/usr/share/octave/packages/signal-1.2.2/iirlp2mb.m is in octave-signal 1.2.2-1build1.
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##
## This program is free software; you can redistribute it and/or modify it under
## the terms of the GNU General Public License as published by the Free Software
## Foundation; either version 3 of the License, or (at your option) any later
## version.
##
## This program is distributed in the hope that it will be useful, but WITHOUT
## ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or
## FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more
## details.
##
## You should have received a copy of the GNU General Public License along with
## this program; if not, see <http://www.gnu.org/licenses/>.
## IIR Low Pass Filter to Multiband Filter Transformation
##
## [Num,Den,AllpassNum,AllpassDen] = iirlp2mb(B,A,Wo,Wt)
## [Num,Den,AllpassNum,AllpassDen] = iirlp2mb(B,A,Wo,Wt,Pass)
##
## Num,Den: numerator,denominator of the transformed filter
## AllpassNum,AllpassDen: numerator,denominator of allpass transform,
## B,A: numerator,denominator of prototype low pass filter
## Wo: normalized_angular_frequency/pi to be transformed
## Wt: [phi=normalized_angular_frequencies]/pi target vector
## Pass: This parameter may have values 'pass' or 'stop'. If
## not given, it defaults to the value of 'pass'.
##
## With normalized ang. freq. targets 0 < phi(1) < ... < phi(n) < pi radians
##
## for Pass == 'pass', the target multiband magnitude will be:
## -------- ---------- -----------...
## / \ / \ / .
## 0 phi(1) phi(2) phi(3) phi(4) phi(5) (phi(6)) pi
##
## for Pass == 'stop', the target multiband magnitude will be:
## ------- --------- ----------...
## \ / \ / .
## 0 phi(1) phi(2) phi(3) phi(4) (phi(5)) pi
##
## Example of use:
## [B, A] = butter(6, 0.5);
## [Num, Den] = iirlp2mb(B, A, 0.5, [.2 .4 .6 .8]);
function [Num,Den,AllpassNum,AllpassDen] = iirlp2mb(varargin)
usage = sprintf(
"%s: Usage: [Num,Den,AllpassNum,AllpassDen]=iirlp2mb(B,A,Wo,Wt[,Pass])\n"
,mfilename());
B = varargin{1}; # numerator polynomial of prototype low pass filter
A = varargin{2}; # denominator polynomial of prototype low pass filter
Wo = varargin{3}; # (normalized angular frequency)/pi to be transformed
Wt = varargin{4}; # vector of (norm. angular frequency)/pi transform targets
# [phi(1) phi(2) ... ]/pi
if(nargin < 4 || nargin > 5)
error("%s",usage)
endif
if(nargin == 5)
Pass = varargin{5};
switch(Pass)
case 'pass'
pass_stop = -1;
case 'stop'
pass_stop = 1;
otherwise
error("Pass must be 'pass' or 'stop'\n%s",usage)
endswitch
else
pass_stop = -1; # Pass == 'pass' is the default
endif
if(Wo <= 0)
error("Wo is %f <= 0\n%s",Wo,usage);
endif
if(Wo >= 1)
error("Wo is %f >= 1\n%s",Wo,usage);
endif
oWt = 0;
for i = 1 : length(Wt)
if(Wt(i) <= 0)
error("Wt(%d) is %f <= 0\n%s",i,Wt(i),usage);
endif
if(Wt(i) >= 1)
error("Wt(%d) is %f >= 1\n%s",i,Wt(i),usage);
endif
if(Wt(i) <= oWt)
error("Wt(%d) = %f, not monotonically increasing\n%s",i,Wt(i),usage);
else
oWt = Wt(i);
