IMPLEMENTING LOUDNESS MODELS IN MATLAB. J. Timoney, T. Lysaght Marc Schoenwiesner L. McManus

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1 IMPLEMENTING LOUDNESS MODELS IN MATLAB J. Timoey, T. Lysaght Marc Schoewieser L. McMaus Dept. of Computer Sciece Dept. of Zoology Dept. of Elec. Egieerig NUI Mayooth, Mayooth, Co. Uiversity of Leipzig, Leipzig, DIT, Dubli, Irelad Kildare, Irelad Germay ABSTRACT I the field of psychoacoustic aalysis the goal is to costruct a trasformatio that will map a time waveform ito a domai that best captures the respose of a huma perceivig soud. A key elemet of such trasformatios is the mappig betwee the soud itesity i decibels ad its actual perceived loudess. A umber of differet loudess models exist to achieve this mappig. This paper examies implemetatio strategies for some of the more well - kow models i the Matlab software eviromet. 1. INTRODUCTION The primary tool i the field of audio for the time - frequecy aalysis of soud is the Spectrogram. It is popular because it is computatioally fast ad its output is well uderstood. However, sice the 1990 s much work has bee carried out o the developmet of better tools for soud aalysis that make more efforts to take ito cosideratio its psychoacoustic properties. This has bee drive by the availability of the techology to fully implemet the results of psychoacoustic research that had bee published over the decades previously combied with the desire for sigificat advaces i the codig of speech ad audio sigals, the MP3 stadard beig a good example. Thus owadays, may algorithms desiged for speech ad audio processig will make referece to pyschoacoustic trasformatios. A importat limitatio of the spectrogram i this regard is maer i which the sigal itesity is displayed, geerally i Decibels SPL. While this provides a measure of objective soud itesity, it does ot properly capture the subjective impressio a soud creates o the listeer i terms of its loudess. To achieve this the sesitivity of the ear to the various soud levels of the frequecy compoets cotaied i the soud must be accouted for. This is the kid of iformatio cotaied i equal loudess curves for the huma ear [1]. These curves show that the ear is less sesitive to low frequecy souds, havig a maximum sesitivity i the regio of 3-4kHz. Employig these curves to modify the db SPL itesity display of the soud trasforms the itesity to the Pho scale, where differet frequecy compoets havig the same Pho value will have the same loudess but will have differet db SPL itesities. Oe disadvatage with the Pho scale is that it is ot directly proportioal to perceived loudess, ad thus a doublig of loudess value i Phos does ot mea a doublig of the soud loudess [2]. To this, the Soe scale was itroduced to provide a liear scale of loudess. The Soe scale ca be related to the pho scale by the equatio [3] L ( i) D i = ( D( i) 40 ) L( i) = 2, if D( i) 40 where ( i) ( i), if D i < 40 L is the perceived loudess of the critical bad i, ad D is the spread critical spectrum i terms of phos i bad i. The coversio from a time domai sigal to a represetatio that describes its loudess i terms of Soe is outlied i Figure 1 Speech waveform Time-Frequecy Decompositio ad Ear Respose Compesatio Figure 1: Block Diagram of loudess modelig procedure [4] Σ Total Loudess Specific Loudess There are various approaches to implemetig the differet stages of the Loudess model i Figure 1. The basic procedure is to first trasform the sigal ito the time-frequecy domai. The frequecy aalysis poits specified will have a relatio to the critical bad resolutio of the ear. Time ad Frequecy maskig may be accouted for ad compesatio carried out for compoets below the threshold of audibility. This stage is followed by a coversio from the itesity levels of each time-frequecy slice to specific loudess levels for each frequecy bad that are the summed to give the overall loudess for each time slice. I this paper, three implemetatio strategies are examied: (1) 1. A direct implemetatio based o a time -frequecy decompositio, a mappig from db SPL to Pho followed by a direct implemetatio of equatio (1). 2. Three implemetatios of Zwicker s model. 3. The Moore ad Glasberg Loudess model DAFX-1

