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 'o' 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' = cos(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 hanning sqD2 2,3,5 -23dB 1.2 1.6 1.2 =cos('sE2') 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): 51–83, 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 2161–2164, 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 211–214, 1992. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Copyright (C) Mike Brookes 2002-2015 Version: $Id: v_windows.m 10429 2018-03-08 07:34:04Z 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. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

- v_windinfo WINDINFO window information and figures of merit X=(W,FS)

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 'o' 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' = cos(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 % hanning sqD2 2,3,5 -23dB 1.2 1.6 1.2 =cos('sE2') 0103 % cos sE2 2,3,5 -23dB 1.2 1.6 1.2 used in MP3 0104 % kaiser(5) boqD2 2 -23dB 1.2 1.7 1.3 used in AAC [4] 0105 % vorbis sE2 2,9,15 -21dB 1.3 1.8 1.4 used in Vorbis 0106 % hamming sE4 3,4,5 -43dB 1.3 1.8 1.4 0107 % hanning sE4 3,4,5 -31dB 1.4 2.0 1.5 0108 % 0109 % References: 0110 % [1] F. J. Harris. On the use of windows for harmonic analysis with the 0111 % discrete fourier transform. Proc IEEE, 66 (1): 51–83, Jan. 1978. 0112 % [2] L. D. Fielder, M. Bosi, G. Davidson, M. Davis, C. Todd, and S. Vernon. 0113 % AC-2 and AC-3: Low-complexity transform-based audio coding. 0114 % In Audio Engineering Society Conference: Collected Papers on Digital Audio Bit-Rate Reduction, May 1996. 0115 % [3] J. Princen, A. Johnson, and A. Bradley. Subband/transform coding using filter 0116 % bank designs based on time domain aliasing cancellation. 0117 % In Proc. IEEE Intl Conf. Acoustics, Speech and Signal Processing, volume 12, 0118 % pages 2161–2164, 1987. doi: 10.1109/ICASSP.1987.1169405. 0119 % [4] T. Sporer, K. Brandenburg, and B. Edler. 0120 % The use of multirate filter banks for coding of high quality digital audio. 0121 % In Proc EUSIPCO, volume 1, pages 211–214, 1992. 0122 % 0123 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 0124 % Copyright (C) Mike Brookes 2002-2015 0125 % Version: $Id: v_windows.m 10429 2018-03-08 07:34:04Z dmb $ 0126 % 0127 % VOICEBOX is a MATLAB toolbox for speech processing. 0128 % Home page: http://www.ee.ic.ac.uk/hp/staff/dmb/voicebox/voicebox.html 0129 % 0130 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 0131 % This program is free software; you can redistribute it and/or modify 0132 % it under the terms of the GNU General Public License as published by 0133 % the Free Software Foundation; either version 2 of the License, or 0134 % (at your option) any later version. 0135 % 0136 % This program is distributed in the hope that it will be useful, 0137 % but WITHOUT ANY WARRANTY; without even the implied warranty of 0138 % MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the 0139 % GNU General Public License for more details. 0140 % 0141 % You can obtain a copy of the GNU General Public License from 0142 % http://www.gnu.org/copyleft/gpl.html or by writing to 0143 % Free Software Foundation, Inc.,675 Mass Ave, Cambridge, MA 02139, USA. 0144 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 0145 0146 kk=[-1 1 1 -1; 0 0 2 -2; 0 1 2 -1; % mode w, h, c [normal windows] 0147 -1 0 1 0; 0 0 2 0; 0 1 2 1; % modes lw, lh, lc 0148 -1 2 1 0; 0 0 2 -2; 0 1 2 -1; % modes rw, rh, rc 0149 -1 1 1 -1; 0 0 2 -2; 0 1 2 -1; % modes bw, bh, bc 0150 -1 1 1 1; 0 0 2 0; 0 1 2 1; % modes nw, nh, nc 0151 -1 1 1 0; 0 0 2 -1; 0 1 2 0;]; % modes sw, sh, sc 0152 0153 