Morphological comparison of signals and its application for estimating signal lag time

Mathematical methods are proposed for comparing the shape of signals with amplitudes distorted by unknown monotonic transformations. These methods are based on the solution of the problems of the best approximation of the signal under analysis to signals of a specified class, and used in estimating the lag time of one fragment of signal with respect to another. Such problems arise, for example, in determining the direction of propagation of a sound wave in the atmosphere. The solution to this problem is based on the assumption that the conditions of the signal detection are different at different spatial points; as a result, the measured signals differ, not only in their lag time but also in nonlinear distortions, such that only the general sape of the signal is preserved: if the amplitude of one signal is the result of a strictly monotonic transformation of the amplitude of another signal, then their shapes are equivalent. In addition, the measurements are accompanied by an additive noise with an unknown dispersion.