endif
endfor
# B(z)
# Inputs B,A specify the low pass IIR prototype filter G(z) = ---- .
# A(z)
# This module transforms G(z) into a multiband filter using the iterative
# algorithm from:
# [FFM] G. Feyh, J. Franchitti, and C. Mullis, "All-Pass Filter
# Interpolation and Frequency Transformation Problem", Proceedings 20th
# Asilomar Conference on Signals, Systems and Computers, Nov. 1986, pp.
# 164-168, IEEE.
# [FFM] moves the prototype filter position at normalized angular frequency
# .5*pi to the places specified in the Wt vector times pi. In this module,
# a generalization allows the position to be moved on the prototype filter
# to be specified as Wo*pi instead of being fixed at .5*pi. This is
# implemented using two successive allpass transformations.
# KK(z)
# In the first stage, find allpass J(z) = ---- such that
# K(z)
# jWo*pi -j.5*pi
# J(e ) = e (low pass to low pass transformation)
#
# PP(z)
# In the second stage, find allpass H(z) = ---- such that
# P(z)
# jWt(k)*pi -j(2k - 1)*.5*pi
# H(e ) = e (low pass to multiband transformation)
#
# ^
# The variable PP used here corresponds to P in [FFM].
# len = length(P(z)) == length(PP(z)), the number of polynomial coefficients
#
# len 1-i len 1-i
# P(z) = SUM P(i)z ; PP(z) = SUM PP(i)z ; PP(i) == P(len + 1 - i)
# i=1 i=1 (allpass condition)
# Note: (len - 1) == n in [FFM] eq. 3
#
# The first stage computes the denominator of an allpass for translating
# from a prototype with position .5 to one with a position of Wo. It has the
# form:
# -1
# K(2) - z
# -----------
# -1
# 1 - K(2)z
#
# From the low pass to low pass tranformation in Table 7.1 p. 529 of A.
# Oppenheim and R. Schafer, Discrete-Time Signal Processing 3rd edition,
# Prentice Hall 2010, one can see that the denominator of an allpass for
# going in the opposite direction can be obtained by a sign reversal of the
# second coefficient, K(2), of the vector K (the index 2 not to be confused
# with a value of z, which is implicit).
# The first stage allpass denominator computation
K = apd([pi * Wo]);
# The second stage allpass computation
phi = pi * Wt; # vector of normalized angular frequencies between 0 and pi
P = apd(phi); # calculate denominator of allpass for this target vector
PP = revco(P); # numerator of allpass has reversed coefficients of P
# The total allpass filter from the two consecutive stages can be written as
# PP
# K(2) - ---
# P P
# ----------- * ---
# PP P
# 1 - K(2)---
# P
AllpassDen = P - (K(2) * PP);
AllpassDen /= AllpassDen(1); # normalize
AllpassNum = pass_stop * revco(AllpassDen);
[Num,Den] = transform(B,A,AllpassNum,AllpassDen,pass_stop);
endfunction
function [Num,Den] = transform(B,A,PP,P,pass_stop)
# Given G(Z) = B(Z)/A(Z) and allpass H(z) = PP(z)/P(z), compute G(H(z))
# For Pass = 'pass', transformed filter is:
# 2 nb-1
# B1 + B2(PP/P) + B3(PP/P)^ + ... + Bnb(PP/P)^
# -------------------------------------------------
# 2 na-1
# A1 + A2(PP/P) + A3(PP/P)^ + ... + Ana(PP/P)^
# For Pass = 'stop', use powers of (-PP/P)
#
na = length(A); # the number of coefficients in A
nb = length(B); # the number of coefficients in B
# common low pass iir filters have na == nb but in general might not
n = max(na,nb); # the greater of the number of coefficients
