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v_windows

PURPOSE ^

WINDOWS Generate a standard windowing function (TYPE,N,MODE,P,H)

SYNOPSIS ^

function w = windows(wtype,n,mode,p,ov)

DESCRIPTION ^

WINDOWS Generate a standard windowing function (TYPE,N,MODE,P,H)

 Inputs:   WTYPE  is a string specifying the window type (see below)
           N      is the number of output points to generate (actually FLOOR(N))
                  and also determines the period of the underlying window [default 256]
           MODE   is a string specifying various options (see below)
           P      is a vector of parameters required for some window types
           OV      is the overlap in samples between succesive windows (must be H<=N/2 and
                  used only for the 'v' option) [default floor(N/2)]

 Outputs:  W(N)   is the output window. If no output is specified, a graph
                  of the window and its frequency response will be drawn.

 The WTYPE input specifies one of the following window types (see [1]):

       Name      Params
    'blackman'
    'cauchy'        1
    'cos'           1      cos window to the power P [default P=1]
    'dolph'         1      Dolph-Chebyshev window with sideband attenuation P dB [default P=50]
    'gaussian'         1      truncated at +-P std deviations [default P=3]
    'hamming'
    'hanning'              also called "hann" or "von hann"
    'harris3'                3-term blackman-harris with 67dB sidelobes
    'harris4'                4-term blackman-harris with 92dB sidelobes
    'kaiser'         1      with parameter P (often called beta) [default P=8]
    'rectangle'
    'triangle'      1      triangle to the power P [default P=1]
    'tukey'         1      cosine tapered 0<P<1 [default P=0.5]
    'vorbis'               perfect reconstruction window from [2] (use mode='sE2')
    'rsqvorbis'            raised squared vorbis with lower sidelobes (use mode='sdD2')

 Window equivalences:

    'hanning'   =    cospow(2) = tukey(1)
    'rectangle' =    tukey(0)
    'reisz'     =    triangle(2)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

 The MODE input determines the scaling and sampling of the window function and
     is a text string with characters whose meanings are given below. The
     default is 'ubw' for window functions whose end points are non-zero and 'unw'
     for window functions whose end points are zero (e.g. hanning window)

         scaling:
                1-9 = set target gain to G = 1/n in scaling options [default n=1 so G=1]
                  u = unscaled  with the peak of the underlying continuous
                      window equalling G. [default]
                  p = scaled to make the actual peak G
                  d = scaled to make DC gain equal to G (summed sample values).
                  D = scaled to make average value equal G
                  e = scaled to make energy = G (summed squared sample values).
                  E = scaled to make mean energy = G (mean squared sample values).
                  q = take square root of the window after scaling

         first and last samples (see note on periodicity below):
                  b [both]    = The first and last samples are at the extreme ends of
                                the window [default for most windows].
                  n [neither] = The first and last samples are one sample away from the ends
                                of the window [default for windows having zero end points].
                  s [shifted] = The first and last samples are half a sample away from the
                                ends of the window .
                  l [left]    = The first sample is at the end of the window while the last
                                is one sample away from the end .
                  r [right]   = The first sample is one sample away from the end while the
                                last is at the end of the window .

         whole/half window (see note on periodicity below):
                  w = The whole window is included [default]
                  c = The first sample starts in the centre of the window
                  h = The first sample starts half a sample beyond the centre

         convolve with rectangle
                  o = convolve w(n) with a rectangle of length N-H [default floor(N/2)]
                      This can be used to force w(n) to satisfy the Princen-Bradley condition

 Periodicity:
     The underlying period of the window function depends on the chosen mode combinations and
     is given in the table below. For overlapping windows with perfect reconstruction choose
     N to be an integer and modes 'ws', 'wl' or 'wr'.

        Whole/half window -->     w         h         c

        End points:       b      N-1      2N-1      2N-2
                          n      N+1      2N+1       2N
                          s       N        2N       2N-1
                          l       N       2N+1       2N
                          r       N       2N-1      2N-2

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 To obtain unity gain for windowed overlap-add processing you can use
 the following. Bandwidths have been multiplied by the window length.
 For perfect reconstruction, you can use any multiple of the overlap factors
 shown assuming the same window is used for both analysis and synthesis.
 These are the Princen-Bradley conditions: fliplr(w)=w, w(i)^2+w(i+n/2)^2=1
 Any symmetric window will satisfy the coinditions with mode 'boqD2' [3].

