Evangelista, Gianpaolo

Polotti, Pietro

Infoscience - École polytechnique fédérale de Lausanne

Dynamic Models of Pseudo-PeriodicityPietro PolottiGianpaolo EvangelistaLaboratoire de Communications Audiovisuelles (LCAV)École Polytechnique Fédérale de Lausanne, Switzerlandpietro.polotti@epfl.ch      gianpaolo.evangelista@epfl.chAbstractVoiced musical sounds have non-zero energy in sidebands of the frequency partials. Our work is based on the as-sumption, often experimentally verified, that the energy distribution of the sidebands is shaped as powers of the in-verse of the distance from the closest partial. The power spectrum of these pseudo-periodic processes is modeled bymeans of a superposition of modulated 1/f components, i.e., by a pseudo-periodic 1/f –like process. Due to the funda-mental selfsimilar character of the wavelet transform, 1/f processes can be fruitfully analyzed and synthesized bymeans of wavelets, obtaining a set of very loosely correlated coefficients at each scale level that can be well approxi-mated by white noise in the synthesis process. Our computational scheme is based on an orthogonal P-band filter bank and a dyadic wavelet transform perchannel. The P channels are tuned to the left and right sidebands of the harmonics so that sidebands are mutuallyindependent. The structure computes the expansion coefficients of a new orthogonal and complete set of HarmonicWavelets. The main point of our scheme is that we need only one parameter in order to model the stochastic fluctua-tion of sounds from a pure periodic behavior.Keywords: wavelets, 1/f-noise, spectral modeling1 IntroductionThe purpose of this work is to introduce a techniquefor the analysis and synthesis of pseudo-periodic signalsbased on a special kind of wavelet packets, i.e., the Har-monic Wavelet Transform. We start from the wavelet-based model for 1/f processes introduced by Wornell [4]and we extend this model to the pseudo-periodic 1/f-likesignals.1/f processes can be employed for representing cha-otic systems that are strongly influenced by their pastbehavior [1-3]. Voiced sounds in speech and music arepseudo-periodic signals and exhibit a long-term correla-tion or more precisely an approximate 1/f behavior in theneighborhood of each harmonic partial.This is due to the chaotic but correlated micro-fluctuations from the periodic behavior of the signalitself. These fluctuations play a relevant role in the emu-lation of naturalness of voiced sounds.Wornell in Theorem 3 in [4] states that a process∑ ∑∞−∞=∞−∞==n mmnn tmxtx )()()( ,ψ ,where )(,tmnψ  form a orthonormal wavelet basis and the)(mxn  are collections of mutually uncorrelated zero-mean coefficients, is nearly 1/f, i.e., its time-averagedpower spectrum( )22 22)( ωσω γ nnnxS Ψ= ∑∞−∞=satisfies the relations γγωσωωσ 2,2, )( qUxqL S ≤≤   (1)for some ∞<≤< σσ 220 ULWe will extend this result to pseudo-periodic signals,introducing the Harmonic Wavelets, which consists of acontinuos-time infinite-channel filter bank and a discretewavelet transformation of each channel.The most important result of this synthesis techniqueis that it allows one to control a highly complex stochas-tic process by means of relatively few parameters.In section 2 we define the pseudo-periodic 1/f noiseby means of a Harmonic Modulation and Demodulationscheme. In section 3 we illustrate the theoretical resultson which our new method of synthesis is based. In sec-tion 4 we briefly review the Discrete Harmonic Waveletsand their properties. Section 5 deals with some applica-tions to music synthesis.2 Pseudoperiodic 1/f –like noiseWe now provide a formal definition of pseudoperi-odic 1/f-like noise based on a general modulation anddemodulation scheme.The frequency spectra of pseudoperiodic signals