IEEE Trans. on Signal Processing, special issue on Theory and Applications of Filter Banks and Wavelets,April 1998, in print.
Abstract
In this paper we extend the definition of dyadic wavelets to include frequency warped wavelets. The new wavelets are generated and the transform computed in discrete-time by alternating the Laguerre transform with perfect reconstruction filterbanks. This scheme provides the unique implementation of orthogonal or biorthogonal warped wavelets by means of rational transfer functions. We show that the discrete-time warped wavelets lead to well-defined continuous-time wavelet bases, satisfying a warped form of the two-scale equation. The shape of the wavelets is not invariant by translation. Rather, the "wavelet translates" are obtained from one another by allpass filtering. We show that the phase of the delay element is asymptotically a fractal. A feature of the warped wavelet transform is that the cutoff frequencies of the wavelets may be arbitrarily assigned, while preserving a dyadic structure. The new transform provides an arbitrary tiling of the time-frequency plane, which can be designed by selecting as little as a single parameter. This feature is particularly desirable in cochlear and perceptual models of speech and music, where accurate bandwidth selection is an issue. As our examples show, by defining pitch-synchronous wavelets based on warped wavelets, the analysis of transients and denoising of inharmonic pseudo-periodic signals is greatly enhanced.
IEEE Trans. on Signal Processing, in print 1998.
Abstract
In this paper we introduce a new generation of perfect-reconstruction filter banks that can be obtained from classical critically sampled filter banks by means of frequency transformations.
The novel filters are Laguerre type IIR that can be directly derived and designed from ordinary orthogonal or biorthogonal filter banks. Generalized downsampling and upsampling operators based on dispersive delay lines are the building blocks of our structures.
By iterating the filter banks we construct new orthogonal and complete sets of wavelets whose pass-bands are not octave-spaced and may be designed by selecting a single parameter.
IEEE Trans. on Signal Processing, vol. 44, no. 7, pp. 1681-1692 July 1996.
Abstract
In a recent paper we introduced the Multiplexed Wavelet Transform (MWT) and pointed out its potential applications to the analysis, synthesis, processing and coding of pseudo-periodic signals such as voiced speech and music. Coders based on the MWT have been shown to outperform the conventional Sub-Band Coders when a reliable pitch parameter can be extracted from data. In this paper we investigate the effects of uniform quantization of the MWT coefficients. We compare the performance of the new coders with that of WT, block-DCT and KLT coders in terms of the coding gain achieved when optimal bit allocation schemes are adopted.
IEEE Trans. on Signal Processing, special issue on Wavelets and Signal Processing, vol. 41, no. 12, pp. 3313-3330, Dec. 1993.
Abstract
A new wavelet representation is explored in this paper. The transform is based on a pitch-synchronous vector representation and it adapts to the oscillatory or aperiodic characteristics of signals. Pseudo-periodic signals are represented in terms of an asymptotically periodic trend and aperiodic fluctuations at several scales. The transform reverts to the ordinary wavelet transform over totally aperiodic signal segments. The Pitch-Synchronous Wavelet Transform is particularly suitable to the analysis, rate-reduction coding and synthesis of speech signals and it may serve as a pre-processing block in automatic speech recognition systems. Feature extraction such as separation of voice from noise in voiced consonants is easily performed by means of partial wavelet expansions. A stochastic model of aperiodic fluctuations is proposed.
Comb and Multiplexed Wavelet Transforms and Their Applications to Signal Processing
Gianpaolo Evangelista
IEEE Trans. on Signal Processing, vol. 42, no. 2, pp. 292-303, Feb. 1994.
Abstract
As an extension of wavelet theory, new wavelet bases are introduced which are discrete in nature and suitable for the analysis and synthesis of pseudo-periodic digital signals. By means of these bases, the signal is represented in terms of a periodic trend and aperiodic fluctuations at several scales. In the frequency domain, the periodic trend lies in bands that are centered on the harmonics while the fluctuations are distributed in several sidebands. Properties of Comb and Multiplexed Wavelet Transforms are examined and the concepts applied to the analysis of real-life signals. In one dimension, the new transforms have interesting applications to speech and music signal processing. Their extension to two dimensions may be useful for the analysis of pseudo-periodic images.
in Representations of Musical Signals, De Poli, Piccialli and Roads Edts., Cambridge, MA: MIT Press, 1991, pp. 119-136.
