Page  00000001 Analysis and Regularization of Inharmonic Sounds via Pitch-Synchronous Frequency Warped Wavelets Gianpaolo Evangelista Sergio Cavaliere Department of Physical Sciences, University "Federico II", Napoli, Italy evangelista @, cavaliere Abstract We present several applications to sound analysis and synthesis of a novel time-frequency representation based on frequency warped wavelets, recently introduced by the authors. These wavelets are obtained from ordinary wavelets via Laguerre expansion. The discrete Laguerre transform is shown to be equivalent to a warping operation on the frequency axis. The amount of warping is controlled by the Laguerre parameter, which can be adapted to the characteristics of the signal. The concept is here applied to the Pitch-Synchronous Wavelet Transform (PSWT), also introduced by one of the authors. In our experiments we found that the inharmonicity characteristics of the piano are well approximated by our warping map family. The suitable Laguerre parameter may be found by means of optimazation techniques. Using the PS-FWWT we were able to achieve a good quality separation of the hammer noise in piano tones. This separation can be useful in sound synthesis both for employing the noisy components as excitation signals or for synthesizing the regular and noisy parts using different techniques. 1 Introduction In a recent paper, one of the authors presented a novel transform, the Pitch-Synchronous Wavelet Transform (PSWT), which proved extremely useful for the separation of inharmonic and noisy components in pseudo-periodic sounds such as the bow noise in a violin tone. However, a large class of sounds from natural instruments exhibit a structural inharmonic behavior. This is the case, e.g., of the lower tones in piano, or vibraphone, cymbals and drums. Stiffness and other phenomena result in dispersion of the wave propagation, resulting in a certain degree of inharmonicity of the sound produced. In this paper we extend the definition of PSWT to a larger class of signals, including those whose partials exhibit detuning. This is achieved by embedding frequency warping in the PSWT. In order to preserve nice properties such as orthogonality, completeness and realizability in digital structures, the frequency warping operation must be carried out in a special form, which may be written in terms of the Laguerre transform. The new transform, the Pitch-Synchronous Frequency Warped Wavelet Transform (PS-FWWT), offers the possibility of regularizing harmonically detuned signals and to separate different aperiodic behaviors at several scales, such as transients, noise and modulation. The embedded Laguerre transform regularizes the signal in order to displace the inharmonic partials to har monic bands, which makes then possible to apply the PSWT method for periodic-aperiodic separation [3]. The regular components of the PS-FWWT have a slowly varying structure and their energy is essentially contained in narrow bands centered on the quasiharmonic frequencies. The noisy components are obtained by subtracting from the signal its regular part. This mechanism is built-in in our transform. In fact, the scaling sequences associated with the PS-FWWT have a comb-like structured frequency spectrum, where the center frequencies of the peaks are not necessarily harmonically related. The PS-FW wavelets are sidebands of this non-uniform comb. 2 Orthogonal Frequency Warping The Laguerre Transform is the building block of the PS-FWWT. It implements the required frequency warping operation in an orthogonal and complete expansion. In this section we will recall the main results of this transformation. Let r (k;b) denote the discrete Laguerre sequence of order r and parameter b (Ibl < 1). Their z-transform is rational [1], with Ar(Z) = b2 (z - b)r (1-bz- )r+1 (1) and satisfies the following recurrence:

Page  00000002 Ar+(z) = A(z)Ar(z) = A(z)r+1Ao(z), r=O,1,... plete and orthogonal set and the transform and its inverse may be computed as illustrated in fig. 2. where (2) (3) z-' -b A(z) = - 1- bz Wi W2 is a first order allpass filter. Consider the expansion of a causal discrete-time signal y(k) over the Laguerre set: y(k) Wi w2 WN VN (a) y(k) y(k) = urr(k;b), r=0 (4) W" VN where ur are the expansion coefficients. It is easy to see that the DTFT of the sequence ur is a filtered frequency warped version of y(k), according to a warping law 9 (0) [4]. The family of warping curves vs. the parameter b is shown in fig. 1, [6]. The Laguerre Transform may be implemented in a cascaded rational filter structure including a dispersive delay line [6]. Due to the time-reversal operation required, the transform may be computed in finite time provided the signal has finite length. Furthermore, the infinite Laguerre series must be truncated to a finite number of terms. This number may be given in terms of the group delay of the Laguerre sequences as in [7]. (b) (a) analysis and Figure. 2: Block diagram of FWWT: (b) synthesis. It can be shown [4] that the FW wavelets are a filtered and frequency warped version of ordinary wavelets. By frequency warping the ordinary wavelets one can adapt the analysis bands to signal features. 