# Accession Number:

## AD0271107

# Title:

## REPRESENTATION AND ANALYSIS OF SIGNALS. PART IX. 'COMPLEMENTARY' SIGNALS AND ORTHOGONALIZED EXPONENTIALS

# Descriptive Note:

# Corporate Author:

## JOHNS HOPKINS UNIV BALTIMORE MD

# Personal Author(s):

# Report Date:

## 1962-01-01

# Pagination or Media Count:

## 1.0

# Abstract:

When a signal is approximated by a finite set of component signals which span a subspace, 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. This analysis concerns the present-instant error, the complementary signal and the complementary filter which are useful concepts in approximating a signal by one-sided exponential components. In particular, it is found that the complementary filter for a given finite exponential basis is an all-pass rational transmittance having zeros in the frequency domin 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. Author