SAMPLING-RATE-AWARE NOISE GENERATION

Poc. of he 14 h In. Confeence on Digial Audio Effecs (DAFx-11), Pais, Fance, Sepembe 19-23, 2011 SAMPLING-RATE-AWARE NOISE GENERATION Henning Thielemann Insiu fü Infomaik, Main-Luhe-Univesiä Halle-Wienbeg, Gemany henning.hielemann@infomaik.uni-halle.de ABSTRACT In his pape we conside he geneaion of discee whie noise. Despie his seems o be a simple poblem, common noise geneao implemenaions do no delive compaable esuls a diffeen sampling aes. Fis we define wha we mean wih compaable esuls. Fom his we conclude, ha he vaiance of he andom vaiables shall gow popoionally o he sampling ae. Evenually we conside how noise behaves unde common signal ansfomaions, such as fequency files, quanisaion and impulse geneaion and we exploe how hese signal ansfomaions mus be designed in ode geneae sampling-ae-awae esuls when applied o whie noise. 1. INTRODUCTION Noise is an ubiquious kind of signal: Ofen i is an annoying aefac of signal ansmission, convesion, o pocessing, bu i is also an essenial pa of sounds like wind, beaking wae waves, wind insumens, dum sounds, and ficaives in speech. In ou pape we exploe he geneaion of he lae kind of noise. 1.1. A moivaing example Imagine a sound designe who woks on a collecion of synhesised insumens ha shall be used in sofwae synhesizes. Fo insance he ies o mach he sound of a panpipe by mixing a sine oscillao and whie noise, ha is fileed by a esonan low-pass file as illusaed in Figue 1. Then he sound designe goes o implemen he simple flow diagam in he sofwae synhesis package Csound as shown in Figue 2. Since he inends o use he insumens in music fo compac discs, he chooses o ende his sound a 16 bi esoluion and 44100 Hz. He achieves his wih he following command line. csound -o panpipe.wav \ panpipe.oc panpipe.sco \ --sample-ae=44100 --conol-ae=441 Ou ais will lae add an envelope, fequency modulaion and ohe enhancemens ha make he sound moe naual. He will also design seveal moe insumens. Afe his collecion of insumens has gown o a consideable size he decides o also pepae peview sounds a a lowe sampling ae fo his web sie. To his end he sas Csound wih he opion --sample-ae=11025. To his supise some insumens sound quie diffeen a he lowe ae. He expeced a wose qualiy, bu he assumed ha insumens would sound essenially he same. He goes back o he simple panpipe pooype algoihm and finds ou ha he noise poion of he sound is consideably loude a he low sampling ae han a he high sampling ae. noise low-pass oscillao Figue 1: Flow diagam (absac signal pocessing algoihm) fo a vey simple sound ha esembles a panpipe. Ochesa file panpipe.oc nchnls = 1 ins 1 iamp = 30000 anoise noise 0.3*iamp, 0 acoloed lowpass2 anoise, 440, 10 aosci oscils 0.5*iamp, 440, 0 ou aosci+acoloed endin Scoe file panpipe.sco i 1 0 2 e Figue 2: The flow diagam in Figue 1 anslaed o a Csound pogam. The line conaining lowpass2 applies a low-pass file wih esonance a 440 Hz o he peviously geneaed whie noise. The opcode oscils geneaes a sine wave also wih fequency 440 Hz. The scoe file specifies ha ou insumen wih numbe 1 sas a second zeo and sops afe wo seconds of playing. Thus he caefully chosen mixing aio of noise and sine wave is los a he low sampling ae. Befoe filing a Csound bug he sound designe wans o find ou, wha is pecisely he poblem. He checks ha he whie noise has he same ampliude a he low and he high sampling ae. The same is ue fo he sine wave. He eplaces he noise geneao by an oscillao and obseves ha fileing a one yields he same esul a boh sampling aes. Tha is, i seems o be he combinaion of noise and fileing ha inoduces he uninended volume dependency on he sampling ae. This is vey sange. Now he becomes cuious whehe ha poblem also occus in ohe sound synhesis sysems. He anslaes his Csound algoihm o he eal-ime synhesizes SupeCollide and ChucK and confims ha hey behave exacly he same way. Finally he gives up and decides o jus downsample he sounds he endeed a + DAF-1

