Complex polynomials in communications: motivation, algorithms, software

Quite recently the polynomial design methods found a new great field of application outside the control area: the algebraic approach has been used successfully in signal processing and mobile communications. In contrast to the control systems synthesis, polynomials and polynomial matrices with complex coefficients are often required when designing filters, equalizers, decouplers and other components of mobile phones for instance. Polynomial Toolbox for Matlab admits complex polynomials in most computations, including Diophantine equations and spectral factorizations. As a result, the toolbox appears a suitable tool for rapid prototyping whenever polynomial design routines with complex coefficients are required. The objective of this report is twofold. First, we explain in a clear and popular manner how the complex coefficients arise in technical practice. Based on this motivation, we present important numerical algorithms for complex polynomials and polynomial matrices and their implementation in the Polynomial Toolbox for Matlab. Finally, the power of the Toolbox is illustrated by selected numerical examples involving complex coefficients.