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spendred.m
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function [enhanced_speech] = spendred(input_speech,fs,algo_params)
% SPENDRED Speech Enhancement and Dereverberation by Doire
%
% Usage : (1) [enhanced_speech] = spendred(corrupted_speech,fs) % performs enhancement in order to denoise and dereverberate the input speech file
% (2) [enhanced_speech] = spendred(corrupted_speech,fs,algo_params) % performs enhancement using the custom parameters specified in 'algo_params'
%
%
% Inputs:
% input_speech Noisy and reverberant input speech signal (single-channel)
% fs Sample frequency in Hz
% algo_params algorithm parameters [optional]
%
%
% Outputs:
% enhanced_speech Enhanced output speech file
%
% Algorithm Parameters:
% The following parameters are defined in 'algo_params'. Default values are shown below.
%
% algo_params.of=6 % Overlap factor [6]
% algo_params.ti=0.005 % Desired frame increment in seconds [0.005]
% algo_params.ri=1 % Set to 1 to round ti to the nearest power of 2 samples [1]
% algo_params.sg=1 % Type of spectral subtraction gain to apply : 1=Wiener Gain, 2=Power Spectral Subtraction, 3=MMSE speech estimate [1]
% algo_params.sc=0.95 % Smoothing constant for computation of the spectral gain [0.95]
% algo_params.sf=1e-5; % Floor for the spectral gain [1e-5]
% algo_params.os=2; % Interference Over-subtraction factor [2]
% algo_params.cl=6; % Number of HMM states to use (minimum is 2 - maximum is 6) [6]
% algo_params.ds=1; % Way of computing posterior distributions : 1 = max track , 2 = weighted sum of tracks
% algo_params.mo=1; % Mode : 'fast' = 1 or 'slow' = 0 [1]
% algo_params.ef=-60; % Energy floor (dB) [-60]
%
%
% References:
%
% [1] C. S. J. Doire, D. M. Brookes, P. A. Naylor, C. M. Hicks, D. Betts, M. A. Dmour, and S. H. Jensen.
% Single-channel online enhancement of speech corrupted by reverberation and noise.
% IEEE Trans. Audio, Speech, Language Processing, 25 (3): 572–587, Mar. 2017. doi: 10.1109/TASLP.2016.2641904.
% [2] B. Cauchi et al.,
% Combination of MVDR beamforming and single-channel spectral processing for enhancing noisy and reverberant speech,
% EURASIP J. Adv. Signal Process., vol. 61, 2015, pp. 1–12.
%
% Author : Clement Doire
% clement.doire11@imperial.ac.uk
%
%
% Revision History:
%
% 0.5 - 16 Feb 2016 - First version, based on my PhD thesis and first draft of [1]
% 0.9 - 13 Mar 2016 - Several tweaks + improved overall performance
% 1.0 - 01 Jul 2016 - Base version used to generate results in the paper
% 1.1 - 19 Sep 2017 - Added algo_params.ef parameter to cope with zero input signals
% 1.2 - 22 Sep 2017 - Changed covariance matrix initialization to cope with zero input signals
%
% Comments:
% - By default the algorithm is in 'fast' mode, which fails in rare
% cases due to numerical errors. This can be overcome by using a Square Root implementation,
% which is done via the 'slow' mode option algo_params.mo=0.
%
% Copyright (C) Clement Doire 2016
% Version: $Id: spendred.m 10131 2017-09-22 18:34:04Z 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.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%% TO DO %%%%
% (1) Deal with other sampling freqs more gracefully
% (2) add paramters for mu_mmse and beta_mmse; see [2] pp 5 & 7 for further information
fs_ori = fs; % save original sample frequency
if (fs ~= 16000)
input_speech = resample(mean(input_speech,2),16000,fs);
fs = 16000;
end
%%%% %%%%
if numel(input_speech)>length(input_speech)
error('Input speech signal must be a vector, not a matrix !');
end
%Default algorithm constants
Csts.of=6; % Overlap factor = (fft length)/(frame increment) [6]
Csts.ti=5e-3; % Desired frame increment [0.005]
Csts.ri=1; % Round ni to the nearest power of 2 [1]
Csts.sg=1; % Type of spectral subtraction gain to apply : 1=Wiener Gain, 2=Power Spectral Subtraction, 3=MMSE speech estimate [1]
Csts.sc=0.95; % Smoothing constant for computation of the spectral gain [0.95]
Csts.sf=1e-5; % Floor for the spectral gain [1e-5]
Csts.os=2; % Interference Over-subtraction factor [2]
Csts.cl=6; % Number of HMM states to use (minimum is 2 - maximum is 6) [6]
Csts.ds=1; % Way of computing posterior distributions : 1 = max track , 2 = weighted sum of tracks
Csts.mo=1; % Mode : 'fast' = 1 or 'slow' = 0 [1]
Csts.ef=-60; % Energy floor (dB) [-60]
if nargin>=3 && ~isempty(algo_params)
qqn=fieldnames(Csts);
for i=1:length(qqn)
if isfield(algo_params,qqn{i})
Csts.(qqn{i})=algo_params.(qqn{i});
end
end
end
%-------------------------------------------------------------------------%
%-------------------------------------------------------------------------%
%load the dictionary
mStates = [-28.4861773669176,-19.9241782718819,-16.4426351907036,...
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covStates=zeros(25,25,6);
covStates(:,:,1)=[55.5505522382743,16.5334640886389,10.9178210580786,13.5974699822659,9.99267788404160,4.65546050187628,-2.81201462169877,-4.68388254847409,-1.74070984578874,...
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0.908686837256395,4.71834252437230,2.56658358594838,-1.04351516389379,-1.45339066300716;-5.55458166577810,-3.37797266880504,-3.54582614084532,-7.43863418366843,-5.95040326389395,7.51922776048866,26.4090813917774,36.4894811171374,45.6795508707621,55.9881212791662,43.6514426662486,9.93469796222488,-13.2832667690708,-15.2969484244819,-12.3359491619840,-10.3304662026549,-4.63756925753954,-2.46321798994156,-5.29194412619753,-8.92651482455280,-6.43200943191655,-1.75789464009488,-0.702979795889193,-2.39982181081864,-5.05836416758705;4.78286109500975,-1.60834775075760,-2.26821098324758,-4.56101523134668,-3.21645466374053,7.13633984234143,19.8222466292767,25.2700241602827,29.5252744555428,43.6514426662486,56.4879305696040,34.0017967352276,1.72485193969848,-8.59895358213287,-7.64272602475186,-9.67278314457806,-7.57048193635153,-1.76542823755722,-3.67359276000700,-11.5811131635602,-11.7605325347021,-7.56180044535635,-3.81790940876935,-3.11883524920612,-6.55269611082009;6.95110229329951,0.795631607365374,0.517292195308823,...