Poc. of he 14 h In. Confeence on Digial Audio Effecs (DAFx-11), Pais, Fance, Sepembe 19-23, 2011 44100 Hz. The downsampled wave files acually sound fine. Howeve ou ais is sill uncomfoable wih he obsevaion ha his signal algoihms depend in a non-obvious manne on he sampling ae. So fa he hough, ha his algoihms absac fom he sampling ae. He had seen his algoihms as an analogy o scalable veco gaphic fomas, such as PosScip, PDF o SVG ha can be endeed a any device a any esoluion while achieving he maximum possible qualiy. He even expeced ha he could use he same algoihm boh fo discee and analogue signal pocessing. Wha is he poin of sound synhesis compaed o sound sampling if no flexible adapaion o changing sound paamees and also o he sampling ae? He is no quie saisfied wih downsampling sounds fom 44100 Hz o 11025 Hz in ode o ge sampled sounds a 11025 Hz. This indiecion means ha he mus inves fou imes of he compuaion ime of endeing immediaely a 11025 Hz plus ime fo esampling. When endeing music diecly a 11025 Hz he could geneae 4 imes as many channels fo spaial effecs o 4 imes of he polyphony of music endeed via he indiecion hough 44100 Hz. Vice vesa: How can he poduce sampled sounds wih maximum possible qualiy a 96000 Hz fom his algoihms ha he designed fo 44100 Hz sampling ae? 1.2. Basic consideaions The noise geneaos found in Csound and ohe packages do wha ceainly eveyone would do in ode o poduce whie noise: They un a sandad pseudo-andom numbe geneao in ode o fill an aay wih andom values fom he ineval [ 1, 1] accoding o a unifom disibuion. Now le us see, wha his acually means when his is pefomed a diffeen sampling aes. In Figue 3 we have whie noise of he same duaion boh a 11025 Hz sampling ae (ende(11025 Hz, noise) in he op-lefcone) and a 44100 Hz sampling ae (ende(44100 Hz, noise) in he op-igh-cone). The noise a he highe sampling ae looks moe dense han ha a lowe ae, of couse. We also hea clealy he addiional high fequencies in he high ae noise. Howeve we have he impession, ha he low fequencies of he noise ae loude in he low ae noise and sofe in he high ae noise. This audioy impession becomes even visual if we apply fequency files o his noise. We have applied a fis ode low-pass file in he second ow of he signal able and a esonan second ode low-pass file in he hid ow of he able. We clealy see ha he signals in he igh column have consideably smalle ampliude han hose in he lef column. Why do he ampliudes of fileed noise depend on he sampling ae? An inuiive answe can be found in he fequency speca ha ae depiced in he boom ow of he able: Since he fequency specum of he high ae signal coves a lage fequency ange, he enegy of he high ae noise is spead ove a lage fequency ineval. In ode o veify, ha he poblem is acually he noise and no he fileing, we have inseed a cene column, whee all signal pocesses ae pefomed on a low ae noise, ha was conveed o a highe ae by simply eplicaing all sample values of he low ae noise fou imes. The audio impession is he same as fo he sounds in he lef column. The impoan diffeence beween he upsampled noise in he cene column and he high ae noise in he igh column is, ha he high ae noise consiss of independen andom values wheeas he andom values in he upsampled noise ae equal wihin blocks of fou values. 1.3. Conibuions Wih ou pape we wan o conibue he following aspecs fo esolving he sampling ae dependence of whie noise: Develop a cieion fo judging whehe a signal geneao o modifie pefoms similaly in diffeen sampling aes in Secion 2.1. Exploe some ways of adaping noise o he sampling ae in Secion 2.2. Conside seveal signal modifies like fequency files, quanises, click geneaos and how hey can be made awae of noise inpu and sampling aes in Secion 2.3. Discuss in Secion 2.4 by wha paamees a sampling-aeawae noise geneao should be conolled. Give a small guide on choosing a andom disibuion in Secion 2.5. 2. MAIN WORK 2.1. Compaabiliy acoss sampling aes In naual sounds hee is no such hing as a ime quanisaion and a sampling ae. Thus naual signals ae commonly modelled by eal funcions. Bu when i comes o signal pocessing in a digial compue we need ime (and value) disceisaion. Noneheless we do no wan o hink abou disceisaion when designing a signal pocessing algoihm. We like o peend ha hee is no sampling and hus an algoihm wihou a efeence o sampling can be used boh fo analogue synhesis and fo digial synhesis a any sampling ae. 1 Definiion (Absac signal pocessing algoihm). We like o call a signal pocessing algoihm absac, if i does no conain any efeence o disceisaion o a sampling ae. All quaniies in such an algoihm shall be physically meaningful, e.g. ime values mus be given in seconds bu no as numbes of sampling peiods. An example is he Csound algoihm in Figue 2. Since an absac signal pocessing algoihm neglecs sampling, we can use i o descibe eal funcions. Inepeing an absac signal pocessing algoihm a a given sampling ae means, ha we appoximae hese eal funcions by discee signals. E.g. if we descibe a fequency file as he soluion of a diffeenial equaion, hen his is an absac algoihm. In he digial compue we compue a coesponding diffeence equaion and his is he inepeaion of he absac algoihm fo a given sampling ae. We measue he qualiy of he diffeence equaion solve by is closeness o he soluion of he accoding diffeenial equaion. Fo invesigaion of noise, eal funcions ae no of much use as a model, since eal funcions wih sochasic values ae neihe coninuous no inegable. In conas o ha, hee is no poblem in compuing diffeences o sums in discee noise. We may be able o model noise using sochasic pocesses, sochasic diffeenial equaions and genealised measuemens of he degee of appoximaion beween a discee signal and sochasic funcion. Bu we hink ha he following appoach is easie: We accep he lack of a disceisaion-fee model ha we can adap ou discee compuaions o. Insead we ask fo compaable esuls, when inepeing he same absac signal pocessing algoihm fo diffeen sampling aes. Tha is, inceasing he sampling ae fo endeing shall impove he audio qualiy bu i shall no ale he imbe of he sound signal. DAF-2

Poc. of he 14 h In. Confeence on Digial Audio Effecs (DAFx-11), Pais, Fance, Sepembe 19-23, 2011 x() x = ende(11025 Hz, noise) x = esample(44100 Hz, ende(11025 Hz, noise)) x() x() x = ende(44100 Hz, noise) x() x() fisodelowpass(x) x() x() x() esonanlowpass(x) x() ˆx(f) ˆx(f) absoluespecum(x) ˆx(f) f f f Figue 3: Table of hee signals and he esul of vaious ansfomaions applied o hem. The hee iniial signals ae noise a sampling ae 11025 Hz, noise a 11025 Hz upsampled o 44100 Hz by consan inepolaion, noise a 44100 Hz. The duaion of he sounds is 50 ms. The iniial signals ae depiced in he fis ow of he able. The ow below conains he esuls of applying a fis ode low-pass file wih cu-off fequency of 500 Hz. The hid ow conains he esuls of a esonan low-pass file fom a sae-vaiable file wih esonance fequency 500 Hz. The las ow conains fequency speca of he iniial sounds. DAF-3

Poc. of he 14 h In. Confeence on Digial Audio Effecs (DAFx-11), Pais, Fance, Sepembe 19-23, 2011 How can we check, whehe a paicula discee inepeaion of an absac signal pocesso geneaes compaable esuls fo diffeen sampling aes? We have o conve beween he sampling aes. We canno add infomaion by upsampling a signal fom a low sampling ae, bu we can discad infomaion by downsampling a signal fom a high o a low sampling ae. Downsampling should ac as a pojecion: I shall mainain he infomaion, ha can be epesened a he lowe ae and i shall discad he emaining infomaion. In ode o wie he compaabiliy equiemen a bi moe fomally, we like o define ende(, A), ha denoes a discee signal, ha is compued fom he absac algoihm A a sampling ae. Think of A being he Csound ochesa definiion in Figue 2, being he numbe we pass o he --sample-ae opion and ende as being he csound command. The sampling ae becomes pa of he geneaed signal, such as i becomes pa of he WAVE file geneaed by csound. Fuhe on we like o denoe he esampling of a signal x fom is associaed sampling ae o anohe sampling ae by esample(, x). Now we can sae: 2 Cieion (compaabiliy acoss sampling aes). The discee inepeaion (expessed by ende) of an absac signal pocessing algoihm A is called compaable acoss sampling aes if 0 1 0 1 ende( 0, A) esample( 0, ende( 1, A)) If ende( 0, A) compues a band-limied vesion of ende( 1, A) and esample pefoms pefec esampling, hen could be eplaced by =. Howeve mos acual implemenaions of discee signal pocessing only appoximae his ideal wold. Fo noise i is even wose, since we can hadly ceae he same noise a diffeen sampling aes. Thus we have o inepe even weake as an equivalence of some sochasic chaaceisics. Alhough he above cieion is in no way mahemaically pecise, i uns ou o be a vey useful guide fo design decisions in he following secions. 