The wavelet transform (WT) has been found to be particularly useful for analyzing signals which can best be described as aperiodic, noisy, intermittent, transient and so on. Its ability to examine the signal simultaneously in both time and frequency in a distinctly different way from the traditional short time Fourier transform (STFT) has spawned a number of sophisticated wavelet-based methods for signal manipulation and interrogation.
Morlet wavelets are the Hamlet's ghost of this work: they make few onstage appearances, but the action could not take place without them.
Their chief benefit, from which all others flow, is that they let one describe wave functions in a way that is localized in time and frequency simultaneously.
They are widely used in applied physics (Morlet was a geophysicist who worked for the French oil company ELF) and in computer graphics (it is not possible to destroy Tokyo in a really satisfying way without wavelets), but are perhaps not that common in the physics literature (but see [Kim-1996]).
We are relying primarily on the discussion in [Kaiser-1994]. Morlet's original reference is [Morlet-1982]. Also helpful are [Addison-2002], [Bratteli-2002], [vandenBerg-1999].
As they are critical for this analysis we review the relevant properties.
Wavelets are generated by starting with a mother wavelet:
We get the general wavelet by scaling the mother wavelet by a scale factor s and displacing her by a displacement d :
As with life, so with wavelets: correct choice of your mother is essential for success. For Morlet wavelets the mother wavelet is given by:
The second term is frankly less interesting but is needed to satisfy the admissibility condition, discussed below.
For us, is usually the quantum time and f is a reference frequency. We keep f a variable to help in calculating the value of the admissibility constant . In the text, we set f equal to one.
Both scale and displacement run from to . Scaling and displacing make the general Morlet wavelet:
Morlet wavelets are (sums of) Gaussians, so work well with path integrals.
The mother wavelet herself is given in this notation as the wavelet with scale factor one, displacement zero:
Any square integrable function may be expressed as a sum over wavelets:
Where is the admissibility constant, given by an integral over the square of the Fourier transform of the mother wavelet:
[ 3]The decomposition will not work if is not finite. For to be finite, we need the zero frequency component of the Fourier transform of the Morlet wavelet mother to be zero:
The Fourier transform of the Morlet mother is:
The Fourier transform of the general Morlet wavelet is:
This may be written in terms of the Fourier transform of the mother:
So for the Fourier transform, the displacement d contributes an exponential factor, while the scale affects the normalization and the frequency.
We can establish the completeness of Morlet wavelet analysis and synthesis using the theory of frames, see [Kaiser-1994]. Here we do this directly.
If we write the analysis and synthesis together we see we want:
We see this will be true if we have:
This looks like a familiar decomposition in terms of a set of states, albeit with an unfamiliar weight function.
If we can show this directly, we have all we need.
To do this, define I by:
We wish to show that this integral gives the function. We write the Morlet wavelets in terms of their Fourier transforms to get:
Then we write the Fourier transforms of the wavelets in terms of the Fourier transform of the mother wavelet:
We recognize the integral over d as a function in w and w ':
Then use this previously disguised function to do the integral over w ':
We change the variable of integration:
And break up the integral into two parts, with scale positive and scale negative:
Then we identify the w integration as yet another function, one which can come outside of the integral:
Getting:
And then by identifying the remaining integral as we conclude:
We do not need the actual value of , only the fact that it is finite.
Now we compute the actual value of the admissibility constant. Explicitly, in terms of the Fourier transform of the mother wavelet:
For convenience, we define:
For f equal to zero, I is zero by inspection.
As w goes to zero, the integrand goes as:
So it is well-behaved in the small w limit. It is obviously convergent as w goes to ; the exponential with argument quadratic in w ensures this.
Therefore we can write I as:
The derivative of I is:
Or:
Giving:
We integrate this, to get an analytic form for :
Where the F 's are generalized hypergeometric functions. For f set to one we have:
Which can be checked by doing the original integral numerically.
Uses of Morlet wavelets in this work:
To avoid errors which can result from the use of non-normalizable wave functions, i.e. plane waves and functions: Wave functions.
To provide a basis for wave functions which works well with path integrals: Feynman path integrals.
To provide convergence in a natural way, without using convergence factors or Wick rotation: Convergence.
To define laboratory time in a manifestly symmetric between time and space way: Geodesic definition of laboratory time.
To argue that our Gaussian gates fully general: Slits in time.
[ 3]There is a difference of from the definition in Kaiser, due to a difference in the definitions of the Fourier transforms.