-0.216773407540774,0.141944034333193,3.24182864983849,7.15767944253354,7.43626844690649,5.16911980987520,9.93469796222488,34.0017967352276,53.2006551201314,37.5152715859299,15.4412487845563,7.48457869608750,1.69725306456590,-1.89147831827456,1.32414050480706,2.23816143063194,-4.01425787953618,-7.55094808541236,-6.64715100035760,-3.30034026108823,-1.40659213221705,-2.07722914612490;0.00879075987156321,...
2.08528253489662,3.14509536372528,3.89543459828660,3.38234829455764,-0.0603781670555945,-2.03667096553156,-4.35997810175267,-9.61007286765810,-13.2832667690708,1.72485193969848,37.5152715859299,59.7155797675980,44.7773950005295,23.6872126249230,15.0419986873052,10.2867275974593,7.44535689098688,8.92865721758249,8.15662452212477,2.92276952642247,-0.928795572640678,-0.214150561175240,1.35299628037947,4.39659721074060;-4.56967576746686,1.60079192483396,3.80829791379001,4.90389717333831,4.89012956399443,0.545855562396198,-2.64966773992657,-5.76338785540435,-10.8114296082780,-15.2969484244819,-8.59895358213287,15.4412487845563,44.7773950005295,56.5751628200309,37.8203840732640,20.3064153793782,17.7523505046127,13.7943682884086,10.9358068943459,11.9597833381183,8.15858880439622,2.20692296620118,0.925687564792643,1.80648630647397,5.51936228519465;4.81515552944994,1.51828664833571,3.32607430412317,4.66127606211765,5.55534520851414,4.27928625063530,0.0770085705865255,-5.10636014339218,-9.75868776317025,-12.3359491619840,-7.64272602475186,7.48457869608750,23.6872126249230,37.8203840732640,50.5567811262763,36.0267512290389,21.5229007099950,21.6212508611315,17.6388359925817,11.9335338722673,8.21479732534940,4.56458312200866,4.33009765711244,5.27098146930465,4.86850728556293;2.62036551470727,1.54391683150779,2.91471516655482,3.47030457098709,3.63735667703317,2.85871315997680,1.07303373835947,-1.97130033648324,...
-6.02803080806971,-10.3304662026549,-9.67278314457806,1.69725306456590,15.0419986873052,20.3064153793782,36.0267512290389,52.6258535141836,37.6429273778615,26.0221565226345,25.2698743276596,20.1154473169106,13.2528440139144,9.04767643570268,8.59623530087994,9.90654514905407,9.81591972530337;-7.29029260769826,0.525534472247695,3.34912935358682,2.74033891407396,3.18406810374213,0.566656257357205,0.291056169731772,0.480151706326002,-0.824738162216459,-4.63756925753954,-7.57048193635153,-1.89147831827456,10.2867275974593,17.7523505046127,21.5229007099950,37.6429273778615,50.9566054526190,36.6093908509986,23.8228848504800,20.5158358369382,12.8996919192979,6.80957915648636,5.82313196344093,6.76698619720571,9.62737791259404;7.60994352313745,1.55170824084267,5.20923042016220,5.72211439376269,7.04176943959671,4.21827660401795,-1.76758780042540,-4.79454189609896,-3.90437152964992,-2.46321798994156,-1.76542823755722,1.32414050480706,7.44535689098688,13.7943682884086,21.6212508611315,26.0221565226345,36.6093908509986,...
52.7467507553735,37.9169199894629,17.3357061745695,7.02185159858676,-0.331036876801312,1.94789829479953,5.02766136665644,4.41374687696034;2.47911656495987,1.33105278816537,4.29738386466969,5.28391368902764,6.21288060544659,3.73962787068916,-1.13793473179759,-4.15180389292841,-5.06206747182193,-5.29194412619753,-3.67359276000700,2.23816143063194,8.92865721758249,10.9358068943459,17.6388359925817,25.2698743276596,23.8228848504800,37.9169199894629,53.8401909646114,38.6507723240261,17.8182427670904,4.88430658725573,1.35573251610327,5.62258771132742,7.23177091210702;-19.0672555520342,-0.281924859658859,2.73025296949213,2.25114917243912,2.52251449794559,-0.753303064804990,-0.106196239349976,0.737869884021180,-2.73036755479272,-8.92651482455280,-11.5811131635602,-4.01425787953618,8.15662452212477,11.9597833381183,11.9335338722673,20.1154473169106,20.5158358369382,17.3357061745695,38.6507723240261,69.0407953227603,50.4721216646804,25.0197633538726,8.68426069638420,8.07850649374198,14.8126523282230;-20.5734048413298,...
-0.573822155925386,1.58166856773255,-0.628514051806710,-0.952991719117629,-1.24908791135452,3.60492394898904,5.24783031692334,0.908686837256395,-6.43200943191655,-11.7605325347021,-7.55094808541236,2.92276952642247,8.15858880439622,8.21479732534940,13.2528440139144,12.8996919192979,7.02185159858676,17.8182427670904,50.4721216646804,75.9968226620634,53.3344079165803,21.0792722143554,10.1835442239764,...
15.5032957956173;-13.2075273562642,1.46664945053241,3.14843379754112,0.228684995216133,-1.84336163229673,-1.60150240414188,4.86921724878281,7.66766914332693,4.71834252437230,-1.75789464009488,-7.56180044535635,-6.64715100035760,-0.928795572640678,2.20692296620118,4.56458312200866,9.04767643570268,6.80957915648636,-0.331036876801312,4.88430658725573,25.0197633538726,53.3344079165803,82.0329350072126,53.0033369734705,21.1505642732596,16.5655771407877;5.91579472308194,5.10400561369775,5.84609591051610,4.69411991158734,2.16951897880099,0.662552139993113,1.46895532474066,2.78565563257161,2.56658358594838,-0.702979795889193,-3.81790940876935,-3.30034026108823,-0.214150561175240,0.925687564792643,4.33009765711244,8.59623530087994,5.82313196344093,1.94789829479953,1.35573251610327,8.68426069638420,21.0792722143554,53.0033369734705,79.5401368685907,49.0744923891711,24.5660118766340;12.5261436233794,7.19978550743818,7.77238354115449,7.23959680984727,5.07738907260155,1.28530643124126,-1.81855900912568,-1.86635253464110,-1.04351516389379,-2.39982181081864,-3.11883524920612,-1.40659213221705,1.35299628037947,1.80648630647397,5.27098146930465,9.90654514905407,6.76698619720571,5.02766136665644,5.62258771132742,8.07850649374198,10.1835442239764,21.1505642732596,49.0744923891711,69.0043607615904,46.0670968831061;0.396845671600897,6.73941584361868,8.43717665704851,7.04380122502727,4.81286211137161,-0.916697712307544,...
