Review of impedance filters

Use Z-transform notation to define a filter R(Z), its input X(Z), and its output Y(Z). Then  

$$Y(Z) \eq R(Z) \ X(Z)$$ (20)

The filter R(Z) is said to be causal if the series representation of R(Z) has no negative powers of Z. In other words, y_t is determined from present and past values of x_t. Additionally, the filter R(Z) is minimum phase if 1/R(Z) has no negative powers of Z. This means that x_t can be determined from present and past values of y_t by straightforward polynomial division in  

$$X(Z) \eq {Y(Z) \over R }(Z)$$ (21)

Given that R(Z) is already minimum phase, it can in addition be an impedance function if positive energy or work is represented by

$$\begin{eqnarray} 0 \ \le\ \rm{work} \ &=& \ \sum_t \ \ \rm{force} \ \times \ \ \... ... X\ R\ X )\ d \omega \eq \int \ \bar X \ X \ \Re \ (R) \ d \omega\end{eqnarray}$$ (22) (23) (24) (25) (26)

Since $\bar X X$ could be an impulse function located at any $\omega$,it follows that $\Re \ [ R ( \omega ) ] \ge 0$ for all real $\omega$.In summary:

$$r_t\ =\ 0\ \ t\ < 0 \mid R(Z)\mid\ <\ \infty\ \ \mid Z\mid\ \leq 1$$ causality

$$\mid 1/R(Z)\mid\ <\ \infty\ \ \mid Z\mid\ \leq 1$$ causal inverse

$$\Re R(\omega)\ =\ R(Z) + \overline{R}(1/Z)\geq 0\quad \rm{real}\ \omega$$ dissipates energy

Adding an impedance to its Fourier conjugate gives a real positive function (the imaginary part of which is zero) like a power spectrum, say,

$$\left(\ r_0 \ +\ r_1 Z \ +\ r_2 Z^2 \ +\ \cdots \right) \ +\... ...uad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad$$

 

$$\quad \quad \ \ \ \ \ \ \ \\ \ \ \ \left( \ \bar r_0 \ +\ \... ... \cdots \right) \ \ \ \ge\ \ \ 0 \ \ \ \hbox{for real} \ \omega$$ (27)

 

$$R(Z)\ +\ \bar R \left( {1 \over Z }\, \right) \ \ \ \ge\ \ \ 0 \ \ \ \hbox{for real} \ \omega$$ (28)

which is the basis for the remarkable fact that every impedance time function is one side of an autocorrelation function.

Impedances also arise in economic theory when X and Y are price and sales volume. I suppose that there the positivity of the impedance means that in the game of buying and selling you are bound to lose!