Is it possible to extend the optimal choice of a basis to signals which are much more irregular than those which are accomodated by a wave packet basis?
Consider the signals accomodated by a seismograph. Two of the most prominent signals are sudden bursts, such as explosions initiated for the purpose of locating petroleum reserves, or precursors to a vulcanic eruption, or earth quakes. Then there is the second type of signals, those which characterize the resonant vibrational or wave motions initiated by such bursts.
It is obvious that the second type is most efficiently analyzed using Fourier or wave packet basis functions. However, a burst-like signal is characterized by variations localized in time. The signal has a finite time duration. It also has a starting edge with a finite temporal thickness which often contains rapid variations (``high frequency structure'') as exemplified in Figure 2.19. Thus under low resolution one would simply measure the amplitude profile of the main body of a pulse of finite duration. But under higher resolution one would also measure the high frequency structure which in Figure 2.19 announces the beginning of that pulse.Given such a signal, how does one represent it in the most efficient way i.e. in compliance with the priciple of unit-economy?
