V_FILTBANKM determine matrix for a linear/mel/erb/bark-spaced v_filterbank [X,IL,IH]=(P,N,FS,FL,FH,W)
Usage:
(1) Calcuate the Mel-frequency Cepstral Coefficients
f=v_rfft(s); % v_rfft() returns only 1+floor(n/2) coefficients
x=v_filtbankm(p,n,fs,0,fs/2,'m'); % n is the fft length, p is the number of filters wanted
z=log(x*abs(f).^2); % multiply the power spectrum by x to get log mel-spectrum
c=dct(z); % take the DCT to get the mel-cepstrum
(2) Calcuate the Mel-frequency Cepstral Coefficients efficiently
f=fft(s); % n is the fft length, p is the number of filters wanted
[x,cf,na,nb]=v_filtbankm(p,n,fs,0,fs/2,'m'); % na:nb gives the fft bins that are needed
z=log(x*(f(na:nb)).*conj(f(na:nb))); % multiply x by the power spectrum
c=dct(z); % take the DCT
(3) Plot the calculated filterbanks as a graph or spectrogram
v_filtbankm(p,n,fs,0,fs/2,'mg'); % use option 'mg' for a graph or 'mG' for a spectrogram
(4) Convert to mel-spectrum and back again
[x,cf,xi]=v_filtbankm(p,n,fs,0,fs/2,'mxXzq'); % n is the fft length, p is the number of filters wanted
f=v_rfft(s); % v_rfft() returns only 1+floor(n/2) coefficients
z=x*abs(f).^2; % multiply the power spectrum by x to get mel-spectrum
gp=xi*z; % multiply by xi to recover the approximate power spectrum
g=v_irfft(sqrt(gp).*exp(1i*angle(f))); % take the inverse DFT using the original phase to recover the time domain signal
Inputs:
p number of filters in filterbank or the filter spacing in k-mel/bark/erb (see 'p' and 'P' options) [ceil(4.6*log10(fs))]
n length of dft
fs sample rate in Hz
fl low end of the lowest filter in Hz (or in mel/erb/bark/log10 with 'h' option) [default = 0Hz or, if 'l' option given, 30Hz]
fh high end of highest filter in Hz (or in mel/erb/bark/log10 with 'h' option) [default = fs/2]
w any sensible combination of the following:
'b' = bark scale
'e' = erb-rate scale
'l' = log10 Hz frequency scale
'f' = linear frequency scale [default]
'm' = mel frequency scale
'n' = round to the nearest FFT bin so each row of x contains only one non-zero entry
'c' = fl specifies centre of low filters instead of low edge
'C' = fh specifies centre of high filter instead of high edge
'h' = fl & fh are in mel/erb/bark/log10 instead of Hz
'H' = give cf outputs in mel/erb/bark/log10 instead of Hz
'x' = lowest filter remains at 1 down to 0 frequency
'X' = highest filter remains at 1 up to nyquist freqency
'p' = input p specifies the number of filters [default if p>=1]
'P' = input p specifies the approximate filter spacing in kHz/kmel/... [default if p<1]
'z' = Treat input power spectrum at 0Hz as an impulse rather than being diffuse
'Z' = Treat input power spectrum at 0Hz as the sum of an impulse and a continuous component with the same amlitude as the adjacent bin
'q' = The first output filter gives the power of the impulse at 0Hz (regardless of the 'D' option). 'zq' ensures exact retention of DC component by xi*x
'd' = input is power spectral density (power per Hz) instead of power
'D' = output is power spectral density (power per Hz) instead of power (option 'dD' makes the rows of x sum to approximately 1)
's' = single-sided input: do not add power from symmetric negative frequencies (i.e. non-DC/Nyquist inputs have already been doubled)
'S' = single-sided output: include power from both positive and negative frequencies (this doubles the non-DC/Nyquist outputs)
'w' = size(x,2)=size(xi,1)=n rather than 1+floor(n/2) although the rightmost half of x is all zeros
'g' = plot filter coefficients as graph
'G' = plot filter coefficients as spectrogram image [default if no output arguments present]
Legacy options, 'yYuU' are mapped as follows: 'y'='xX', 'Y'='x', 'yY'='X', 'u'='dD', 'U'='D'
Outputs: x(p,k) a sparse matrix containing the v_filterbank amplitudes
If the il and ih output arguments are included then k=ih-il+1 otherwise k=1+floor(n/2)
Note that, with the 'S' option, the peak filter values equal 2 to account for the energy in the negative FFT frequencies.
cf(p) the v_filterbank centre frequencies in Hz (or in mel/erb/bark/log10 with 'H' option)
xi(k,p) [optional] sparse matrix that is an approximate inverse of x
il,ih the lowest and highest fft bins with non-zero coefficient 1<=il,ih<=1+n/2 (Note: you must specify *both* il and ih or neither)
The input power will be preserved if the options 'xXS' are given
The output of the routine is a sparse filterbank matrix. The vector output of the filterbank can then be obtained
by pre-multiplying an input power spectrum vector (as a column vector) by the filterbank matrix. The input and
output vectors can optionally be in either the power domain or the power spectral density domain.