2.2. Adap noise o sampling ae A simple way o povide noise ha behaves simila acoss diffeen sampling aes, is o upsample noise fom a low ae. Say, we ae saisfied wih he ange of fequencies conained in discee noise a 11025 Hz sampling ae. Fo sounds a 44100 Hz we can jus upsample ha noise fom 11025 Hz o 44100 Hz. This appoach ivially geneaes compaable noise signals fo all sampling aes above 11025 Hz, even wih eplaced by = in Cieion 2 when we use pseudo-andom numbes wih he same seed fo all sampling aes. Bu hee ae wo disadvanages: This way we canno geneae compaable noise fo aes below 11025 Hz. We wan o incease endeing qualiy by inceasing he sampling ae. Fo noise, we like o ead highe qualiy o mean a lage ange of andom fequencies. Howeve, wih he upsampling appoach we do no auomaically ge highe fequency poions in he noise, when we swich o highe sampling aes. If we geneae pseudo-andom numbes a he age sampling ae, hen we auomaically fill he enie available fequency space. Howeve, as we have seen in he inoducion, we have o somehow adap he noise ampliude in ode o povide equal fequency ampliudes. We sill have o live wih he dawback, ha his kind of sampling-ae-awae noise is compaable acoss sampling aes only wih espec o sochasic paamees bu no in ems of acual appoximaions. In he following secions we will deive he necessay ampliude adjusmen and we will see how ohe signal pocesses mus be adaped in ode o wok nicely wih noise. 2.3. How o fuhe pocess noise 2.3.1. Fequency File An impoan way of modifying whie noise is fequency fileing. In analogy o elecomagneic oscillaions of ligh, fileed noise is called coloued noise. Pink noise, i.e. low-pass fileed noise, can be used as conol cuve. Whie noise fileed by esonan low-pass files can poduce sounds of wind, echo sounding, o ficaives. We wan o invesigae how o adap noise o sampling aes such ha i behaves simila wih espec o fequency files. Tha is, accoding o Cieion 2 we wan o achieve 0 1 0 1 ende( 0, file(noise)) esample( 0, ende( 1, file(noise))). (1) Le us sa wih he simple example of a moving aveage file, whee he aihmeic mean of w successive values is compued. We model whie noise as a sequence of andom vaiables, ha all have he expecaion value 0 and he same vaiance. The echnical em fo such a sequence is discee sochasic pocess. The expecaion value coesponds o he diec cuen offse, wheeas he sandad deviaion (oo of he vaiance) is he measue of he noise volume. We sa wih fileing whie noise low a sampling ae 11025 Hz. ende(11025 Hz, file(noise)) : Y low,k = 1 w k+w 1 low,j j=k In ode o pefom he same file a he highe sampling ae 44100 Hz and an accoding whie noise high, we have o incease he numbe of aveaged values o 4w. ende(44100 Hz, file(noise)) : Y high,k = 1 4w We obseve ha k+4w 1 high,j j=k σ(y low,0) = 1 w σ(low,0) σ(y high,0) = 1 2 w σ(high,0) ha is, fo equal sandad deviaions of he whie noises he sandad deviaions of he fileed noises ae no equal. Fom σ( low,0) = σ( high,0) i follows σ(y low,0) = 2 σ(y high,0). How o esolve his inconsisency? Fo he fileed noise he esample opeaion in (1) is essenially a mae of keeping evey fouh value. Tha is we can equie Y low,k Y high,4k. We like o ead his as k σ(y low,k ) = σ(y high,4k ) DAF-4

Poc. of he 14 h In. Confeence on Digial Audio Effecs (DAFx-11), Pais, Fance, Sepembe 19-23, 2011 To achieve his, we have o se he sandad deviaion of k popoional o he squae oo of he sampling ae. Noe, ha downsampling of whie noise canno be done simply by picking values a a coase gid, since his skips he necessay limiaion of he fequency band. σ 2 ( k ) Now we move on o geneal fequency files. They become mos simple in he fequency specum (jus a weighing of he specal values) and also downsampling is only mae of shoening he specum. Thus we like o inepe in (1) as compaing he fequency speca. Le be a sequence of n andom vaiables, ha all have he expecaion value 0 and he vaiance y 2. The noise sampling ae is and i may have a physical uni such as Hz. The discee fequency specum DFT -1 () is defined by DFT -1 () k = 1 n 1 j exp 2πi j k «n j=0 We have o inepe in (1) as he equaliy of he sandad deviaions of he FOURIER coefficiens, because we canno expec similaiy of obseved fequency ampliudes. Since he andom vaiables in ae independen, hei vaiance is addiive. σ 2 (DFT -1 () k ) = 1 n 1 σ 2 ( j) 2 j=0 = n y2 (2) 2 σ(dft -1 () k ) = n y Since n depends on he sampling-ae via n = l, and we mus compae signals of he same lengh l, we have o subsiue n. σ(dft -1 l () k ) = y Tha is, if noise of duaion l a sampling ae shall have specal values wih sandad deviaion c (i.e. c = σ(dft -1 () k )), hen we have o choose y = l c. (3) The ampliude of he noise is popoional o he squae oo of he sampling ae. The pa in (3) ha does no depend on he sampling ae is c l. We like o call ha he noise volage specal densiy value. Usually specal densiy is a funcion defined fo eal signals. In he following definiions we wan o adap he equied ems fom eal signals o discee ones. 3 Definiion (Wide-sense saionay discee sochasic pocess). A discee sochasic (o andom) pocess, whee all elemens have expeced value 0 is called saionay in a wide sense if he covaiance beween is elemens depends only on he disance bu no on he ime poin. Expessed in fomulas: k E( k ) = 0 k d E( 0 d ) = E( k k+d ) The whie noise signals, ha we conside in his pape, and also fileed whie noise signals ae always discee sochasic pocesses in a wide sense.. 