2 The sources for some of the implemetatios discussed are speech quality measuremet strategies. Specifically, the time -frequecy decompositios ad loudess coversio from the EMBSD [3], PSQM [10] ad PEAQ [12] measures are ivestigated. 2.1 Calibratio 2. MODEL IMPLEMETATIONS I all implemetatios the first, ad possibly most crucial, stage is the calibratio of the iput sigal. I Matlab souds typically are read i from wav files ormalize the amplitude levels of the soud to lie betwee 1 ad 1. However, this will either reflect the true recordig or playback levels of the soud. The ampl itude of soud ca be scaled to give the soud a desired value of db SPL. Whe usig db SPL to set s soud level, a value for the referece level must be chose. For air, the referece level is usually chose as 20 micropascals [5]. If the actual db SPL used whe recordig the soud is ukow, i the case of speech, if it is at a ormal level, it is reasoable to assume a coversatioal level of betwee 65 ad 70 db SPL. Thus, to scale the sigal vector y to a level of 70dB i Matlab [6], SPLmeas=70; Pref = 20e-6; y_refscaled= (y./pref); RMS=sqrt(mea(y_refscaled.^2)); SPLmat=20*log10(RMS); % dbspl i matlab c=10^((splmeas-splmat)/20); ycal=c*(y_refscaled); If the HUTear toolbox is istalled, it is also possible to use the fuctio [7], ret=pascalize(y,70); 2.2 Direct Implemetatio of Loudess Represetatio The algorithm for the direct implemetatio is take from the EMBSD speech quality measure. The sigal is separated ito frames, each oe widowed with a Haig fuctio ad the power spectral desity obtaied. Each power spectrum is partitioed ito critical bads of width oe bark, with a upper frequecy limit of 3.4 khz. I [3] Schroeder s spreadig fuctio model is applied to iclude the effects of frequecy maskig across the critical bads. The loudess level of each critical bad i uits of pho is obtaied usig a set of equal-loudess cotours take from the literature ad db itesity values that lie i betwee the published cotours are iterpolated to get the correct loudess level [3]. These loudess levels are the coverted to soe usig equatio (1). NFFT=1024;NOVERLAP=0; Bf=1:18; [Yxx,f] = psd(ycal,nfft,fs,nfft,0); Yxx_scale=(2.*Yxx)./NFFT; [B_XX,bark]=bk_frq02(Bf,f,Yxx_scale); C_XX=spread_ew(Bf,B_XX); P_XX=dbtopho(C_XX); S_XX=phtos(P_XX); N_mbsd(l)=sum(S_XX); The Matlab programs are as give i [3]. However, it was foud that it was ecessary to make adjustmets to the program dbtopho.m. First of all, i the program code a file amed equal.mat is called as it holds the trascribed equal loudess cotours. However, the C program versio i the thesis also cotais the cotour values i a array, which ca be copied for use with Matlab. Furthermore, the lies below, which were foud to cause errors o occasio, j = 1; while T(i) >= eqlco(j,i) j = j + 1; if j == 16 fpritf(1,'error\') if j == 1 P_XX(i) = phos(1); ca be replaced with [I]=fid(T(i)<=equalco(:,i)); if mi(i)==1 P_XX(i)=phos(1); 2.3 Implemetatios based o Zwicker s model Possibly the most well-kow ad popular model of loudess is the oe proposed by Zwicker. It has formed part of a iteratioal stadard [8], ad has bee adopted as for use i a umber of ITU stadards o speech ad audio quality. However, differeces exist i the implemetatios Implemetatio of the DIN 45631/ISO532B Loudess Model This Matlab program was a direct coversio from the basic program provided i [6]. This implemetatio uses a filterbak of oe-third-octave filters for the spectral decompositio of the sigal, however, a drawback is that this yields oly a rough approximatio to the shape of the auditory filters ad the locatio of their ceter frequecies. The equatio for the specific loudess N i Soe/Bark of a the db SPL soud level L i a oe-third-octave bad is give G by [9] 0.025L 1 ETQ 0.1 LG ao LETQ N = (2) 4 The trasmissio of freefield soud to our hearig system through the head ad the outer ear is described as atteuatio a. The 0 excitatio threshold i quiet is. Values for these two L ETQ parameters were give i [8]. To ru the implemetatio give i [6], the sequece of Matlab commads is [Yxx,f]=PowSpec(Pref.*ycal,fs,df); [YdB, err]=covert2db(yxx, 1); I this implemetatio, the power spectral desity of the sigal Yxx is foud without the factor Pref take ito accout, ad DAFX-2

