EP2717599A1 - Method for processing an audio signal with modelling of the overall response of the electro-dynamic loudspeaker - Google Patents

Abstract

The method involves determining a state vector by application of voltage and current measurements to a predictive filter estimator incorporating a representation of a dynamic model of an electrodynamic loudspeaker, where a predictive filter is an extended Kalman filter (16). The filter is adapted to operate a prediction of the vector and readjust the prediction by calculation of an estimate of a voltage (Uestn) based on the state vector, current measurement and comparison of the estimate to the voltage measurement. A processing function of the vector is applied to an audio signal.

Description

The invention relates to a technique for processing an audio signal based on the estimation of the overall response of a loudspeaker intended to reproduce this audio signal, that is to say taking into account all the parameters electrical, mechanical and acoustic characteristics characterizing this response.

It is a question of modeling the physical behavior of the loudspeaker to simulate the operation when the audio signal is applied to it after amplification, so as to be able to operate upstream various corrective treatments of this audio signal in order to optimize the quality of the final acoustic reproduction returned to the listener.

In particular, it is common to reinforce the low frequencies to compensate for the fact that the speakers dedicated to this register or woofers, which are generally installed in open enclosures ( vented system) or closed, are always more or less limited in the restitution of the most serious frequencies, the low limit (so-called cutoff frequency of the speaker) depending on the size of the speaker, the volume of the speaker and the type of editing used.

However, if the level of the low frequency electric signal is increased by appropriate analog or digital filtering, the excursion of the loudspeaker diaphragm, that is, the amplitude of its displacement by relative to its equilibrium position, quickly becomes too important, with the risk of damage to the loudspeaker and, at the very least, the introduction of excessive distortion, clipping and saturation values which rapidly degrade rapidly. the quality of playback of the audio signal.

The knowledge of the overall response of the loudspeaker makes it possible to anticipate this risk, to limit if necessary the level of the signal to be reproduced in order to avoid excessive excursions or nonlinearities generating distortions.

Another type of treatment that can be envisaged consists in applying to the audio signal a specific compensation filtering of the non-linearities introduced by the loudspeaker, in order to reduce the audio distortions and to provide a better quality of listening.

It is then, independently of any limitation of the maximum excursion, to make the displacement of the speaker's membrane the most linearly possible, especially for the most serious frequencies, by compensating for the physical limitations of the loudspeaker response in this register in the vicinity of and below the acoustic cut-off frequency of the loudspeaker / speaker assembly.

Knowledge of the parameters modeling the overall response of the loudspeaker is essential to operate such treatments.

These parameters are conventionally those of "Thiele and Small" (T / S), which describe a modeling of an electrodynamic loudspeaker taking into account the various electrical, mechanical and acoustic phenomena involved in the reproduction of the signal, as well as the electro-mechanical and mechano-acoustic conversions. The response of the loudspeaker, especially in the low frequencies, can be described by a set of parameters, referenced uniformly by the loudspeaker manufacturers.

• en premier lieu, ils sont susceptibles de dériver au cours du temps, en fonction par exemple du vieillissement du haut-parleur, de l'échauffement en cours d'utilisation, etc. ;
• en second lieu, si l'on souhaite disposer d'une modélisation précise et réaliste du comportement du haut-parleur, il faut tenir compte de ce que certains de ces paramètres ne sont pas linéaires, c'est-à-dire que leurs valeurs ne sont pas fixes mais varient constamment en fonction de l'excursion instantanée, c'est-à-dire de la position à un instant donné de la bobine mobile et de la membrane du haut-parleur par rapport à la position centrale d'équilibre. Tel est notamment le cas de l'inductance électrique, de la raideur mécanique totale du système (la raideur de la membrane augmentant au fur et à mesure que celle-ci s'éloigne de sa position d'équilibre) et du "facteur de force" d'entrainement de la membrane (lié au champ magnétique dans l'entrefer de la bobine, il décroit au fur et à mesure que la bobine s'éloigne de la position d'équilibre).

These T / S parameters are not constant in time, nor linear.

• in the first place, they are likely to drift over time, depending for example on the aging of the loudspeaker, the heating during use, etc. ;
• secondly, if one wishes to have a precise and realistic modeling of the behavior of the loudspeaker, one has to take into account that some of these parameters are not linear, that is to say that their values are not fixed but vary constantly depending on the instantaneous excursion, ie the position at a given moment of the voice coil and the speaker diaphragm relative to the central equilibrium position . This is particularly the case of the electrical inductance, the total mechanical stiffness of the system (the stiffness of the membrane increasing as it moves away from its equilibrium position) and the "force factor""Training of the membrane (related to the magnetic field in the air gap of the coil, it decreases as the coil moves away from the equilibrium position).

The EP 1 799 013 A1 describes a technique for predicting the behavior of a loudspeaker, based on the T / S parameters, to compensate for non-linearities of the loudspeaker and to reduce the audio distortions introduced into the acoustic signal delivered to the user.

The T / S parameters are, however, considered as invariants, known a priori, so that the modeling of the response is fixed and can not take into account either the slow evolution of the parameters, due for example to their drift over time because of the aging of the components.

