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qp_kaiser.m

## Copyright (C) 2002 André Carezia
##
## 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 2 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/>.

## Usage:  qp_kaiser (nb, at, linear)
##
## Computes a finite impulse response (FIR) filter for use with a
## quasi-perfect reconstruction polyphase-network filter bank. This
## version utilizes a Kaiser window to shape the frequency response of
## the designed filter. Tha number nb of bands and the desired
## attenuation at in the stop-band are given as parameters.
##
## The Kaiser window is multiplied by the ideal impulse response
## h(n)=a.sinc(a.n) and converted to its minimum-phase version by means
## of a Hilbert transform.
##
## By using a third non-null argument, the minimum-phase calculation is
## ommited at all.

## $Id: qp_kaiser.m 4585 2008-02-04 13:47:45Z adb014 $
##
## Author: André Carezia <andre@carezia.eng.br>
## Description:  Coefficients for a PPN filter bank

function h = qp_kaiser (nb, at, linear)

  if (nargin < 2)
    usage ("qp_kaiser (nb, at)");
  endif

  if (nargin < 3)
    linear = 0;
  endif

  if !(isscalar (nb) && (nb == round(nb)) && (nb >= 0))
    error ("qp_kaiser: nb has to be a positive integer");
  endif

  if !(isscalar (at) && (at == real (at)))
    error ("qp_kaiser: at has to be a real constant");
  endif

                        # Bandwidth
  bandwidth = pi/nb;

                        # Attenuation correction (empirically
                        # determined by M. Gerken
                        # <mgk@lcs.poli.usp.br>)
  corr = (1.4+0.6*(at-20)/80)^(20/at);
  at = corr * at;

                        # size of window (rounded to next odd
                        # integer)
  N = (at - 8) / (2.285*bandwidth);
  M = fix(N/2);
  N = 2*M + 1;

                        # Kaiser window
  if (at>50)
    beta = 0.1102 * (at - 8.7);
  elseif (at>21)
    beta = 0.5842 * (at - 21)^0.4 + 0.07886 * (at - 21);
  else
    beta = 0;
  endif
  w = kaiser(N,beta);
                        # squared in freq. domain
  wsquared = conv(w,w);

                        # multiplied by ideal lowpass filter
  n = -(N-1):(N-1);
  hideal = 1/nb * sinc(n/nb);
  hcomp = wsquared .* hideal;

                        # extract square-root of response and
                        # compute minimum-phase version
  Ndft = 2^15;
  Hsqr = sqrt(abs(fft(hcomp,Ndft)));
  if (linear)
    h = real(ifft(Hsqr));
    h = h(2:N);
    h = [fliplr(h) h(1) h];
  else
    Hmin = Hsqr .* exp(-j*imag(hilbert(log(Hsqr))));
    h = real(ifft(Hmin));
    h = h(1:N);
  endif
                        # truncate and fix amplitude scale
                        # (H(0)=1)
  h = h / sum(h);

endfunction

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