Relation of amplitude to phase

As we learned from equation (19), a minimum-phase function is determined completely from its spectrum. Thus its phase is determinable from its spectrum. Likewise, we will see that, except for a scale, the spectrum is determinable from the phase.

So far we have not discussed the fact that spectral factorization implicitly uses Hilbert transformation. Somehow we simply generated a phase. To see how the phase arose, recall equation (18) and (19):  

$$S_k \ = \ e^{\ln S_k} \ = \ e^{U_k} \ = \ e^{ (U_k - i\Phi... ... } \ = \ e^{\overline{C}_k} e^{C_k} \ = \ \overline{B}_k B_k$$ (20)

Where did $\Phi_k$ come from? We took U_k + i 0 to the time domain, obtaining u_t. Then we multiplied u_t by a real-valued step function of time. This multiplication in the time domain is what created the phase, because multiplication in the time domain implies a convolution in the frequency domain. Recall that the Fourier transform of a real-valued step function arises with Hilbert transform. Multiplying in time with a step means that, in frequency, U_k has been convolved with $\delta_{k =0} +i \times (90^{\circ}$ phase-shift filter). So U_k is unchanged and a phase, $\Phi_k$, has been generated. This explanation will be somewhat clearer if you review the Z-transform approach discussed at the beginning of the chapter, because there we can see both the frequency domain and the time domain in one expression.

To illustrate different classes of discontinuity, pulse, step, and slope, Figure 13 shows another Hilbert-transform pair.

 

hilb2
Figure 13 A Hilbert-transform pair.

EXERCISES:

1. What is the meaning of minimum-phase waveform if the roles of the time and frequency domains are interchanged?
2. Show how to do the inverse Hilbert transform: given $\phi$, find u. What is the interpretation of the fact that we cannot get u₀?
3. Consider a model of a portion of the earth where x is the north coordinate, +z represents altitude above the earth, and magnetic bodies are distributed in the earth, creating no component of magnetic field in the east-west direction. We can show that the magnetic field h above the earth is represented by

   $$\left[ \begin{array} {cc} h_x (x, z) \ h_z (x, z) \end{arr... ...t k\vert \end{array} \right] \, e^{ikx - \vert k\vert z} \, dk$$

   Here F(k) is some spatial frequency spectrum.

   (a)

   By using Fourier transforms, how do you compute h_x(x, 0) from h_z (x, 0) and vice versa?

   (b)

   Given h_z(x, 0), how do you compute h_z(x, z)?

   (c)

   Notice that, at z = 0,

   $$f(x) = h_z(x) + ih_x(x) = \int^{+\infty}_{-\infty} e^{ikx} F(k)\, (\vert k\vert + k)\, dk$$

   and that F(k) (|k| + k) is a one-sided function of k. With a total field magnetometer we observe that

   $$h^2_x (x) + h^2_z(x) = w(x) \bar w (x)$$

   What can you say about obtaining F(k) from this?

   (d)

   How unique are h_x(x) and h_z(x) if $f(x) \bar f(x)$ is given?

4. Test this idea: write code to factor X(Z) into X(Z)=A(Z)B(Z), where B(Z) is minimum-phase and A(Z) is maximum-phase. Maximum-phase means that Z^N A(1/Z) is minimum-phase. First compute $U(\omega)= \ln X(\omega)$.Then remove a linear trend in the phase of $U(\omega)$ to get N. Then split U with its trend removed into causal and anticausal parts U(Z) = C^-(1/Z) + C⁺(Z). Finally, form $B(Z)=\exp C^+(Z)$ and $Z^N A(1/Z)=\exp (C^-(Z))$.