Design of deconvolution filters

The estimation of the measurand in the analysis of dynamic measurements typically corresponds to a deconvolution problem. Therefore, a digital filter can be designed whose input is the measured system output signal and whose output is an estimate of the measurand. This module implements methods for the design of such filters given an array of frequency response values with associated uncertainties for the measurement system.

This module contains functions to carry out a least-squares fit of a digital filter to the reciprocal of a given complex frequency response.

This module contains the following functions * LSFIR: Least-squares fit of a digital FIR filter to the reciprocal of a given frequency response. * LSFIR_unc: Design of FIR filter as fit to reciprocal of frequency response values with uncertainty * LSFIR_uncMC: Design of FIR filter as fit to reciprocal of frequency response values with uncertainty via Monte Carlo * LSIIR: Design of a stable IIR filter as fit to reciprocal of frequency response values * LSIIR_unc: Design of a stable IIR filter as fit to reciprocal of frequency response values with uncertainty

PyDynamic.deconvolution.fit_filter.LSFIR(H, N, tau, f, Fs, Wt=None)[source]

Least-squares fit of a digital FIR filter to the reciprocal of a given frequency response.

Parameters:
  • H (np.ndarray of shape (M,) and dtype complex) – frequency response values
  • N (int) – FIR filter order
  • tau (float) – delay of filter
  • f (np.ndarray of shape (M,)) – frequencies
  • Fs (float) – sampling frequency of digital filter
  • Wt (np.ndarray of shape (M,) - optional) – vector of weights
Returns:

bFIR – filter coefficients

Return type:

np.ndarray of shape (N,)

References

PyDynamic.deconvolution.fit_filter.LSFIR_unc(H, UH, N, tau, f, Fs, wt=None, verbose=True, trunc_svd_tol=None)[source]

Design of FIR filter as fit to reciprocal of frequency response values with uncertainty

Least-squares fit of a digital FIR filter to the reciprocal of a frequency response for which associated uncertainties are given for its real and imaginary part. Uncertainties are propagated using a truncated svd and linear matrix propagation.

Parameters:
  • H (np.ndarray of shape (M,)) – frequency response values
  • UH (np.ndarray of shape (2M,2M)) – uncertainties associated with the real and imaginary part
  • N (int) – FIR filter order
  • tau (float) – delay of filter
  • f (np.ndarray of shape (M,)) – frequencies
  • Fs (float) – sampling frequency of digital filter
  • wt (np.ndarray of shape (2M,) - optional) – array of weights for a weighted least-squares method
  • verbose (bool, optional) – whether to print statements to the command line
  • trunc_svd_tol (float) – lower bound for singular values to be considered for pseudo-inverse
Returns:

  • b (np.ndarray of shape (N+1,)) – filter coefficients of shape
  • Ub (np.ndarray of shape (N+1,N+1)) – uncertainties associated with b

References

PyDynamic.deconvolution.fit_filter.LSIIR(Hvals, Nb, Na, f, Fs, tau, justFit=False, verbose=True)[source]

Design of a stable IIR filter as fit to reciprocal of frequency response values

Least-squares fit of a digital IIR filter to the reciprocal of a given set of frequency response values using the equation-error method and stabilization by pole mapping and introduction of a time delay.

Parameters:
  • Hvals (np.ndarray of shape (M,) and dtype complex) – frequency response values.
  • Nb (int) – order of IIR numerator polynomial.
  • Na (int) – order of IIR denominator polynomial.
  • f (np.ndarray of shape (M,)) – frequencies corresponding to Hvals
  • Fs (float) – sampling frequency for digital IIR filter.
  • tau (float) – initial estimate of time delay for filter stabilization.
  • justFit (bool) – if True then no stabilization is carried out.
Returns:

b,a – IIR filter coefficients, int tau – time delay (in samples)

Return type:

np.ndarray

References

PyDynamic.deconvolution.fit_filter.LSIIR_unc(H, UH, Nb, Na, f, Fs, tau=0)[source]

Design of stabel IIR filter as fit to reciprocal of given frequency response with uncertainty

Least-squares fit of a digital IIR filter to the reciprocal of a given set of frequency response values with given associated uncertainty. Propagation of uncertainties is carried out using the Monte Carlo method.

Parameters:
  • H (np.ndarray of shape (M,) and dtype complex) – frequency response values.
  • UH (np.ndarray of shape (2M,2M)) – uncertainties associated with real and imaginary part of H
  • Nb (int) – order of IIR numerator polynomial.
  • Na (int) – order of IIR denominator polynomial.
  • f (np.ndarray of shape (M,)) – frequencies corresponding to H
  • Fs (float) – sampling frequency for digital IIR filter.
  • tau (float) – initial estimate of time delay for filter stabilization.
Returns:

  • b,a (np.ndarray) – IIR filter coefficients
  • tau (int) – time delay (in samples)
  • Uba (np.ndarray of shape (Nb+Na+1, Nb+Na+1)) – uncertainties associated with [a[1:],b]

References

PyDynamic.deconvolution.fit_filter.LSFIR_uncMC(H, UH, N, tau, f, Fs, wt=None, verbose=True)[source]

Design of FIR filter as fit to reciprocal of frequency response values with uncertainty

Least-squares fit of a FIR filter to the reciprocal of a frequency response for which associated uncertainties are given for its real and imaginary parts. Uncertainties are propagated using a Monte Carlo method. This method may help in cases where the weighting matrix or the Jacobian are ill-conditioned, resulting in false uncertainties associated with the filter coefficients.

Parameters:
  • H (np.ndarray of shape (M,) and dtype complex) – frequency response values
  • UH (np.ndarray of shape (2M,2M)) – uncertainties associated with the real and imaginary part of H
  • N (int) – FIR filter order
  • tau (int) – time delay of filter in samples
  • f (np.ndarray of shape (M,)) – frequencies corresponding to H
  • Fs (float) – sampling frequency of digital filter
  • wt (np.ndarray of shape (2M,), optional) – vector of weights
  • verbose (bool, optional) – whether to print statements to the command line
Returns:

  • b (np.ndarray of shape (N+1,)) – filter coefficients of shape
  • Ub (np.ndarray of shape (N+1, N+1)) – uncertainties associated with b

References