'Complementary' Signals and Orthogonalized Exponentials

When a signal is approximated by a finite set of component signals which span a subspace
\Phi
, the least-square approximation may be interpreted geometrically in signal space as the projection of the true signal vector upon this finite dimensional subspace. In case the component signals are one-sided exponentials, the projection operators may be realized by simple physical filters following Kautz' procedure for constructing orthogonalized exponentials. The purpose of this paper is to describe the 'present-instant' error, the 'complementary' signal and the 'complementary' filter which are useful concepts in approximating a signal by one-sided exponential components. An attempt is made to interpret directly in the time domain Kautz' procedure using the 'present-instant' error concept. Some important properties of the 'complementary' signals which prove to be of value in simplifying the process of error energy evaluation and synthesis of the approximating signal, are derived and discussed. In particular, it is found that the 'complementary' filter for a given finite exponential basis is simply an all-pass rational transmittance having zeros in the frequency domain which match the exponents of the basis. This familiar all-pass filter indeed represents an orthogonal transformation which preserves the energy of the signal under transformation. A signal to be approximated is transformed by this filter into the 'complementary' signal which can be separated in time domain into two parts, namely a 'complementary' approximating signal and a 'complementary' error signal. The actual approximating signal and error signal may be recovered independently from them by making certain physical filtering operations.