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Identify discrete-time filter parameters from frequency response data

`[b,a] = invfreqz(h,w,n,m)`

[b,a] = invfreqz(h,w,n,m,wt)

[b,a] = invfreqz(h,w,n,m,wt,iter)

[b,a] = invfreqz(h,w,n,m,wt,iter,tol)

[b,a] = invfreqz(h,w,n,m,wt,iter,tol,'trace')

[b,a] = invfreqz(h,w,'complex',n,m,...)

`invfreqz`

is the inverse
operation of `freqz`

; it finds
a discrete-time transfer function that corresponds to a given complex
frequency response. From a laboratory analysis standpoint, `invfreqz`

can
be used to convert magnitude and phase data into transfer functions.

`[b,a] = invfreqz(h,w,n,m)`

returns
the real numerator and denominator coefficients in vectors `b`

and `a`

of
the transfer function

$$H(z)=\frac{B(z)}{A(z)}=\frac{b(1)+b(2){z}^{-1}+\cdots +b(n+1){z}^{-n}}{a(1)+a(2){z}^{-1}+\cdots +a(m+1){z}^{-m}}$$

whose complex frequency response is given in vector `h`

at
the frequency points specified in vector `w`

. Scalars `n`

and `m `

specify
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
of `w`

. `invfreqz`

uses `conj`

`(h)`

at `-w`

to
ensure the proper frequency domain symmetry for a real filter.

`[b,a] = invfreqz(h,w,n,m,wt)`

weights
the fit-errors versus frequency, where `wt`

is a
vector of weighting factors the same length as `w`

.

`[b,a] = invfreqz(h,w,n,m,wt,iter)`

and

`[b,a] = invfreqz(h,w,n,m,wt,iter,tol)`

provide
a superior algorithm that guarantees stability of the resulting linear
system and searches for the best fit using a numerical, iterative
scheme. The `iter`

parameter tells `invfreqz`

to
end the iteration when the solution has converged, or after `iter`

iterations,
whichever comes first. `invfreqz`

defines convergence
as occurring when the norm of the (modified) gradient vector is less
than `tol`

, where `tol`

is an optional
parameter that defaults to 0.01. To obtain a weight vector of all
ones, use

invfreqz(h,w,n,m,[],iter,tol)

`[b,a] = invfreqz(h,w,n,m,wt,iter,tol,'trace')`

displays
a textual progress report of the iteration.

`[b,a] = invfreqz(h,w,'complex',n,m,...)`

creates
a complex filter. In this case no symmetry is enforced, and the frequency
is specified in radians between -π and π.

By default, `invfreqz`

uses an equation error
method to identify the best model from the data. This finds `b`

and `a`

in

$$\underset{b,a}{\mathrm{min}}{\displaystyle \sum _{k=1}^{n}wt(k){\left|h(k)A(w(k))-B(w(k))\right|}^{2}}$$

by creating a system of linear equations and solving them with
the MATLAB^{®} `\`

operator. Here *A*(ω(*k*))
and *B*(ω(*k*)) are the Fourier
transforms of the polynomials `a`

and `b`

,
respectively, at the frequency ω(*k*), and *n* is
the number of frequency points (the length of `h`

and `w`

).
This algorithm is a based on Levi [1].

The superior (“output-error”) algorithm uses the damped Gauss-Newton method for iterative search [2], 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.

$$\underset{b,a}{\mathrm{min}}{\displaystyle \sum _{k=1}^{n}wt(k){\left|h(k)-\frac{B(w(k))}{A(w(k))}\right|}^{2}}$$

[1] Levi, E. C. “Complex-Curve Fitting.” *IRE
Transactions on Automatic Control*. Vol. AC-4,
1959, pp. 37–44.

[2] Dennis, J. E., Jr., and R. B. Schnabel. *Numerical
Methods for Unconstrained Optimization and Nonlinear Equations*.
Englewood Cliffs, NJ: Prentice-Hall, 1983.

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