Identify discrete-time filter parameters from frequency response data
invfreqz is the inverse
freqz; it finds
a discrete-time transfer function that corresponds to a given complex
frequency response. From a laboratory analysis standpoint,
be used to convert magnitude and phase data into transfer functions.
the real numerator and denominator coefficients in vectors
the transfer function
whose complex frequency response is given in vector
the frequency points specified in vector
the desired orders of the numerator and denominator polynomials.
Frequency is specified in radians between 0 and π,
and the length of
h must be the same as the length
ensure the proper frequency domain symmetry for a real filter.
the fit-errors versus frequency, where
wt is a
vector of weighting factors the same length as
a superior algorithm that guarantees stability of the resulting linear
system and searches for the best fit using a numerical, iterative
iter parameter tells
end the iteration when the solution has converged, or after
whichever comes first.
invfreqz defines convergence
as occurring when the norm of the (modified) gradient vector is less
tol is an optional
parameter that defaults to 0.01. To obtain a weight vector of all
a textual progress report of the iteration.
a complex filter. In this case no symmetry is enforced, and the frequency
is specified in radians between -π and π.
Convert a simple transfer function to frequency response data and then back to the original filter coefficients. Sketch the zeros and poles of the function.
a = [1 2 3 2 1 4]; b = [1 2 3 2 3]; [h,w] = freqz(b,a,64); [bb,aa] = invfreqz(h,w,4,5) zplane(bb,aa)
bb = 1.0000 2.0000 3.0000 2.0000 3.0000 aa = 1.0000 2.0000 3.0000 2.0000 1.0000 4.0000
aa are equivalent to
a, respectively. However, the system is unstable because it has poles outside the unit circle. Use
invfreqz's iterative algorithm to find a stable approximation to the system. Verify that the poles are within the unit circle.
[bbb,aaa] = invfreqz(h,w,4,5,,30) zplane(bbb,aaa)
bbb = 0.2427 0.2788 0.0069 0.0971 0.1980 aaa = 1.0000 -0.8944 0.6954 0.9997 -0.8933 0.6949
invfreqz uses an equation error
method to identify the best model from the data. This finds
by creating a system of linear equations and solving them with
\ operator. Here A(ω(k))
and B(ω(k)) are the Fourier
transforms of the polynomials
respectively, at the frequency ω(k), and n is
the number of frequency points (the length of
This algorithm is a based on Levi .
The superior ("output-error") algorithm uses the damped Gauss-Newton method for iterative search , with the output of the first algorithm as the initial estimate. This solves the direct problem of minimizing the weighted sum of the squared error between the actual and the desired frequency response points.
 Levi, E. C. "Complex-Curve Fitting." IRE Transactions on Automatic Control. Vol. AC-4, 1959, pp. 37–44.
 Dennis, J. E., Jr., and R. B. Schnabel. Numerical Methods for Unconstrained Optimization and Nonlinear Equations. Englewood Cliffs, NJ: Prentice-Hall, 1983.