# Obtaining a high sample rate digital approximation to an analog filter from lower frequency measurements

Consider a set of dense, but generally irregularly-spaced frequency response measurements of some real low-pass analog filter. Denote the maximum frequency for which a frequency response measurement is available as F_a_max.

I would like to create a digital filter model for this analog filter. The sample rate of this filter F_s needs to be upsampled with respect to the rate implied by the measurements, i.e., F_s > 2 * F_a_max. I have used Matlab's invfreqz function to obtain the transfer function of the digital filter from the frequency response measurements. In principle, I can associate an arbitrarily high sample rate with the frequency response measurements. However, I'm concerned about what the response of the digital filter will look like for frequencies greater than F_a_max. While I don't know the exact desired response for such high frequencies (since I don't have frequency measurements for such high frequencies), I don't want the response to go "crazy" in that high frequency band.

An alternative may be using invfreqs to obtain coefficients of an analog transfer function and then use, e.g., a bilinear transform at the desired sample rate to obtain a digital filter. However, the same concern exists.

Are there standard approaches to this problem? I have seem some recommend Lagrange interpolation of the invfreqz output where F_s is first set to 2*F_a_max, and then Lagrange interpolation would resample the response at the higher rate.

You mention $$2F_{a_{max}}$$, which makes me think that you're trying to invoke the Nyquist-Shannon sampling theorem. But that theorem doesn't deal with any "highest measured values". It deals with that process of getting a signal from the continuous-time domain (or some higher-rate sampled domain) to a sampled-time domain.