This is an introductory paper on wavelet transforms and their applications to sound analysis and synthesis.
in Musical Signal Processing, Piccialli, De Poli, Roads and Pope Edts., Swets & Zeitlinger, Amsterdam, 1997.
Abstract
Thanks to the ever increasing number of contributions of many scientists, wavelet theory has become a mature field. Along with the extension of the theory, several interesting applications have emerged, shedding new light on the relevance of the transform in signal processing. This renovated interest has led me to write the present tutorial in which I have tried to include a number of novel perspectives. The paper is intended as an introductory tutorial for the non-specialist reader. Due to the vastness of the subject, I have chosen to present, in informal style, only those results that, in my opinion, are most relevant to the analysis, synthesis and manipulation of sound. For more general overviews, rigorous and extensive presentations of the results, the reader is referred to the tutorials and original papers cited in the -- necessarily partial -- bibliography.
Proc. of DAFX98, Barcelona, Spain
Abstract
The aim of this paper is to present results on digital processing of sounds by means of both dispersive delay lines and pitch-synchronous transforms in a unified framework. The background on frequency warping is detailed and applications of this technique are pointed out with reference to the existing literature. These include transient extraction, pitch shifting, harmonic detuning and auditory modeling.
Proc. of ICASSP'98, Seattle (Washington)
Abstract
In this paper we consider an extension of the wavelet transform leading to the construction of wavelets with arbitrary bandwidth. The new wavelets are complete, orthonormal and dyadic; nevertheless their bandwidth is not constrained to be one octave, rather it may be designed by selecting a set of parameters. The construction of the new bases starts in the discrete-time domain, exploiting properties of the Laguerre transform. Furthermore, we provide a procedure to define continuous-time warped wavelets. Flexibility of the bandwidth allocation allows for more and improved applications of the wavelet transform, such as signal coding, the design of auditory model based filterbanks and transient detection in pseudoperiodic signals, pointed out in the paper.
Proc. of EUSIPCO'98, Rodi(Greece)
Abstract
In this paper we present an approach to auditory modeling based on frequency warped wavelet transforms. This set of wavelets overcomes the limitations of ordinary wavelets given by octave band resolution or by rational sam-pling rate filterbanks. The frequency bandwidths of the basis sequences may be arbitrarily designed by selecting a single parameter for each scale level. This parameter controls the pole position of a chain of identical first order all-pass filters, which is the computational structure of the Laguerre trans-form. The Laguerre parameter determines the amount of warping necessary at each scale in order to match the pre-scribed cutoff frequencies. Bark, mel or other filterbanks based on perceptual criteria can be exactly implemented by means of our transform, taking full advantage of the ot-hogonality and completeness of the transform.
Proc. of EUSIPCO'98, Rodi(Greece)
Abstract
In this paper we discuss a novel representation of pseudoperiodic signals based on a frequency warped pitch-synchronous transform. The unitary warping operation is performed by means of the discrete Laguerre transform and controlled by the Laguerre parameter. This parameter can be adapted to the characteristics of the signal. In particular, for a large class of signals, by means of frequency warping one can regularize the spacing of the partials, so that the resulting signal is pseudo-harmonic. By applying the Pitch-Synchronous Wavelet Transform to the regularized signal one can achieve an interesting separation of noise and transients from the resonant components. These concepts are integrated in a single unitary transform. The Pitch-Synchronous Frequency Warped Wavelet Transform has applications in sound analysis, coding and synthesis.
Proc. of ICASSP 1997, Munich, Germany.