The US 2003/0142832 A1 describes a technique for adaptively estimating the parameters of a loudspeaker, including non-linear parameters, from the measurement of the current flowing through this loudspeaker, with implementation of a gradient descent algorithm. This method requires a pre-determination of the parameters during a static calibration phase: during this calibration, the T / S parameters are calculated for different values of the position of the membrane (offset or offset with respect to the equilibrium position). ), with measurement of the impedance. Then, a measurement of the current is compared to an estimate of the same current (squared and filtered by a low-pass filter) to calculate the derivative of the error with respect to each parameter. The technique also implements a gradient descent algorithm of the least mean square type (LMS).

However, this method has the disadvantage of requiring a prior calibration phase with impedance measurements and application of a predetermined signal, which excludes a re-estimation of subsequent parameters, at least by a consumer user. On the other hand, the simple gradient descent LMS algorithms do not take into account the measurement noises, which are inevitable, which makes the estimator rather inefficient in real cases of use.

The US 2008/0189087 A1 describes another technique for estimating the parameters of a loudspeaker, also of LMS type by gradient descent. More particularly, the method processes separately the estimation of the linear part and that of the nonlinear part. For this, the error signal used by the LMS algorithm (difference between the measured signal and the predicted signal) is processed in order to decorrelate the linear part and the nonlinear part. This document also proposes to implement the estimator by applying as input a particular audio signal, modified by a comb filter selectively eliminating certain selected frequencies.

This technique has the same drawbacks as the previous one, in particular the need for a calibration from a modified input signal likely to alter the listening comfort of the user, which does not allow the user to operate. estimate during a musical listening, transparently for the user.

Yet another process is described in the academic dissertation of Marcus Arvidsson and Daniel Karlsson, Department of Electrical Engineering, Linköpings Universitet (SE), dated 18.06.2012, XP055053802 . This method is based on an observation vector comprising only electrical parameter measurements (voltage and current), which are applied to an extended Kalman predictive filter estimator. This estimator provides the prediction of a state vector whose components include the value of the excursion and the value of the current in the loudspeaker. But this method does not allow to estimate on the fly the parameters of both linear and non-linear speaker response and then apply appropriate corrective audio processing.

• qui prenne en compte de la façon la plus fidèle et la plus précise l'ensemble des non-linéarités de cette réponse, ainsi que les dérives éventuelles des paramètres, par une réévaluation périodique de ces paramètres ;
• qui n'introduise aucune modification ni dégradation du signal d'entrée qui pourrait altérer le confort d'écoute de l'utilisateur ;
• qui ne nécessite pour sa mise en oeuvre aucune calibration préalable ni application d'un signal spécifique (bruit blanc, etc.) ;
• qui soit immédiatement fonctionnel à partir de n'importe quel type de signal musical, par utilisation de ce signal "à la volée" pour le réajustement des paramètres de l'estimateur - en d'autres termes, qui puisse fonctionner de manière transparente pour l'utilisateur, l'estimateur opérant pendant que la musique est jouée et sur la base de cette musique, sans qu'il soit nécessaire de demander à l'utilisateur de jouer un type particulier de signal pour mettre en oeuvre l'algorithme d'estimation des paramètres du haut-parleur ; et
• qui, pour être compatible avec des produits grand public, ne nécessite que la mesure de paramètres électriques immédiatement accessibles (tension aux bornes du haut-parleur et intensité dans la bobine) et soit utilisable avec des haut-parleurs conventionnels, dépourvus de capteur électromécanique (capteur de déplacement, de pression acoustique, etc.) - en d'autres termes, où le déplacement mécanique de la membrane (excursion) reste une "variable cachée", non mesurée, de l'estimateur.

The problem of the invention is to have an estimator of the overall response of an electrodynamic loudspeaker:

• which takes into account in the most faithful and the most precise way all the non-linearities of this answer, as well as the possible drifts of the parameters, by a periodic reassessment of these parameters;
• that does not introduce any modification or degradation of the input signal that could affect the listening comfort of the user;
• which does not require for its implementation any prior calibration or application of a specific signal (white noise, etc.);
• which is immediately functional from any type of musical signal, by using this "on the fly" signal for readjustment of the parameters of the estimator - in other words, which can function transparently for user, the estimator operating while the music is playing and on the basis of this music, without the need to ask the user to play a particular type of signal to implement the estimation algorithm speaker settings; and
• which, to be compatible with consumer products, only requires the measurement of immediately accessible electrical parameters (voltage across the loudspeaker and current in the coil) and can be used with conventional loudspeakers, without electromechanical sensors ( displacement sensor, sound pressure sensor, etc.) - in other words, where the mechanical displacement of the diaphragm (excursion) remains an unmeasured "hidden variable" of the estimator.

1. a) la détermination d'un vecteur d'observation ne comprenant que des mesures de paramètres électriques, avec : une mesure de la tension aux bornes du haut-parleur, et une mesure du courant traversant le haut-parleur ;
2. b) la détermination d'un vecteur d'état par application des mesures de tension et de courant à un estimateur à filtre prédictif incorporant une représentation d'un modèle dynamique du haut-parleur,
   ce filtre prédictif étant un filtre de Kalman étendu apte à : opérer une prédiction du vecteur d'état à partir des mesures de tension et d'intensité, et recaler cette prédiction par calcul d'une estimée de la tension et comparaison de cette estimée à la mesure de la tension ; et
3. c) l'application au signal audio d'un traitement fonction dudit vecteur d'état.

For this purpose, the invention proposes a method for processing a digital audio signal of the general type disclosed by the aforementioned Arvidsson and Karlsson specification, namely a method comprising:

1. a) determining an observation vector comprising only measurements of electrical parameters, with: a measurement of the voltage across the loudspeaker, and a measurement of the current flowing through the loudspeaker;
2. b) determining a state vector by applying the voltage and current measurements to a predictive filter estimator incorporating a representation of a dynamic model of the loudspeaker,
   this predictive filter being an extended Kalman filter able to: make a prediction of the state vector from the voltage and intensity measurements, and readjust this prediction by calculating an estimate of the voltage and comparing this estimate with the measurement of the voltage; and
3. c) applying to the audio signal a processing function of said state vector.

• · des valeurs de paramètres linéaires de réponse du haut-parleur compris dans le groupe : résistance électrique et résistance mécanique, et
• · des coefficients polynomiaux de paramètres non-linéaires de réponse du haut-parleur compris dans le groupe : facteur de force, raideur équivalente et inductance électrique.

In a characteristic manner of the invention, the components of the state vector comprise:

• · Linear response parameter values of the loudspeaker included in the group: electrical resistance and mechanical resistance, and
• Polynomial coefficients of non-linear loudspeaker response parameters included in the group: force factor, equivalent stiffness and electrical inductance.

The processing applied to the audio signal may in particular be a compensation processing of the non-linearities of the loudspeaker response, as determined from the state vector delivered by the predictive filter estimator.

Alternatively or additionally, the processing applied to the audio signal may comprise: c1) calculating a current value of the loudspeaker excursion as a function of i) an amplification gain of the audio signal and ii) the loudspeaker response as determined from the state vector outputted by the predictive filter estimator; c2) comparing the current excursion value thus calculated with a maximum excursion value; and c3) calculating a possible attenuation of the amplification gain in case the current excursion value exceeds the maximum excursion value. Moreover, the state vector components may include additional acoustic parameter values representative of the loudspeaker response associated with a rear cavity provided with a decompression vent.

Very advantageously, the determination of the state vector of step b) and carried out on the fly from the current audio signal object of the treatment of step c) and reproduced by the loudspeaker, by collecting the electrical parameters at speaker terminals while playing this audio signal.

The method may then comprise the following steps: storing a sequence of samples of the audio signal for a predetermined duration; analyzing the sequence to calculate an energy parameter of the stored audio signal; if the calculated energy parameter is greater than a predetermined threshold, enabling the estimation by the predictive filter; if not, inhibit prediction filter estimation and retain the previously estimated state vector values.

• La Figure 1 est un schéma équivalent d'un haut-parleur électrodynamique faisant intervenir les différents paramètres T/S modélisant la réponse globale de celui-ci.
• La Figure 2 illustre, sous forme de schéma par blocs, les principales étapes de traitement du procédé de l'invention.
• La Figure 3 illustre plus précisément le fonctionnement de l'estimateur à filtre de Kalman étendu.

An embodiment of the invention will now be described with reference to the appended drawings in which the same reference numerals designate elements that are identical or functionally similar from one figure to another.

• The Figure 1 is an equivalent diagram of an electrodynamic loudspeaker involving the various T / S parameters modeling the overall response of the latter.
• The Figure 2 illustrates, in block diagram form, the main processing steps of the process of the invention.
• The Figure 3 illustrates more precisely the operation of the extended Kalman filter estimator.

Modeling the overall response of a speaker (Thiele and Small parameters)

We will first expose, with reference to the Figure 1 , the various parameters and equations describing the response of an electrodynamic loudspeaker HP, subjected to electrical excitation by a generator G and delivering a pressure signal on an acoustic load CH.

The left half schematizes the electrical part of the loudspeaker, to which is applied a measurable excitation voltage, Umes, from an amplifier producing a current i, also measurable, passing through the coil of the loudspeaker. The first report transformer BI schematizes the electrical conversion into mechanical force applied to the coil. Finally, the report gyrator Sd schematizes the mechanical conversion (displacement of the speaker membrane) in acoustic pressure.

