Function File: [B,A] = invfreqz(H,F,nB,nA)
: [B,A] = invfreqz(H,F,nB,nA,W)
: [B,A] = invfreqz(H,F,nB,nA,W,iter,tol,'trace')

Fit filter B(z)/A(z)to the complex frequency response H at frequency points F.

A and B are real polynomial coefficients of order nA and nB. Optionally, the fit-errors can be weighted vs frequency according to the weights W.

Note: all the guts are in invfreq.m

H: desired complex frequency response

F: normalized frequency (0 to pi) (must be same length as H)

nA: order of the denominator polynomial A

nB: order of the numerator polynomial B

W: vector of weights (must be same length as F)

Example: 

    [B,A] = butter(4,1/4);
    [H,F] = freqz(B,A);
    [Bh,Ah] = invfreq(H,F,4,4);
    Hh = freqz(Bh,Ah);
    disp(sprintf('||frequency response error||= %f',norm(H-Hh)));

Demonstration 1

The following code

 order = 9;  # order of test filter
 # going to 10 or above leads to numerical instabilities and large errors
 fc = 1/2;   # sampling rate / 4
 n = 128;    # frequency grid size
 [B0, A0] = butter(order, fc);
 [H0, w] = freqz(B0, A0, n);
 Nn = (randn(size(w))+j*randn(size(w)))/sqrt(2);
 [Bh, Ah, Sig0] = invfreqz(H0, w, order, order);
 [Hh, wh] = freqz(Bh, Ah, n);
 [BLS, ALS, SigLS] = invfreqz(H0+1e-5*Nn, w, order, order, [], [], [], [], "method", "LS");
 HLS = freqz(BLS, ALS, n);
 [BTLS, ATLS, SigTLS] = invfreqz(H0+1e-5*Nn, w, order, order, [], [], [], [], "method", "TLS");
 HTLS = freqz(BTLS, ATLS, n);
 [BMLS, AMLS, SigMLS] = invfreqz(H0+1e-5*Nn, w, order, order, [], [], [], [], "method", "QR");
 HMLS = freqz(BMLS, AMLS, n);
 plot(w,[abs(H0) abs(Hh)])
 xlabel("Frequency (rad/sample)");
 ylabel("Magnitude");
 legend('Original','Measured');
 err = norm(H0-Hh);
 disp(sprintf('L2 norm of frequency response error = %f',err));

Produces the following output

L2 norm of frequency response error = 0.000000

and the following figure

Figure 1

Package: signal