Abstract
In this paper we show that the dyadic wavelet transform may be generalized to include non-octave spaced frequency resolution. We introduce orthogonal and complete wavelets and wavelet packets whose set of cutoff frequencies may be adapted, in the simplest case, by changing a single parameter. The novel wavelets and the FWWT transform computational structure are obtained via an intermediate Laguerre representation of the signal. The warped wavelets are related to the ordinary wavelets by means of frequency transformations and orthogonalizing filtering.
The classical sampled filter bank theory is extended to include frequency dependent upsampling and downsampling operators and dispersive delay lines. The FWWT frequency band flexibility may be exploited in order to adapt the wavelet transform to signals.
Proc. of the IEEE Internat. Symposium on Time-Frequency and Time-Scale (TFTS'96), pp. 121-124, Paris, France, June 1996.
Abstract
This paper presents a new model for the synthesis of pseudo-periodic signals, based on self-similar noise processes that can be generated by computing the inverse Multiplexed Wavelet Transform (MWT) of seed and characteristic sequences. These processes, which exhibit an approximate 1/f behavior near the harmonics of the fundamental frequency, are shown to be samples of a self-similar random field. They provide an extension of the Wornell-Oppenheim class of dy-homogeneous signals. Applications to speech and music are pointed out, where the defined processes may be used as excitation signals in order to approximate the fine structure of the frequency spectrum typical of voiced sounds .
Proc. dell' XI Colloquio d' Informatica Musicale, pp. 85-88, Bologna, Italy, Nov. 1995.
Abstract
The Karplus and Strong (KS) algorithm plays a central role in the synthesis of string instruments and percussions, making it possible to obtain excellent results at a reasonable computational cost. Often it is desirable to extract the KS synthesis parameters from natural sounds. However, the accurate estimate of the synthesis parameters is still an open problem. In this paper we try to provide new solutions by first introducing a closely related synthesis algorithm, the ICC technique, together with an analysis procedure, and then generalizing the ICC analysis method to the KS algorithm.
Proc. of IX Colloquio d' Informatica Musicale, pp. 303-313, Genova, Italy, Nov. 1991.
Abstract
The accurate synthesis of musical sounds based on the analysis of few sample tones of a given natural instrument is still an open problem. Several synthesis techniques rest either upon models of sound sources or on general signal representations which are unrelated to both the source and the listener. It may be interesting to explore algorithms which are based on models of human hearing. Time-scale representations provide a useful mathematical tool working in that direction.
in Wavelets and Applications, Y. Meyer, Ed., Springer-Verlag, 1992, pp. 396-412.
Abstract
The close relationship between quadrature mirror filter banks and wavelet transforms is exploited to provide a parametrization of wavelets and devise efficient lattice filter structures to compute the transform. To any quadrature mirror filter pair a scattering matrix can be associated and factored. Inverting this procedure, one can build any quadrature mirror filter pair out of elementary factors of the scattering matrix. We were able to generate new orthonormal wavelet sets from both recursive and non-recursive filters. As an example we introduce for the first time the class of wavelets generated by Butterworth pairs which is extremely well-behaved in both frequency and time domains.
invited paper at IEEE-ISCAS'90, International Symposium on Circuits and Systems, Proceedings of ISCAS'90, New Orleans, LA, May 1990.
Abstract
Inspired by continuous-time wavelet transforms, new transforms for the expansion of discrete-time signals are here derived. Biorthogonal discrete wavelet expansions are exactly implemented in critically sampled quadrature mirror filter banks. Scale-variant pseudo-wavelet transforms are defined as a natural generalization. Lattice parametrization of both finite and infinite response quadrature mirror filters leads to efficient implementation of orthogonal discrete wavelet transforms and allows for optimization techniques for the choice of basis.
Proc. of the 23-rd Asilomar Conference, Pacific Grove, CA, November 1989.
Abstract
A new class of othogonal basis functions has been recently introduced which can be relevant to signal processing. Wavelet bases are constructed from a single function y(t), the wavelet., by considering its translates and dilates on a dyadic grid of points. Any signal can be represented by the set of its expansion coefficients which can in principle be computed with a critically sampled pruned binary tree structured QMF (quadrature mirror filter) bank. We introduce a discrete version of wavelet bases suitable for the expansion of sampled signals. The transform is actually implemented in a QMF bank.