The different components of this equivalent diagram (resistors, inductances and capacitance) model electrical, mechanical (for example the mass of the mobile coil / membrane) or acoustic (the air volume of the rear cavity of the speaker).

$$\mathrm{Bl}\left(\mathrm{x}\right)*\mathrm{i}\left(\mathrm{t}\right)+\mathrm{d}{\mathrm{L}}_{\mathrm{e}}\left(\mathrm{x}\left(\mathrm{t}\right)\right)/\mathrm{dx}*{\mathrm{i}}^{\mathrm{2}}\left(\mathrm{t}\right)={\mathrm{M}}_{\mathrm{ms}}*{\mathrm{d}}^{\mathrm{2}}\mathrm{x}/{\mathrm{dt}}^{\mathrm{2}}+{\mathrm{R}}_{\mathrm{eq}}*\mathrm{dx}/\mathrm{dt}+{\mathrm{K}}_{\mathrm{eq}}\left(\mathrm{x}\right)*\mathrm{x}$$

u étant la tension appliquée aux bornes du haut-parleur,
i étant le courant traversant la bobine,
x étant le déplacement de la membrane,
R_e étant la résistance électrique du système,
M_ms étant une masse équivalente modélisant la masse de l'équipage mobile totale du système,
R_eq étant une résistance équivalente modélisant les frottements et pertes mécaniques du système,
Le étant l'inductance électrique du système,
BI étant le facteur de force motrice (le produit du champ magnétique dans l'entrefer par la longueur de la bobine), et
K_eq étant une raideur équivalente modélisant la raideur globale de la suspension (spider, suspension externe et cavité).The system is governed by the following related equations (for a loudspeaker in the open air or mounted in a closed back cavity): $$\mathrm{u}\left(\mathrm{t}\right)={\mathrm{R}}_{\mathrm{e}}*\mathrm{i}\left(\mathrm{t}\right)+\mathrm{bl}\left(\mathrm{x}\right)*\mathrm{dx}/\mathrm{dt}+\mathrm{d}\left({\mathrm{The}}_{\mathrm{e}}\left(\mathrm{x}\left(\mathrm{t}\right)\right)*\mathrm{i}\left(\mathrm{t}\right)\right))/\mathrm{dt}$$

$$\mathrm{bl}\left(\mathrm{x}\right)*\mathrm{i}\left(\mathrm{t}\right)+\mathrm{d}{\mathrm{The}}_{\mathrm{e}}\left(\mathrm{x}\left(\mathrm{t}\right)\right)/\mathrm{dx}*{\mathrm{i}}^{\mathrm{2}}\left(\mathrm{t}\right)={\mathrm{M}}_{\mathrm{ms}}*{\mathrm{d}}^{\mathrm{2}}\mathrm{x}/{\mathrm{dt}}^{\mathrm{2}}+{\mathrm{R}}_{\mathrm{eq}}*\mathrm{dx}/\mathrm{dt}+{\mathrm{K}}_{\mathrm{eq}}\left(\mathrm{x}\right)*\mathrm{x}$$

u being the voltage applied across the loudspeaker,
i being the current flowing through the coil,
x being the displacement of the membrane,
R _e being the electrical resistance of the system,
M _ms being an equivalent mass modeling the mass of the total mobile equipment of the system,
R _eq being an equivalent resistance modeling the friction and mechanical losses of the system,
The being the electrical inductance of the system,
BI being the driving force factor (the product of the magnetic field in the air gap by the length of the coil), and
K _eq being an equivalent stiffness modeling the overall stiffness of the suspension (spider, external suspension and cavity).

The first three parameters (R _e , M _ms and R _eq ) are linear parameters, the equivalent mass M _ms being even an invariant, assumed to be known according to the manufacturer's specifications. On the other hand, R _e and R _eq , which can be considered constant over a short period (the time of their estimation) are parameters that can drift progressively over time as a function of the rise in temperature of the voice coil. aging of components, etc. and they must therefore be re-evaluated at regular intervals.

$${\mathrm{K}}_{\mathrm{eq}}\left(\mathrm{x}\right)={\mathrm{K}}_{\mathrm{eq}\mathrm{0}}+{\mathrm{K}}_{\mathrm{eq}1}\mathrm{x}+{\mathrm{K}}_{\mathrm{eq}\mathrm{2}}{\mathrm{x}}^{\mathrm{2}}$$

$${\mathrm{L}}_{\mathrm{e}}\left(\mathrm{x}\right)={\mathrm{L}}_{\mathrm{e}\mathrm{0}}+{\mathrm{L}}_{\mathrm{e}\mathrm{1}}\mathrm{x}+{\mathrm{L}}_{\mathrm{e}\mathrm{2}}{\mathrm{x}}^{\mathrm{2}+}{\mathrm{L}}_{\mathrm{e}\mathrm{3}}\mathrm{x}\mathrm{3}+{\mathrm{L}}_{\mathrm{e}\mathrm{4}}{\mathrm{x}}^{\mathrm{4}}$$

The last three parameters (Le, BI and K _eq ) are nonlinear parameters, which depend on the instantaneous value of the displacement x of the membrane. They can be approximated by polynomial models: $$\mathrm{bl}\left(\mathrm{x}\right)={\mathrm{bl}}_{\mathrm{0}}+{\mathrm{<figure-callout\; id="1"\; label="bl"\; filenames="imgf0001.png"\; state="\{\{state\}\}">bl</figure-callout>}}_{\mathrm{1}}\mathrm{x}+{\mathrm{bl}}_{\mathrm{2}}{\mathrm{x}}^{\mathrm{2}}$$