arecharacterized by harmonically spaced peak at frequenciesPTkkpiω 2= , where TP is the average period of the signal.In order to separate the contribution of each of the har-monic bands one can devise a set of ideal narrow-bandfilters of bandwidth PTpiω =∆  each fitting a single side-band of the harmonics. The magnitude frequency re-sponse of these filters is given by[ [] ]<≥=++0)(0)()()1()1(,,qqHPTqPTqPTqPTqqωωωpipipipiχχ    ,  q=0,±1,±2,...,where[ [  <≤=otherwise      0 if      1)(,BABAωωχis the characteristic function of the interval [A,B[. Theoutputs of these filters may be baseband shifted, accord-ing to a suitable demodulation scheme. This results in thedemodulation scheme reported in Fig. 1.2piTP−2piTPpiTP−piTPpiTP−piTPωωω0R+R- L- L+L+ L-R+R-(a)(b)(c)Fig. 1: Baseband shift of harmonic sidebands: (b) side-bands of the 2nd harmonic; (a) demodulation of the leftsidebands; (c) demodulation of the right sidebands.Demodulation of a signal x(t) in 2L  may be de-scribed by the scalar productsτττ dxtKtw qq )(),()( ∫∞∞−= ,   ,...1,0=q ,where the kernels of the set of linear operators K q ,q=0,1,... ,  have  the form( ) ( )K tTt qTtTq PqPqP( , ) cos sincτ τ pi β τ= − − + +−1 1 2 12 2    (2)where β βq q= − −1  are arbitrary phase factors.The operators (2) have support in∪=+−+−pppp TqTqTqTqqWpipipipi )1()1(,,and perform a harmonic demodulation to baseband] [PP TTpipi,− of the signal subband Wq. The operator K q  isinvertible, with inverseK t K t K tq q q−= =1( , ) ( , ) ( , )τ τ τ† ,where the symbol † denotes the adjoint. Thus K q  isunitary and the operators K q−1 perform a harmonicmodulation from baseband to Vq where Vq is the 2Lsubspace of signals bandlimited to Wq. We can thus model acoustic pseudoperiodic signalswith fundamental frequency piω 2/00 =f  by means of asuperposition of harmonic modulated bandlimited 1/fprocesses. Each process has bandwidth equal to half theharmonic spacing 0ω and contributes to a single sideband of each of the harmonics.Denoting by k the harmonics index and by L and Rthe left and right sideband, respectively, the averagespectrum of a pseudoperiodic 1/f -like process has thefollowing form for 0≥ω :( )[ [( )( )[ [( )ωωωσωωωσωωωγωωγχχ00,00, ,2/102,02/1 ,02,                             )(qqLqqqqRqLqRqqqS−∞=+−++−= ∑,   (3)where 2,Rkσ  and 2,Lkσ  are the amplitudes and γ k R,  andγ k L,  the decay parameters.Defining an ideal bandpass filter[ ] [ ] )()()( 2/0 ,- ,2/0)( ωωω ωεεω χχε += −H ,       (4)where ε is arbitrarily small, we arrive at the followingDefinition 1. A stochastic process x(t) is said to be a 1/f -like pseudoperiodic noise if there exists a TP>0 such thatwhen x(t) is operated by qK in (2), yields a collection ofprocessesτττ dxtKtw qq )(),()( ∫∞∞−= ,   ,...1,0=q      (5)such that, when filtered through )()( ωεH , withPTpiω 20 = , they become wide-sense stationary and ban-dlimited with power spectrum <<=otherwise  ifSqqqw               02/    /)( 02 ωωεωσωγ ,   (6)for some γ q andσq .The operations involved in (5) are equivalent to fil-tering the single sidebands of each of the harmonics,separately for the positive and negative frequencies, andproperly baseband shifting the result.Comparing (6) with the model spectrum in (3), wecan make the following associations:Lqq ,12 γγ =− ,  Rqq ,2 γγ = ,  Lqq ,12 σσ =− ,  Rqq ,2 σσ = .Since the resulting processes )(twq  in Definition 1are bandlimited to [ ]2/ , 2/ 00 ωω− , they can be sampledwith sampling rate PT120=piω.For our purposes we introduce then the set of func-tions { }Z∈= kqkq tg  ,...;1,0, )( ,)()( 0,, Pqkq kTtgtg −= , (7)with +=PPPq TttTqTtg2sinc 212 cos 1)(0, pi  ,which is easily shown to form an orthonormal basis. It iseasy to see that the scalar product (5) followed by sam-pling are equivalent up to a