-3.24303790466327,-1.48161111587469,-1.45339066300716,-5.05836416758705,-6.55269611082009,-2.07722914612490,4.39659721074060,5.51936228519465,4.86850728556293,9.81591972530337,9.62737791259404,4.41374687696034,7.23177091210702,14.8126523282230,15.5032957956173,16.5655771407877,24.5660118766340,46.0670968831061,63.1772269464555];
%load transition probabilities
trans_probs = [0.500000000000000,0.0953909984666727,0.0936862992694146,0.104581942815911,0.0849463335437900,0.121394425904212;...
0.102287865624095,0.500000000000000,0.0917752678830003,0.102558161984748,0.0842648904334395,0.119113814074718;0.101311261421621,...
0.0948241762588505,0.500000000000000,0.101766148490962,0.0846386614453542,0.117459752383213;0.103803845526540,0.0960747358692304,...
0.0942443957159167,0.500000000000000,0.0871948677986988,0.118682155089614;0.100744217514483,0.0920229185166392,0.0896338111548057,...
0.0993593809463402,0.500000000000000,0.118239671867732;0.109138242310916,0.0989022503062738,0.0960571281191566,0.106954026694178,0.0889483525694758,0.500000000000000];
%load observation noise data
kappa =[0.327973912574545;0.303909349115727;0.282744198518673;0.257369355096232;0.239081601404447;0.217386355882733;0.199072758206112;0.183132887880788;0.165871243388662;...
0.150781712435529;0.137445214975176;0.126283512235535;0.114402394802476;0.104466246018628;0.0947204546876257;0.0862050943518338;0.0782510877463125;0.0707492641810876;...
0.0644826158612981;0.0585904141685921;0.0529404263834298;0.0480940839999611;0.0436761803436944;0.0398349470828149;0.0359906621644362];
kappa_s = ((10./log(10)).^2).*(kappa+0.1);
kappa = repmat(kappa_s,1,(Csts.cl)^2);
%-------------------------------------------------------------------------%
%-------------------------------------------------------------------------%
%take care of the HMM states
switch Csts.cl
case 6
%do nothing
change_tp = 0;
case 5
mStates(:,6)=[];
covStates(:,:,6)=[];
trans_probs(6,:) = [];trans_probs(:,6) = [];
change_tp = 1;
case 4
mStates(:,6)=[];
covStates(:,:,6)=[];
trans_probs(6,:) = [];trans_probs(:,6) = [];
mStates(:,1)=[];
covStates(:,:,1)=[];
trans_probs(1,:) = [];trans_probs(:,1) = [];
change_tp = 1;
case 3
mStates(:,6)=[];
covStates(:,:,6)=[];
trans_probs(6,:) = [];trans_probs(:,6) = [];
mStates(:,1)=[];
covStates(:,:,1)=[];
trans_probs(1,:) = [];trans_probs(:,1) = [];
mStates(:,2)=[];
covStates(:,:,2)=[];
trans_probs(2,:) = [];trans_probs(:,2) = [];
change_tp = 1;
case 2
mStates(:,6)=[];
covStates(:,:,6)=[];
trans_probs(6,:) = [];trans_probs(:,6) = [];
mStates(:,1)=[];
covStates(:,:,1)=[];
trans_probs(1,:) = [];trans_probs(:,1) = [];
mStates(:,2)=[];
covStates(:,:,2)=[];
trans_probs(2,:) = [];trans_probs(:,2) = [];
mStates(:,2)=[];
covStates(:,:,2)=[];
trans_probs(2,:) = [];trans_probs(:,2) = [];
change_tp = 1;
otherwise
error('The number of states you have selected is not permitted !');
end
if change_tp ==1
for i=1:Csts.cl
trans_probs(i,i)=0.5;
trans_probs(i,union(1:i-1,i+1:Csts.cl)) = 0.5*trans_probs(i,union(1:i-1,i+1:Csts.cl))./sum(trans_probs(i,union(1:i-1,i+1:Csts.cl)));
end
end
%-------------------------------------------------------------------------%
%Prepare the STFT
if Csts.ri
ni=pow2(nextpow2(Csts.ti*fs*sqrt(0.5)));
else
ni=round(Csts.ti*fs); % frame increment in samples
end
tinc=ni/fs; % true frame increment in time
no=round(Csts.of); % integer overlap factor
nf=ni*no; % fft length (size of analysis window)
w_synth=hann(nf+1)'; w_synth(end)=[]; % hamming window for synthesis
w=w_synth/sqrt(sum(w_synth(1:ni:nf).^2));
%-------------------------------------------------------------------------%
%Mel scale transformation matrices
[ThO_Tr,~]=filtbankm(25,nf,fs,[],[],'m'); %forward transformation matrix)
reco_mat = interpofiltbankm(25,nf,fs); %inverse transformation matrix (interpolation of the gain from mel bands to full STFT spectrum - see function below)
%-------------------------------------------------------------------------%
[DFTc,~]=enframe(input_speech,w,ni,'z'); %zero-padding the end of the signal to match the number of frames
C=rfft(DFTc,nf,2);
[nrows,ncols] = size(C);
gt_YP_full=(C.'.*conj(C.')./(ncols*sum(w.^2)));
gt_YP = 10.*log10(ThO_Tr*gt_YP_full);
gt_YP = max(gt_YP,max(gt_YP(:))+Csts.ef); % clip to 60 dB range avoid negative infinities
Energy = 10.^(gt_YP./10);
%-------------------------------------------------------------------------%
%various initialisations
K = Csts.cl; %Number of states in our HMM speech model
nfc = size(gt_YP,1);
nb_frames = size(gt_YP,2);
if Csts.mo ==0
%UDU decomposition of the covariance of each state distribution
U_state = zeros(nfc,nfc,K);
D_state = zeros(nfc,nfc,K);
for i=1:K
[tmpU,tmpD] = udu(covStates(:,:,i));
U_state(:,:,i) = tmpU;
D_state(:,:,i) = tmpD;
end
end
%init clean speech posterior mean and covariance with priors
M_speech = mStates;
if Csts.mo ==0
U_speech = U_state;
D_speech = D_state;
else
Cov_speech = covStates;
end
%probs of each path
probs = log((1/(K)).*ones(K,1)); %initialise the probabilities
%our state space representation --> [gain;reverb_power_in_subbands;noise_power_in_subbands]
X = zeros(2*nfc+1,K);
X(1,:) = -12.*ones([1,K]);
X(2:nfc+1,:) = repmat((mean(gt_YP(:,1:12),2)),[1,K]); %initialise reverberation energy to the mean observed power of the first frames (same as noise)
X(nfc+2:2*nfc+1,:) = repmat(mean(gt_YP(:,1:12),2),[1,K]);%initialise noise level to the mean of the 1st few frames
%initialize the reverb parameters
alpha = 10.^((-6.*tinc)./[0.625814328889286,0.635814328889286,0.559669908460684,0.533597692539940,0.523597692539940,0.519657090507566,...
