We will establish the following theorems about exponentials, logarithms, and powers of Fourier transforms of filters:

- The exponential of a causal filter is causal.
- The exponential of a causal filter is a minimum-phase filter.
- The logarithm of a minimum-phase filter is causal.
- The Fourier domain representation of a minimum-phase filter is a curve that does not enclose the origin of the complex plane.
- Any power of a minimum-phase filter is minimum phase.
- Any real fractional power of an impedance function is an impedance function.

To establish theorem 1, define the *Z*-transform of an arbitrary
causal function

(51) |

(52) |

To establish theorem 2, that the exponential is not just causal but also minimum phase, consider

(53) | ||

(54) |

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:

(55) | ||

(56) | ||

(57) |

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

(58) | ||

(59) |

On to theorem 5, which says that any power of a minimum-phase function is minimum phase. Consider

(60) |

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.

10/31/1997