   Window     Mode Overlap-Factor Sidelobe  3dB-BW  6dB-BW Equiv-noise-BW
   rsqvorbis  sqD2     2           -26dB     1.1      1.5      1.1
   hamming    sqD2     2,3,5       -24dB     1.1      1.5      1.1
   cos        sE2      2,3,5       -23dB     1.2      1.6      1.2 used in MP3
   kaiser(5)  boqD2    2           -23dB     1.2      1.7      1.3 used in AAC [4]
   vorbis     sE2      2,9,15      -21dB     1.3      1.8      1.4 used in Vorbis
   hamming    sE4      3,4,5       -43dB     1.3      1.8      1.4
   hanning    sE4      3,4,5       -31dB     1.4      2.0      1.5

 References:
  [1]  F. J. Harris. On the use of windows for harmonic analysis with the
       discrete fourier transform. Proc IEEE, 66 (1): 5183, Jan. 1978.
  [2]    L. D. Fielder, M. Bosi, G. Davidson, M. Davis, C. Todd, and S. Vernon.
       AC-2 and AC-3: Low-complexity transform-based audio coding.
       In Audio Engineering Society Conference: Collected Papers on Digital Audio Bit-Rate Reduction, May 1996.
  [3]    J. Princen, A. Johnson, and A. Bradley. Subband/transform coding using filter
       bank designs based on time domain aliasing cancellation.
       In Proc. IEEE Intl Conf. Acoustics, Speech and Signal Processing, volume 12,
       pages 21612164, 1987. doi: 10.1109/ICASSP.1987.1169405.
  [4]    T. Sporer, K. Brandenburg, and B. Edler.
       The use of multirate filter banks for coding of high quality digital audio.
       In Proc EUSIPCO, volume 1, pages 211214, 1992.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
      Copyright (C) Mike Brookes 2002-2015
      Version: $Id: v_windows.m 6808 2015-09-13 11:22:27Z dmb $

   VOICEBOX is a MATLAB toolbox for speech processing.
   Home page: http://www.ee.ic.ac.uk/hp/staff/dmb/voicebox/voicebox.html

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
   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 can obtain a copy of the GNU General Public License from
   http://www.gnu.org/copyleft/gpl.html or by writing to
   Free Software Foundation, Inc.,675 Mass Ave, Cambridge, MA 02139, USA.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

CROSS-REFERENCE INFORMATION ^

This function calls: This function is called by:

SOURCE CODE ^

0001 function w = windows(wtype,n,mode,p,ov)
0002 %WINDOWS Generate a standard windowing function (TYPE,N,MODE,P,H)
0003 %
0004 % Inputs:   WTYPE  is a string specifying the window type (see below)
0005 %           N      is the number of output points to generate (actually FLOOR(N))
0006 %                  and also determines the period of the underlying window [default 256]
0007 %           MODE   is a string specifying various options (see below)
0008 %           P      is a vector of parameters required for some window types
0009 %           OV      is the overlap in samples between succesive windows (must be H<=N/2 and
0010 %                  used only for the 'v' option) [default floor(N/2)]
0011 %
0012 % Outputs:  W(N)   is the output window. If no output is specified, a graph
0013 %                  of the window and its frequency response will be drawn.
0014 %
0015 % The WTYPE input specifies one of the following window types (see [1]):
0016 %
0017 %       Name      Params
0018 %    'blackman'
0019 %    'cauchy'        1
0020 %    'cos'           1      cos window to the power P [default P=1]
0021 %    'dolph'         1      Dolph-Chebyshev window with sideband attenuation P dB [default P=50]
0022 %    'gaussian'         1      truncated at +-P std deviations [default P=3]
0023 %    'hamming'
0024 %    'hanning'              also called "hann" or "von hann"
0025 %    'harris3'                3-term blackman-harris with 67dB sidelobes
0026 %    'harris4'                4-term blackman-harris with 92dB sidelobes
0027 %    'kaiser'         1      with parameter P (often called beta) [default P=8]
0028 %    'rectangle'
0029 %    'triangle'      1      triangle to the power P [default P=1]
0030 %    'tukey'         1      cosine tapered 0<P<1 [default P=0.5]
0031 %    'vorbis'               perfect reconstruction window from [2] (use mode='sE2')
0032 %    'rsqvorbis'            raised squared vorbis with lower sidelobes (use mode='sdD2')
0033 %
0034 % Window equivalences:
0035 %
0036 %    'hanning'   =    cospow(2) = tukey(1)
0037 %    'rectangle' =    tukey(0)
0038 %    'reisz'     =    triangle(2)
0039 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
0040 %
0041 % The MODE input determines the scaling and sampling of the window function and
0042 %     is a text string with characters whose meanings are given below. The
0043 %     default is 'ubw' for window functions whose end points are non-zero and 'unw'
0044 %     for window functions whose end points are zero (e.g. hanning window)
0045 %
0046 %         scaling:
0047 %                1-9 = set target gain to G = 1/n in scaling options [default n=1 so G=1]
0048 %                  u = unscaled  with the peak of the underlying continuous
0049 %                      window equalling G. [default]
0050 %                  p = scaled to make the actual peak G
0051 %                  d = scaled to make DC gain equal to G (summed sample values).
0052 %                  D = scaled to make average value equal G
0053 %                  e = scaled to make energy = G (summed squared sample values).
0054 %                  E = scaled to make mean energy = G (mean squared sample values).
0055 %                  q = take square root of the window after scaling
0056 %
0057 %         first and last samples (see note on periodicity below):
0058 %                  b [both]    = The first and last samples are at the extreme ends of
0059 %                                the window [default for most windows].
0060 %                  n [neither] = The first and last samples are one sample away from the ends
0061 %                                of the window [default for windows having zero end points].
0062 %                  s [shifted] = The first and last samples are half a sample away from the
0063 %                                ends of the window .
0064 %                  l [left]    = The first sample is at the end of the window while the last
0065 %                                is one sample away from the end .
0066 %                  r [right]   = The first sample is one sample away from the end while the
0067 %                                last is at the end of the window .
0068 %
0069 %         whole/half window (see note on periodicity below):
0070 %                  w = The whole window is included [default]
0071 %                  c = The first sample starts in the centre of the window
0072 %                  h = The first sample starts half a sample beyond the centre
0073 %
0074 %         convolve with rectangle