4 Pitch-Synchronous Frequency Warped Wavelets The PS-FWWT is obtained by combining the PSWT [3] with Laguerre warping. Its block diagram may be obtained from that shown in fig. 2, by replacing the ordinary WT and IWT blocks, respectively, by PSWT and IPSWT blocks. In the constant pitch P case, the PS-FW wavelets are given by,n,m,q(k) = EZln,m(l)4q+lP(k), (5) Figure 1: Family of frequency warping curves (1<b<1). 3 Frequency Warped Wavelets The Frequency Warped Wavelet Transform (FWWT) is obtained by embedding Laguerre warping in the ordinary Wavelet Transform. The sequence of Laguerre coefficients is projected onto a set of dyadic wavelets In,k (mn), [4]. The FW wavelets form a com where ] n,k (m) are ordinary wavelets. The scaling sequences associated with the PS-FWWT have a comblike structured frequency spectrum, where the center frequencies of the peaks can be displaced from the harmonics by an amount controlled by the Laguerre parameter b. A positive value of the parameter results both in an increased spacing of the higher center-band frequencies and in a dilation of the corresponding frequency bands. The opposite effect is obtained by selecting a negative value of b. The PS-FW wavelets are sidebands of this non-uniform comb, as shown in fig. 3. The same scheme may be adopted when the pitch is time-varying. In this case, equation (5) is replaced by the corresponding time-varying expression, similar to that found in [3].

Page  00000003 SIWaveletn=l I SIWaveletn=2 I iWavelet n=3 Scaling n=3 0 500 1000 1500 2000 2500 3000 3500 4000 Figure 3: Fourier Transform of typical FW-PS wavelets. 5 Analysis and Synthesis of PseudoPeriodic Sounds via PS-FWWT Due to stiffness of the vibrating medium or other phenomena, a large class of sounds such as piano tones in the bass register, sounds of plates and other percussive instruments exhibit a remarkable inharmonicity of the partials. While complicating their analysis and synthesis, this phenomenon together with period waveform variability contributes to the richness and naturalness of the sound produced. The PSWT based method for separating the periodic component from the aperiodic or noisy components is not directly applicable to this class of sounds since a large portion of the energy of the partials is trapped in the wavelet projections. The PS-FWWT representation allows us to relocate the bands of interest in order to match the distribution of the partials. This is achieved by adapting the Laguerre parameter b to the dispersion characteristics. If adaptation is successful, the signal is projected onto a "regular" part (scaling component), containing information on its quasi-periodic behavior, and "noisy" components (wavelet projections), containing information on the fluctuations from the quasi-periodic behavior, i.e., transients and modulation, at several scales. The power of this representation depends on how well the inverse warping curve matches the spectrum dispersion. In order to assess the validity of the PS-FWWT method, we performed several experiments on string instruments. The dispersion curve of a piano tone with fundamental frequency 27.3 Hz is shown in fig. 4. There, the X marks represent the spacing of the partial frequencies vs. the index of the partial. These data were obtained by means of a peak picking algorithm, using an adaptive window length. In order to compensate for the frequency dependent spacing of the pseudoharmonics, one has to match the data points against the derivative of one of the Laguerre warping curves (see fig. 1). This is achieved by adapting the parameter b by means of a non-linear optimization algorithm. Once the best-fit inverse warping curve is obtained, the optimum value of b, with its sign reversed, is used as the Laguerre parameter in the PS-FWWT representation. Notice that the pitch parameter P of the PS-FWWT must be adapted to the pitch of the signal after warping. Equivalently, one can set this value to the actual pitch period as transformed by the warping curve. Inharmonicity Characteristics 38.6 36.9 - 35.1 - 33.3 31.6 29.8 26.3 10 20 30 Partial Index 40 50 Figure 4: Diagram showing the frequency spacing of partials in a piano tone (f = 27.3 Hz) and the derivative of the best-fit inverse frequency warping curve (solid). The close match of the Laguerre warping curve against the partial frequency data may be also justified from a theoretical point of view, by solving the fourth order PDEs describing the vibration of stiff strings, rods and plates. The eigenfrequencies distribute on a curve closely approximated by a Laguerre warping curve corresponding to a suitable parameter b. The approximation is actually the same as that involved in digital waveguide models of stiff strings [8]. The results of the decomposition of the piano tone in regular and noisy components are reported in figs. 5 and 6. The hammer excitation noise prevails in the noisy component, while the regular component represents the piano tone deprived of its attack. This separation is much more accurate than that achieved by means of PSWT.