multiplicative con-stant PT to the projection of x(t) over the basis.3 Synthesis of pseudoperiodic 1/f-likenoise by means of Harmonic WaveletTransformIn the Introduction we recalled Wornell's resultsabout the synthesis of 1/f processes by means of Waveletbasis. We will provide now an equivalent result for thepseudoperiodic case. We need to prove a Lemma first.In order to do this we define a new continuous- timeMultiwavelet basis which we will call Harmonic WaveletTransform (HWT):∑=rrqmnqmn tgrt )()()( ,,,, ψξ  (8)where the )(,rmnψ  form an ordinary Discrete Waveletbasis and the )(,tg rq  are defined according to (4).The Fourier transforms of Harmonic Wavelets are:)()()( 0,,,, ωωω qmnqmn HPΨ=Ξ .   (9)The action of filtering is essentially that of selecting asingle sideband of the harmonics. Then we can prove thefollowing:Lemma 2A signal x(t) such that∑∑ ∑∑ ∑ ∞= =∞−∞=∞=∞−∞===0 1,,2/0,)()()()()(qNn mqmnnqnqq kkqq tmtgktx ξνβν ,where the )(,,tqmnξ  are defined in the (8),  the { })(kqν  areWSCS processes of the kind of:∑ ∑=∞−∞==Nn mmnnqnqq rmr1,2/ )()()( ψνβν ,       (10)the )(mnqν  are mutually uncorrelated coefficients andqqqγσβ 22= , is cyclostationary with period 2NTP. Thesame result holds for the scale residue of the scale-truncated expansion.For the synthesis we derived the followingProposition 3The random-process∑ ∑∞=∞−∞==0,)()()(q mmqq tgmts ν  ,             (11)where the { }Z∈= kqkq tg  ,...;1,0, )(  form an orthonormal set offunctions, as defined in (7) and the )(mqν are, up to amultiplicative constant PT , the samples of approxi-mately 1/f  processes synthesized by means of a DWTfilter bank, yields an average power spectrum of theform:Φ+Ψ×=∑∑=∞=20,120,020,2)(2)(2            )(1)(PnqNNnPnqnqqqPTTGTSωωωσωγγ (12)In the ideal case+= +−+− )()()( )1(,,)1(20, ωωω pipipipi χχPPPP TqTqTqTqqGand (12) is approximately 1/f near each harmonic Pk pi2 ,with  +=21qk  ,  q=0,...,P-1 ; i.e., for   ( )   212PkPk piωpi ≤≤−  if q is odd,or for ( )PkPpiωpi 12k2   +≤≤ if q is even,                     (13)we haveσωpiωσωpiγ γL qNU qkPSkPq q, ,( )2 22 2−≤ ≤−for some ∞<≤< σσ 2 ,2 ,0 qUqL .For the proofs of Lemma 2 and Proposition 3 see [8].4 Discrete Harmonic WaveletsThe discrete counterpart of (7) is given by the basisassociated with an ideal P band filter bank. We want toobtain an efficient scheme for the analysis and synthesisof pseudoperiodic 1/f noise. Thus we consider an ap-proximation of the ideal filter bank granting perfect re-construction. In particular, we consider the class of TypeIV cosine modulated bases:)()( 0,, rPlhlh qrq −=  ,       Z∈−= rPq      ;1,....,0( ) −−−+=−4210, )1(212cos)()( pipi qMq lPqlWlh  ,where the lowpass prototype impulse response )(lW  oflength M satisfies some technical conditions [8]. It iseasy to prove that the set is orthogonal and complete.In order to synthesize the samples of 1/f processes)(kwq we adopt the scheme devised by Wornell [4],which consists of an ordinary discrete wavelet synthesisstructure, with white noise inputs. The overall structureis realized by introducing the Discrete Harmonic Wave-lets [9]. These constitute a special type of multiwavelets,generalizing the Pitch-Synchronous Wavelet Transformclass [6,7], defined by∑=rrqmnqmn khrk )()()( ,,,, ψξ               (15)where )(,rmnψ  are discrete-time ordinary wavelets [5].The Fourier transforms of the basis elements (15) areshown in Fig. 2.In the analysis structure of the Discrete HarmonicWavelet Transform, the signal is sent to a P channelfilter bank and each output is Wavelet transformed (WTblock). Signal reconstruction is achieved by separatelyinverse Wavelet transforming the Harmonic Waveletcoefficients and passing these sequences through theinverse P channel filter bank. This technique generalizesthe one