State-space representation for digital waveguide networks of lossy flared acoustic pipes

This paper deals with digital waveguide modeling of wind instruments. It presents the application of state-space representations to the acoustic model of Webster-Lokshin. This acoustic model describes the propagation of longitudinal waves in axisymmetric acoustic pipes with a varying cross-section, visco-thermal losses at the walls, and without assuming planar or spherical waves. Moreover, three types of discontinuities of the shape can be taken into account (radius, slope and curvature), which can lead to a good fit of the original shape of pipe. The purpose of this work is to build low-cost digital simulations in the time domain, based on the Webster-Lokshin model. First, decomposing a resonator into independent elementary parts and isolating delay operators lead to a network of input/output systems and delays, of Kelly-Lochbaum network type. Second, for a systematic assembling of elements, their state-space representations are derived in discrete time. Then, standard tools of automatic control are used to reduce the complexity of digital simulations in time domain. In order to validate the method, simulations are presented and compared with measurements

Puzzles in pipes with negative curvature: from the Webster PDE to stable numerical simulation in real time

Minimal realizations of a class of delay-differential systems are derived for the digital simulation of waveguides, modelled by the Webster horn equation. Studying their stability is an interesting issue, since negative curvatures could lead to unstable systems. Spectral properties of Toeplitz matrix play a key role in this work

On the singularities of fractional differential systems, using a mathematical limiting process based on physical grounds

Fractional systems are associated with irrational transfer functions for which nonunique analytic continuations are available (from some right-half Laplace plane to a maximal domain). They involve continuous sets of singularities, namely cuts, which link fixed branching points with an arbitrary path. In this paper, an academic example of the 1D heat equation and a realistic model of an acoustic pipe on bounded domains are considered. Both involve a transfer function with a unique analytic continuation and singularities of pole type. The set of singularities degenerates into uniquely defined cuts when the length of the physical domain becomes infinite. From a mathematical point of view, both the convergence in Hardy spaces of some right-half complex plane and the pointwise convergence are studied and proved

Digital waveguide modeling for wind instruments: building a state-space representation based on the Webster-Lokshin model

This paper deals with digital waveguide modeling of wind instruments. It presents the application of state-space representations for the refined acoustic model of Webster-Lokshin. This acoustic model describes the propagation of longitudinal waves in axisymmetric acoustic pipes with a varying cross-section, visco-thermal losses at the walls, and without assuming planar or spherical waves. Moreover, three types of discontinuities of the shape can be taken into account (radius, slope and curvature). The purpose of this work is to build low-cost digital simulations in the time domain based on the Webster-Lokshin model. First, decomposing a resonator into independent elementary parts and isolating delay operators lead to a Kelly-Lochbaum network of input/output systems and delays. Second, for a systematic assembling of elements, their state-space representations are derived in discrete time. Then, standard tools of automatic control are used to reduce the complexity of digital simulations in the time domain. The method is applied to a real trombone, and results of simulations are presented and compared with measurements. This method seems to be a promising approach in term of modularity, complexity of calculation and accuracy, for any acoustic resonators based on tubes

Stable Realization of a Delay System Modeling a Convergent Acoustic Cone

This paper deals with the physical modeling and the digital time simulation of acoustic pipes. We will study the simplified case of a single convergent cone. It is modeled by a linear system made of delays and a transfer function which represents the wave reflection at the entry of the cone. According to [1], the input/output relation of this system is causal and stable whereas the reflection function is unstable. In the continuous time-domain, a first state space representation of this delay system is done. Then, we use a change of state to separate the unobservable subspace and its orthogonal complement, which is observable. Whereas the unobservable part is unstable, it is proved that the observable part is stable, using the D-Subdivision method. Thus, isolating this latter observable subspace, to build the minimal realization, defines a stable system. Finally, a discrete-time version of this system is derived and is proved to be stable using the Jury criterion. The main contribution of this work is neither the minimal realization of the system nor the proofs of stability, but it is rather the solving of an old problem of acoustics which has heen achieved using standard tools of automatic control