3 oly whe this quatity is coverted to db is it icluded, i.e. iside Covert2dB.m YdB=10*log10((cal^2)*Yxx/(Pref^2)); The other iput argumets to PowSpec are fs ad df, which are the samplig frequecy ad the frequecy resolutio respectively. The third-octave-bad filters are geerated usig the code below, the filter desig is by a program obtaied from the mathworks called Oct2dsg.m. f is a output of PowSpec. Scalig of the filter resposes is carried out to esure o eergy gai i itroduced ito the sigal. The filtered bads of the sigal power spectrum is give as Lt. %% Filter [H, err]=geeratefilters_16000(f); H= *H; for ik=1:24 Lt(ik)=10*log10(sum((10.^(YdB/10)).*(abs(H(ik,: ).^2)))); The fial fuctio call returs the total loudess N ad specific loudess vector Ns. The iput MS defies the soud field ad by default is set to a free field, i.e., MS = 'f'. %% Calculate Loudess [N, Ns, err]=din45631_16000(lt, MS); Zwicker s Loudess model as used i PSQM The PSQM algorithm was adopted by the ITU for speech quality assessmet [10], but oly recetly has it bee replaced by the PESQ algorithm [11]. The equatio used for the implemetatio of Zwicker s model is differet to (2). The specific loudess LX ( f ) is give by γ γ P 0 f PPX f LX f = S (3) l 0.5 P0 f where f is the absolute hearig threshold S is a scalig factor, l P 0 ( f ) at frequecy f ad PPX is the Pitch Power Desity at frequecy f for frame. γ is a costat. The Pitch power desities are the power spectrum of a sigal frame warped to the bark scale with a resolutio of Bark, with scalig relative to the badwidth i Hertz. Accordig to the PSQM documet [10], S = ad γ = , l [Yxx,f]=psd(ycal,NFFT,fs,NFFT,NOVERLAP); deltaz=0.312; for i=2:57 deltaf=ffreqs(i)-ffreqs(i-1); scal=(deltaf./deltaz); idice=(ffreqs(i-1) <= f & f < Ffreqs(i)); idex=fid(idice>0); idex_first=idex(1); idex_last=idex(); PPX(i-1)=(scal./(idex_lastidex_first+1)).*sum(Yxx(Ffreqs(i-1) <= f & f < Ffreqs(i))); Lx=Sl.*(P0).^gamma.*((( *PPX./P0).^gamma)-1); Lx(fid(Lx<0))=0; N_pseq=sum(Lx);%total loudess Zwicker s Loudess model as used i PEAQ This is the most sophisticated of the psycho-acoustic decompositios [12]. The power spectrum of each frame is weighted by the frequecy respose of the outer ad middle ear derived from a model. The power spectral eergies are the grouped ito Critical bads, spaced at 0.25 Bark. A offset is the added to the Critical bad eergies to compesate for iteral oise geerated i the ear. A triagular (i db) spreadig fuctio is used to implem et spreadig i the frequecy domai. E ~ is the spread excitatio SR patter. Ulike the PSQM algorithm the values for the excitatio threshold i PEAQ was computed usig a model descriptio [12], with c a costat that is set to , 0.23 ~ 0.23 E t f s f ESR f (4) LX f = 1 + c s f 1 s f E0 Et f where, i terms of db, the threshold idex is give by 2 1 f 1 f s f = ta 0.75 ta (5) db the excitatio threshold is 0.8 E tdb ( f ) = 3.64( f 1000) (6) A complete implemetatio of this fuctio is give i [13]. The variables X2 are the power spectrum of the frame, Eb is the bark warped spectrum, E is the spectrum followig the applicatio of spreadig, ad Lx ad Ntot are the specific loudess ad total loudess respectively. The fuctios amed i the code below are the same as described i [13] but with the additioal iput parameters of sigal legth le ad samplig frequecy Fs. X2=PQDFTFrame(ycal,le); Eb=PQgroupCB(X2,'Basic',le,Fs); E=PQspreadCB(Eb,'Basic',Fs); [Ntot,LX]=PQLoud(E,'Basic','FFT',Fs); N_tot(l)=Ntot; 2.4 Loudess Model of Moore ad Glasberg The model of Moore ad Glasberg [14] is differet to that of Zwicker i that the auditory frequecy scale used is the equivalet rectagular badwidth (ERB) ad the equatio for the specific loudess i a filter bad is α α N ' C E sig E (7) where bad, Esig = ThQ is the excitatio patter withi a particular frequecy E is the excitatio at the hearig threshold, ad C ad ThQ α are costats. A Matlab implemetatio was proposed by [15]. It relied o the HUTear toolbox [7] to calculate the excitatio patters. Assumig a gammatoe filterbak with 128 filters, the suggested model iput parameters were model.fs=fs; [f,b,cetfreq]=make_cgtbak(128,fs,200,6); DAFX-3