4 Definiion (Auocovaiance funcion). Fo a wide-sense discee sochasic pocess we define he auocovaiance funcion R (ofen called auocoelaion) as he covaiances beween signal values depending on hei disance. R (d) = E( 0 d ) This capues all possible values of covaiances beween signal values, because he wide-sense saionaiy waans imeinvaiance of he covaiances. 5 Definiion (Noise powe specal densiy). The noise specal densiy of a wide-sense discee sochasic pocess is he specum of he auocovaiance funcion of. NSD() = DFT -1 (R ) Since he signal values of whie noise ae independen, he auocovaiance funcion is an impulse a ime poin zeo wih heigh σ 2 ( 0). Is specum is a consan funcion wih value σ2 ( 0 ). Accoding o (2) ha is equal o c2. We like o call his value l he noise powe specal densiy value of whie noise. The volage specal densiy is he squae oo of he powe specal densiy. We wan o use his as he paamee, ha descibes he ampliude of whie noise in a sampling-ae-awae way, and hus give i a symbol, namely VSD. 2.3.2. Quanisaion VSD = σ(0) (4) Quanising noise in ime diecion is a way, o give noise a pich chaaceisic, when using small quanisaion peiods, and is useful as conol cuve fo lage quanisaion peiods. Fo easons of simpliciy we will conside quanisaion wih fixed quanisaion peiods ha ae inegal muliples of he sampling peiod. Fo a discee inpu signal x wih sampling ae and quanisaion peiod, and d being he quanisaion peiod measued in unis of he sampling peiod, ha is d =, d N, we could simply define quanise(x) k = x k (k mod d). (5) This would yield a consan ampliude of quanise(x) fo consan quanisaion peiod and vaying sampling ae if he ampliude (sandad deviaion) of x would no depend on. Howeve if we quanise sampling-ae-awae noise as descibed in Secion 2.3.1 his way, hen he ampliude of he quanised noise wih espec o a consan quanisaion peiod will incease popoional o he squae oo of he sampling ae. In his espec an ampliude ha inceases wih he sampling ae is no good, since he quanisaion peiod acs like an aificial low sampling ae epesened a a high sampling ae, and his quanisaion peiod does no depend on he acual sampling ae. We can avoid a gowing ampliude by aveaging ove he quanisaion peiod. quanise(x) κ = q κ/d (6) wih q k = 1 (k+1) d 1 d j=k d x j DAF-5

Poc. of he 14 h In. Confeence on Digial Audio Effecs (DAFx-11), Pais, Fance, Sepembe 19-23, 2011 In he following poof we show ha he ampliude of quanised sampling-ae-awae whie noise wih specal densiy as in (4) does no depend on he sampling ae. Tha is, we check he cieion in Cieion 2 whee we inepe as compaing he sandad deviaion (= he ampliude) of he quanised noise. Q k = 1 (k+1) d 1 d j=k d j σ 2 (Q k ) = d d 2 σ2 ( k d ) = 1 VSD2 σ (Q k ) = VSD We see ha he ampliude of he quanised noise gows popoionally o he squae oo of he quanisaion fequency 1. This is complian wih ou sampling-ae-awae whie noise geneaion, whee we wan ha noise wih moe fequency conen is also loude. In fac quanisaion can be seen as downsampling, ha includes an appopiae low-pass file, wih subsequen upsampling by consan inepolaion. A disadvanage of aveaging quanisaion as in (6) is ha in eal-ime pocessing i delays he signal by one quanisaion peiod, wheeas he simple quanisaion is in (5) does no cause such a delay. 2.3.3. Random clicks (impulse noise) Thee is anohe impoan kind of sounds ha is based on andomness: Randomly occuing impulses. By subsequen pocesses like fequency files we can change he chaaceisic o seveal naual sounds. Examples ae he sound of aindops, hail, o he GEIGER- MÜLLER-coune fo measuemen of ionising adiaion. A simple appoach o geneae andom impulses fom whie noise is as follows: Fom whie noise a sampling ae wih samples ha ae unifomly disibued beween y and y we wan o obain andom impulses wih a fequency f. We geneae an impulse in he oupu signal wheneve he whie noise sample is in he ineval ˆ yf,. yf This appoach is vey simple bu i has seveal dawbacks: I is bound o inpu