octave-signal 1.3.0-1 / usr / share / octave / packages / signal-1.3.0 / iirlp2mb.m

## Copyright (C) 2011 Alan J. Greenberger <alanjg@ptd.net>
##
## 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