Design of orthogonal pulse shapes for communications via semidefinite programming

In digital communications, orthogonal pulse shapes are often used to represent message symbols for transmission through a channel. In this paper, the design of such pulse shapes is formulated as a convex semidefinite programming problem, from which a globally optimal pulse shape can be efficiently found. The formulation is used to design filters that achieve (a) the minimal bandwidth for a given filter length; (b) the minimal filter length for a given bandwidth; (c) the maximal robustness to timing error for a given bandwidth and filter length. Bandwidth is measured either in spectral energy concentration terms or with respect to a spectral mask. The effectiveness of the method is demonstrated by the design of waveforms with substantially improved performance over the "chip" waveforms specified in standards for digital mobile telecommunications.