0.498220803534608,0.488220803534608,0.470970193330601,0.460004032902958,0.445158336751369,0.436812074651546,...
0.400699712568631,0.395830867067534,0.399005310050930,0.401204573100827,0.417815999144577,...
0.417892069024374,0.419406546706839,0.422573783237992,0.417601620276530,0.408528394010926,...
0.403943250672072,0.354372247720777,0.316876109784076])'; %average subbands T60 values from measurements on a variety of RIRs (inaccurate at low freqs)
f = (1-alpha)./(10.^(linspace(-0.2,0.8,25)')); %corresponds to DRR = [-2dB --> 8dB]
Rp = [10.*log10(alpha./(1-alpha))+1.5;10.*log10(f./(1-f))];
%Initialisation of some arrays used in the main body of the function
X_rev = X(:,1);
Speech_rev = mStates(:,1);
%Set up the time update covariance matrices
Qx=[1/350000,zeros([1,2*nfc]);...
zeros([nfc,1]),diag(repmat(1/1500,[nfc,1])),zeros(nfc);...
zeros([nfc,nfc+1]),diag(repmat(1/7550,[nfc,1]))];
Qr=[diag(repmat(1/1700,[nfc,1])),zeros(nfc);...
zeros(nfc),diag(repmat(1/700,[nfc,1]))];
Qx = 15.*Qx + 1e-5*eye(2*nfc+1)*trace(Qx);
Qr = 15.*Qr + 1e-5*eye(2*nfc)*trace(Qr);
Covx = Qx.*1.5;
Covr = Qr.*15;
%Initialise the noise covariance matrix with the observation from the first few frames
% Covx(nfc+2:end,nfc+2:end) = cov(gt_YP(:,1:10).');
Covx(nfc+2:end,nfc+2:end) = Covx(nfc+2:end,nfc+2:end)+cov(gt_YP(:,1:10).'); % modified to prevent errors when initial frames are silent
if Csts.mo == 0 %UDU decomposition of the state space covariance matrix (SR-EKF implementation)
Uq = eye(2*nfc+1);
[tU,tD] = udu(Covx);
Up = repmat(tU,[1,1,K]);
Dp = repmat(tD,[1,1,K]);
Um = zeros(2*nfc+1,2*nfc+1,K);
Dm = Um;
else
Cov = cell(K,1);
for i=1:K
Cov{i} = Covx;
end
end
%matrices used in the computation of the prediction and update stages (vectorisation purposes)
pdm = [zeros(nfc,1),eye(nfc),zeros(nfc),eye(nfc),zeros(nfc,2*nfc);...
zeros(nfc,2*nfc+1),eye(nfc),zeros(nfc,2*nfc);...
ones(nfc,1),zeros(nfc,3*nfc),eye(nfc),eye(nfc);...
zeros(nfc,3*nfc+1),eye(nfc),zeros(nfc)];
pdm2 = [zeros(nfc,1);ones(nfc,1);zeros(nfc,1);ones(nfc,1)];
pdm2_aug = repmat(pdm2,1,K);
pdm3 = [ones(nfc,1),zeros(nfc,2*nfc),eye(nfc);...
zeros(nfc,1),eye(nfc),zeros(nfc,2*nfc);...