Proceedings of the 2-nd Workshop on Representations of Musical Sounds, Capri, Italy, Oct. 1992.
Abstract
Finite-scale Discrete Wavelet Transforms lead to a decomposition of the signal into a low-pass trend plus fluctuations at several scales. In this paper, I consider a new wavelet representation in which signals are decomposed into an asymptotically periodic trend and aperiodic fluctuations at several scales. The transform may be realized by multiplexing signals over a number of channels and wavelet transforming each channel individually. The number of multiplexing channels may be adaptively selected in order to take into account the fluctuations of the pitch. The transform reverts to the ordinary wavelet transform over totally aperiodic signal segments. Multiplexed Wavelet Transforms are particularly suitable to the analysis, rate-reduction coding and synthesis of music signals that are pseudo-periodic. Feature extraction such as separation of harmonic oscillations from inharmonic information is easily performed by means of partial wavelet expansions. Inharmonic components may be modeled as a superposition of independent stochastic processes, each representing a fluctuation from periodicity at a given scale.
Ph.D. dissertation at Univ. of California, Irvine, June 1990.
Abstract
Integral wavelet transforms and continuous-time orthogonal wavelet systems have been recently defined. However, most of modern signal processing is performed in discrete time. We define both orthogonal and non-orthogonal wavelet sequences which are directly suitable for the expansion of discrete-time signals. The resulting wavelets are not related to their continuous-time counterpart by ordinary Shannon's theorem. Discrete-time wavelet transforms are directly realized in multirate quadrature mirror filterbanks. Efficient lattice implementations of quadrature mirror filters lead to parametrizations of wavelet sets which can be exploited for the design of analyzing wavelets. We provide sufficient conditions for the completeness of discrete-time wavelet sets in the space of finite energy sequences. We generalize the concept of wavelet transforms to non-orthogonal and scale-varying transforms. We define wavelet seqences which are complete and orthogonal with respect to a given autocorrelation kernel. We illustrate some of the applications and limitations of wavelet expansions in the context of speech and music compression by subband coding.
Proc. of the 12-th International Computer Music Conference, pp. 293-297, The Hague, Holland, 1986.
Abstract
Various techniques have been developed for the synthesis of audio signals, as additive , frequency modulation, waveshaping, and others. The aim of the above techniques is that of representing (coding) complex sounds by a reduced set of parameters and that of obtaining time varying spectra by a small set of controlling parameters. Both FM and waveshaping (which can be identified under certain conditions) meet the above requests. Anew it can be said that they cannot grant easily: A) a required harmonic content B) "continuous" frequency bands as in many real instruments C) monotonicity of single partials during spectral evolution Synthesis by Formants meets some of the above conditions and achieving an high compression ratio in the representation of acoustical events (both for coding and synthesis).
Proc. of the 1986 IEEE Workshop on Applications of Signal Processing to Audio and Acoustics, pp. 179-184, La Paltz, NY, 1986.
Abstract
Synthesis by formants is widely used for the production of acoustical signals, due to the broad range of sounds (both vocal and instrumental) that it can produce. Here we introduce a basic unit made up of a digital resonator, realized by a simple FIR filter, together with the control of the resonant frequency, the pass-band and the amplitude. The output of this unit is obtained by convolution of the pulse response of the pass-band filter with a sequence of pulses, equispaced and enveloped by a slow time varying amplitude. The number of operations (generally only sums in an appropriate implementation) is very limited, thus resulting in an efficient technique of data and computation reduction. Complex sounds are realized by additive synthesis of such defined formants, each with it’s amplitude envelope: the low rate of controlling parameters allows producing efficiently dynamically sounds. Finally we propose an hardware implementation of a modular unit for the synthesis of sound by the control of four formants.
Proc. of Signal Processing IV, EURASIP '88 , North Holland, pp.903-906, 1988.