The routine implements the filterbank in two conceptual stages (which are merged in the practical implementation):
Stage 1:
The discrete input spectrum is converted to a continuous power spectral density using linear interpolation in frequency.
Each element of the input spectrum influences a frequency interval of width 2d where d is the input frequency bin width.
The DC component of the input is treated specially in one of three ways: (a) it can be treated as a normal element that
influences an interval (-d,+d) like the other elements [default]; (b) it can be treated as an impulse at DC ['z' option];
(c) it can be treated as a mixture of an impulse and a normal component whose value equals that of the adjacent frequency
bin ['Z' option].
Stage 2:
The filterbank outputs are calculated by integrating the product of the continuous spectrum and a filter weight that is
triangular in the frequency domain. Optionally, the first filterbank preserves the DC impulse component of the continuous
spectrum ['q' option].
References:
[1] S. S. Stevens, J. Volkman, and E. B. Newman. A scale for the measurement
of the psychological magnitude of pitch. J. Acoust Soc Amer, 8: 185-190, 1937.
[2] S. Davis and P. Mermelstein. Comparison of parametric representations for
monosyllabic word recognition in continuously spoken sentences.
IEEE Trans Acoustics Speech and Signal Processing, 28 (4): 357-366, Aug. 1980.0001 function [x,cf,xi,il,ih]=v_filtbankm(p,n,fs,fl,fh,w) 0002 %V_FILTBANKM determine matrix for a linear/mel/erb/bark-spaced v_filterbank [X,IL,IH]=(P,N,FS,FL,FH,W) 0003 % 0004 % Usage: 0005 % (1) Calcuate the Mel-frequency Cepstral Coefficients 0006 % 0007 % f=v_rfft(s); % v_rfft() returns only 1+floor(n/2) coefficients 0008 % x=v_filtbankm(p,n,fs,0,fs/2,'m'); % n is the fft length, p is the number of filters wanted 0009 % z=log(x*abs(f).^2); % multiply the power spectrum by x to get log mel-spectrum 0010 % c=dct(z); % take the DCT to get the mel-cepstrum 0011 % 0012 % (2) Calcuate the Mel-frequency Cepstral Coefficients efficiently 0013 % 0014 % f=fft(s); % n is the fft length, p is the number of filters wanted 0015 % [x,cf,na,nb]=v_filtbankm(p,n,fs,0,fs/2,'m'); % na:nb gives the fft bins that are needed 0016 % z=log(x*(f(na:nb)).*conj(f(na:nb))); % multiply x by the power spectrum 0017 % c=dct(z); % take the DCT 0018 % 0019 % (3) Plot the calculated filterbanks as a graph or spectrogram 0020 % 0021 % v_filtbankm(p,n,fs,0,fs/2,'mg'); % use option 'mg' for a graph or 'mG' for a spectrogram 0022 % 0023 % (4) Convert to mel-spectrum and back again 0024 % 0025 % [x,cf,xi]=v_filtbankm(p,n,fs,0,fs/2,'mxXzq'); % n is the fft length, p is the number of filters wanted 0026 % f=v_rfft(s); % v_rfft() returns only 1+floor(n/2) coefficients 0027 % z=x*abs(f).^2; % multiply the power spectrum by x to get mel-spectrum 0028 % gp=xi*z; % multiply by xi to recover the approximate power spectrum 0029 % g=v_irfft(sqrt(gp).*exp(1i*angle(f))); % take the inverse DFT using the original phase to recover the time domain signal 0030 % 0031 % Inputs: 0032 % p number of filters in filterbank or the filter spacing in k-mel/bark/erb (see 'p' and 'P' options) [ceil(4.6*log10(fs))] 0033 % n length of dft 0034 % fs sample rate in Hz 0035 % fl low end of the lowest filter in Hz (or in mel/erb/bark/log10 with 'h' option) [default = 0Hz or, if 'l' option given, 30Hz] 0036 % fh high end of highest filter in Hz (or in mel/erb/bark/log10 with 'h' option) [default = fs/2] 0037 % w any sensible combination of the following: 0038 % 0039 % 'b' = bark scale 0040 % 'e' = erb-rate scale 0041 % 'l' = log10 Hz frequency scale 0042 % 'f' = linear frequency scale [default] 0043 % 'm' = mel frequency scale 0044 % 0045 % 'n' = round to the nearest FFT bin so each row of x contains only one non-zero entry 0046 % 0047 % 'c' = fl specifies centre of