noise wih unifomly disibued sample values. fo given fequency f depends on The heshold value yf he sampling ae. If he inpu whie noise is samplingae-awae and unifomly disibued, hen is sample values cove [ k, k ] (i.e. y = k ) fo sampling-ae independen k and he heshold mus be kf. Fo smooh inpu signals (no noise) we ge cluses of impulses, wha in discee signal pocessing means, ha we ge signals consising of consan pieces ahe han sepaaed impulses. We can only conol he oveall fequency of impulses bu no he degee of andomness. All of hese poblems can be solved using Σ-modulaion as in Figue 4. We inegae whie noise wih a posiive diec cuen offse unil i exceeds a heshold. A his ime poin we emi an impulse in he oupu signal and hen sa inegaing wih cleaed accumulao again. We epea his pocedue in an endless loop. In he discee implemenaion of he Σ-convee he inegao R is a cumulaive sum, he compaao > y emis an impulse wih a heigh elaed o y, if he inpu exceeds he heshold y R delay > y Figue 4: Σ-modulaion. and zeo ohewise, he delay delays by one sampling peiod in ode o make he feedback possible, and he subacs he fedback impulse signal fom he inpu, such ha he inegao is ese afe evey emied impulse. The expecaion value of he inpu whie noise, i.e. he diec cuen offse, deemines he fequency of peaks in he oupu, wheeas he vaiance of he noise deemines he degee of andomness of peak disibuion. We wan o pove, ha impulse geneaion fom whie noise via Σ-modulaion yields compaable fequency and andomness of impulses acoss sampling aes. The noise geneaion and he inegaion ae he only opeaions ha adap o he sampling ae. Tha is, i suffices o show ha he inegal ove a fixed duaion of a sequence of idenically disibued andom vaiables wih sandad deviaions as in (4) has an expeced value and a vaiance ha does no depend on he sampling ae. Fo simpliciy shall be an inege. Z 0 = 1 1 k=0 k Z «E = E( k ) 0 Z «σ 2 = 0 2 VSD2 Z «σ = VSD 0 We have sill no answeed he quesion, wha kind of impulses he Σ-modulao shall geneae. If we use impulses wih one sampling peiod as duaion, hen he heigh of he impulses mus be chosen, such ha when fed back i eses he accumulao in he inegao. To his end le us conside he involved physical unis: Le he inpu signal have ime uni s and ampliude uni V. Then he inegaed signal has ampliude uni Vs and so he heshold in he compaao mus have his uni, oo. Thus he impulse, ha he compaao geneaes, mus have an aea equal o he heshold y, in ode o clea he accumulao. Since is widh is 1, is heigh mus be y. This way he impulses have sizes such ha hey epesen he aea of he inpu signal ove he pauses beween he impulses. This means, ha smoohing he impulse ain yields a signal simila o he inpu signal afe smoohing. This popey is acually he key fo using Σ-modulao in digial-analogue convees. An alenaive appoach fo geneaing andom impulses is o choose he pauses beween he impulses accoding o pseudoandom numbes wih expecaion value aveage silence duaion beween impulses and vaiance degee of andomness. Tha is, sicly spoken i is no necessay o geneae andom impulses DAF-6

Poc. of he 14 h In. Confeence on Digial Audio Effecs (DAFx-11), Pais, Fance, Sepembe 19-23, 2011 fom whie noise. Howeve, on he one hand we waned o show, ha andom impulse geneaion fom sampling-ae-awae whie noise is possible in a way, ha is iself sampling-ae-awae. On he ohe hand we waned o poin ou ha he use of a compaao (as in he beginning of his secion) can lead o signal algoihms, ha depend on he sampling ae by acciden. 2.4. Noise paamees In pinciple a sampling-ae-awae whie noise geneao wih ampliude uni V and ime uni s mus be conolled by a paamee wih uni V/ Hz o V s, ha we called volage specal densiy value VSD (see (4)). Howeve ha is boh uninuiive and unsuppoed by he usual implemenaions of physical dimensions in pogamming languages ([1, 2]) whee exponens of unis mus be ineges. I is uninuiive, because i is no simple o choose a numbe ha yields a easonable ampliude. E.g. we mus choose 1 V 44100 Hz VSD = 4.67 mv s in ode o ge sandad deviaion 1 V when endeing a sampling ae 44100 Hz. We avoid he facional powes in unis using he squaed paamee, ha is he powe specal densiy value wih uni V 2 s. This is even less inuiive, since doubling he noise ampliude means using fou imes of he powe densiy value. In ou expeience a vey inuiive soluion is o use wo paamees y and f wih he unis V and Hz. They mean ha a sampling ae f he vaiance shall be y and he