$${\mathrm{K}}_{\mathrm{eq}}\left(\mathrm{x}\right)={\mathrm{K}}_{\mathrm{eq}\mathrm{0}}+{\mathrm{K}}_{\mathrm{eq}1}\mathrm{x}+{\mathrm{K}}_{\mathrm{eq}\mathrm{2}}{\mathrm{x}}^{\mathrm{2}}$$

$${\mathrm{The}}_{\mathrm{e}}\left(\mathrm{x}\right)={\mathrm{The}}_{\mathrm{e}\mathrm{0}}+{\mathrm{The}}_{\mathrm{e}\mathrm{1}}\mathrm{x}+{\mathrm{The}}_{\mathrm{e}\mathrm{2}}{\mathrm{x}}^{\mathrm{2}+}{\mathrm{The}}_{\mathrm{e}\mathrm{3}}\mathrm{x}\mathrm{3}+{\mathrm{The}}_{\mathrm{e}\mathrm{4}}{\mathrm{x}}^{\mathrm{4}}$$

The complete knowledge of the model therefore requires the determination of the linear parameters R _e and R _eq , and that of the polynomial coefficients of the nonlinear parameters BI, K _eq and Le.

The set of these parameters will be called later "state vector" X, with X = [R _e , R _eq , Bl ₀ , Bl ₁ , Bl ₂ , K _eq0 , K _eq1 , K _eq2 , L _e0 , L _e1 , L _e2 , L _e3 , L _e4 ] ^T.

The displacement x, which is an unmeasured parameter, will be a hidden variable of the estimator.

$$\mathrm{Bl}\left({\mathrm{x}}_{\mathrm{n}}\right)*{\mathrm{i}}_{\mathrm{n}}+\mathrm{Leʹ}\left({\mathrm{x}}_{\mathrm{n}}\right)*{{\mathrm{i}}_{\mathrm{n}}}^{\mathrm{2}}={\mathrm{M}}_{\mathrm{ms}}*{\mathrm{F}}_{\mathrm{s}}*\left({\mathrm{v}}_{\mathrm{n}+\mathrm{1}}-{\mathrm{v}}_{\mathrm{n}}\right)+{\mathrm{R}}_{\mathrm{eq}}*{\mathrm{v}}_{\mathrm{n}}+{\mathrm{K}}_{\mathrm{eq}}\left({\mathrm{x}}_{\mathrm{n}}\right)*{\mathrm{x}}_{\mathrm{n}}$$

où v_n = F_s*(x_n+1-X_n) représente la vitesse de déplacement de la membrane, F_s étant la fréquence d'échantillonnage et j_n = F_s*(i_n+1-i_n) étant la dérivée du courant.Since the previous equations are written in continuous time, if we want to pass in discrete time (corresponding to a numerical sampling), we use the Euler transform, which gives: $${\mathrm{u}}_{\mathrm{not}}={\mathrm{R}}_{\mathrm{e}}*{\mathrm{i}}_{\mathrm{not}}+{\mathrm{The}}_{\mathrm{e}}\mathrm{\text{'}}\left({\mathrm{x}}_{\mathrm{not}}\right)*{\mathrm{v}}_{\mathrm{not}}*{\mathrm{i}}_{\mathrm{not}}+{\mathrm{The}}_{\mathrm{e}}\left({\mathrm{x}}_{\mathrm{not}}\right)*{\mathrm{j}}_{\mathrm{not}}+\mathrm{bl}\left({\mathrm{x}}_{\mathrm{not}}\right)*{\mathrm{v}}_{\mathrm{not}}$$

$$\mathrm{bl}\left({\mathrm{x}}_{\mathrm{not}}\right)*{\mathrm{i}}_{\mathrm{not}}+\mathrm{The}\left({\mathrm{x}}_{\mathrm{not}}\right)*{{\mathrm{i}}_{\mathrm{not}}}^{\mathrm{2}}={\mathrm{M}}_{\mathrm{ms}}*{\mathrm{F}}_{\mathrm{s}}*\left({\mathrm{v}}_{\mathrm{not}+\mathrm{1}}-{\mathrm{v}}_{\mathrm{not}}\right)+{\mathrm{R}}_{\mathrm{eq}}*{\mathrm{v}}_{\mathrm{not}}+{\mathrm{K}}_{\mathrm{eq}}\left({\mathrm{x}}_{\mathrm{not}}\right)*{\mathrm{x}}_{\mathrm{not}}$$

where v _n = F _s * (x _{n + 1} -X _n ) represents the speed of displacement of the membrane, F _s being the sampling frequency and j _n = F _s * (i _{n + 1} -i _n ) being the derivative of the current.

