The Mellin Matched Filter

In propagation channels signals are undergone translation and dilation changes. The Fourier and Mellin transforms are natural foundations of the analysis of wideband signals. Three interconnected problems, where the Mellin transform plays a key role, are considered in the paper: i) synthesis of a linear matched filter that is invariant to signal scale; ii) estimation of the signal scale; and iii) the wideband ambiguity function and its properties. The filter that is invariant to the signal scale and maximizes the output SNR is synthesized in the paper using the Mellin transform. The filter is named the Mellin matched filter. Introduction of the Mellin matched filter allows to give a new interpretation of the signal scale estimation and the wideband ambiguity function. Concept of the Mellin matched filter facilitates to produce a physical interpretation of the resolution of wideband signals and the time-frequency-scale uncertainties.