presented in [10] since independent control toeach subband is allowed.0    pi /8  pi /4  3 pi /8 pi /2  5 pi /8 3 pi /4 7 pi /8  pi   01 02 03 04 05 06 07 08 0Fig. 2 : Magnitude Fourier transform of HarmonicWavelets (P=5).5 Applications to Music SynthesisOur synthesis technique requires the estimation ofthree parameters per each harmonic partial k: σk , γ k,R ,γ k,L . The parameters σ  may be estimated from the fre-quency spectrum by means of a peak-picking algorithm.The estimation of the parameters γ k is may be based onthe HWT analysis. It is necessary to perform a linearregression according to the following law:const))((log,,2 += nxVar qqmn γ                   (16)with  21+= qk .Our experimental results show that HWT analysissupports the hypothesis that the shape of the side-bandsof voiced sounds have an approximate 1/f behavior. Infact by linear regression we obtain correlation coeffi-cients in the range 0.8 - 0.9 for the most relevant har-monics. Estimation improves if the bootstrap algorithm isemployed to simulate a larger data set. We have thenconsidered different samples of real instruments. Thereproduction of the energy of the harmonic bands and ofthe sidebands in the synthetic sounds is well performedby our method, as shown in Figs. 3 and Fig. 4.6 ConclusionsIn this paper we introduced a new method for soundsynthesis that allows us to control and reproduce themicro fluctuations present in real life voiced sounds. Thismethod is a sort of additive synthesis where one addsmodulated 1/f signals instead of pure sinusoidal func-tions. We defined a new class of stochastic processes, i.e.,pseudoperiodic 1/f-like noise. We introduced a specialtype of multiwavelet transform, i.e., the HarmonicWavelet Transform. We devised an efficient analy-sis/synthesis scheme able to perform parameter estima-tion and generate pseudoperiodic 1/f -like noise.1000 2000 3000 4000 5000 6000 7000-100-90-80-70-60-50-40-30-20-100Frequency (Hz)Magnitude (dB)Fig. 3 Frequency spectrum of real-life oboe (287.5 Hz)1000 2000 3000 4000 5000 6000 7000-100-90-80-70-60-50-40-30-20-100Frequency (Hz)Magnitude (dB)Fig. 4 Frequency spectrum of synthesized oboe.References[1] D. T. Gillespie, “The Mathematics of Brownian Motionand Johnson Noise,” Am. J. Phys., Vol. 64, pp. 225-240,March 1996.[2] P. Flandrin “Wavelet Analysis and Synthesis of FractionalBrownian Motion,” IEEE Trans. on Information Theory,Vol. 38, No 2, pp. 910-917, March 1992.[3] M. Deriche and A.H. Tewfik, “Signal Modeling withFiltered Discrete Fractional Noise Processes,” IEEETrans. on Signal Processing,  Vol. 41, No 9, Sept. 1993,pp. 2839-2849.[4] G. W. Wornell, “Wavelet-Based Representations for the1/f  Family of Fractal Processes,” Proc. IEEE, Vol. 81,No. 10, pp. 1428-1450, Oct. 1993.[5] I. Daubechies, Ten Lectures on Wavelets, SIAM CBMSseries, Apr. 1992.[6] G. Evangelista, “Comb and Multiplexed Wavelet Trans-forms and Their Applications to Signal Processing,” IEEETrans. on Signal Processing, vol. 42, no. 2, pp. 292-303,Feb. 1994.[7] G. Evangelista, “Pitch Synchronous Wavelet Representa-tions of Speech and Music Signals,” IEEE Trans. on Sig-nal Processing, special issue on Wavelets and SignalProcessing, vol. 41, no.12, pp. 3313-3330, Dec. 1993.[8] P. Polotti, G. Evangelista, “Fractal Synthesis of Speechand Music by means of Harmonic Wavelets,” technical re-port EPFL-LCAV.99ds[9] G. Evangelista, P. Polotti, “Analysis and synthesis ofpseudoperiodic 1/f-like noise by means of multibandwavelets,” Proc. of XII CIM, pp.35-38, Sept. 1998.[10] G. Evangelista, “Pseudo-Periodic 1/f-Like Noise,” Proc. ofTFTS’96, pp. 121-124, June 1996.

Dynamic models of pseudo-periodicity

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Infoscience - École polytechnique fédérale de Lausanne