zeros(nfc,nfc+1),eye(nfc),zeros(nfc)];
stat_mat = reshape(repmat(mStates,K,1),nfc,K^2);
%Declaration of Jacobians
Fx = eye(2*nfc+1);
Fu = zeros(2*nfc+1,nfc);
if Csts.mo ==0
Hx = zeros(nfc,2*nfc+1);
else
Hx = cell(K,K);
Hu = cell(K,K);
end
%Initialise Gain
SpecGain = zeros(ncols,nb_frames+1); SpecGain(:,1) = 0.00001.*ones(ncols,1);
%%%%% extra stuff we need if we're after MMSE estimator of clean speech %%%%%
if Csts.sg == 3
Xi = 0.00001.*ones(nfc,1);
p0 = 0.5;
pInf = 1;
mu_mmse = 0.5; % shape parameter of the chi distribution for clean speech magnitude
beta_mmse = 0.5; % compression factor
gammaFactor = (gamma(mu_mmse+beta_mmse/2)./gamma(mu_mmse));
end
%-------------------------------------------------------------------------%
%Main loop, frame by frame processing
for idx=2:nb_frames+1
%%%%% first do the prediction stage for each track %%%%%
if Csts.mo ==0
tmpX = X;
for k = 1 : K %for each track
tmpaug = [X(:,k);Rp;M_speech(:,k)]; %create the augmented state
tmp = reshape(exp((pdm*tmpaug).*0.2302585093)+pdm2,[nfc,4]); %0.2302585093 = log(10)/10
tmp_sum = tmp(:,1)./tmp(:,2) + tmp(:,3)./tmp(:,4);
%compute the reverb power part of the output state
tmpX(2:nfc+1,k) = 10.*log10(tmp_sum);
%compute the jacobians
Fx(2:nfc+1,1:nfc+1) = [(tmp(:,3)./tmp(:,4))./tmp_sum,diag((tmp(:,1)./tmp(:,2))./tmp_sum)];
Fu(2:nfc+1,:) = diag(Fx(2:nfc+1,1));
%compute the covariance matrix (SR-EKF)
[Um(:,:,k),Dm(:,:,k)] = mwgs_factor([Uq,Fx*Up(:,:,k),Fu*U_speech(:,:,k)],diag([diag(Qx);diag(Dp(:,:,k));diag(D_speech(:,:,k))])); %Covariance matrix of the prediction stage in UDU form
end
else
tmpaug = [X;repmat(Rp,1,K);M_speech];%create the augmented state
tmp = exp((pdm*tmpaug).*0.2302585093)+pdm2_aug;
tmp_sum1 = tmp(1:nfc,:)./tmp(nfc+1:2*nfc,:);
tmp_sum2 = tmp(2*nfc+1:3*nfc,:)./tmp(3*nfc+1:end,:);
tmp_sum = tmp_sum1 + tmp_sum2;
%compute the reverb power part of the output state
X(2:nfc+1,:) = 10.*log10(tmp_sum);
%Jacobian comp
Ftemp = [tmp_sum2./tmp_sum;tmp_sum1./tmp_sum];
for k = 1 : K %for each track
%compute the jacobians
Fx(2:nfc+1,1:nfc+1) = [Ftemp(1:nfc,k),diag(Ftemp(nfc+1:end,k))];
Fu(2:nfc+1,:) = diag(Ftemp(1:nfc,k));
%compute the covariance matrix
Cov{k} = Fu*Cov_speech(:,:,i)*Fu' + Fx*Cov{k}*Fx' + Qx;
end
end
new_probs = repmat(probs,1,K) + trans_probs;
%%%%% now the update stage with K^2 possibilities %%%%%
%some init of various results we need to store
err = zeros(nfc,K,K);
Sk = zeros(nfc,nfc,K,K);
lkl = zeros(K,K);
if Csts.mo ==0
vx = zeros(2*nfc+1,nfc,K,K);
vu = zeros(nfc,nfc,K,K);
for k1 = 1:K
for k2 = 1:K
%get mean of the augmented state
m_tilde = [tmpX(:,k1);mStates(:,k2)];
%compute jacobians
tmp2 = reshape(exp((pdm3*m_tilde).*0.2302585093),[nfc,3]);
tmp_sum2 = sum(tmp2,2);
Hx(:,1) = tmp2(:,1)./tmp_sum2;
Hx(:,2:nfc+1) = diag(tmp2(:,2)./tmp_sum2);
Hx(:,nfc+2:2*nfc+1) = diag(tmp2(:,3)./tmp_sum2);
Hu = diag(Hx(:,1));
%compute predicted output and error
zk = 10.*log10(tmp_sum2);
err(:,k1,k2) = gt_YP(:,idx-1)-zk; %error between update stage and observed log-power
R = diag(kappa_s+((10/log(10))^2).*log(1+2.*(tmp2(:,1).*tmp2(:,2)+tmp2(:,1).*tmp2(:,3)+tmp2(:,2).*tmp2(:,3))./tmp_sum2));
%marginal on the output
vx(:,:,k1,k2) = (Um(:,:,k1)')*Hx';
vu(:,:,k1,k2) = (U_state(:,:,k2)')*Hu';
[Usk,Dsk] = mwgs_factor([eye(nfc),vx(:,:,k1,k2)',vu(:,:,k1,k2)'],diag([diag(R);diag(Dm(:,:,k1));diag(D_state(:,:,k2))])); %Covariance matrix of the prediction stage in UDU form
Sk(:,:,k1,k2) = Usk*Dsk*Usk'; %covariance matrix of the observation
%compute the likelihood
lkl(k1,k2) = gaussmixp(gt_YP(:,idx-1)',zk',Sk(:,:,k1,k2)); %gaussmixp returns log-probability
end
end
else
%get mean of the augmented state
m_tilde = [repmat(X,1,K);stat_mat];
tmp2 = exp((pdm3*m_tilde).*0.2302585093);
tmp2_sum2 = tmp2(1:nfc,:) + tmp2(nfc+1:2*nfc,:) + tmp2(2*nfc+1:end,:);
%predicted output
zk = 10.*log10(tmp2_sum2);
%observation noise
R = kappa+((10/log(10))^2).*2.*(tmp2(1:nfc,:).*tmp2(nfc+1:2*nfc,:)+tmp2(1:nfc,:).*tmp2(2*nfc+1:end,:)+tmp2(nfc+1:2*nfc,:).*tmp2(2*nfc+1:end,:))./tmp2_sum2;
%Jacob comp
Htemp = tmp2./repmat(tmp2_sum2,3,1);
for k1 = 1:K
for k2 = 1:K
%compute jacobians
Hx{k1,k2} = [Htemp(1:nfc,K*(k2-1)+k1),diag(Htemp(nfc+1:2*nfc,K*(k2-1)+k1)),diag(Htemp(2*nfc+1:end,K*(k2-1)+k1))];
Hu{k1,k2} = diag(Htemp(1:nfc,K*(k2-1)+k1));
%compute predicted error
err(:,k1,k2) = gt_YP(:,idx-1)-zk(:,K*(k2-1)+k1); %error between update stage and observed log-power
%marginal on the output
Sk(:,:,k1,k2) = Hx{k1,k2}*Cov{k1}*Hx{k1,k2}' + Hu{k1,k2}*covStates(:,:,k2)*Hu{k1,k2}' + diag(R(:,K*(k2-1)+k1));
