How to synthesize network from driving point impedance

Every other coefficient is the value of a parallel, then series, component. The value of that term (x, 1/x or constant) determines which kind of component it is (L, C, R), including if it is a pole-zero pair (all three).

I suppose you'll have to figure out if each term is itself a simple pole, or a more complex network subject to the same decomposition.

Tim

The established route is to compute the complex feed point impedance by analysing the network, then make your driving network the complex conjugate of that. This is usually most tractable using the Laplace transforms, but I never got good at them.

Off the top of my head it sounds like a problem that could be accomplished systematically via ABCD two-port parameters and dynamic programming/divide and conquer.

The small-signal driving point impedance of the network can be represented by the ratio of the B' and C' components of the A'B'C'D' two port matrix representing the entire unknown ladder network.

Then somewhat similar to the matrix optimization example in the link, divide and conquer the problem into subproblems set up such that the only information you need to optimize the first matrix of the entire two-port is information about the driving point impedance of the second matrix that the "whole" matrix is composited from by multiplying together. Then recurse each half of those subproblems into subproblems that so on and so forth to the base case of indivisible single components, number depending on the order of polynomial ratio you need. Then reassemble.

You extract them one at time, usually... If it represents a transmission zero, then you do a partial extraction. It's all laid out in many textbooks. For example:

There is also Analog Filters by Su.