in some sense optimal since each basis function has the highest energy concentration within a certain subspace of ℒ2. Furthermore, this basis can be employed to analyze and design communication signals for which the energy concentration within the essential duration and essential bandwidth is important and some other optimum characteristics would also be required.