%compute the likelihood
try
lkl(k1,k2) = gaussmixp(gt_YP(:,idx-1)',zk(:,K*(k2-1)+k1)',Sk(:,:,k1,k2)); %gaussmixp returns log-probability
catch
error('Covariance matrix not positive definite - To avoid this, try a higher energy floor (e.g. algo_params.ef=-50) or running the algorithm in slow mode using algo_params.mo = 0');
end
end
end
end
joint_lkl = new_probs + lkl;
%now pick the best track arriving at each HMM state and rearrange
[max_val,max_idx] = max(joint_lkl);
probs = max_val';
%Compute the posterior densities for the best tracks only
if Csts.mo == 0
for k = 1 : K
%Compute posterior means
K_n = (Um(:,:,max_idx(k))*Dm(:,:,max_idx(k))*vx(:,:,max_idx(k),k))/Sk(:,:,max_idx(k),k); %Kalman Gain for state space (Square root EKF)
K_n_u = (U_state(:,:,k)*D_state(:,:,k)*vu(:,:,max_idx(k),k))/Sk(:,:,max_idx(k),k); %Kalman Gain for clean speech (Square root EKF)
X(:,k) = tmpX(:,max_idx(k)) + K_n*err(:,max_idx(k),k);
M_speech(:,k) = mStates(:,k) + K_n_u*err(:,max_idx(k),k);
%Compute posterior covariances
[B,Dp(:,:,k)] = udu(Dm(:,:,max_idx(k)) - (((Dm(:,:,max_idx(k))*vx(:,:,max_idx(k),k))/Sk(:,:,max_idx(k),k))*(vx(:,:,max_idx(k),k).')*Dm(:,:,max_idx(k)))); %UDU update posterior covariance matrix state space (SR-EKF)
[Bs,D_speech(:,:,k)] = udu(D_state(:,:,k) - (((D_state(:,:,k)*vu(:,:,max_idx(k),k))/Sk(:,:,max_idx(k),k))*(vu(:,:,max_idx(k),k).')*D_state(:,:,k))); %UDU update posterior covariance matrix clean speech (SR-EKF)
Up(:,:,k) = Um(:,:,max_idx(k))*B;
U_speech(:,:,k) = U_state(:,:,k)*Bs;
end
else
Cov_back = Cov;
for k = 1 : K
%Compute posterior means
K_n = (Cov{max_idx(k)}*Hx{max_idx(k),k}')/Sk(:,:,max_idx(k),k); %Kalman Gain for state space (Square root EKF)
K_n_u = (covStates(:,:,k)*Hu{max_idx(k),k}')/Sk(:,:,max_idx(k),k); %Kalman Gain for clean speech (Square root EKF)
X(:,k) = X(:,max_idx(k)) + K_n*err(:,max_idx(k),k);
M_speech(:,k) = mStates(:,k) + K_n_u*err(:,max_idx(k),k);
%Compute posterior covariances
Cov{k} = Cov_back{max_idx(k)} - K_n*Sk(:,:,max_idx(k),k)*K_n';
Cov_speech(:,:,k) = covStates(:,:,k) - K_n_u*Sk(:,:,max_idx(k),k)*K_n_u';
end
end
%get the weighted_sum state space
weights = exp(max_val-max(max_val))';
weights= weights/sum(weights);
Weighted_X = sum(reshape(X,[2*nfc+1,K]).*repmat(weights',2*nfc+1,1),2);
Weighted_Speech = sum(M_speech.*repmat(weights',nfc,1),2);
if Csts.ds == 2 %If the user decides to do the enhancement using the weighted sum of the densities
Weighted_cov = 0;
Weighted_cov_speech = 0;
if Csts.mo ==0
for k=1:K
Weighted_cov = Weighted_cov + weights(k).*...
(Up(:,:,k)*Dp(:,:,k)*(Up(:,:,k)')) + weights(k).*((X(:,k)-...
Weighted_X)*(X(:,k)-Weighted_X)');
Weighted_cov_speech = Weighted_cov_speech + ...
weights(k).*(U_speech(:,:,k)*D_speech(:,:,k)*(U_speech(:,:,k)')) + ...
weights(k).*((M_speech(:,k)-Weighted_Speech)*(M_speech(:,k)-Weighted_Speech)');
end
else
for k=1:K
Weighted_cov = Weighted_cov + weights(k).*...
Cov{k} + weights(k).*((X(:,k)-...
Weighted_X)*(X(:,k)-Weighted_X)');
Weighted_cov_speech = Weighted_cov_speech + ...
weights(k).*Cov_speech(:,:,k) + ...
weights(k).*((M_speech(:,k)-Weighted_Speech)*(M_speech(:,k)-Weighted_Speech)');
end
end
end
%%%%% Update the reverb parameters estimate %%%%%
%model prediction stage : no movement/random walk
Covr = Covr + Qr;
%update stage : the observation is the new posterior on reverb power
tmpaugpr = [X_rev;Rp;Speech_rev];
tmppr = reshape(exp((pdm*tmpaugpr).*0.2302585093)+pdm2,[nfc,4]); %0.2302585093 = log(10)/10
tmp_sumpr = tmppr(:,1)./tmppr(:,2) + tmppr(:,3)./tmppr(:,4);
outrev = 10.*log10(tmp_sumpr);
Gx = [diag((tmppr(:,1)./(tmppr(:,2).^2))./tmp_sumpr),diag((tmppr(:,3)./(tmppr(:,4).^2))./tmp_sumpr)];
pred_err = Weighted_X(2:nfc+1)-outrev;
RevSk = Gx*Covr*Gx' + trace(Covr).*eye(nfc)./5; %add some artificial observation noise as our model is approximate and to avoid big jumps in the reverb parameters as a result
RevK = (Covr*(Gx'))/RevSk; %Kalman Gain
Rp = Rp + RevK*pred_err;
Covr = Covr - RevK*RevSk*(RevK');
%prepare for next time frame
X_rev = Weighted_X;
Speech_rev = Weighted_Speech;
%Transform instantaneous best powers for gain computation in the power domain (processing with minimum latency)
if Csts.ds == 1
[~,max_idx2] = max(max_val); %get the instantaneous best index
if Csts.mo ==0
GainRevNoise = exp((log(10)/10).*X(:,max_idx2) - 0.5.*diag(((log(10)/10)^2).*(Up(:,:,max_idx2)*...
Dp(:,:,max_idx2)*Up(:,:,max_idx2)')));
SpeechPost = GainRevNoise(1).*exp((log(10)/10).*M_speech(:,max_idx2) - 0.5.*diag(((log(10)/10)^2).*(U_speech(:,:,max_idx2)*...
