Generating nonorthogonal bases for signal representation

The representation of functions in a basis function expansionz(t)= k=1/=,a_k>x_k(t) is straightforward when the basis functionsx_k(t) are orthogonal. There has been very little work up to this time in determining how to use nonorthogonal bases in signal representation. On the other hand, applications in data compression and signal synthesis often require using specific tailor-made bases. Presented here is a method for constructing very general nonorthogonal bases.Orthogonality has often been used to show that a basis spans the set of functions of interest and to calculate the coefficients of the representation. In this paper, both of these fundamental aspects are addressed for nonorthogonal bases. A new basis {y_k(t)} is obtained by performing a linear transformation on a known existing basis {x_k(t)}. This transformation is constructed such that the coefficients of signal representation on the new basis are readily found. Then, a useful and sufficient condition is placed upon the new basis such that representations converge.The fundamental methods are applied to the standard examples of signal representation. The complex sinusoids, the Rademacher functions, the orthogonal polynomials, and the decaying exponentials are used as the original basis {x_k(t)} from which a new basis {y_k(t)} is generated. Two examples are given to illustrate general applications: one in signal synthesis and one in signal analysis.