if nargin<2 0154 n=256; 0155 end 0156 if nargin<3 || isempty(mode) || ~ischar(mode) 0157 mode='uw'; 0158 end; 0159 mm=zeros(1,length(mode)+1); 0160 ll='hc lrbns'; 0161 for i=1:8 0162 mm(mode==ll(i))=i-3; 0163 end 0164 wtype=lower(wtype); 0165 k=1+3*max(mm)-min(mm); 0166 if k<4 0167 switch wtype % check if window goes all the way to zero 0168 case {'hanning','triangle','blackman','cos','tukey','vorbis'} 0169 k=k+12; 0170 end 0171 end 0172 if any(mode=='o') % need to convolve with rectangle 0173 if nargin<5 || ~numel(h) 0174 ov=floor(n/2); 0175 end 0176 n=n-ov+1; % shorten baseline window 0177 else 0178 ov=0; 0179 end 0180 0181 % determine the sample points 0182 % the number of points corresponding to a full period is (kk(k,3)*n+kk(k,4)) 0183 fn=floor(n); 0184 kp=(kk(k,3)*n+kk(k,4)); 0185 ks=kk(k,1)*fn+kk(k,2); 0186 v=((0:2:2*fn-2)+ks)/kp; 0187 0188 % now make the window 0189 np=0; 0190 switch wtype 0191 case 'hanning' 0192 w = 0.5+0.5*cos(pi*v); 0193 0194 case 'cos' 0195 if nargin<4, p=1; end; 0196 np=1; % number of parameters = 1 0197 w = cos(0.5*pi*v).^p(1); 0198 0199 case 'dolph' 0200 if nargin<4, p=50; end; 0201 np=1; % number of parameters = 1 0202 if rem(ks+kp,2) % for shifted windows, we generate twice as many points 0203 w=chebwin(2*kp+1,abs(p(1))); 0204 w=w((1:2:2*fn)+round(ks+kp)); 0205 else 0206 w=chebwin(kp+1,abs(p(1))); 0207 w=w((1:fn)+round((ks+kp)/2)); 0208 end 0209 0210 case 'tukey' 0211 if nargin<4, p=0.5; end; 0212 np=1; % number of parameters = 1 0213 if p(1)>0 0214 p(1)=min(p(1),1); 0215 w = 0.5+0.5*cos(pi*max(1+(abs(v)-1)/p(1),0)); 0216 else 0217 w = ones(size(v)); 0218 end 0219 0220 case 'cauchy' 0221 if nargin<4, p=1; end; 0222 np=1; % number of parameters = 1 0223 w = (1+(p(1)*v).^2).^-1; 0224 0225 case 'rectangle' 0226 w = ones(size(v)); 0227 0228 case 'triangle' 0229 if nargin<4, p=1; end; 0230 np=1; % number of parameters = 1 0231 w = 1-abs(v).^p(1); 0232 0233 case 'gaussian' 0234 if nargin<4, p=3; end; 0235 w=exp(-0.5*p(1)^2*(v.*v)); 0236 np=1; 0237 0238 case 'kaiser' 0239 if nargin<4, p=8; end; 0240 w=besseli(0,p*sqrt(1-v.^2))/besseli(0,p(1)); 0241 np=1; 0242 0243 case 'hamming' 0244 w = 0.54+0.46*cos(pi*v); 0245 0246 case 'blackman' 0247 w = 0.42+0.5*cos(pi*v) + 0.08*cos(2*pi*v); 0248 0249 case 'harris3' 0250 w = 0.42323 + 0.49755*cos(pi*v) + 0.07922*cos(2*pi*v); 0251 0252 case 'harris4' 0253 w = 0.35875 + 0.48829*cos(pi*v) + 0.14128*cos(2*pi*v) + 0.01168*cos(3*pi*v); 0254 0255 case 'vorbis' 0256 w = sin(0.25*pi*(1+cos(pi*v))); 0257 0258 case 'rsqvorbis' 0259 w = 0.571-0.429*cos(0.5*pi*(1+cos(pi*v))); 0260 0261 otherwise 0262 error(sprintf('Unknown window type: %s', wtype)); 0263 end; 0264 % now convolve with rectangle 0265 if ov 0266 % w=filter(ones(1,ov),1,w); % more adds but might be just as efficient as cumsum 0267 w=cumsum(w); 0268 w(n+1:n+ov-1)=w(n)-w(n-ov+1:n-1); 0269 w(ov+1:n)=w(ov+1:n)-w(1:n-ov); 0270 n=n+ov-1; % restore original value of n 0271 end 0272 % scale if required 0273 mk=find(mode>='1' & mode<='9',1); 0274 if numel(mk) 0275 g=1/(mode(mk)-'0'); 0276 else 0277 g=1; 0278 end 0279 if any(mode=='d') 0280 w=w*(g/sum(w)); 0281 elseif any(mode=='D') || any(mode=='a') 0282 w=w*(g/mean(w)); 0283 elseif any(mode=='e') 0284 w=w*sqrt(g/sum(w.^2)); 0285 elseif any(mode=='E') 0286 w=w*sqrt(g/mean(w.^2)); 0287 elseif any(mode=='p') 0288 w=w*(g/max(w)); 0289 end 0290 if any(mode=='q') 0291 w=sqrt(w); 0292 end 0293 0294 if ~nargout 0295 v_windinfo(w,n); 0296 if np>0 0297 title(sprintf('%s (%s ) window - mode=''%s''',wtype,sprintf(' %g',p(1:np)),mode)); 0298 else 0299 title(sprintf('%s window - mode=''%s''',wtype,mode)); 0300 end 0301 end; 0302

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