How to transform a Fractional Order Laplace Transfer Function into a digital filter?

I'm working with loudspeaker impedance analysis. Electrical behavior of loudspeakers can be modeled with RLC networks. But real loudspeakers have components, that exhibit some non-linear and frequency dependent behaviors, that make them difficult to model with simple LTI systems.

One of the problems with loudspeakers is, the voice coil inductance decreases with frequency. (ignoring low frequency behavior, for this question, and focusing on higher freq's). Understanding impedance Z(s) as a Laplace Transfer Function between current and voltage signals, V(s) = Z(s) * I(s). Speaker impedance does not follow the simple formula of a RL series circuit, Z(s) = Re + (L * s). Neither behaves like a plain resistance Z(s) = Re .

Instead it follows a formula where the s is raised to a non-integer exponent. So it would be Z(s) = Re + (L * (s^a)).

Impedance rises with frequency, not proportionally with f, but proportionally with f^a. Where f = frequency, and a is a real fractional number.

in practice this number called a is around 0.7 (depending on spkr). This phenomenon occurs because frequency dependency of magnetic permeability of the iron core.

following picture describes the issue:

See at 5KHz, impedance is 17 Ohms. At twice this frequency, 10KHz. With a simple inductance, a doubled impedance would be expected, 34 Ohms. But that does not occur. Impedance is increased but not doubled. It is around 27 Ohms. Which is a smaller value than expected with a plain inductance.

Now, what I want to do is to transform the Laplace TF Z = Re + L * (s^0.7), to a discrete-time, Z-transform TF, and then to a IIR digital filter. That would allow to see and analyze the current waveform, from a given voltage signal. Voltage signal is a music MP3 file. Because audio amps outputs are voltage controlled.

With integer exponents in the Laplace TF, is very easy to transform with the Match-Z or Tustin methods. But I have no clue how to do it with fractional exponents. Suppose I want to do Match-Z, how I find the roots? Suppose Tustin Method, replacing s with ((z-1)/(z+1)), How can I raise the ((z-1)/(z+1)) term to a 0.7 exponent???

I know this is a bit hard. Thanks in advance.

• Unfortunately I haven't time to respond properly to your question, but you should I know that the numerical design of Fractional Delay Filters has been adressed in the literature, e.g. in users.spa.aalto.fi/vpv/publications/vesan_vaitos/… (see part 3.2) – Klaz Dec 20 '17 at 9:30
• Why do you insist on a recursive filter implementation? Using a truncated finite impulse response with a non-recursive implementation is a bit brute force, but should work well enough for your problem. – Jazzmaniac Dec 20 '17 at 10:23
• If for some reason not mentioned you need a recursive system, you can expand your Laplace domain function into a rational approximation. Complex analysis provides all the tools you need for that. The Padé approximation should be your first stop. – Jazzmaniac Dec 20 '17 at 10:26
• $s$ is a differentiation operator, so it’s a fractional derivative which are calculated by integration. You might try a numerical integration approach directly – Stanley Pawlukiewicz Dec 20 '17 at 14:28
• Thanks guys, I forgot to post back here, I found a close solution, Oustaloup Method, described in section 2.2.1.3 in following paper google.com/… Gives a pretty close Laplace TF, with a second grade Rational Function, it is very exact at least in the audio frequency band (20Hz-20KHz). And is easy to convert to a IIR – Leandro Alsina Aug 26 '18 at 15:31