low filters instead of low edge 0048 % 'C' = fh specifies centre of high filter instead of high edge 0049 % 'h' = fl & fh are in mel/erb/bark/log10 instead of Hz 0050 % 'H' = give cf outputs in mel/erb/bark/log10 instead of Hz 0051 % 0052 % 'x' = lowest filter remains at 1 down to 0 frequency 0053 % 'X' = highest filter remains at 1 up to nyquist freqency 0054 % 0055 % 'p' = input p specifies the number of filters [default if p>=1] 0056 % 'P' = input p specifies the approximate filter spacing in kHz/kmel/... [default if p<1] 0057 % 0058 % 'z' = Treat input power spectrum at 0Hz as an impulse rather than being diffuse 0059 % 'Z' = Treat input power spectrum at 0Hz as the sum of an impulse and a continuous component with the same amlitude as the adjacent bin 0060 % 'q' = The first output filter gives the power of the impulse at 0Hz (regardless of the 'D' option). 'zq' ensures exact retention of DC component by xi*x 0061 % 0062 % 'd' = input is power spectral density (power per Hz) instead of power 0063 % 'D' = output is power spectral density (power per Hz) instead of power (option 'dD' makes the rows of x sum to approximately 1) 0064 % 0065 % 's' = single-sided input: do not add power from symmetric negative frequencies (i.e. non-DC/Nyquist inputs have already been doubled) 0066 % 'S' = single-sided output: include power from both positive and negative frequencies (this doubles the non-DC/Nyquist outputs) 0067 % 'w' = size(x,2)=size(xi,1)=n rather than 1+floor(n/2) although the rightmost half of x is all zeros 0068 % 0069 % 'g' = plot filter coefficients as graph 0070 % 'G' = plot filter coefficients as spectrogram image [default if no output arguments present] 0071 % 0072 % Legacy options, 'yYuU' are mapped as follows: 'y'='xX', 'Y'='x', 'yY'='X', 'u'='dD', 'U'='D' 0073 % 0074 % Outputs: x(p,k) a sparse matrix containing the v_filterbank amplitudes 0075 % If the il and ih output arguments are included then k=ih-il+1 otherwise k=1+floor(n/2) 0076 % Note that, with the 'S' option, the peak filter values equal 2 to account for the energy in the negative FFT frequencies. 0077 % cf(p) the v_filterbank centre frequencies in Hz (or in mel/erb/bark/log10 with 'H' option) 0078 % xi(k,p) [optional] sparse matrix that is an approximate inverse of x 0079 % il,ih the lowest and highest fft bins with non-zero coefficient 1<=il,ih<=1+n/2 (Note: you must specify *both* il and ih or neither) 0080 % 0081 % The input power will be preserved if the options 'xXS' are given 0082 % 0083 % The output of the routine is a sparse filterbank matrix. The vector output of the filterbank can then be obtained 0084 % by pre-multiplying an input power spectrum vector (as a column vector) by the filterbank matrix. The input and 0085 % output vectors can optionally be in either the power domain or the power spectral density domain. 0086 % The routine implements the filterbank in two conceptual stages (which are merged in the practical implementation): 0087 % 0088 % Stage 1: 0089 % The discrete input spectrum is converted to a continuous power spectral density using linear interpolation in frequency. 0090 % Each element of the input spectrum influences a frequency interval of width 2d where d is the input frequency bin width. 0091 % The DC component of the input is treated specially in one of three ways: (a) it can be treated as a normal element that 0092 % influences an interval (-d,+d) like the other elements [default]; (b) it can be treated as an impulse at DC ['z' option]; 0093 % (c) it can be treated as a mixture of an impulse and a normal component whose value equals that of the adjacent frequency 0094 % bin ['Z' option]. 0095 % 0096 % Stage 2: 0097 % The filterbank outputs are calculated by integrating the product of the continuous spectrum and a filter weight that is 0098 % triangular in the frequency domain. Optionally, the first filterbank preserves the DC impulse component of the continuous 0099 % spectrum ['q' option]. 