4 save gt128_test f b CetFreq; model.cochlea.fb.file='gt128_test.mat'; % model.cochlea.asymmcomp=1; model.haircell.rcf.r='half'; model.haircell.rcf.c=0.7; model.haircell.rcf.f='1khz'; model.eural.fuctio='mea'; For each frame the excitatio patter of the sigal was geerated usig the AudMod fuctio from the toolbox [7], Esig=AudMod(frame(:,l),model); Similarly, to geerate the excitatio patter at the hearig threshold, [CrctLiPwr, frqnpts, CrctdB] = OutMidCrct2('MAF',128,Fs); MAF = iterp1_ext(frqnpts,crctdb,cetfreq,'liear','ex trap'); for i=1:legth(cetfreq) toes(i,:)=pascalize(si(2.*pi.*(0:frame_le- 1).*CetFreqs(i)./Fs),MAF(i)); ; Ethq=AudMod(sum(toes),model); Oce the excitatio patters are kow, the specific ad total loudess ca be calculated. To fid the total loudess from the specific loudess, scalig is applied based o the badwidth of the ERB filters before summig. N=C.*(Estim.^alpha-Ethq.^alpha); N(fid(N<0))=0; EarQ = 1/ ; mibw = 24.7; order = 1; b = 1.019; ERBwidth = ((CetFreq/EarQ).^order + mibw^order).^(1/order); totalloudess=sum((n.*erbwidth)'); 3. OUTPUT CALIBRATION AND TESTING I the cases of the MBSD loudess model, the PSQM loudess model ad the Moore ad Glasberg loudess model calibratio was foud to be ecessary. The Matlab fuctio lsqcurvefit.m from the optimizatio toolbox was used. Oe of its requiremets is a fuctio ame withi its iput argumets, ad usig the MBSD loudess model as a example, it was be writte i the form fuctio [l]= mbsd_cal(coef,s_xx) l=diag(sqrt(coef.*s_xx)*sqrt(coef.*s_xx)'); where coef is the calibratio parameter, S_XX is excitatio used to compute the specific loudess ad l is the total loudess. 512-poit Siewaves of frequecy 1000Hz ad samplig frequecy 16kHz that were calibrated to be { 40,50,60,70,80} db SPL that should have a total loudess of { 1,2,4,8,16} were used. I the case of the MBSD loudess model S_XX eeds to be called by a factor For the PSQM model, 4 S l = ad γ = For the Moore ad Glasberg model C = ad α = Model 40db 50dB 60dB 70dB 80dB MBSD DIN PSQM PEAQ MooreGlas Table 1: Iput Siusoid SPL Values ad Models outputs i Soes The total loudess i Soe produced by each model for these siewaves is give i Table 1. It ca be see from the table that oe of the measures produce the exact figure for total loudess but that all are approximately close to the expected value. 3. CONCLUSIONS This paper has preseted Matlab implemetatios of a umber of loudess models. Furthermore, where ecessary the issue of model calibratio was addressed. Fially, results were preseted to demostrate the model output for a siewaves of various db SPL levels. 4. REFERENCES [1] Gelfad, S.A., Hearig: A Itroductio to Psychological ad Physiological Acoustics, Marcel Dekker, [2] [3] Woho, Y., Ehaced modified bark spectral distortio (EMBSD):A objective speech quality measure based o audible distortio ad cogitio model, Ph.D. thesis, Temple Uiversity, Ft. Washigto, USA, [4] Appell, J., et al., Review of loudess models for ormal ad hearig-impaired listeers based o the model proposed by Zwicker, Audiologische Akustik, 40, No.(2), [5] wer.pdf [6] [7] [8] Zwicker, E., Fastl, H., ad Dallmayr, C., BASIC program for calculatig the loudess of souds from their 1/3 oct bad spectra accordig to ISO 532 B, Acustica 55, 1984, pp [9] Quast, H., Absolute Perceived Loudess of Speech, Proceedigs of the 7th Joit Symposium o Neural Computatio, USC, [10] ITU Recommatio, P.861 Objective Measuremet of Telephoe Bad ( Hz) Speech Codecs (PSQM) [11] ITU Recommatio, P.862 Perceptual Evaluatio of Speech Quality (PESQ), the New ITU Stadard for Ed-to- Speech Quality Assessmet, [12] Thiede, T., Perceptual Audio Quality Assessmet usig a No- Liear Filter Bak, Ph.D. thesis, Techische Uiversitat Berli, Berli, Germay, [13] Kabal, P., A Examiatio ad Iterpretatio of ITU -R BS.1387: Perceptual Evaluatio of Audio Quality, TSP Lab DAFX-4

5 Techical Report, Dept. Electrical & Computer Egieerig, McGill Uiversity, May [14] Moore, B., Glasberg, B., Baer, T., A model for the predictio of thresholds, loudess ad partial loudess, J. Audio Eg. Soc. 45, 1997, pp [15] DAFX-5