Level-phase functions

I define a ``level-phase function'' to be one for which the phase of the Fourier transform at minus the Nyquist frequency is the same as the phase at plus Nyquist frequency.

An example of a function that is not ``level-phase'' is the delayed impulse $Z=e^{i\omega\Delta t}$.The phase of this function is $\omega\Delta t$so at the minus Nyquist frequency it is $-\pi \Delta t$ while at the plus Nyquist frequency it is $\pi \Delta t$.(The complex function $Z=e^{i\omega\Delta t}$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 $\omega$ 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 $i - \nabla^2$.In one dimension, it is $i - \partial_{xx}$.Expressing it in finite differences, it is an autocorrelation function (-1,2,-1) with the imaginary number i added to the zero lag getting (-1,2+i,-1). Expressing $i - \partial_{xx}$ as a Z transform, it is $i + (2-\cos(k_x\Delta x))/\Delta x^2$.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 $e^{ik_x\Delta x}$.Thus $i - \partial_{xx}$ is level-phase and using the logic of the helix, the operator $i - \nabla^2$is also level-phase.

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

$$U(Z) = \overline{B}(1/Z) + A(Z)$$ (1)

Let us exponentiate this sum:  

$$X(Z) = e^{U(Z)} = e ^ { \overline{B}(1/Z) + A(Z) }$$ (2)

The heart of the matter is that the phase of X(Z) is the imaginary part of U(Z), and the imaginary part of U(Z) is a convergent sum of sines (and maybe cosines). A sum of sines is periodic with one period going from minus to plus Nyquist. As a phase, it fits the definition of ``level phase.'' Thus an arbitrary function U(Z) always constructs a level-phase function X(Z).

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). Evidently, any crosscorrelation function can be shifted to become a level-phase function. Thus we now have a representation for any crosscorrelation function in terms of two minimum-phase wavelets and a delay.