# n-1
# Multiply top and bottom by P^ yields:
#
# n-1 n-2 2 n-3 nb-1 n-nb
# B1(P^ ) + B2(PP)(P^ ) + B3(PP^ )(P^ ) + ... + Bnb(PP^ )(P^ )
# ---------------------------------------------------------------------
# n-1 n-2 2 n-3 na-1 n-na
# A1(P^ ) + A2(PP)(P^ ) + A3(PP^ )(P^ ) + ... + Ana(PP^ )(P^ )
# Compute and store powers of P as a matrix of coefficients because we will
# need to use them in descending power order
global Ppower; # to hold coefficients of powers of P, access inside ppower()
np = length(P);
powcols = np + (np-1)*(n-2); # number of coefficients in P^(n-1)
# initialize to "Not Available" with n-1 rows for powers 1 to (n-1) and
# the number of columns needed to hold coefficients for P^(n-1)
Ppower = NA(n-1,powcols);
Ptemp = P; # start with P to the 1st power
for i = 1 : n-1 # i is the power
for j = 1 : length(Ptemp) # j is the coefficient index for this power
Ppower(i,j) = Ptemp(j);
endfor
Ptemp = conv(Ptemp,P); # increase power of P by one
endfor
# Compute numerator and denominator of transformed filter
Num = [];
Den = [];
for i = 1 : n
# n-i
# Regenerate P^ (p_pownmi)
if((n-i) == 0)
p_pownmi = [1];
else
p_pownmi = ppower(n-i,powcols);
endif
# i-1
# Regenerate PP^ (pp_powim1)
if(i == 1)
pp_powim1 = [1];
else
pp_powim1 = revco(ppower(i-1,powcols));
endif
if(i <= nb)
Bterm = (pass_stop^(i-1))*B(i)*conv(pp_powim1,p_pownmi);
Num = polysum(Num,Bterm);
endif
if(i <= na)
Aterm = (pass_stop^(i-1))*A(i)*conv(pp_powim1,p_pownmi);
Den = polysum(Den,Aterm);
endif
endfor
# Scale both numerator and denominator to have Den(1) = 1
temp = Den(1);
for i = 1 : length(Den)
Den(i) = Den(i) / temp;
endfor
for i = 1 : length(Num)
Num(i) = Num(i) / temp;
endfor
endfunction
function P = apd(phi) # all pass denominator
# Given phi, a vector of normalized angular frequency transformation targets,
# return P, the denominator of an allpass H(z)
lenphi = length(phi);
Pkm1 = 1; # P0 initial condition from [FFM] eq. 22
for k = 1 : lenphi
P = pk(Pkm1, k, phi(k)); # iterate
Pkm1 = P;
endfor
endfunction
function Pk = pk(Pkm1, k, phik) # kth iteration of P(z)
# Given Pkminus1, k, and phi(k) in radians , return Pk
#
# From [FFM] eq. 19 : k
# Pk = (z+1 )sin(phi(k)/2)Pkm1 - (-1) (z-1 )cos(phi(k)/2)PPkm1
# Factoring out z
# -1 k -1
# = z((1+z )sin(phi(k)/2)Pkm1 - (-1) (1-z )cos(phi(k)/2)PPkm1)
# PPk can also have z factored out. In H=PP/P, z in PPk will cancel z in Pk,
# so just leave out. Use
# -1 k -1
# PK = (1+z )sin(phi(k)/2)Pkm1 - (-1) (1-z )cos(phi(k)/2)PPkm1
# (expand) k
# = sin(phi(k)/2)Pkm1 - (-1) cos(phi(k)/2)PPkm1
#
# -1 k -1
# + z sin(phi(k)/2)Pkm1 + (-1) z cos(phi(k)/2)PPkm1
Pk = zeros(1,k+1); # there are k+1 coefficients in Pk
sin_k = sin(phik/2);
cos_k = cos(phik/2);
for i = 1 : k
Pk(i) += sin_k * Pkm1(i) - ((-1)^k * cos_k * Pkm1(k+1-i));
#
# -1
# Multiplication by z just shifts by one coefficient
Pk(i+1) += sin_k * Pkm1(i) + ((-1)^k * cos_k * Pkm1(k+1-i));
endfor
# now normalize to Pk(1) = 1 (again will cancel with same factor in PPk)
Pk1 = Pk(1);
for i = 1 : k+1
Pk(i) = Pk(i) / Pk1;
endfor
endfunction
function PP = revco(p) # reverse components of vector
l = length(p);
for i = 1 : l
PP(l + 1 - i) = p(i);
endfor
endfunction
function p = ppower(i,powcols) # Regenerate ith power of P from stored PPower
global Ppower
if(i == 0)
p = 1;
else
p = [];
for j = 1 : powcols
if(isna(Ppower(i,j)))
break;
endif
p = horzcat(p, Ppower(i,j));
endfor
endif
endfunction
function poly = polysum(p1,p2) # add polynomials of possibly different length
n1 = length(p1);
n2 = length(p2);
if(n1 > n2)
# pad p2
p2 = horzcat(p2, zeros(1,n1-n2));
elseif(n2 > n1)
# pad p1
p1 = horzcat(p1, zeros(1,n2-n1));
endif
poly = p1 + p2;
endfunction
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