Abstract
In the following paper I report about the realization of a new generation real-time computer for digital processing of signals in the acoustical range. Main feature of the described architecture is the ability to perform as a general purpose signal processing architecture which could be a flexible testbed for various approaches to parallel processing of digital signals. In fact, recalling mostly the "signal flow graph" approach, the reconfigurable hardware will be able to emulate also systolic architectures as well as connection architectures. Referring to the specific application for which it is designed, final goal for the described architecture will the ability to produce and process a very high number of audio channels, up to the complexity of a number of different voices (up to 16-32-64, depending on the chosen configuration).
Proc. of the International Computer Music Conference, San Jose, CA, pp. 348-351, 1992.
Abstract
The X20 is an ASIC (Application Specific Integrated Circuit) conceived and designed in IRIS. It was intended to be used for research developments in new synthesis algorithms and architectures. The X20 is a fully programmable DSP with parallel operators, high interconnection freedom and address capability. It includes some useful facilities for musical applications. Two X20 with a general purpose controller (Motorola MC68302) , their respective memories and glue logic useful for connection between them, lay on the SM1000 board, representing the core of the Mars workstation. The SM1000 communicates by a MIDI port, an RS232 interface, a parallel port, a serial IIS standard port with the external world.
Proc of ICMC '97, Thessaloniki, Greece, 1997, pp. 51-54.
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.
Proc of ISMA'97, Edinburgh, U.K., pp. 219-224.
We present an extension of the Karplus-Strong algorithm to systems in which the stiffness of the material is not negligible. This generalization involves frequency warping implemented in a dispersive delay line given by a chain of all-pass filters. The filter coefficients are obtained from the expression of the solution of the equation of the stiff string in the frequency domain, with the help of an optimization algorithm. We derive the relationships between integrating constants and initial conditions and detail the consequences of fulfilling homogeneous boundary conditions.
in Musical Signal Processing, Piccialli, De Poli, Roads and Pope Edts., Swets and Zeitlinger, Amsterdam, 1997
...............In what follows we describe existing techniques for granular synthesis, showing the signal processing implications of these techniques. In particular we focus on pitch-synchronous techniques on which our research is based, including methods of analysis, synthesis and parameter control, and finally implementation on signal processing machines in terms of computational units programmed for real-time synthesis.
S. Cavaliere
Description
In this paper we address the problem of the incresing computational power required by signal processing algorythms. Architectural as well as technological issues are described, with the emphasis on new generation architectures.Parallelism both at grain and at larger scales is analyzed. Pipelining issues are also adressed. Finally some typical parallel architecture are described and analyzed. Some large multiprocessor systems, data flow architectures as well as systolic and wavefront arrays are discussed, starting from the signal flow graph representation of DSP algorythms.
Computer Music Journal, Cambridge, MA: MIT Press, vol 6, no.4, Winter 1982, pag.22-35.
Description
The paper describes the realization of an automaton purposely designed and realized for theatre perfomance, presented at the Festival of Two Worlds, Spoleto 1981. The overall system consisted in a system of controls and feedback in which all elements of the mise-en-scène are regulated by a logic of interrelations, structured on a computer music score. The interactive system is constructed from a calculator of the mini/personal computer class, which controls a complex of devices purposely designed. This computer updates, by program, the parameters to be sent to the external world: stage motors, synthesis of sound effects, reproduction system, lighting, etc., often connecting events or sequences of events with the elaboration of optical and acoustic signals drawn from the action and movement on the stage. Central, in the sistem is a movement-to-sound system made of a videocamera sending its signal to a digitizing device and a special purpose analyzing board, witch was aable to compute in real time parameters from the scene, to be used as commands for sound synthesis and manipulation.
Annals of IUN, vol. XLV-XLVI, pp. 235-241, 1976-77.