vaiance fo ohe sampling aes shall be adjused accodingly. Given hese paamees he noise geneao mus compue samples of andom vaiables wih 2.5. Random disibuion E() = 0 σ() = y f So fa we did no need o conside paicula andom disibuions, because we only needed addiiviy of he vaiance of andom vaiables. The choice of he andom disibuion does no have an effec on he shape of he fequency specum. If he andom vaiables of a noise signal ae independen fom each ohe, hen all specal values have he same vaiance. Howeve, since he human ea pefoms somehing moe like a sho-ime FOURIER ansfom, a andom disibuion consideably diffeen fom nomal disibuion may geneae single clicks, ha can be head. Because in signal pocessing many opeaions like fequency fileing, inegaion, mixing involve addiion, i is likely ha he Cenal Limi Theoem applies. The esul of applying signal algoihms o whie noise ae likely o yield andom vaiables wih andom disibuions close o nomal disibuion. Fo easons of consisency we may hus pefe nomal disibuions fom he beginning. A vey simple way o appoximae nomally disibued andom vaiables wih vaiance 1 is o add hee unifomly [ 1, 1]- disibued andom vaiables. The acual disibuion has he shape of a quadaic B-spline. If speed maes, hen whie noise wih unifomly disibued andom vaiables is he bes choice. This is wha pseudo-andom numbe geneaos ceae. 3. RELATED WORK A wide ange of he lieaue consides noise ha aises as an undesiable aefac of signal pocessing. This pa of he lieaue. idenifies popeies ha allows o compae he behaviou of eleconic cicuis wih espec o noise and o sepaae noise and nonnoise poions of a signal. In his lieaue he noion of he noise specal densiy is well-known. [3, Chape 2] Inended geneaion of noise is no equally popula. A noable excepion is [4], whee he auhos consuc noise by mixing sine waves a andom fequencies. Consequenly hey use a cusom definiion of noise specal densiy, whee he densiy is he numbe of sine waves divided by he widh of he fequency band. By using he same aio of pesen fequencies pe band acoss sampling aes hey can ceae sampling-ae-awae noise in a ivial way. The sinusoidal model allows o conol he noise colou in an inuiive way, bu fo whie noise, i is compuaionally moe inensive, even when using a Fas Fouie Tansfom, han ou appoach of jus adaping he ampliude of a andom-numbe sequence. Alhough he noion of he noise specal densiy is wellknown in he lieaue, we could no find he conclusion, ha discee whie noise should be geneaed wih a vaiance popoional o he sampling ae. As menioned in he inoducion i is also no implemened in common sofwae synhesizes. We have esed Csound-5.10.1 [5], SupeCollide-3.3.1 [6], ChucK-1.2.0.8 [7]. None of hese packages adveises o be sample-ae-awae, alhough he use of physically moivaed paamees sugges ha hey ae. Howeve physical paamees such as ime in seconds and fequency in Hez ae mixed wih low-level paamees like plain digial file paamees, e.g. Csound:noise:kbea, Supe- Collide:OnePole:coef, ChucK:BiQuad:a0. In Csound he noise opcode wih disabled smoohing (paamee kbea=0) geneaes whie noise. I does no adap is ampliude o he sampling ae. This applies o all ohe of Csound s whie noise geneaos, ha povide diffeen disibuions of he andom vaiables (opcodes gauss, uniand, linand, cauchy,... ). I also applies o he whie noise geneaos in SupeCollide (WhieNoise) and ChucK (Noise). If we wan o ge coloued noise, we can call specialised noise geneaos in hose packages. E.g. hee is he Csound opcode pinkish in he defaul mode and he SupeCollide uni geneao PinkNoise, ha geneae pink noise following a muliscale scheme (MOORE/VOSS-MCCARTNEY mehod). Alhough no sicly compaable acoss diffeen sampling aes, because a lowe sampling aes hee ae moe low-fequency componens, he geneaed noise is almos compaable acoss diffeen sampling aes. To ou supise we have no found ime quanisaion in Csound, SupeCollide and ChucK. Thus hese packages have no poblem in combining a noise geneao wih a quanisaion. Howeve Csound:andh, SupeCollide:LFNoise0 and ChucK:SubNoise povide quanised noise a a given ae. By design hese poduce compaable esuls acoss diffeen sampling aes. We could also no find dela-sigma modulaion in Csound, SupeCollide and ChucK. Since SupeCollide does no allow sho-ime feedback, dela-sigma modulaion canno be build fom ohe componens. Noneheless geneaion of andom impulses is possible wih Csound:mpulse wih and inpu and SupeCollide:Dus. The fequency of impulses is sample-ae-awae, bu he aea of geneaed impulses vaies acoss sampling aes. Thee ae also sofwae synhesizes like Timidiy and Fluid- Synh ha ae designed fo SoundFons. SoundFon is a foma fo achiving sampled sounds ogehe wih loop poins, envelopes and pos-pocessing feaues like fequency files. Since SoundFon-2 DAF-7

