Muir's rules for combining impedances

For every physical system that conserves or dissipates energy there is an impedance function. Impedance functions are special combinations of differential operators and positive-valued physical constants. We will see just what combinations are allowed.

To ensure stable computations, it is important to be able to ensure that a supposed impedance function really is an impedance function. A difficulty in applied geophysics is this: Although you might require results only over a limited range of frequencies, and you might make approximations that are reasonable within that range, if the calculated impedance becomes negative outside the applicable range (it often happens near the Nyquist frequency), then the impedance filter will yield a numerically divergent output. So even though the impedance is almost correct, it is not usable.

Francis Muir provided three rules for combining simple impedances to get more complicated ones. These rules are especially useful because we can start from the discrete-time forms of the differentiation and integration operators. Let R ' denote a new impedance function generated from known impedance functions R, R₁, and R₂. These three ways of combining impedance are

$$R'\ =\ a\ R$$ 1.    Multiplication by positive scalar a

$$R'\ =\ \displaystyle {\strut 1\over R}$$ 2. Inversion

$$R'\ =\ R_1 + R_2$$ 3. Addition

These rules do not include multiplication. Multiplication is not allowed because squaring, for example, doubles the phase angle, and thus may destroy the positivity of the real part. Since these rules do not include multiplication, but only scaling, summation, and inversion, the impedance functions that occur in nature will often be represented mathematically as continued fractions.

The first two of Muir's rules are so obvious we will not prove them. The third rule deserves more careful attention. To prove any rule, we need to show three things about R ', namely, it is causal, it is PR (the Fourier transform has a positive real part), and it has an inverse. This last part is the hard part with Muir's third rule, namely, that the sum of two impedances has a causal inverse. Proof of this fact will take about two pages, and introduce several additional concepts.