UWB orthogonal pulse design using Sturm–Liouville boundary value problem

The problem of designing UWB pulses which meet specific spectrum requirements is usually treated by filtering common pulses such as Gaussian doublets, modified Hermite polynomials and wavelets. When there is the need to have a number of orthogonal pulses (e.g., in a multiuser scenario), a naive approach is to filter all the members of an orthogonal set, which is likely to destroy their orthogonality property. In this paper, we study the design of a set of pulses that simultaneously satisfy the orthogonality property and spectrum requirements. Our design is based on the eigenfunctions of Sturm-Liouville boundary value problems. Indeed, we introduce Sturm-Liouville differential equations for which the eigenfunctions meet the FCC mask constraints. Computer simulation results show that all such waveforms occupy almost 55% of the allowed spectrum (utilization efficiency). A comparison of the proposed method with some conventional techniques of orthogonal UWB pulse generation will demonstrate the advantages of the new proposal.