Identify continuous-time filter parameters from frequency response data
invfreqs is the inverse operation of
It finds a continuous-time transfer function that corresponds to a
given complex frequency response. From a laboratory analysis standpoint,
useful in converting magnitude and phase data into transfer functions.
the real numerator and denominator coefficient 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.
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.
invfreqs 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.
a = [1 2 3 2 1 4]; b = [1 2 3 2 3]; [h,w] = freqs(b,a,64); [bb,aa] = invfreqs(h,w,4,5)
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
aa has poles with positive real part. View the poles of
Use the iterative algorithm of
invfreqs to find a stable approximation to the system.
[bbb,aaa] = invfreqs(h,w,4,5,,30)
bbb = 0.6816 2.1015 2.6694 0.9113 -0.1218
aaa = 1.0000 3.4676 7.4060 6.2102 2.5413 0.0001
Verify that the system is stable by plotting the new poles.
Generate two vectors,
phase, that simulate magnitude and phase data gathered in a laboratory. Also generate a vector,
w, of frequencies.
rng('default') fs = 1000; t = 0:1/fs:2; mag = periodogram(sin(2*pi*100*t)+randn(size(t))/10,,,fs); phase = randn(size(mag))/10; w = linspace(0,fs/2,length(mag))';
invfreqs to convert the data into a continuous-time transfer function. Plot the result.
[b,a] = invfreqs(mag.*exp(1j*phase),w,2,2,,4); freqs(b,a)
When building higher order models using high frequencies, it
is important to scale the frequencies, dividing by a factor such as
half the highest frequency present in
w, so as
to obtain well conditioned values of
This corresponds to a rescaling of time.
invfreqs 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(w(k))
are the Fourier transforms of the polynomials
respectively, at the frequency w(k),
and n is the number of frequency points (the length
w). This algorithm
is based on Levi . Several variants
have been suggested in the literature, where the weighting function
less attention to high frequencies.
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 Trans. 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.