Abstract
We have constructed a special purpose fast processor capable of realizing 64 oscillators and 64 amplitude modulators in real time. This system has been operating for two years in musical applications, in particular, in the additive synthesis of acoustic signals. The system is presently controlled by a Digital PDP 15, but the interface is easely adaptable to a smaller minicomputer or to a microprocessor. The frequency stability, depending on a crystal, is 1/1000000. The sampling rate is 64 kHz, and the arithmetic unit is of 24 bits. The frequency of each oscillator can be varied from 0 to 32 kHz in steps of 4 mHz, while the amplitude can be varied from 0 to the maximum value in 2048 steps. The oscillators can generate waves of arbitrary waveform.
Signal processing, North-Holland, vol. 4, nos. 5-6,pp.375-385, October 1982.
Abstract
Here we propose a novel realization of digital filter for noise measurements used in the European telephone networks. Our proposal is based on the idea of encoding the filter coefficients with the same PCM A-law that is used for signals on the telephone channels to minimize the word length. The hardware is reduced as a result of expressing the coefficients with 8 bits. A further advantage is to use a stored table multiplier (ROM) for carrying out multiplication without any PCM/linear converter that are used normally to filter PCM coded signals (A-law).
Computer Music Journal, Cambridge, MA: MIT Press, vol.12, no.1, pp. 29-42, Spring 1988.
Proc. of DAFX99, Trondheim , Norway. Conference on Digital Audio Effects November 1999
Abstract
Dispersive tapped delay lines are attractive structures for altering
the frequency content of a signal. In previ-ous papers we showed that in
the case of a homogeneous line with first order all-pass sections the signal
formed by the output samples of the chain of delays at a given time is
equivalent to compute the Laguerre transform of the input signal. However,
most musical signals require a time-varying frequency modification in order
to be properly processed. Vibrato in musical instruments or voice intonation
in the case of vocal sounds may be mod-eled as small and slow pitch variations.
Simulations of these effects require techniques for time-varying pitch
and/or brightness modification that are very useful for sound processing.
In our experiments the basis for time-varying frequency warping is a time-varying
version of the Laguerre transformation. The corresponding imple-mentation
structure is obtained as a dispersive tapped delay line, where each of
the frequency dependent delay element has its own phase response. Thus,
time-varying warping results in a space-varying, inhomogeneous, propagation
structure. We show that time-varying frequency warping may be associated
to expansion over bior-thogonal sets generalizing the discrete Laguerre
basis. Slow time-varying characteristics lead to slowly varying parameter
sequences. The corresponding sound transformation does not suffer from
discontinuities typical of delay lines based on unit delays.
Keywords: signal transformations, frequency warping
IASTED International Conference Signal and
Image Processing, October 1999 Nassau, Bahamas.
Abstract
In this paper we study asymptotic properties of frequency warped wavelets defined by means of iterated warping. These wavelets are based on the iteration of a structure consisting of a two-channel perfect reconstruction, orthogonal filter bank and a Laguerre transform block. In particular, we show that 1) the sequence of Laguerre parameters determined by the exponential cutoff choice (n=an( converges for 0<a<1; 2) the normalized map converges to a differentiable function uniformly on compact sets and 3) the asymptotic warping map satisfies Schröder's equation. We provide an extension of Königs's model to parametric maps on the unit circle. In our construction, we exploit the resulting conjugacy properties to define continuous-time warped wavelets and to show that these wavelets correspond to scale-a wavelets. We also show that the characteristic warping function is related to a self-similar map.
Keywords: Wavelets, Frequency Warping, Laguerre Transform, Schröder's Equation, Königs's Model.
Introduzione
Il concetto di trasformazione di coordinate è un concetto di
uso molto generale e ricco di svariate applicazioni nell’elaborazione numerica
dei segnali e delle immagini. Una trasformazione di coordinate permette
di lavorare in un dominio ‘trasformato’ in cui alcune elaborazioni risultano
più semplici ed immediate, permettendo così di descrivere,
analizzare ed elaborare un segnale, modificando i suoi parametri nello
spazio della rappresentazione. L’esempio piu’ immediato di questo concetto
è la rappresentazione di Fourier, che permette di passare dal dominio
del tempo, in cui sono le caratteristiche temporali del segnale ad essere
in evidenza, al dominio della frequenza, in cui sono in evidenza diretta
le frequenze contenute nel segnale, la loro ampiezza e fase: lavorare in
questo dominio permette di modificare direttamente le frequenze, per poi
ritornare allo spazio di partenza delle funzioni del tempo.