D_speech(:,:,max_idx2)*U_speech(:,:,max_idx2)')));
else
GainRevNoise = exp((log(10)/10).*X(:,max_idx2) - 0.5.*diag(((log(10)/10)^2).*Cov{k}));
SpeechPost = GainRevNoise(1).*exp((log(10)/10).*M_speech(:,max_idx2) - 0.5.*diag(((log(10)/10)^2).*Cov_speech(:,:,k)));
end
else
GainRevNoise = exp((log(10)/10).*Weighted_X - 0.5.*diag(((log(10)/10)^2).*Weighted_cov));
SpeechPost = GainRevNoise(1).*exp((log(10)/10).*Weighted_Speech - 0.5.*diag(((log(10)/10)^2).*Weighted_cov_speech));
end
%compute the gain
switch Csts.sg
case 1
%Wiener gain
SpecGain(:,idx) = Csts.sc .* SpecGain(:,idx-1) + (1-Csts.sc) .* (reco_mat*(SpeechPost./(SpeechPost + Csts.os.*(GainRevNoise(2:nfc+1) + GainRevNoise(2+nfc:2*nfc+1)))));
case 2
%power spectral gain
SpecGain(:,idx) = Csts.sc .* SpecGain(:,idx-1) + (1-Csts.sc) .* sqrt(reco_mat*(SpeechPost./(SpeechPost + Csts.os.*(GainRevNoise(2:nfc+1) + GainRevNoise(2+nfc:2*nfc+1)))));
case 3
%mmse estimate of clean speech
Xi = Csts.sc .* Xi + (1-Csts.sc) .* (SpeechPost./(Csts.os.*(GainRevNoise(2:nfc+1) + GainRevNoise(2+nfc:2*nfc+1))));
Gamma_kl = Energy(:,idx-1)./(Csts.os.*(GainRevNoise(2:nfc+1) + GainRevNoise(2+nfc:2*nfc+1)));
nu_kl = (Gamma_kl.*Xi)./(mu_mmse+Xi);
aHat0 = sqrt(Xi./(mu_mmse+Xi)) .* (gammaFactor).^(1/beta_mmse) .* (1./sqrt(Gamma_kl));
SpecGain(:,idx) = reco_mat*((1./(1+nu_kl)).^p0 .* aHat0 + (nu_kl./(1+nu_kl)).^pInf .* Xi./(mu_mmse+Xi));
end
end
%-------------------------------------------------------------------------%
FinalGain = max(SpecGain(:,2:end),Csts.sf);
%-------------------------reconstruct audio signal -----------------------%
ze_post_sub=(irfft((C.*FinalGain.').',nf).').*repmat(w,nrows,1); % Inverse DFT and apply output window
enhanced_speech=zeros(ni*(nrows+no-1),no); % Allocate space for overlapped output speech
for i=1:no
nm=nf*(1+floor((nrows-i)/no)); % Number of samples in this set
enhanced_speech(1+(i-1)*ni:nm+(i-1)*ni,i)=reshape(ze_post_sub(i:no:nrows,:)',nm,1);
end
enhanced_speech=sum(enhanced_speech,2);
enhanced_speech=enhanced_speech(1:length(input_speech)); %make sure they're the same length
enhanced_speech = enhanced_speech.*10^((activlev(input_speech, fs, 'rd') - activlev(enhanced_speech, fs, 'rd'))/20); %normalise levels
if (fs_ori ~= 16000)
enhanced_speech = resample(enhanced_speech,fs_ori,16000); %put back to original sampling frequency
end
end
%-------------------------------------------------------------------------%
%-------------------------------------------------------------------------%
%-------------------------------------------------------------------------%
%-------------------------------------------------------------------------%
%%%%% Functions needed to compute the main body of code %%%%%
%-------------------------------------------------------------------------%
%-------------------------------------------------------------------------%
%-------------------------------------------------------------------------%
%-------------------------------------------------------------------------%
function [u,d] = mwgs_factor(W,D) %needed for slow mode
% UD factorization using the Modified Weighted Gram-Schmidt method
% Reference : Catherine Thornton, 'Triangular Covariance
% Factorizations for Kalman Filtering', PhD Thesis, 1976.
[n,m] = size(W);
[k1,k2] = size(D);
if ((n>m)||(m~=k1)||(k1~=k2))
error('!! Error !! - Check input matrices dimensions');
end
u = eye(n);
d = zeros(n);
w = W; %copy to work on this one instead
for j=n:-1:1
d(j,j) = w(j,:)*D*(w(j,:).');
for i=1:j-1
u(i,j) = (1/d(j,j)).*(w(i,:)*D*(w(j,:).'));
w(i,:) = w(i,:) - u(i,j).*w(j,:);
end
end
end
%-------------------------------------------------------------------------%
%%%%%% %%%%% %%%%% %%%%% %%%%% %%%%% %%%%% %%%%% %%%%%% %%%%% %%%%% %%%%% %%%%%
%-------------------------------------------------------------------------%
function [U,D]=udu(P) %needed for slow mode
% UDU Decomposition ---> P=U*D*U'
% Reference : Gerald J. Bierman, 'Factorization methods for discrete
% sequential estimation', Mathematics in science and engineering,
% Volume 128, 1977.
