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@na.infn.it, cavaliere@na.infn.it Abstract We present several applications to sound analysis and synthesis of a novel time-frequency representa- tion based on frequency warped wavelets, recently introduced by the authors. These wavelets are ob- tained 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 ap- plied 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 tech- niques. 1 Introduction In a recent paper, one of the authors presented a novel transform, the Pitch-Synchronous Wavelet Trans- form (PSWT), which proved extremely useful for the separation of inharmonic and noisy components in pseudo-periodic sounds such as the bow noise in a vio- lin 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 propaga- tion, resulting in a certain degree of inharmonicity of the sound produced. In this paper we extend the defini- tion 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, com- pleteness and realizability in digital structures, the fre- quency warping operation must be carried out in a spe- cial form, which may be written in terms of the Laguerre transform. The new transform, the Pitch-Synchronous Fre- quency Warped Wavelet Transform (PS-FWWT), of- fers 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 sig- nal 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 quasi- harmonic frequencies. The noisy components are ob- tained 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 har- monically 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 ex- pansion. In this section we will recall the main results of this transformation. Let λ r kb ( ; ) denote the discrete Laguerre sequence of order r and parameter b ( b < 1 ). Their z-transform is rational [1], with Λ r r r z b z b bz () ( ) ( ) = - - - - - + 1 1 2 1 1 1 , (1) and satisfies the following recurrence: