Suppose we have measured the frequency response of a system that is known to be all-pole; measuring impulse response is not possible. What are the methods available, if any, to estimate the coefficients of the underlying IIR filter?

EDIT: Frequency response exists only as measured data, so closed mathematical expression is not available, nor is the phase information.

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    $\begingroup$ Does the measured frequency response include phase information? $\endgroup$ – Olli Niemitalo Sep 17 '15 at 8:27
  • $\begingroup$ Do you have any closed form expression for the frequency response? or you have only samples of the frequency response? $\endgroup$ – Oliver Sep 17 '15 at 8:47

Half of the information about a system is in the phase response, and half of it in the amplitude response. There's no way to reconstruct a filter if you know only either half, unless you strongly restrict the poles mathematically.

  • $\begingroup$ Supposing one did have the phase information available, what would be the method to reconstruct the filter? $\endgroup$ – sarasvati119 Sep 19 '15 at 8:41
  • $\begingroup$ 1. restrict how many poles you can have. 2. find out whether your filter is stable for harmonic excitation (poles all within unit circle?), 3. be $y[n] = x[n]*h[n]$ (discrete filter $h$ convoluted with signal $x$), calc the z-transform of both sides of that equation, divide by $X$. 4. inverse z-Transform of $H$ $\endgroup$ – Marcus Müller Sep 19 '15 at 10:30

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