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

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

Delayed Dynamical Systems: Networks, Chimeras and Reservoir Computing

We present a systematic approach to reveal the correspondence between time delay dynamics and networks of coupled oscillators. After early demonstrations of the usefulness of spatio-temporal representations of time-delay system dynamics, extensive research on optoelectronic feedback loops has revealed their immense potential for realizing complex system dynamics such as chimeras in rings of coupled oscillators and applications to reservoir computing. Delayed dynamical systems have been enriched in recent years through the application of digital signal processing techniques. Very recently, we have showed that one can significantly extend the capabilities and implement networks with arbitrary topologies through the use of field programmable gate arrays (FPGAs). This architecture allows the design of appropriate filters and multiple time delays which greatly extend the possibilities for exploring synchronization patterns in arbitrary topological networks. This has enabled us to explore complex dynamics on networks with nodes that can be perfectly identical, introduce parameter heterogeneities and multiple time delays, as well as change network topologies to control the formation and evolution of patterns of synchrony

Optimal adaptive filter realizations for sampled stochastic processes with an unknown parameter

Optimal adaptive filter for sampled stochastic processes with unknown paramete