0100 % 0101 % References: 0102 % 0103 % [1] S. S. Stevens, J. Volkman, and E. B. Newman. A scale for the measurement 0104 % of the psychological magnitude of pitch. J. Acoust Soc Amer, 8: 185-190, 1937. 0105 % [2] S. Davis and P. Mermelstein. Comparison of parametric representations for 0106 % monosyllabic word recognition in continuously spoken sentences. 0107 % IEEE Trans Acoustics Speech and Signal Processing, 28 (4): 357-366, Aug. 1980. 0108 0109 % Bugs/Suggestions 0110 % (1) default frequencies won't work if the h option is specified 0111 % (2) low default frequency is invalid if the 'l' option is specified 0112 % (3) Add option to choose the domain in which linear interpolation is performed 0113 0114 % Copyright (C) Mike Brookes 1997-2024 0115 % Version: $Id: v_filtbankm.m $ 0116 % 0117 % VOICEBOX is a MATLAB toolbox for speech processing. 0118 % Home page: http://www.ee.ic.ac.uk/hp/staff/dmb/voicebox/voicebox.html 0119 % 0120 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 0121 % This program is free software; you can redistribute it and/or modify 0122 % it under the terms of the GNU General Public License as published by 0123 % the Free Software Foundation; either version 2 of the License, or 0124 % (at your option) any later version. 0125 % 0126 % This program is distributed in the hope that it will be useful, 0127 % but WITHOUT ANY WARRANTY; without even the implied warranty of 0128 % MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the 0129 % GNU General Public License for more details. 0130 % 0131 % You can obtain a copy of the GNU General Public License from 0132 % http://www.gnu.org/copyleft/gpl.html or by writing to 0133 % Free Software Foundation, Inc.,675 Mass Ave, Cambridge, MA 02139, USA. 0134 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 0135 0136 % Notes: 0137 % (1) In the comments, "FFT bin_0" assumes DC = bin 0 whereas "FFT bin_1" means DC = bin 1nfout 0138 % (2) "input" and "output" need to be interchanged if the 'i' option is given 0139 0140 if nargin<6 || isempty(w) % if no mode option, w, is specified 0141 w='f'; % default mode option: 'f' = linear output frequency scale 0142 end 0143 wr=max(any(repmat('lebm',length(w),1)==repmat(w',1,4),1).*(1:4)); % output warping: 0=linear,1=log,2=erbrate,3=bark,4=mel 0144 ww=any(repmat('ncChHxXyYpPzZqdDuUsSgGw',length(w),1)==repmat(w',1,23),1); % decode all other options 0145 % ww elements: 1=n,2=c,3=C,4=h,5=H,6=x,7=X,8=y,9=Y,10=p,11=P,12=z,13=Z,14=q,15=d,16=D,17=u,18=U,19=s,20=S,21=g,22=G,23=w 0146 % Convert legacy option codes: 'y'='xX', 'Y'='x', 'yY'='X', 'u'='dD', 'U'='D' 0147 ww(6)=ww(6) || (ww(8) ~= ww(9)); % convert 'y' or 'Y' (but not both) to 'x'; extend low frequencies 0148 ww(7)=ww(7) || ww(8); % convert 'y' to 'X'; extend high frequencies 0149 ww(15)=ww(15) || ww(17); % convert 'u' to 'd'; input is also in power spectral density 0150 ww(16)=ww(16) || ww(17) || ww(18); % convert 'u' or 'U' to 'd'; output is in power spectral density 0151 flhconv=repmat(wr>0 && ~ww(4),1,2); % flag indicating need to convert filterbank limits from Hz to mel/erb/bark/log10 0152 if nargin < 4 || isempty(fl) 0153 fl=30*(wr==1); % lower limit is 0 Hz unless 'l' option specified, in which case it is 30 Hz 0154 flhconv(1)=wr>0; 0155 end 0156 if nargin < 5 || isempty(fh) 0157 fh=0.5*fs; % max freq is the nyquist frequency 0158 flhconv(2)=wr>0; 0159 end 0160 % 0161 % Sort out input frequency bins 0162 % 0163 if numel(n)>1 0164 error('non-standard input frequency spacing no longer supported'); 0165 else % n gives dft length 0166 nf=1+floor(n/2); % number of input positive-frequency bins from DFT 0167 df=fs/n; % input frequency bin spacing 0168 end 0169 % 0170 % Sort out output frequency bins 0171 % 0172 