An example of a function that is not ``level-phase'' is the delayed impulse .The phase of this function is so at the minus Nyquist frequency it is while at the plus Nyquist frequency it is .(The complex function has a real part that is a cosine and an imaginary part that is a sine; in the complex plane this function is a circle, looping around the origin as increases. The phase is the arctangent of the ratio of the imaginary to the real part, and it steadily increases. The average phase is not level but tilted.)

Examples of functions that are level-phase are those causal wavelets with a causal inverse, known in the geophysical world as ``minimum-phase wavelets.''

Unlike a minimum-phase wavelet,
a level-phase function need not be causal.
I have been noticing level-phase functions for some time,
but only recently recognized their essential features.
Consider for example
.In one dimension, it is
.Expressing it in finite differences,
it is an autocorrelation function
(-1,2,-1)
with the
imaginary number *i* added to the zero lag: (-1,2+*i*,-1).
Expressing
as a *Z* transform, it is
.This function is the positive imaginary constant *i* plus
a positive real spectrum.
Thus for all real values of the frequency *k*_{x},
its phase angle stays in the upper right quarter of the complex plane
so it cannot wrap around the origin,
as does the phase of .Thus
is level-phase and
using the logic of the helix
Claerbout (1998)
the operator
is also level-phase.

Let us add a causal wavelet *A*(*Z*) to an anticausal wavelet
.Using *Z*-transform polynomials in positive powers of *Z*
the sum can be denoted as

(1) |

(2) |

Although *u*_{t} is an arbitrary time function
from which we could always construct
another time function *x*_{t},
the reverse is not true.
There exist time functions *x*_{t}
for which there is no corresponding *u*_{t}.
The example that we have seen is *X*(*Z*)=*Z*.

The reason we cannot always construct a *U*(*Z*)
from any possible *X*(*Z*)
is that we cannot always take logarithms.
When poles and zeros are in the wrong place in the complex plane,
the power series for logarithm diverges.

There is no requirement on *X*(*Z*) other than that it be level-phase.
This is so because convergent Fourier sums can represent almost
any analytic function.
Since they are periodic,
the one thing they cannot make is a function whose
value at minus Nyquist differs from that at plus Nyquist.
In summary, for *x*_{t}
to be represented as an exponential with (2),
the necessary and sufficient condition
is that it be a level-phase function.

For a while I mistakenly thought
that *X*(*Z*) could be taken to be an arbitrary crosscorrelation function.
Now we see that this is not so because *Z* is
a cross-correlation function
(a white signal crosscorrelated with itself delayed).
Any crosscorrelation function can be shifted
to become a level-phase function.
(This is because we can use integration
to find the phase difference between and .Dividing by tells us how many pixels to shift.)
Thus we now have a representation for
any crosscorrelation function in terms of two
minimum-phase wavelets and a delay.

7/5/1998