où xp (qui sera une seconde variable cachée de l'estimateur) représente le déplacement de la masse d'air contenue dans l'évent, et M_pm, R_boxm, K_boxm et R_pm sont des paramètres connus dépendant de la taille de l'évent et de la cavité arrière.It should be noted that this system of equations can also be extended to the estimation of the response of a loudspeaker mounted with a rear cavity having an outward vent, for example of the "bass-reflex" type. It is then necessary to add to the model a third equation: $${\mathrm{xp}}_{\mathrm{not}}=\mathrm{2}*{\mathrm{xp}}_{\mathrm{not}-\mathrm{1}}-{\mathrm{xp}}_{\mathrm{not}-\mathrm{2}}+\left(-{\mathrm{F}}_{\mathrm{s}}*\left({\mathrm{R}}_{\mathrm{boxm}}+{\mathrm{R}}_{\mathrm{pm}}\right)*\left({\mathrm{xp}}_{\mathrm{not}-\mathrm{1}}-{\mathrm{xp}}_{\mathrm{not}-\mathrm{2}}\right)-{\mathrm{K}}_{\mathrm{boxm}}*\left({\mathrm{xp}}_{\mathrm{not}}+{\mathrm{x}}_{\mathrm{not}}\right)-{\mathrm{R}}_{\mathrm{boxm}}*{\mathrm{F}}_{\mathrm{s}}*\left({\mathrm{x}}_{\mathrm{not}+\mathrm{1}}-{\mathrm{x}}_{\mathrm{not}}\right)\right)/\left({{\mathrm{F}}_{\mathrm{s}}}^{\mathrm{2}}*{\mathrm{M}}_{\mathrm{pm}}\right)$$

where xp (which will be a second hidden variable of the estimator) represents the displacement of the mass of air contained in the vent, and M _pm , R _boxm , K _boxm and R _pm are known parameters depending on the size of the the vent and the rear cavity.

Application of an extended Kalman filter to estimate the response of a loudspeaker

With reference to Figures 2 and 3 , the method of the invention will now be described, making it possible to estimate the various parameters of the loudspeaker in order to apply to the audio signal appropriate processing taking into account the modeling of the response thereof.

It will be noted that, although these diagrams are presented in the form of interconnected circuits, the implementation of the various functions is essentially software, this representation having no illustrative character. The software can in particular be implemented within a dedicated digital signal processing chip of the DSP type.

In concrete terms, the processes that will be described are performed on previously digitized signals, the algorithms being executed iterative at the sampling frequency for the successive signal frames, for example frames of 1024 samples.

In a characteristic way, the present invention implements a Kalman filtering, and more precisely an extended Kalman filtering (EKF), of which we will re-outline the main lines below.

Basic principles of the extended Kalman filter

The "Kalman filter", which is based on a widely known algorithm, is a state estimator comprising an infinite impulse response (IIR) filter that estimates the states of a dynamic system from a set of equations describing the behavior of the system and a series of observed measures.

Such a filter makes it possible in particular to determine a "hidden state", which is a parameter not observed but essential for the estimation.

• le système dynamique est la réponse du haut-parleur ;
• les équations décrivant le comportement du système sont les Équations (1), (2) et éventuellement (3) ci-dessus ;
• les mesures observées appliquées en entrée du filtre sont la tension appliquée aux bornes du haut-parleur et le courant traversant la bobine de celui-ci ; et
• l'état caché est l'excursion instantanée, à savoir le déplacement physique de la membrane par rapport à sa position d'équilibre, qui est un paramètre essentiel pour l'estimation des paramètres non linéaires de la réponse du haut-parleur, comme exposé plus haut.

In the present case :

• the dynamic system is the response of the speaker;
• the equations describing the behavior of the system are equations (1), (2) and possibly (3) above;
• the observed measurements applied at the input of the filter are the voltage applied across the loudspeaker and the current flowing through the coil thereof; and
• the hidden state is the instantaneous excursion, ie the physical displacement of the membrane with respect to its equilibrium position, which is an essential parameter for the estimation of the non-linear parameters of the loudspeaker response, as explained upper.

• 1°) une phase de prédiction, effectuée à chaque itération du filtre : cette phase consiste à prédire la réponse du haut-parleur à l'instant courant par rapport à l'instant précédent selon une équation d'évolution ; et
• 2°) une phase de recalage, qui consiste à corriger la prédiction en utilisant les mesures courantes (tension, courant) : la modélisation de la réponse est alors adaptée et mise à jour pour tenir compte notamment des erreurs de mesure systématique.

The Kalman filter operates in two phases, with successively:

• 1 °) a prediction phase, performed at each iteration of the filter: this phase consists in predicting the response of the loudspeaker at the current time with respect to the previous instant according to an evolution equation; and
• 2 °) a resetting phase, which consists in correcting the prediction using current measurements (voltage, current): the response modeling is then adapted and updated to take account of systematic measurement errors.

Applying the extended Kalman filter to estimate speaker response

x _k étant le vecteur d'état, représentant l'état à l'instant k,
F _k étant la matrice de transition (définie à la conception du filtre) qui détermine l'évolution de l'état k-1 au nouvel état k,
B _k étant un vecteur de bruit (bruit gaussien engendré par les capteurs),
u _k étant un vecteur de contrôle (paramètre en entrée du filtre), et
w _k étant un état représentant le bruit à l'instant k.In general, if we adopt the formalism of the state representation, the first equation of the Kalman process is the "equation of evolution" of the model: $${\mathrm{x}}_k={\mathbf{F}}_k{\mathrm{x}}_{k-1}+{\mathbf{B}}_k{\mathbf{u}}_k+{\mathbf{w}}_k$$

x _k being the state vector, representing the state at time k,
F _k being the transition matrix (defined at the design of the filter) which determines the evolution of the state k-1 to the new state k,
B _k being a noise vector (Gaussian noise generated by the sensors),
u _k being a control vector (parameter at the input of the filter), and
w _k being a state representing noise at time k .