[~,n]=size(P);
for j=n:-1:2
D(j,j)=P(j,j);
alpha=1/D(j,j);
for k=1:1:j-1
beta=P(k,j);
U(k,j)=alpha*beta;
for i=1:1:k
P(i,k)=P(i,k)-beta*U(i,j);
end
end
end
D(1,1)=P(1,1);
for i=1:1:n
U(i,i)=1;
end
end
%-------------------------------------------------------------------------%
%%%%%% %%%%% %%%%% %%%%% %%%%% %%%%% %%%%% %%%%% %%%%%% %%%%% %%%%% %%%%% %%%%%
%-------------------------------------------------------------------------%
function [x]=interpofiltbankm(p,n,fs)
%INTERPOFILTBANKM determines interpolation matrix for a MEL to STFT bins transformation
%
% VERY heavily inspired by FILTBANKM from Mike Brookes' voicebox MATLAB toolbox
% (basically just changed the beginning to do the reverse operation,
% the rest of the code is pretty much the same)
%
% Inputs:
% p number of filters in filterbank or the filter spacing in k-mel/bark/erb [ceil(4.6*log10(fs))]
% n length of fft
% fs sample rate in Hz
%
% Outputs: x a matrix containing the filterbank amplitudes
% size(x)=[p,1+floor(n/2)]
%
w='m';
wr ='m';
fh=0.5*fs; % max freq is the nyquist
fl=0;
f1=0;
nf=1+floor(n/2); % number of input frequency bins
df=fs/n; % input frequency bin spacing
cf=f1+(0:nf)*df; % input frequency bins
mflh=[fl fh];
mflh=frq2mel(mflh); % convert frequency limits into mel
melrng=mflh*(-1:2:1)'; % mel/erb/... range
% fn2=floor(n/2); % bin index of highest positive frequency (Nyquist if n is even)
melinc=melrng/(p+1);
fin0 = mflh(1)+(0:p+1)*melinc; % centre frequencies in mel/erb/... including dummy ends
fin0(2:end)=max(fin0(2:end),0); % only the first point can be negative
fin0 = mel2frq(fin0);
cf = [cf(1)-df,cf];
p = length(cf) - 2;
mb = cf;
% first sort out 2-sided input frequencies
fin=fin0;
%fin(end+1)=fin(end)+df; % add on a dummy point at the high end
fin=[-fin(end:-1:2) fin];
nfin=length(fin); % length of extended input frequency list
nf = (nfin - 3)/2;
% now sort out the interleaving
fout=mb; % output frequencies in Hz
lowex=any(w=='y')~=any(w=='Y'); % extend to 0 Hz
highex=any(w=='y') && (fout(end-1)<fin(end)); % extend at high end
if lowex
fout=[0 0 fout(2:end)];
end
if highex
fout=[fout(1:end-1) fin(end) fin(end)];
end
mfout=length(fout);
if any(w=='u') || any(w=='U')
gout=fout(3:mfout)-fout(1:mfout-2);
gout=2*(gout+(gout==0)).^(-1); % Gain of output triangles
else
gout=ones(1,mfout-2);
end
if any(w=='u')
gin=ones(1,nfin-2);
else
gin=fin(3:nfin)-fin(1:nfin-2);
gin=2*(gin+(gin==0)).^(-1); % Gain of input triangles
end
msk=fin(2:end-1)==0;
if lowex
gin(msk)=2*gin(msk); % double DC input to preserve its power
end
foutin=[fout fin];
nfall=length(foutin);
wleft=[0 fout(2:mfout)-fout(1:mfout-1) 0 fin(2:nfin)-fin(1:nfin-1)]; % left width
wright=[wleft(2:end) 0]; % right width
ffact=[0 gout 0 0 gin(1:min(nf,nfin-nf-2)) zeros(1,max(nfin-2*nf-2,0)) gin(nfin-nf-1:nfin-2) 0]; % gain of triangle posts
% ffact(wleft+wright==0)=0; % disable null width triangles shouldn't need this if all frequencies are distinct
[fall,ifall]=sort(foutin);
jfall=zeros(1,nfall);
infall=1:nfall;
jfall(ifall)=infall; % unsort->sort index
ffact(ifall([1:max(jfall(1),jfall(mfout+1))-2 min(jfall(mfout),jfall(nfall))+2:nfall]))=0; % zap nodes that are much too small/big
nxto=cumsum(ifall<=mfout);
nxti=cumsum(ifall>mfout);
nxtr=min(nxti+1+mfout,nfall); % next input node to the right of each value (or nfall if none)
nxtr(ifall>mfout)=1+nxto(ifall>mfout); % next post to the right of opposite type (unsorted indexes)
nxtr=nxtr(jfall); % next post to the right of opposite type (converted to unsorted indices) or if none: nfall/(mfout+1)
% each triangle is "attached" to the node at its extreme right end
% the general result for integrating the product of two trapesiums with
% heights (a,b) and (c,d) over a width x is (ad+bc+2bd+2ac)*w/6
%
% integrate product of lower triangles
msk0=(ffact>0);
msk=msk0 & (ffact(nxtr)>0); % select appropriate triangle pairs (unsorted indices)
ix1=infall(msk); % unsorted indices of leftmost post of pair
jx1=nxtr(msk); % unsorted indices of rightmost post of pair
vfgx=foutin(ix1)-foutin(jx1-1); % length of right triangle to the left of the left post
yx=min(wleft(ix1),vfgx); % integration length
wx1=ffact(ix1).*ffact(jx1).*yx.*(wleft(ix1).*vfgx-yx.*(0.5*(wleft(ix1)+vfgx)-yx/3))./(wleft(ix1).*wleft(jx1)+(yx==0));
% integrate product of upper triangles
nxtu=max([nxtr(2:end)-1 0],1);
msk=msk0 & (ffact(nxtu)>0);
ix2=infall(msk); % unsorted indices of leftmost post of pair
jx2=nxtu(msk); % unsorted indices of rightmost post of pair
vfgx=foutin(ix2+1)-foutin(jx2); % length of left triangle to the right of the right post
yx=min(wright(ix2),vfgx); % integration length
yx(foutin(jx2+1)<foutin(ix2+1))=0; % zap invalid triangles
wx2=ffact(ix2).*ffact(jx2).*yx.^2.*((0.5*(wright(jx2)-vfgx)+yx/3))./(wright(ix2).*wright(jx2)+(yx==0));
% integrate lower triangle and upper triangle that ends to its right
nxtu=max(nxtr-1,1);
msk=msk0 & (ffact(nxtu)>0);
ix3=infall(msk); % unsorted indices of leftmost post of pair
jx3=nxtu(msk); % unsorted indices of rightmost post of pair
vfgx=foutin(ix3)-foutin(jx3); % length of upper triangle to the left of the lower post
yx=min(wleft(ix3),vfgx); % integration length
yx(foutin(jx3+1)<foutin(ix3))=0; % zap invalid triangles
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));
% integrate upper triangle and lower triangle that starts to its right
nxtu=[nxtr(2:end) 1];
msk=msk0 & (ffact(nxtu)>0);
ix4=infall(msk); % unsorted indices of leftmost post of pair
jx4=nxtu(msk); % unsorted indices of rightmost post of pair
vfgx=foutin(ix4+1)-foutin(jx4-1); % length of upper triangle to the left of the lower post
yx=min(wright(ix4),vfgx); % integration length
wx4=ffact(ix4).*ffact(jx4).*yx.^2.*(0.5*vfgx-yx/3)./(wright(ix4).*wleft(jx4)+(yx==0));
% now create the matrix
iox=sort([ix1 ix2 ix3 ix4;jx1 jx2 jx3 jx4]);
msk=iox(2,:)<=(nfall+mfout)/2;
iox(2,msk)=(nfall+mfout+1)-iox(2,msk); % convert negative frequencies to positive
if highex
iox(1,iox(1,:)==mfout-1)=mfout-2; % merge highest two output nodes
end
if lowex
iox(1,iox(1,:)==2)=3; % merge lowest two output nodes
end
x=sparse(iox(1,:)-1-lowex,max(iox(2,:)-nfall+nf+1,1),[wx1 wx2 wx3 wx4],p,nf);
end