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## Review of impedance filters

Use Z-transform notation to define a filter R(Z), its input X(Z), and its output Y(Z). Then
 (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, yt is determined from present and past values of xt. Additionally, the filter R(Z) is minimum phase if 1/R(Z) has no negative powers of Z. This means that xt can be determined from present and past values of yt by straightforward polynomial division in
 (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
 (22) (23) (24) (25) (26)
Since could be an impulse function located at any ,it follows that for all real .In summary:

 2|c| 2|c| Definition of an Impedance 2|c| causality for i.e. for causal inverse for dissipates energy 2

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

 (27)

 (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!

Next: Causal integration Up: IMPEDANCE Previous: Z - transform
Stanford Exploration Project
10/31/1997