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

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

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

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

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

An Introduction to Model Selection: Tools and Algorithms

Model selection is a complicated matter in science, and psychology is no exception. In particular, the high variance in the object of study (i.e., humans) prevents the use of Poppers falsification principle (which is the norm in other sciences). Therefore, the desirability of quantitative psychological models must be assessed by measuring the capacity of the model to fit empirical data. In the present paper, an error measure (likelihood), as well as five methods to compare model fits (the likelihood ratio test, Akaikes information criterion, the Bayesian information criterion, bootstrapping and cross-validation), are presented. The use of each method is illustrated by an example, and the advantages and weaknesses of each method are also discussed

Understanding statistical power using noncentral probability distributions: Chi-squared, G-squared, and ANOVA

This paper presents a graphical way of interpreting effect sizes when more than two groups are involved in a statistical analysis. This method uses noncentral distributions to specify the alternative hypothesis, and the statistical power can thus be directly computed. This principle is illustrated using the chi-squared distribution and the F distribution. Examples of chi-squared and ANOVA statistical tests are provided to further illustrate the point. It is concluded that power analyses are an essential part of statistical analysis, and that using noncentral distributions provides an argument in favour of using a factorial ANOVA over multiple t tests

Common and Distinct Functional Brain Networks for Intuitive and Deliberate Decision Making

Reinforcement learning studies in rodents and primates demonstrate that goal-directed and habitual choice behaviors are mediated through different fronto-striatal systems, but the evidence is less clear in humans. In this study, functional magnetic resonance imaging (fMRI) data were collected whilst participants ( n = 20) performed a conditional associative learning task in which blocks of novel conditional stimuli (CS) required a deliberate choice, and blocks of familiar CS required an intuitive choice. Using standard subtraction analysis for fMRI event-related designs, activation shifted from the dorso-fronto-parietal network, which involves dorsolateral prefrontal cortex (DLPFC) for deliberate choice of novel CS, to ventro-medial frontal (VMPFC) and anterior cingulate cortex for intuitive choice of familiar CS. Supporting this finding, psycho-physiological interaction (PPI) analysis, using the peak active areas within the PFC for novel and familiar CS as seed regions, showed functional coupling between caudate and DLPFC when processing novel CS and VMPFC when processing familiar CS. These findings demonstrate separable systems for deliberate and intuitive processing, which is in keeping with rodent and primate reinforcement learning studies, although in humans they operate in a dynamic, possibly synergistic, manner particularly at the level of the striatum.Peer reviewedFinal Published versio