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fracshift.m

## Copyright (C) 2008 Eric Chassande-Mottin, CNRS (France)
##
## 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 .

## -*- texinfo -*-
## @deftypefn {Function File}  {[@var{y} @var{h}]=} fracshift(@var{x},@var{d})
## @deftypefnx {Function File} {@var{y} =} fracshift(@var{x},@var{d},@var{h})
## Shift the series @var{x} by a (possibly fractional) number of samples @var{d}.
## The interpolator @var{h} is either specified or either designed with a Kaiser-windowed sinecard.
## @end deftypefn
## @seealso{circshift}

## Ref [1] A. V. Oppenheim, R. W. Schafer and J. R. Buck, 
## Discrete-time signal processing, Signal processing series,
## Prentice-Hall, 1999
##
## Ref [2] T.I. Laakso, V. Valimaki, M. Karjalainen and U.K. Laine
## Splitting the unit delay, IEEE Signal Processing Magazine, 
## vol. 13, no. 1, pp 30--59 Jan 1996

function  [y, h] = fracshift( x, d, h )

if nargchk(2,3,nargin)
  usage("fracshift.m: [y, h] = fracshift( x, d[, h] )");
endif;


## if the delay is an exact integer, use circshift
if d==fix(d)
  y=circshift(x,d);
  return
endif;

## filter design if required

if (nargin < 4)

  ## properties of the interpolation filter
  
  log10_rejection = -3.0;
  stopband_cutoff_f = 1.0 / 2.0;
  roll_off_width = stopband_cutoff_f / 10;
  
  ## determine filter length
  ## use empirical formula from [1] Chap 7, Eq. (7.63) p 476
  
  rejection_dB = -20.0*log10_rejection;
  L = ceil((rejection_dB-8.0) / (28.714 * roll_off_width));
  
  ## ideal sinc filter
  
  t=(-L:L)';
  ideal_filter=2*stopband_cutoff_f*sinc(2*stopband_cutoff_f*(t-(d-fix(d))));  
  
  ## determine parameter of Kaiser window
  ## use empirical formula from [1] Chap 7, Eq. (7.62) p 474
  
  if ((rejection_dB>=21)&(rejection_dB<=50))
    beta = 0.5842 * (rejection_dB-21.0)^0.4 + 0.07886 * (rejection_dB-21.0);
  elseif (rejection_dB>50)
    beta = 0.1102 * (rejection_dB-8.7);
  else
    beta = 0.0;
  endif
  
  ## apodize ideal (sincard) filter response
  
  m = 2*L;
  t = (0 : m)' - (d-fix(d));
  t = 2 * beta / m * sqrt (t .* (m - t));
  w = besseli (0, t) / besseli (0, beta);
  h = w.*ideal_filter;

endif

## check if input is a row vector
isrowvector=false;
if ((rows(x)==1)&(columns(x)>1))
   x=x(:);
   isrowvector=true;
endif

## check if filter is a vector
if ~isvector(h)
  error("fracshift.m: the filter h should be a vector");
endif

Lx = length(x);
Lh = length(h);
L = ( Lh - 1 )/2.0;
Ly = Lx;

## pre and postpad filter response
hpad = prepad(h,Lh);
offset = floor(L);
hpad = postpad(hpad,Ly + offset);

## filtering
xfilt = upfirdn(x,hpad,1,1);
y = xfilt(offset+1:offset+Ly,:);

y=circshift(y,fix(d));

if isrowvector,
   y=y.';
endif

endfunction

%!test
%! N=1024;
%! d=1.5;
%! t=(0:N-1)-N/2;
%! tt=t-d;
%! err=zeros(N/2,1);
%! for n = 0:N/2-1,
%!   phi0=2*pi*rand;
%!   f0=n/N;
%!   sigma=N/4;
%!   x=exp(-t'.^2/(2*sigma)).*sin(2*pi*f0*t' + phi0);
%!   [y,h]=fracshift(x,d);
%!   xx=exp(-tt'.^2/(2*sigma)).*sin(2*pi*f0*tt' + phi0);
%!   err(n+1)=max(abs(y-xx));
%! endfor;
%! rolloff=.1;
%! rejection=10^-3;
%! idx_inband=1:ceil((1-rolloff)*N/2)-1;
%! assert(max(err(idx_inband))<rejection);

%!test
%! N=1024;
%! d=7/6;
%! t=(0:N-1)-N/2;
%! tt=t-d;
%! err=zeros(N/2,1);
%! for n = 0:N/2-1,
%!   phi0=2*pi*rand;
%!   f0=n/N;
%!   sigma=N/4;
%!   x=exp(-t'.^2/(2*sigma)).*sin(2*pi*f0*t' + phi0);
%!   [y,h]=fracshift(x,d);
%!   xx=exp(-tt'.^2/(2*sigma)).*sin(2*pi*f0*tt' + phi0);
%!   err(n+1)=max(abs(y-xx));
%! endfor;
%! rolloff=.1;
%! rejection=10^-3;
%! idx_inband=1:ceil((1-rolloff)*N/2)-1;
%! assert(max(err(idx_inband))<rejection);

%!test
%! N=1024;
%! p=6;
%! q=7;
%! d1=64;
%! d2=d1*p/q;
%! t=128;
%! n=zeros(N,1);
%! n(N/2+(-t:t))=randn(2*t+1,1);
%! [b a]=butter(10,.25);
%! n=filter(b,a,n);
%! n1=fracshift(n,d1);
%! n1=resample(n1,p,q);
%! n2=resample(n,p,q);
%! n2=fracshift(n2,d2);
%! err=abs(n2-n1);
%! rejection=10^-3;
%! assert(max(err)<rejection);

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