# Signal reconstruction of time domain data via transfer function of a quadripole

Dear signal processing community,

I hope my question finds you all well.

I have an electrical network, consisting of three complex impedances. These impedances basically form a simple voltage divider. The network can thus be seen as a quadripole with the input voltage Vin and the output voltage Vout.

What I wanna do is, I wanna model this electrical network with a transfer function or similar, such that I can reconstruct the input signal Vin(t) from a measured output signal Vout(t) or vice versa.

From measurements of this electrical network, I have all frequency-dependent complex impedance values available (Z1(f), Z2(f), Zm(f)). The dataset contains 801 samples each, in a frequencyrange between 10Hz and 50MHz (samples distributed on the frequency scale in a logarithmic fashion, not linearly).

From this dataset I derived a transfer function by simply applying the voltage divider rule. The obtained transfer function also has 801 samples and is defined in a frequency range between 10Hz and 50MHz.

Since the measured impedance values are very noisy in the lower frequency range (measurement error), I flatened / idealized the frequency response in the lower frequency range. The frequency response of the transfer function looks like the following:

In addition to the impedance measurements, I have voltage measurements of input voltage and output voltage available in the time domain which I wanna use to verify the model. The measurements were recorded with a sampling rate of 3.125GS/s and have a length of 8125 samples. The measurements in the time domain look like the following:

My take on the reconstruction of the signals / validation of the model would now be to transform the input voltage from the time domain into the frequency domain by a DFT. By multiplying the transformed signal with the transfer function, I obtain the output signal in the frequency domain. Transforming this signal by an IDFT should finally give me the output voltage signal in the time domain, which should more or less precisely match the measured output signal in a comparison.

HOWEVER: Since the transfer function which I derived from my impedance measurements is not fully defined for the frequency range of my input/output signals (and has less samples) as seen below, I for sure will run into problems when multiplying the TF with my frequency domain input signal.

Therefore I suppose further processing is required. On the one hand side I assume I need to cut off the input and output signal at 50MHz (upper limit frequency of TF for which it is still defined). On the other hand I somehow need to take care of the lower frequency components by maybe interpolation? Interpolation would then be anyway required to match the length of the TF vector with my frequency domain input signal vector.

My question to you is, how would you accomplish this? How would you manipulate the transfer function data sets and/or the input/output voltage signals to allow precise reconstruction of time domain signals?

It somehow should be possible to create a valid model of such a simple electrical network based on the data I have but as you can see my skills in signal processing are pretty mediocre :)

Any sort of help is much appreciated. Thank you!