0075 %                  o = convolve w(n) with a rectangle of length N-H [default floor(N/2)]
0076 %                      This can be used to force w(n) to satisfy the Princen-Bradley condition
0077 %
0078 % Periodicity:
0079 %     The underlying period of the window function depends on the chosen mode combinations and
0080 %     is given in the table below. For overlapping windows with perfect reconstruction choose
0081 %     N to be an integer and modes 'ws', 'wl' or 'wr'.
0082 %
0083 %        Whole/half window -->     w         h         c
0084 %
0085 %        End points:       b      N-1      2N-1      2N-2
0086 %                          n      N+1      2N+1       2N
0087 %                          s       N        2N       2N-1
0088 %                          l       N       2N+1       2N
0089 %                          r       N       2N-1      2N-2
0090 %
0091 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
0092 % To obtain unity gain for windowed overlap-add processing you can use
0093 % the following. Bandwidths have been multiplied by the window length.
0094 % For perfect reconstruction, you can use any multiple of the overlap factors
0095 % shown assuming the same window is used for both analysis and synthesis.
0096 % These are the Princen-Bradley conditions: fliplr(w)=w, w(i)^2+w(i+n/2)^2=1
0097 % Any symmetric window will satisfy the coinditions with mode 'boqD2' [3].
0098 %
0099 %   Window     Mode Overlap-Factor Sidelobe  3dB-BW  6dB-BW Equiv-noise-BW
0100 %   rsqvorbis  sqD2     2           -26dB     1.1      1.5      1.1
0101 %   hamming    sqD2     2,3,5       -24dB     1.1      1.5      1.1
0102 %   cos        sE2      2,3,5       -23dB     1.2      1.6      1.2 used in MP3
0103 %   kaiser(5)  boqD2    2           -23dB     1.2      1.7      1.3 used in AAC [4]
0104 %   vorbis     sE2      2,9,15      -21dB     1.3      1.8      1.4 used in Vorbis
0105 %   hamming    sE4      3,4,5       -43dB     1.3      1.8      1.4
0106 %   hanning    sE4      3,4,5       -31dB     1.4      2.0      1.5
0107 %
0108 % References:
0109 %  [1]  F. J. Harris. On the use of windows for harmonic analysis with the
0110 %       discrete fourier transform. Proc IEEE, 66 (1): 5183, Jan. 1978.
0111 %  [2]    L. D. Fielder, M. Bosi, G. Davidson, M. Davis, C. Todd, and S. Vernon.
0112 %       AC-2 and AC-3: Low-complexity transform-based audio coding.
0113 %       In Audio Engineering Society Conference: Collected Papers on Digital Audio Bit-Rate Reduction, May 1996.
0114 %  [3]    J. Princen, A. Johnson, and A. Bradley. Subband/transform coding using filter
0115 %       bank designs based on time domain aliasing cancellation.
0116 %       In Proc. IEEE Intl Conf. Acoustics, Speech and Signal Processing, volume 12,
0117 %       pages 21612164, 1987. doi: 10.1109/ICASSP.1987.1169405.
0118 %  [4]    T. Sporer, K. Brandenburg, and B. Edler.
0119 %       The use of multirate filter banks for coding of high quality digital audio.
0120 %       In Proc EUSIPCO, volume 1, pages 211214, 1992.
0121 %
0122 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
0123 %      Copyright (C) Mike Brookes 2002-2015
0124 %      Version: $Id: v_windows.m 6808 2015-09-13 11:22:27Z dmb $
0125 %
0126 %   VOICEBOX is a MATLAB toolbox for speech processing.
0127 %   Home page: http://www.ee.ic.ac.uk/hp/staff/dmb/voicebox/voicebox.html
0128 %
0129 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
0130 %   This program is free software; you can redistribute it and/or modify
0131 %   it under the terms of the GNU General Public License as published by
0132 %   the Free Software Foundation; either version 2 of the License, or
0133 %   (at your option) any later version.
0134 %
0135 %   This program is distributed in the hope that it will be useful,
0136 %   but WITHOUT ANY WARRANTY; without even the implied warranty of
0137 %   MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
0138 %   GNU General Public License for more details.
0139 %
0140 %   You can obtain a copy of the GNU General Public License from
0141 %   http://www.gnu.org/copyleft/gpl.html or by writing to
0142 %   Free Software Foundation, Inc.,675 Mass Ave, Cambridge, MA 02139, USA.
0143 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
0144 
0145 kk=[-1 1 1 -1; 0 0 2 -2; 0 1 2 -1;   % mode  w,  h,  c  [normal windows]
0146     -1 0 1 0; 0 0 2 0; 0 1 2 1;       % modes lw, lh, lc
0147     -1 2 1 0; 0 0 2 -2; 0 1 2 -1;     % modes rw, rh, rc
0148     -1 1 1 -1; 0 0 2 -2; 0 1 2 -1;    % modes bw, bh, bc
0149     -1 1 1 1; 0 0 2 0; 0 1 2 1;       % modes nw, nh, nc