Page  00000004 U -1 -1.5 n I I I I I I I I I -0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Figure 5: Quasi-periodic component of a piano tone (f=27.3 Hz) obtained by means of PS-FWWT analysis. 0.8 0.6 0.4 0.2 0.2 0.4 0.6 the resonant transfer function characteristic of the instrument. References [1] P. W. Broome, "Discrete Orthonormal Sequences," J. Assoc. Comput. Machinery, vol. 12, no. 2, pp. 151-168, 1965. [2] G. Evangelista, "Comb and Multiplexed Wavelet Transforms and Their Applications to Signal Processing," IEEE Trans. on Signal Processing, vol. 42, no. 2, pp. 292-303, Feb. 1994. [3] G. Evangelista, "Pitch Synchronous Wavelet Representations of Speech and Music Signals," IEEE Trans. on Signal Processing, special issue on Wavelets and Signal Processing, vol. 41, no.12, pp. 3313-3330, Dec. 1993. [4] G. Evangelista and S. Cavaliere, "Frequency Warped Filter Banks and Wavelet Transforms: A Discrete-Time Approach Via Laguerre Expansion," submitted to IEEE Trans. on Signal Processing, July 1996. [5] G. Evangelista and S. Cavaliere, "Discrete Frequency Warped Wavelets: Theory and Applications," submitted to IEEE Trans. on Signal Processing, special issue on Theory and Applications of Filter Banks and Wavelets. January 1997. [6] A.V. Oppenheim and D. H. Johnson, "Discrete Representation of Signals," Proc. IEEE, vol. 60, pp.681-691, June 1972. [7] G. Evangelista and S. Cavaliere, "The DiscreteTime Frequency Warped Wavelet Transforms," Proc. of ICASSP 1997: IEEE International Conference on Acoustics, Speech, and Signal Processing, Munich, April 1997. [8] I. Testa, S. Cavaliere and G. Evangelista, "A Physical Model of Stiff Strings," Proc. of Int. Symposium on Musical Acoustics ISMA'97, Edimburgh, August 1997, in print. [9] S. A. Van Duyne and J. O. Smith, "A Simplified Approach to Modeling Dispersion Caused by Stiffness in Strings and Plates," ICMC Proc., 1994. -0.8 I I I I I I I I I I 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Figure 6: Noisy component of a piano tone (f=27.3 Hz) obtained by adding together the PS-FWWT wavelet projections. 6 Conclusions The PS-FWWT is an useful generalization of the PSWT for the analysis, synthesis and coding of pseudoperiodic sounds whose partials are non-uniformly distributed. The parameters of the transform may be adapted to the signal characteristics as in the case of dispersion in stiff strings. In the analysis, this may be useful for separating transients and noise from the quasi-periodic component. In the synthesis, distinct models for the transform coefficients may be introduced at each scale. The inverse transform will provide a sound with desired inharmonicity, matching, e.g., those produced by natural instruments. By combining PSWT synthesis with frequency warping at different scales one can add inharmonicity to harmonic sounds while preserving the noisy components. This scheme is useful in order to add naturalness to existing sound synthesis techniques. As a further application, inharmonic 1/f-like pseudoperiodic signals may be generated by modeling the PS-FWWT coefficients as properly scaled white noise sources. The resulting signals sound extremely natural when filtered by