Functional analysis

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 $-1 \le \ \rho\ \le\ 1$ of an impedance function is an impedance function.

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

$$U (Z) \ \ \ =\ \ \ u_0\ +\ u_1\ Z\ +\ u_2\ Z^2 \ +\ \cdots$$ (51)

and substitute it into the familiar power series for the exponential function:  

$$B (Z) \ \ \ =\ \ \ e^U \eq 1\ +\ U\ +\ { U^2 \over 2 !}\ +\ ... ...r 3 !}\ + \cdots \ \ \ \ \ \ \ \ ( \vert U \vert \ < \ \infty )$$ (52)

No negative powers of Z can be found in the right side of (52), so B(Z) will have no negative powers of Z. Also, the factorials in the denominator assure us that (52) always converges, so b_t is always causal.

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

$$\begin{eqnarray} B_+ \ \ \ &=&\ \ \ e^{{+U}} \\ B_- \ \ \ &=&\ \ \ e^{{-U}}\end{eqnarray}$$ (53) (54)

Clearly both B₊ and B_- are causal, and they are inverses of one another. A minimum-phase filter is defined to be causal with a causal inverse. So B₊ and B_- are minimum phase.

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:

$$\begin{eqnarray} U \ \ \ &=&\ \ \ \ln \ B \\ {dU \over dZ} \ \ \ &=&\ \ \ u_1\ +... ...\cdots \\ {dU \over dZ} \ \ \ &=&\ \ \ {1 \over B }\ {dB \over dZ}\end{eqnarray}$$ (55) (56) (57)

Since B was assumed to be minimum phase, both 1/B and dB/dZ on the right of (57) are causal. Since the product of two causals is causal, dU/dZ is causal. But dU/dZ cannot be causal unless U is causal. That proves theorem 3, disregarding the remote danger that B might converge while dB/dZ diverges.

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 $[x( \omega ) , y ( \omega ) ] = [\Re\, B(Z),\,\Im\,B(Z)]$.The phase angle $\phi ( \omega )$ is defined by the arctangent of the ratio y/x. For example, the causal, non-minimum-phase filter $U(Z) = Z = e^{{i} \omega}$ gives the parametric equations $x=\cos\omega$ and $y=\sin\omega$ which define a circle surrounding the origin. Notice that the phase of $Z=e^{{i} \omega}$ is $\phi ( \omega ) = \omega$, which is a monotonically increasing function of $\omega$.In the minimum phase case, $\phi ( \omega =0 ) = \phi ( \omega = 2 \pi )$.In the non-minimum-phase case, the curve loops the origin, so $\phi ( \omega =0 ) \ =\phi ( \omega = 2 \pi )+ 2 \pi$.Theorem 3 allows us to say that a general formula for minimum-phase filters is

$$\begin{eqnarray} \ \ \ B \eq e^{U(Z)} \ \ \ &=&\ \ \ \exp \left( \ \sum_{{k=0}... ...\right) \\ \ &=&\ \ \ \exp [r( \omega ) \ +\ i \ \phi ( \omega )]\end{eqnarray}$$ (58) (59)

The phase $\phi ( \omega )$, being a sum of periodic functions, is itself a periodic function of $\omega$,which means that in the plane of $( \Re\,B,\ \Im\,B)$ the curve representing $B ( \omega )$ does not enclose the origin.

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

$$B^r \eq \left( e^{ \ln \ B } \right)^r \eq e^{{r}\ \ln\ B}$$ (60)

Since B is assumed to be minimum phase, by theorem 3, $\ln \, B$ will be causal. Scaling by a real or complex constant r does not change causality. Exponentiating shows, by theorem 2, that B^r is minimum phase.

Finally the proof of theorem 6, that an impedance function can be raised to any real fractional power $-1\ \le\ \rho\ \le\ + 1$ 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 $\phi$ lies in the range $- \pi / 2\ < \ \phi\ < \ + \pi / 2$.Raising the impedance function to the $\rho$ power will compress the range to $- \pi \rho / 2 \ < \ \phi \ < \ \pi \rho / 2$.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.