The scale representation

We consider “scale” a physical attribute of a signal and develop its properties. We present an operator which represents scale and study its characteristics and representation. This allows one to define the scale transform and the energy scale density spectrum which is an indication of the intensity of scale values in a signal. We obtain explicit expressions for the mean scale, scale bandwidth, instantaneous scale, and scale group delay. Furthermore, we derive expressions for mean time, mean frequency, duration, frequency bandwidth in terms of the scale variable. The short-time transform is defined and used to obtain the conditional value of scale for a given time. We show that as the windows narrows one obtains instantaneous scale. Convolution and correlation theorems for scale are derived. A formulation is devised for studying linear scale-invariant systems. We derive joint representations of time-scale and frequency-scale. General classes for each are presented using the same methodology as for the time-frequency case. As special cases the joint distributions of Marinovich-Altes and Bertrand-Bertrand are recovered. Also, joint representations of the three quantities, time-frequency-scale are devised. A general expression for the local scale autocorrelation function is given. Uncertainty principles for scale and time and scale and frequency are derived.