mflh=[fl fh]; % low and high limits of filterbank triangular filters 0173 if any(flhconv) % convert mflh from Hz to mel/erb/... unless already converted via 'h' option 0174 switch wr 0175 case 1 % 'l' = log scaled 0176 if fl<=0 0177 error('Low frequency limit must be >0 for ''l'' log10-frequency option'); 0178 end 0179 mflh(flhconv)=log10(mflh(flhconv)); % convert frequency limits into log10 Hz 0180 case 2 % 'e' = erb-rate scaled 0181 mflh(flhconv)=v_frq2erb(mflh(flhconv)); % convert frequency limits into erb-rate 0182 case 3 % 'b' = bark scaled 0183 mflh(flhconv)=v_frq2bark(mflh(flhconv)); % convert frequency limits into bark 0184 case 4 % 'm' = mel scaled 0185 mflh(flhconv)=v_frq2mel(mflh(flhconv)); % convert frequency limits into mel 0186 end 0187 end 0188 melrng=mflh(2)-mflh(1); % filterbank range in Hz/mel/erb/... 0189 if isempty(p) 0190 p=ceil(4.6*log10(2*(nf-1)*df)); % default number of output filters 0191 end 0192 puc=ww(11) || (p<1) && ~ww(10); % input p specifies the filter spacing rather than the number of filters 0193 if puc 0194 p=round(melrng/(p*1000))+ww(2)+ww(3)-1+ww(14); % p now gives the number of filters (excluding DC impulse) 0195 end 0196 melinc=melrng/(p+ww(2)+ww(3)+1-ww(14)); % inter-filter increment in mel 0197 mflh=mflh+[-ww(2) ww(3)]*melinc; % update mflh to include the full width of all filters 0198 % 0199 % Calculate the output centre frequencies in Hz including dummy end points 0200 % 0201 pmq=p-ww(14); % number of filters excluding the one for the DC impulse 0202 cf=mflh(1)+(0:pmq+1)*melinc; % centre frequencies in mel/erb/... including dummy ends 0203 cf(2:end)=max(cf(2:end),0); % only the first point can be negative [*** doesn't make sense for log scale ***] 0204 switch wr % convert centre frequencies to Hz from mel/erb/... 0205 case 1 % 'l' = log scaled 0206 mb=10.^(cf); 0207 case 2 % 'e' = erb-rate scaled 0208 mb=v_erb2frq(cf); 0209 case 3 % 'b' = bark scaled 0210 mb=v_bark2frq(cf); 0211 case 4 % 'm' = mel scaled 0212 mb=v_mel2frq(cf); 0213 otherwise % [default] = linear scaled; no conversion needed 0214 mb=cf; 0215 end 0216 % 0217 % sort out 2-sided input frequencies 0218 % 0219 fin=(-nf:nf)*df; % reflect negative frequencies excluding DC 0220 nfin=length(fin); % length of extended input frequency list [nfin=2*nf+1] 0221 % 0222 % now sort out the list of output frequencies 0223 % 0224 fout=mb; % output centre frequencies in Hz including dummy values at each end 0225 highex=ww(7) && (fout(end-1)<fin(end)); % extend at high end if 'X' specified and final centre frequency < Nyquist 0226 if ww(6) % ww(6)='x': extend first filter at low end to DC 0227 fout=[0 0 fout(2:end)]; % ... add two dummy values at DC instead of previous single dummy value 0228 end 0229 if highex % extend last filter at high end to Nyquist 0230 fout=[fout(1:end-1) fin(end) fin(end)]; % ... add two dummy values at Nyquist instead of previous single dummy value 0231 end 0232 fout=min(fout,fs/2); % limit output filters to Nyquist frequency 0233 nfout=length(fout); % number of output filters including one or two dummy points at each end 0234 foutin=[fout fin]; 0235 nfall=length(foutin); % = nfout + nfin 0236 wleft=[0 fout(2:nfout)-fout(1:nfout-1) 0 fin(2:nfin)-fin(1:nfin-1)]; % width of lower triangle attached to each node 0237 wright=[wleft(2:end) 0]; % width of upper triangle attached to each node 0238 ffact=[0 ones(1,nfout-2) 0 0 ones(1,2*nf-1) 0]; % valid triangle posts 0239 ffact(wleft+wright==0)=0; % disable null width triangles (*** probably unnecessary if all frequencies are distinct ***) 0240 [fall,ifall]=sort(foutin); % fall is sorted frequencies with fall=foutin(ifall) 0241 jfall=zeros(1,nfall); % create inverse index ... 0242 infall=1:nfall; % ... 