In this case, the state vector x _k is the vector composed of the parameters of the loudspeaker model: $${\mathbf{x}}_{\mathit{k}}={\left[\mathrm{Re}{\mathrm{R}}_{\mathrm{eq}}{\mathrm{bl}}_{\mathrm{0}}{\mathrm{<figure-callout\; id="1"\; label="bl"\; filenames="imgf0001.png"\; state="\{\{state\}\}">bl</figure-callout>}}_{\mathrm{1}}{\mathrm{bl}}_{\mathrm{2}}{\mathrm{K}}_{\mathrm{eq}\mathrm{0}}{\mathrm{K}}_{\mathrm{eq}\mathrm{1}}{\mathrm{K}}_{\mathrm{eq}\mathrm{2}}{\mathrm{The}}_{\mathrm{e}\mathrm{0}}{\mathrm{The}}_{\mathrm{e}\mathrm{1}}{\mathrm{The}}_{\mathrm{e}\mathrm{2}}{\mathrm{The}}_{\mathrm{e}\mathrm{3}}{\mathrm{The}}_{\mathrm{e}\mathrm{4}}\right]}^{\mathrm{T}}$$

z _k étant le vecteur d'observation à l'instant k (mesures de tension et de courant),
H _k étant la matrice de mesure à l'instant k, c'est-à-dire la matrice d'observation reliant l'état à la mesure, déterminée à la conception du filtre, et v _k étant le vecteur de bruit de la mesure à l'instant k. The second equation of the Kalman process is the "measurement equation": $${\mathbf{z}}_k={\mathbf{H}}_k{\mathbf{x}}_k+{\mathbf{v}}_k$$

z _k being the observation vector at instant k (voltage and current measurements),
H _k being the measurement matrix at time k, that is to say the observation matrix connecting the state to the measurement, determined at the design of the filter, and v _k being the noise vector of the measure at the instant k.

Covariance de prédiction (a priori) $${\mathbf{P}}_{k|k-1}={\mathbf{F}}_k{\mathbf{P}}_{k-1|k-1}{\mathbf{F}}_k^{\mathrm{T}}+{\mathbf{Q}}_k$$

The first step is the prediction of the model at time k, from the state at time k-1, given by the following equations: Prediction ( a priori ) of the estimated state $${\hat{\mathbf{x}}}_{k|k-1}={\mathbf{F}}_k{\hat{\mathbf{x}}}_{k-1|k-1}+{\mathbf{B}}_k{\mathbf{u}}_k$$

Covariance of prediction ( a priori ) $${\mathbf{P}}_{k|k-1}={\mathbf{F}}_k{\mathbf{P}}_{k-1|k-1}{\mathbf{F}}_k^{\mathrm{T}}+{\mathbf{Q}}_k$$

Covariance de l'innovation $${\mathbf{S}}_k={\mathbf{H}}_k{\mathbf{P}}_{k|k-1}{\mathbf{H}}_k^{\mathbf{T}}+{\mathbf{R}}_k$$

Gain de Kalman optimal $${\mathbf{K}}_k={\mathbf{P}}_{k|k-1}{\mathbf{H}}_k^{\mathrm{T}}{\mathbf{S}}_k^{-1}$$

Mise à jour (a posteriori) de l'état estimé $${\tilde{\mathbf{x}}}_{k|k}={\hat{\mathbf{x}}}_{k|k-1}+{\mathbf{K}}_k{\tilde{\mathbf{y}}}_k$$

Mise à jour (a posteriori) de la covariance $${\mathbf{P}}_{k|k}=\left(I-{\mathbf{K}}_k{\mathbf{H}}_k\right){\mathbf{P}}_{k|k-1}$$

The second step is the updating of the model, thanks to the observation of the measurement at instant k, by the following system of equations:
Innovation or measurement residue $${\widetilde{\mathbf{there}}}_k={\mathbf{z}}_k-{\mathbf{H}}_k{\hat{\mathbf{x}}}_{k|k-1}$$

Covariance of innovation $${\mathbf{S}}_k={\mathbf{H}}_k{\mathbf{P}}_{k|k-1}{\mathbf{H}}_k^{\mathbf{T}}+{\mathbf{R}}_k$$

Optimal Kalman gain $${\mathbf{K}}_k={\mathbf{P}}_{k|k-1}{\mathbf{H}}_k^{\mathrm{T}}{\mathbf{S}}_k^{-1}$$

Update ( a posteriori ) of the estimated state $${\tilde{\mathbf{x}}}_{k|k}={\hat{\mathbf{x}}}_{k|k-1}+{\mathbf{K}}_k{\widetilde{\mathbf{there}}}_k$$

Update ( a posteriori) of the covariance $${\mathbf{P}}_{k|k}=\left(I-{\mathbf{K}}_k{\mathbf{H}}_k\right){\mathbf{P}}_{k|k-1}$$

In the case of a linear system, the Kalman estimate is optimal in the least squares sense of the hidden model.