Poc. of he 14 h In. Confeence on Digial Audio Effecs (DAFx-11), Pais, Fance, Sepembe 19-23, 2011 Ochesa file panpipe-adap.oc nchnls = 1 ins 1 iamp = 30000 anoise noise 0.3*iamp*sq(s/44100), 0 acoloed lowpass2 anoise, 440, 10 aosci oscils 0.5*iamp, 440, 0 ou aosci+acoloed endin Figue 5: Modified vesion of he Csound pogam in Figue 2 ha auomaically adaps o he sampling ae. does no seem o suppo a noise geneao, noise mus be povided as a sampled sound. If esampling is implemened popely (i.e. including band-limiaion), hen noise is auomaically adaped o he sampling ae. Of couse he ange of fequencies conained in noise can neve be lage han he noise conained in he soed sampled sound. Digial hadwae synhesizes have a fixed sampling ae and hus hey do no need o adap o diffeen sampling aes. The same applies o analogous synhesizes ha do no have a sampling ae a all. In [8] he auhos descibe an eleconic cicui called Noise Manipulao ha ceaes andom impulses fom whie noise. I uses he following signal algoihm: monoflop(compaao(y, pinknoise)) This means: Pink noise is conveed o a ecangula signal using a compaao wih heshold y. Then he monoflop conves low high jumps o impulses. The fequency and andomness of he impulses depend on he heshold y and he specum of he pink noise in a non-obvious way. In analogue signal pocessing hee is no infiniesimally sho DIRAC impulse and no sampling peiod. Tha is we mus explicily assign a duaion and a heigh o impulses. I is ceainly woh o also geneae impulses wih a definie duaion in discee signal pocessing. 4. CONCLUSIONS AND FUTURE WORK In ou pape we found ha i is useful o adap he ampliude of whie noise popoionally o he squae oo of he sampling ae in ode o achieve a consisen audio impession acoss diffeen sampling aes. To speak in ems of he sound designe in he inoducion (Secion 1): He exends he ampliude paamees of all of his noise geneaos by he faco p as in Figue 5 44100 Hz whee is he sampling ae. This way he noise ampliudes a 44100 Hz emain as he found hem in he couse of developing he signal algoihms. Fo ohe sampling aes he ampliude gows popoionally o he squae oo of he sampling ae. This solves he poblem fo he sound designe. Howeve he signal algoihm now conains a efeence o he sampling ae. Tha is, accoding o Definiion 1 i is no longe absac. E.g. i would no be possible o anslae he algoihm and is paamees o an analogue synhesize. To his end we would need a noise geneao wih he noise volage specal densiy as paamee as in Secion 2.4. Addiionally we have checked, ha fuhe pocessing seps of he noise like fileing, quanisaion and impulse geneaion, can mainain he audioy expeience acoss sampling aes when implemened popely. As seen in secion Secion 2.3.3 i is sill possible o accidenally develop signal algoihms ha poduce consideably diffeen esuls acoss sampling aes. Especially non-linea opeaions like compaaos ae poblemaic. We have o fuhe invesigae how o educe ha isk while emaining able o implemen all ineesing signal pocessing algoihms. 5. ACKNOWLEDGMENTS Many hanks go o my colleague Alexande Hinnebug fo poof eading and discussing he daf of he pape. 6. REFERENCES [1] Don Syme, Adam Ganicz, and Anonio Cisenino, Expe F#, Apess, 2007. [2] Bjon Buckwale, dimensional: Saically checked physical dimensions., hp://hackage.haskell.og/ package/dimensional-0.8.0.1, June 2010. [3] Wal Kese, Ed., The Daa Convesion Handbook, Analog Devices Inc. Newnes, 3d ediion, 2005. [4] Piee Hanna, Anhony Beuivé, and Myiam Desaine- Caheine, Real-ime noise synhesis wih conol of he specal densiy, in Poceedings of he 5h Confeence on Digial Audio Effecs (DAF 02), 2002, pp. 151 156. [5] Bay Vecoe, CSound, hp://www.csounds.com/, 2009. [6] James McCaney, Supe Collide, hp://www. audiosynh.com/, Mach 1996. [7] Ge Wang and Pey Cook, Chuck: a pogamming language fo on-he-fly, eal-ime audio synhesis and mulimedia, in MULTIMEDIA 04: Poceedings of he 12h annual ACM inenaional confeence on Mulimedia, New Yok, NY, USA, 2004, pp. 812 815, ACM. [8] Hans-Jochen Schulze and Geog Engel, Modene Musikelekonik, Miliävelag de Deuschen Demokaischen Republik, Belin, 1s ediion, 1989. DAF-8