0243 jfall(ifall)=infall; % ... inverse-index satisfying foutin=fall(jfall) 0244 ffact(ifall([1:max(jfall(1),jfall(nfout+1))-2 min(jfall(nfout),jfall(nfall))+2:nfall]))=0; % zap input nodes that lie outside the output filters 0245 nxto=cumsum(ifall<=nfout); % next output node to the right (or equal) to each node 0246 nxti=cumsum(ifall>nfout); % number of input nodes to the left (or equal) to each node 0247 nxtr=min(nxti+1+nfout,nfall); % next input node to the right of each value (or nfall if none) 0248 nxtr(ifall>nfout)=1+nxto(ifall>nfout); % next post to the right of opposite input/output type (using sorted indexes) 0249 nxtr=nxtr(jfall); % next post to the right of opposite input/output type (converted to unsorted indices) or if none: nfall or (nfout+1) 0250 % 0251 % The interpolated spectrum at any frequency can be expressed as the sum of the values at the adjacent input bins 0252 % multiplied by triangular weights that decreases from 1 to 0 between the two bins. The value at an output bin 0253 % is equal to the integral of the interpolated spectrum multiplied by a triangular weight that decreases from 0254 % 1 to 0 either side of the output bin. Thus, if all input/output bins are sorted into ascending order, the 0255 % interval between two adjacent bins contains four partial triangles (a.k.a. trapeziums): two "lower" triangles 0256 % that increase with frequency and two "upper" triangles that decrease with frequency. We need to integrate the 0257 % resultant four input-output trapezium products and add the integrals onto the sum for the appropriate output bins. 0258 % Each triangle has a "post" at one end and is zero at the other end; we enumerate the triangle pairs by pairing 0259 % all input and output triangles with the first available triangle of the other type (i.e. output or input) whose 0260 % rightmost node is to the right of the entire first triangle. 0261 % 0262 % The general result for integrating the product of two trapesiums with 0263 % heights (a,b) and (c,d) over a width x is (ad+bc+2bd+2ac)*x/6 0264 % 0265 % integrate product of lower triangles whose posts (and rightmost nodes) are ix1 and jx1 0266 % 0267 msk0=(ffact>0); % posts with a non-zero magnitude 0268 msk=msk0 & (ffact(nxtr)>0); % select triangle pairs with both posts having non-zero magnitudes 0269 ix1=infall(msk); % unsorted indices of leftmost post of pair 0270 jx1=nxtr(msk); % unsorted indices of rightmost post of pair 0271 vfgx=foutin(ix1)-foutin(jx1-1); % portion of triangle attached to rightmost post that lies to the left of the leftmost post 0272 yx=min(wleft(ix1),vfgx); % integration length. Maybe more efficient: dfall=diff(fall); yx=dfall(jfall(ix1)-1) 0273 wx1=ffact(ix1).*ffact(jx1).*yx.*(wleft(ix1).*vfgx-yx.*(0.5*(wleft(ix1)+vfgx)-yx/3))./(wleft(ix1).*wleft(jx1)+(yx==0)); 0274 0275 % integrate product of upper triangles whose posts are ix2 and jx2 and whose rightmost nodes are ix2+1 and jx2+1 0276 0277 nxtu=max([nxtr(2:end)-1 0],1); % post of the upper triangle of opposite type whose rightmost end is to the right of this triangle's rightmost end 0278 msk=msk0 & (ffact(nxtu)>0); % select triangle pairs with both posts having non-zero magnitudes 0279 ix2=infall(msk); % unsorted indices of leftmost post of pair 0280 jx2=nxtu(msk); % unsorted indices of rightmost post of pair 0281 vfgx=foutin(ix2+1)-foutin(jx2); % length of left triangle to the right of the right post 0282 yx=min(wright(ix2),vfgx); % integration length 0283 yx(foutin(jx2+1)<foutin(ix2+1))=0; % zap invalid triangles where the rightmost ends are in the wrong order 0284 wx2=ffact(ix2).*ffact(jx2).*yx.^2.*((0.5*(wright(jx2)-vfgx)+yx/3))./(wright(ix2).