I wrote four new LP to LP/HP/BP/BS functions that work directly with the transfer function coefficients.  I call these lp3lp, lp3hp, lp3bp and lp3bs.  The lp2* functions do the transformations in state-space and produce results similar to the MATLAB functions of the same name.

If the lowpass prototype filter is described by its coefficients, the lp3* functions appear to produce more accurate results than the lp2* functions.  The lp3* functions rely on a new function called tftrans to do the coefficient transformations.  It is similar to sftrans and sstrans which do transformations in pole-zero and state-space domains.

All of the lp3* functions include demo code to illustrate their usage.

I will include the lp3* functions in this comment and the tftrans function in the next one.

Here is a simple script to show the differences between lp2bp and lp3bp when doing a lowpass-to-bandpass transformation on a second order lowpass Butterworth filter.  Theoretically this should produce a fourth order bandpass filter with all (four) zeros at the origin.  The lp2bp function creates zeros that are slightly displaced from the origin.  Increase the value of n to see the differences for higher order filters.

% Nth order Butterworth Lowpass
n = 2;
Wo = 1000; Bw = 200;
[b a] = butter(n, 1, "s");
[bt2, at2] = lp2bp(b, a, Wo, Bw);
[bt3, at3] = lp3bp(b, a, Wo, Bw);

[bt2' bt3']
[at2' at3']


(file #51270, file #51271, file #51272, file #51273)

I wrote a new function called sstrans that does the various transformations (lp, hp, bp, bs) in state space.  It is modeled after sftrans which does the transformations in pole-zero space.  I then modified lp2lp, lp2hp, lp2bp, and lp2bs to call sstrans.  Since I am limited to four file uploads at a time, I will upload the four lp2* functions first and then upload sstrans in the next comment.

(file #51258, file #51259, file #51260, file #51261)

After sleeping on this, I wonder if my original implementations (files 51223-51226) might be more robust than the newer ones that use sftrans (files 51227-51230).  They really aren't much uglier than the implementation of sftrans.  I could see moving all of the coefficient transformation code into a common function (kind of like sftrans) so that the implementations of lp2lp, etc are cleaner.

The newer functions use tf2zp to transform from the transfer function (coefficient) representation to the zero-pole representation, sftrans is then used to transform the zero-poles of the prototype filter to those of the desired filter (lp, hp, etc), then zp2tf is used to transform the zeros and poles back to the transfer function (coefficients) of desired filter. The original implementations directly transform the coefficients of the prototype filter to those of the desired filter using polynomial multiplication (conv) operations.  I would think that the direct transformation in the original code would be more accurate than the multiple transformations in the newer code, but I have not tested this at all.  (It seems as if the newer code could also introduce extraneous poles and zeros, but I am not sure.)  The newer code uses existing Octave functions that have been tested over time while the original functions have undergone very little testing.

I would guess that when there are two different algorithms that can be used to implement an Octave function a decision is based on accuracy and robustness (speed wouldn't appear to be much of a concern for these functions).  I'll try to do some more testing of the original functions (maybe refactor and clean up the code a bit) and compare results to the newer functions.

My previous functions were so ugly I was motivated to explore using sftrans to implement lp2lp, lp2hp, lp2bp, and lp2bs.  These implementations are much more elegant.  I did run into some issues with sftrans.  It seems to expect the input arguments to be row vectors but tf2zp now produces column vectors.

(file #51227, file #51228, file #51229, file #51230)

Attached are two Octave m-files, phi2p.m and Hp2g.m that implement frequency transformations using the algorithm of Figure 6.7.3 of "Digital Signal Processing" R.A. Roberts and C.T. Mullis Addison-Wesley ISBN 0-201-16350-0. The script ellip5_test.m does some examples. I would be happy to try and work this up into a format acceptable for the toolbox.

(file #35505)