0150     -1 1 1 0; 0 0 2 -1; 0 1 2 0;];    % modes sw, sh, sc
0151 
0152 if nargin<2
0153     n=256;
0154 end
0155 if nargin<3 || isempty(mode) || ~ischar(mode)
0156     mode='uw';
0157 end;
0158 mm=zeros(1,length(mode)+1);
0159 ll='hc lrbns';
0160 for i=1:8
0161     mm(mode==ll(i))=i-3;
0162 end
0163 wtype=lower(wtype);
0164 k=1+3*max(mm)-min(mm);
0165 if k<4
0166     switch wtype                % check if window goes all the way to zero
0167         case {'hanning','triangle','blackman','cos','tukey','vorbis'}
0168             k=k+12;
0169     end
0170 end
0171 if any(mode=='o') % need to convolve with rectangle
0172     if nargin<5 || ~numel(h)
0173         ov=floor(n/2);
0174     end
0175     n=n-ov+1; % shorten baseline window
0176 else
0177     ov=0;
0178 end
0179 
0180 % determine the sample points
0181 % the number of points corresponding to a full period is (kk(k,3)*n+kk(k,4))
0182 fn=floor(n);
0183 kp=(kk(k,3)*n+kk(k,4));
0184 ks=kk(k,1)*fn+kk(k,2);
0185 v=((0:2:2*fn-2)+ks)/kp;
0186 
0187 % now make the window
0188 np=0;
0189 switch wtype
0190     case 'hanning'
0191         w = 0.5+0.5*cos(pi*v);
0192         
0193     case 'cos'
0194         if nargin<4, p=1; end;
0195         np=1;               % number of parameters = 1
0196         w = cos(0.5*pi*v).^p(1);
0197         
0198     case 'dolph'
0199         if nargin<4, p=50; end;
0200         np=1;               % number of parameters = 1
0201         if rem(ks+kp,2)     % for shifted windows, we generate twice as many points
0202             w=chebwin(2*kp+1,abs(p(1)));
0203             w=w((1:2:2*fn)+round(ks+kp));
0204         else
0205             w=chebwin(kp+1,abs(p(1)));
0206             w=w((1:fn)+round((ks+kp)/2));
0207         end
0208         
0209     case 'tukey'
0210         if nargin<4, p=0.5; end;
0211         np=1;               % number of parameters = 1
0212         if p(1)>0
0213             p(1)=min(p(1),1);
0214             w = 0.5+0.5*cos(pi*max(1+(abs(v)-1)/p(1),0));
0215         else
0216             w = ones(size(v));
0217         end
0218         
0219     case 'cauchy'
0220         if nargin<4, p=1; end;
0221         np=1;               % number of parameters = 1
0222         w = (1+(p(1)*v).^2).^-1;
0223         
0224     case 'rectangle'
0225         w = ones(size(v));
0226         
0227     case 'triangle'
0228         if nargin<4, p=1; end;
0229         np=1;               % number of parameters = 1
0230         w = 1-abs(v).^p(1);
0231         
0232     case 'gaussian'
0233         if nargin<4, p=3; end;
0234         w=exp(-0.5*p(1)^2*(v.*v));
0235         np=1;
0236         
0237     case 'kaiser'
0238         if nargin<4, p=8; end;
0239         w=besseli(0,p*sqrt(1-v.^2))/besseli(0,p(1));
0240         np=1;
0241         
0242     case 'hamming'
0243         w = 0.54+0.46*cos(pi*v);
0244         
0245     case 'blackman'
0246         w = 0.42+0.5*cos(pi*v) + 0.08*cos(2*pi*v);
0247         
0248     case 'harris3'
0249         w = 0.42323 + 0.49755*cos(pi*v) + 0.07922*cos(2*pi*v);
0250         
0251     case 'harris4'
0252         w = 0.35875 + 0.48829*cos(pi*v) + 0.14128*cos(2*pi*v) + 0.01168*cos(3*pi*v);
0253         
0254     case 'vorbis'
0255         w = sin(0.25*pi*(1+cos(pi*v)));
0256         
0257     case 'rsqvorbis'
0258         w = 0.571-0.429*cos(0.5*pi*(1+cos(pi*v)));
0259         
0260     otherwise
0261         error(sprintf('Unknown window type: %s', wtype));
0262 end;
0263 % now convolve with rectangle
0264 if ov
0265     %     w=filter(ones(1,ov),1,w); % more adds but might be just as efficient as cumsum
0266     w=cumsum(w);
0267     w(n+1:n+ov-1)=w(n)-w(n-ov+1:n-1);
0268     w(ov+1:n)=w(ov+1:n)-w(1:n-ov);
0269     n=n+ov-1; % restore original value of n
0270 end
0271 % scale if required
0272 mk=find(mode>='1' & mode<='9',1);
0273 if numel(mk)
0274     g=1/(mode(mk)-'0');
0275 else
0276     g=1;
0277 end
0278 if any(mode=='d')
0279     w=w*(g/sum(w));
0280 elseif any(mode=='D') || any(mode=='a')
0281     w=w*(g/mean(w));
0282 elseif any(mode=='e')
0283     w=w*sqrt(g/sum(w.^2));
0284 elseif any(mode=='E')
0285     w=w*sqrt(g/mean(w.^2));
0286 elseif any(mode=='p')
0287     w=w*(g/max(w));
0288 end
0289 if any(mode=='q')
0290     w=sqrt(w);
0291 end
0292 
0293 if ~nargout
0294     v_windinfo(w,n);
0295     if np>0
0296         title(sprintf('%s (%s ) window  - mode=''%s''',wtype,sprintf(' %g',p(1:np)),mode));
0297     else
0298         title(sprintf('%s window - mode=''%s''',wtype,mode));
0299     end
0300 end;
0301

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