*wright(jx2)+(yx==0)); 0285 0286 % integrate lower triangle and upper triangle that ends to its right 0287 0288 nxtu=max(nxtr-1,1); % post of the upper triangle of opposite type whose rightmost end is to the right of this triangle's post 0289 msk=msk0 & (ffact(nxtu)>0); % select triangle pairs with both posts having non-zero magnitudes 0290 ix3=infall(msk); % unsorted indices of lower triangle 0291 jx3=nxtu(msk); % unsorted indices of upper triangle 0292 vfgx=foutin(ix3)-foutin(jx3); % length of upper triangle to the left of the lower post 0293 yx=min(wleft(ix3),vfgx); % integration length 0294 yx(foutin(jx3+1)<foutin(ix3))=0; % zap invalid triangles where the rightmost ends are in the wrong order 0295 wx3=ffact(ix3).*ffact(jx3).*yx.*(wleft(ix3).*(wright(jx3)-vfgx)+yx.*(0.5*(wleft(ix3)-wright(jx3)+vfgx)-yx/3))./(wleft(ix3).*wright(jx3)+(yx==0)); 0296 0297 % integrate upper triangle and lower triangle that starts to its right 0298 0299 nxtu=[nxtr(2:end) 1]; 0300 msk=msk0 & (ffact(nxtu)>0); % select triangle pairs with both posts having non-zero magnitudes 0301 ix4=infall(msk); % unsorted indices of upper triangle 0302 jx4=nxtu(msk); % unsorted indices of lower triangle 0303 vfgx=foutin(ix4+1)-foutin(jx4-1); % length of upper triangle to the left of the lower post 0304 yx=min(wright(ix4),vfgx); % integration length 0305 wx4=ffact(ix4).*ffact(jx4).*yx.^2.*(0.5*vfgx-yx/3)./(wright(ix4).*wleft(jx4)+(yx==0)); 0306 % 0307 % now assemble the matrix 0308 % 0309 iox=sort([ix1 ix2 ix3 ix4;jx1 jx2 jx3 jx4]); % iox(1,:) are output posts, iox(2,:) are input posts 0310 msk=iox(2,:)<=(nfall+nfout)/2; % find references to negative input frequencies 0311 iox(2,msk)=(nfall+nfout+1)-iox(2,msk); % convert negative frequencies to positive 0312 % 0313 % Sort out output gains: 0314 % If output is power then output gain is 1; if output is power/Hz then output gain is 1/area of output filter 0315 % 0316 if ww(6) % ww(6)='x': if lowest filter extended to DC, we added a dummy point at 0Hz, so 0317 iox(1,iox(1,:)==2)=3; % merge lowest two output nodes 0318 end 0319 if highex % if highest filter extended, we added a dummy point at Nyquist, so 0320 iox(1,iox(1,:)==nfout-1)=nfout-2; % merge highest two output nodes 0321 end 0322 x=sparse(iox(1,:)-1-ww(6),max(iox(2,:)-nfout-nf,1),[wx1 wx2 wx3 wx4],pmq,nf); % forward transformation matrix without input/output gains 0323 gout=full(sum(x,2)); % area of each output integral 0324 goutd=sparse(1:pmq,1:pmq,(gout+(gout==0)).^(-1)); % create sparse diagonal matrix of output gains 0325 gouti=full(sum(x(:,1+ww(12):end),2)); % area of each output integral excluding DC if 'z' option given 0326 goutid=sparse(1:pmq,1:pmq,(gouti+(gouti==0)).^(-1)); % create sparse diagonal matrix of output gains 0327 % 0328 % Sort out input gains: 0329 % If input is power then input gain is 1/area; if input is power/Hz then input gain is 1 0330 % 0331 gin=fin(3:nfin)-fin(1:nfin-2); % full width of input interpolation filters 0332 gin=2*(gin+(gin==0)).^(-1); % input gain equals 1/area 0333 ginsi=repmat(1+ww(19),1,nf-2); % 's' option means all inputs except DC and Nyquist have been doubled 0334 ginsd=sparse(1:nf,1:nf,[1-ww(12) ginsi.^(-1) 1]); % ... cancel this out with additional input scaling for forward transform 0335 ginsid=sparse(1:nf,1:nf,[2*(1-ww(12)) ginsi 2]); % and back again for inverse transform 0336 gind=sparse(1:nf,1:nf,gin(end-nf+1:end)); % input gains 0337 % 0338 % Now create the x and xi matrices 0339 % 0340 switch 2*ww(16)+ww(15) 0341 case 0 % '': input and output are both power 0342 xi=ginsid*x'*goutid; 0343 x=x*(gind*ginsd); 0344 case 1 % 'd': input is power/Hz, output is power 0345 xi=(ginsid*gind)*x'*goutid; 0346 x=x*ginsd; 0347 case 2 % 'D': input is power, output is power/Hz 0348 xi=ginsid*x'; 0349 x=goutd*x*(gind*ginsd); 0350 case 3 % 'dD': input and output are both power/Hz 0351 xi=(ginsid*gind)*x'; 0352 x=goutd*x*ginsd; 0353 end 0354 if ww(20) % 'S': double outputs to include negative frequency energy 0355 x=2*x; 0356 xi=0.5*xi; 0357 end 0358 if ww(13) % 'Z': DC input is an impulse plus a diffuse component 0359 x(:,2)=x(:,2)+x(:,1)*ginsd(2,2); % power of diffuse component at DC is