Finding the Transfer Function of a Multiplicative Distortion in MATLAB

Question

I have the frequency response of two signals shown above. One is uncontaminated, and the other is the same signal contaminated by a multiplicative distortion with two poles and two zeros. I need to find/estimate the transfer function values of the contaminating filter in order to equalize the contamination. How do I accomplish this in MATLAB? Thanks.

Answer 1

Basically the transfer function is given by (Continuous time case):

$$ H \left( j \omega \right) = K \frac{ \left( j \omega - {q}_{1} \right) \left( j \omega - {q}_{2} \right) }{ \left( j \omega - {p}_{1} \right) \left( j \omega - {p}_{2} \right) } $$

Now, given your frequency data sampled an $ N $ points $ {\left\{ {\omega}_{i} \right\}}_{i = 1}^{N} $ we need to use the discrete-time form of the transfer function over the Z domain (Z Transform).
Hence the model becomes:

$$ H \left( z \right) {\mid}_{z = {e}^{j \omega}} = K \frac{ \left( z - {q}_{1} \right) \left( z - {q}_{2} \right) }{ \left( z - {p}_{1} \right) \left( z - {p}_{2} \right) }{\Bigg|}_{z = {e}^{j \omega}} = K \frac{ \left( {e}^{j \omega} - {q}_{1} \right) \left( {e}^{j \omega} - {q}_{2} \right) }{ \left( {e}^{j \omega} - {p}_{1} \right) \left( {e}^{j \omega} - {p}_{2} \right) } $$

you may set a system of $ N $ complex equations and use non linear solver to find the 5 parameters ($ K $, $ {q}_{1} $, $ {q}_{2} $, $ {p}_{1} $, $ {p}_{2} $) you need.

In case the measurements are noisy you may use Least Squares.

Remark
The above solves the problem from the data of @jmcd in frequency domain. In practice, using the data domain one could use simpler System Identification (See System Identification: An Overview) methods to get the equivalent ARMA model.
Regarding Frequency Domain based methods for System Identification have a look at Rik Pintelon - System Identification: A Frequency Domain Approach.