Esempi più recenti di trasformazioni sono le rappresentazioni
congiunte di segnale, che includono la rappresentazione tempo-frequenza
e tempo-scala, in particolare la Trasformata Wavelet.
Non tutti i segnali possono essere analizzati mediante queste trasformazioni
di segnale, che, viceversa, sono adatte a classi specifiche di segnale;
per rendere comunque utile l’uso di queste trasformazioni si può
procedere in vari modi: si può ottimizzare il nucleo della trasformata,
in modo da adattarlo al segnale ottenendo quindi la trasformata ottimale
rispetto al segnale o alla classe di segnali in esame oppure si può
modificare il segnale in modo da adattarlo alla trasformata e vedremo come.
Una classe di rapresentazioni è particolarmente semplice da utilizzare
ed è quella delle trasformazioni cosiddette unitarie, che, nello
spazio euclideo ad n dimensioni corrispondono alle semplici rotazioni (o
riflessioni intorno ad un asse), e che conservano sia l’energia che il
prodotto interno.
Quetse trasformazioni godono della proprietà che l’unitarietà
della rappresentazione si conserva se si mettono in cascata due trasformazioni
unitarie, conservando così tutte le proprietà relative, oltre
a quelle della completezza ed invertibilità.
La strategia quindi può essere di utilizzare delle trasformate
di segnale standard, modificando però il segnale mediante una trasformazione
appunto unitaria, in modo tale che il segnale così trasformato venga
bene analizzato nel dominio finale. Si conservano così la completezza
e l’invertibiltà della trasformazione ed inoltre si preserva l’energia
nel passaggio da un dominio all’altro. Tutte queste proprietà sono
insostituibili quando si vogliano correttamente analizzare e modificare
i segnali in modo controllato. Infine dalla cascata delle due trasformazioni
deriva una nuova trasformazione, il prodotto delle due, che può
dare luogo ad implementazioni molto interessanti ed utili, come nel caso
che vedremo.
Il caso che studiamo in questa presentazione è il caso dell’uso
di una trasformata di segnale, trasformata discreta di Laguerre, combinata
con la trasformata wavelet classica; dalla combinazione delle due è
possibile definire una nuova trasformata la Warped Wavelet Transform, e
definire una classe nuova di wavelet, le Frequency Warped Wavelets che
estendono significativamente il range di definizione delle trasformate
wavelet. In particolare risulta superata la limitazione sulla divisione
delle bande di analisi e sintesi, che, nelle wavelet diadiche ordinarie
procedono per potenze di due, mentre nelle warped wavelets, sono liberamente
scelte mediante i parametri della trasformazione di Laguerre.
XII CIM Colloquio di Informatica Musicale- Gorizia Settembre 1998.
Abstract
Unitary transforms recently introduced by the authors, of which the
Frequency Warped Wavelet Transform in its basic version and its Pitch Synchronous
version are special cases, are explored as a new means for char-acterizing
a large class of sounds. In these sounds the stiffness of the medium in
which oscillations propagate results in a frequency dependent velocity
and hence in a dispersive characteristic for the higher partials. . The
frequency warping induced by a Laguerre Transforma-tion modifies the structure
of the partials, reverting the source sound pseudoharmonic features into
a 'perfectly harmonic' sound, provided that a suitable choice of the controlling
parameter is adopted. Besides the theoreti-cal interest for the classification
and definition of pseudo-periodic sounds, the adopted technique results
in improvements in the characterization and separation of the periodic
part from the aperiodic component of these sounds. This is performed in
the domain of the frequency warped pitch synchronous wavelet trans-form,
where the signal appears as a perfectly harmonic signal. In this paper
we present a new method for matching the warping parameter to signals.
This method is based on the adaptation of a normalized warped notch comb-filter
choosing as a criterion that of minimizing the output energy of the filtered
signal.