equal to that opf adjacent bin corrected for 's' option 0360 x(:,1)=0; % Eliminate references to DC input in forward transform only 0361 end 0362 if ww(14) % 'q': we need an extra output that replicates the DC component 0363 if ww(12) % 'z': DC input is an impulse 0364 x=[sparse(1,1,1,1,nf); x]; 0365 xi=[sparse(1,1,1,nf,1) xi]; 0366 elseif ww(13) % 'Z': DC input is an impulse plus a diffuse component 0367 x=[sparse([1 1],[1 2],[1 -ginsd(2,2)],1,nf); x]; % impulse component is DC input minus adjacent bin corrected for 's' option 0368 xi=[sparse(1,1,1,nf,1) xi]; 0369 else 0370 x=[sparse(1,nf); x]; % '': DC input is diffuse as normal 0371 xi=[sparse(nf,1) xi]; 0372 end 0373 end 0374 % 0375 % sort out the output argument options 0376 % 0377 if ~ww(5) % output cf in Hz instead of mel/erb/... 0378 cf=[zeros(1,ww(14)) mb(2:pmq+1)]; % ... and include an initial 0 if 'q' option (ww(14)==1) 0379 else % keep cf in mel/erb/... 0380 if ww(14) % 'q' (ww(14)==1): we need an extra output for the DC component 0381 if wr==1 % log-scaled so ... 0382 cf=[-Inf cf(2:p)]; % ... DC corresponds to -Inf 0383 else % not log-scaled ... 0384 cf=[0 cf(2:p)]; % ... DC corresponds to 0 0385 end 0386 else % no 'q' option (ww(14)==0) ... 0387 cf=cf(2:p+1); % ... just remove dummy end frequencies 0388 end 0389 end 0390 if ww(1) % round outputs to the centre of gravity bin 0391 sx2=sum(x,2); % sum of each row 0392 msk=full(sx2~=0); 0393 vxc=zeros(pmq,1); 0394 vxc(msk)=round((x(msk,:)*(1:nf)')./sx2(msk)); % find centre of gravity of each row 0395 x=sparse(1:pmq,vxc,sx2,pmq,nf); % put all the weight into the centre of gravity bin 0396 end 0397 il=1; % default range is entire x maxtrix 0398 ih=nf; 0399 if nargout > 3 % if il and/or ih output arguments are specified ... 0400 if nargout==4 % xi has been omitted ... 0401 msk=full(any(x>0,1)); % find non-zero columns in x 0402 else % xi output included 0403 msk=full(any(x>0,1) | any(xi>0,2)'); % find non-zero columns in x or rows in xi 0404 end 0405 il=find(msk,1); % il is first non-zero column 0406 if ~numel(il) % if x is all zeros ... 0407 il=1; % ... set il and ih to 1 0408 ih=1; 0409 elseif nargout >3 0410 ih=find(msk,1,'last'); % ih is last non-zero column 0411 end 0412 x=x(:,il:ih); % remove redundant columns from x 0413 if nargout==4 % xi has been omitted ... 0414 xi=il; % shift the il and ih outputs up by one position 0415 il=ih; 0416 else 0417 xi=xi(il:ih,:); % remove redundant rows from xi 0418 end 0419 elseif ww(23) % ww(23)='w': use whole dft 0420 x=[x sparse(p,n-nf)]; % append zeros onto x 0421 xi=[xi; xi(n-nf+1:-1:2,:)]; % reflect elements other than the DC and Nyquist 0422 end 0423 % 0424 % plot results if no output arguments or 'g','G' options given 0425 % 0426 if ~nargout || ww(21) || ww(22) % plot idealized filters 0427 ww(22)=~ww(21); % 'G' option is the default unless 'g' is specified 0428 finax=(il-1:ih-1)*df; % input frequency axis 0429 newfig=0; 0430 if ww(21) 0431 plot(finax,x(:,il:ih)'); 0432 hold on 0433 plot(finax,sum(x,1),'--k'); 0434 v_axisenlarge([-1 -1.05]); 0435 plot(repmat(mb(2:end-1),2,1),get(gca,'ylim'),':k'); 0436 hold off 0437 title(['filtbankm: mode = ''' w '''']); 0438 xlabel(['Frequency (' v_xticksi 'Hz)']); 0439 ylabel('Weight'); 0440 newfig=1; 0441 end 0442 if ww(22) 0443 if newfig 0444 figure; 0445 end 0446 imagesc(finax,1:pmq,x); 0447 axis 'xy' 0448 colorbar; 0449 hold on 0450 ylim=get(gca,'ylim'); 0451 plot(repmat(mb(2:end-1),2,1),ylim,':w'); 0452 hold off 0453 v_cblabel('Weight'); 0454 switch wr 0455 case 1 0456 type='Log-spaced'; 0457 case 2 0458 type='Erb-spaced'; 0459 case 3 0460 type='Bark-spaced'; 0461 case 4 0462 type='Mel-spaced'; 0463 otherwise 0464 type='Linear-spaced'; 0465 end 0466 ylabel([type ' Filter']); 0467 xlabel(['Frequency (' v_xticksi 'Hz)']); 0468 title(['filtbankm: mode = ''' w '''']); 0469 end 0470 0471 end