Digital waveguide simulation of convex acoustic pipes

This work deals with the physical modelling of acoustic pipes for real-time simulation, using the “Digital Waveguide Network” approach and the horn equation. With this approach, a piece of pipe is represented by a two-port system with a loop which involves two delays for wave propagation, and some subsystems without internal delay. A well-known form of this system is the “Kelly-Lochbaum” framework. It allows the reduction of the computation complexity and it gives a physically meaningful interpretation of the involving subsystems. In this paper, we focus this work on the simulation of pipes with a convex profile, for which a curvature coefficient is constant and negative. In the literature, it has been shown that such pipes have trapped modes. With the formalism of automatic control, adapted for “Waveguides”, we meet some substates of the system which do not take effect on the outputs. But, using the “Kelly-Lochbaum” framework with the horn equation, two problems occur: first, even if the outputs are bounded, some substates have their values which diverge; second, there is an infinite number of such substates. The system is then unstable and cannot be simulated as such. The solution of this problem is obtained with two steps. First, we show that there is a simple standard form compatible with the “Waveguide” approach, for which there is an infinite number of solutions which preserve the input/output relations. Second, we look for one solution which guarantees the stability of the system and which makes easier the approximation in order to get a low-cost simulation

Simulation en guides d'ondes numériques stables pour des tubes acoustiques à profil convexe

This work deals with the physical modelling of convex acoustic pipes for real-time simulation. The main purpose of this paper is the use the automatic control to solve some problems of stability

DDSP-Piano: A Neural Sound Synthesizer Informed by Instrument Knowledge

Instrument sound synthesis using deep neural networks has received numerous improvements over the last couple of years. Among them, the Differentiable Digital Signal Processing (DDSP) framework has modernized the spectral modeling paradigm by including signal-based synthesizers and effects into fully differentiable architectures. The present work extends the applications of DDSP to the task of polyphonic sound synthesis, with the proposal of a differentiable piano synthesizer conditioned on MIDI inputs. The model architecture is motivated by high-level acoustic modeling knowledge of the instrument, which, along with the sound structure priors inherent to the DDSP components, makes for a lightweight, interpretable, and realistic-sounding piano model. A subjective listening test has revealed that the proposed approach achieves better sound quality than a state-of-the-art neural-based piano synthesizer, but physical-modeling-based models still hold the best quality. Leveraging its interpretability and modularity, a qualitative analysis of the model behavior was also conducted: it highlights where additional modeling knowledge and optimization procedures could be inserted in order to improve the synthesis quality and the manipulation of sound properties. Eventually, the proposed differentiable synthesizer can be further used with other deep learning models for alternative musical tasks handling polyphonic audio and symbolic data

Considerations on travelling waves in the horn equation and energetic aspects

International audienceThe digital waveguide synthesis of wind resonators and of the vocal tract is based on decompositions into travelling waves. Typical ones are planar waves in straight pipes and spherical waves in conical pipes. However, approximating a bore by cascading such basic segments introduce unrealistic discontinuities on the radius R or the slope R' (with acoustic consequences). It also can generate artificial instabilities in time-domain simulations, e.g. for non convex junctions of cones. In this paper, we investigate the case of the "conservative curvilinear horn equation" for segments such that the flaring parameter R''/R is constant, with which smooth profiles can be built. First, acoustic states that generalize planar waves and spherical waves are studied. Examining the energy balance and the passivity for these travelling waves allows to characterize stability domains. Second, two other definitions of travelling waves are studied: (a) one locally diagonalizes the wave propagation operator, (b) one diagonalizes the transfer matrix of a segment. The propagators obtained for (a) are known to efficiently factorize computations in simulations but are not stable if the flaring parameter is negative. A study in the Laplace domain reveals that propagators (b) are stable for physically meaningful configurations

From a model of lossy flared pipes to a general framework for simulation of waveguides

This paper deals with the theory and application of waveguide modeling of lossy flared acoustic pipes, relying on the Webster-Lokshin equation. This model describes the propagation of longitudinal waves in axisymmetric acoustic pipes with a varying cross-section, visco-thermal losses at the walls, and without assuming planar waves or spherical waves. Solving this model for a piece of pipe leads to a two-port system made of four transfer functions which mimic the global acoustic effects. Moreover, introducing some relevant physical interpretations makes it possible to separate elementary effects due to the geometry of the piece of pipe (section, slope, and curvature) and isolate corresponding elementary transfer functions. From this decomposition a framework is obtained which allows to recover some digital waveguide models introduced earlier in the literature. This work contributes to the standardization of some different waveguide models, and brings a higher level of refinement that is visco-thermal losses combined with curvature effects