Frequency/duration relation.

The concepts of Fourier frequency and wavelet duration are obviously related. To translate results (e.g. spectra) from one to the other, the relation needs to be established at a quantitative level. The result is somewhat unsettling, because the correspondence will depend on the wavelet shape.

Among many sensible definitions, we chose to require that the peak of the wavelet mean spectrum should be matched with the singularity of the Fourier transform of the cosine wave. Then, it can be shown that the frequency of the cosine is related to the wavelet duration by the relation

Analytically, the peak of the mean wavelet spectrum is obtained by cancelling the derivative of the spectrum with respect to k: HR>

HR> gives the above relation.

The ambiguous definition of the duration of an event is common to the sizing of any object of irregular shape. Say, if an object is a perfect hard sphere, nobody will argue taking its diameter as THE size (although its radius and circumference could also be adopted). But what is the size of a spheroid, such as the Earth? What is the size of a snowball? of a football? of a cotton ball? A mean effective diameter might make sense if the total volume occupied is the primary concern - but what if the mass of material is (think of the cotton ball)? what if the spheroid is to fit in a hole with tight tolerances?

Similarly, many definitions of `duration' (and an associated dominant frequency) emerge from our analysis. Each wavelet shape will provide a slightly different number, and should be used accordingly with caution.