We will establish the following theorems about exponentials, logarithms, and powers of Fourier transforms of filters:
To establish theorem 1, define the Z-transform of an arbitrary causal function
To establish theorem 2, that the exponential is not just causal but also minimum phase, consider
Now we will establish the converse of theorem 2--namely, theorem 3--which states that the logarithm of a minimum-phase filter is causal. Take the logarithm of (52) and form the Z-derivative:
Theorem 4 refers to the Fourier domain representation of the minimum-phase filter. In the complex plane, the filter gives parametric equations for a curve, say .The phase angle is defined by the arctangent of the ratio y/x. For example, the causal, non-minimum-phase filter gives the parametric equations and which define a circle surrounding the origin. Notice that the phase of is , which is a monotonically increasing function of .In the minimum phase case, .In the non-minimum-phase case, the curve loops the origin, so .Theorem 3 allows us to say that a general formula for minimum-phase filters is
On to theorem 5, which says that any power of a minimum-phase function is minimum phase. Consider
Finally the proof of theorem 6, that an impedance function can be raised to any real fractional power and the result will still be an impedance function. An impedance function is defined to be a minimum-phase function with the additional property that the real part of its Fourier transform is positive. This means that the phase angle lies in the range .Raising the impedance function to the power will compress the range to .This will keep the real part of the impedance function positive. Theorem 5 states that any power of a minimum-phase function is causal, which is more than we need to be certain that a fractional real power of an impedance function will be causal.