However, it has been seen above that the dynamic model of response of the loudspeaker used is not a linear model, so that the Kalman filtering that has just been described is not applicable to the present invention. .

For this reason, the method used will be that known under the name of "extended Kalman filtering" or EKF.

$${\mathbf{z}}_k=h\left({\mathbf{x}}_k\right)+{\mathbf{v}}_k$$

f et h étant des fonctions non linéaires, mais différentiables.The equation of evolution of the model and the equation of measure are in the form: $${\mathbf{x}}_k=f\left({\mathbf{x}}_{k-1}{\mathbf{u}}_k\right)+{\mathbf{w}}_k$$

$${\mathbf{z}}_k=h\left({\mathbf{x}}_k\right)+{\mathbf{v}}_k$$

f and h being nonlinear but differentiable functions.

$${\mathbf{P}}_{k|k-1}={\mathbf{F}}_{k-1}{\mathbf{P}}_{k-1|k-1}{\mathbf{F}}_{k-1}^{\mathrm{T}}+{\mathbf{Q}}_{k-1}$$

et :
Innovation ou résidu de mesure $${\tilde{\mathbf{y}}}_k={\mathbf{z}}_k-h\left({\hat{\mathbf{x}}}_{k|k-1}\right)$$

Covariance de l'innovation $${\mathbf{S}}_k={\mathbf{H}}_k{\mathbf{P}}_{k|k-1}{\mathbf{H}}_k^{\mathrm{T}}+{\mathbf{R}}_k$$

Gain de Kalman presque-optimal $${\mathbf{K}}_k={\mathbf{P}}_{k|k-1}{\mathbf{H}}_k^{\mathrm{T}}{\mathbf{S}}_k^{-1}$$

Mise à jour (a posteriori) de l'état estimé $${\tilde{\mathbf{x}}}_{k|k}={\hat{\mathbf{x}}}_{k|k-1}+{\mathbf{K}}_k{\tilde{\mathbf{y}}}_k$$

Mise à jour (a posteriori) de la covariance $${\mathbf{P}}_{k|k}=\left(I-{\mathbf{K}}_k{\mathbf{H}}_k\right){\mathbf{P}}_{k|k-1}$$

Extended Kalman filtering consists of approximating these functions f and h by their partial derivatives during the computation of the covariance matrices (prediction matrix and update matrix), in order to locally linearize the model and to apply to it at each point the systems of Kalman filter prediction and update equations discussed above. These systems of equations become, respectively:
Prediction ( a priori ) of the estimated state
Covariance of prediction ( a priori ) $${\hat{\mathbf{x}}}_{k|k-1}=f\left({\hat{\mathbf{x}}}_{k-1|k-1}{\mathbf{u}}_{k-1}\right)$$

$${\mathbf{P}}_{k|k-1}={\mathbf{F}}_{k-1}{\mathbf{P}}_{k-1|k-1}{\mathbf{F}}_{k-1}^{\mathrm{T}}+{\mathbf{Q}}_{k-1}$$

and
Innovation or measurement residue $${\widetilde{\mathbf{there}}}_k={\mathbf{z}}_k-h\left({\hat{\mathbf{x}}}_{k|k-1}\right)$$

Covariance of innovation $${\mathbf{S}}_k={\mathbf{H}}_k{\mathbf{P}}_{k|k-1}{\mathbf{H}}_k^{\mathrm{T}}+{\mathbf{R}}_k$$

Near-optimal Kalman gain $${\mathbf{K}}_k={\mathbf{P}}_{k|k-1}{\mathbf{H}}_k^{\mathrm{T}}{\mathbf{S}}_k^{-1}$$

Update ( a posteriori ) of the estimated state $${\tilde{\mathbf{x}}}_{k|k}={\hat{\mathbf{x}}}_{k|k-1}+{\mathbf{K}}_k{\widetilde{\mathbf{there}}}_k$$

Update (posterior) of the covariance $${\mathbf{P}}_{k|k}=\left(I-{\mathbf{K}}_k{\mathbf{H}}_k\right){\mathbf{P}}_{k|k-1}$$

$${\mathbf{H}}_k=\frac{\partial h}{\partial \mathbf{x}}{|}_{{\hat{\mathbf{x}}}_{k|k-1}}$$

The transition matrix and the observation matrix are the following Jacobian matrices (partial derivative matrices): $${\mathbf{F}}_{k-1}=\frac{\partial f}{\partial \mathbf{x}}{|}_{{\hat{\mathbf{x}}}_{k-1|k-1},{\mathbf{u}}_{k-1}}$$

$${\mathbf{H}}_k=\frac{\partial h}{\partial \mathbf{x}}{|}_{{\hat{\mathbf{x}}}_{k|k-1}}$$

Practical implementation of the Kalman filter extended to the processing of an audio signal reproduced by a loudspeaker

The procedure that has just been described can be implemented in the manner illustrated schematically on the Figure 2 .

A digitized audio signal E coming from a media player is reproduced acoustically by a loudspeaker 10 after digital / analog conversion (block 12) and amplification (block 14).

The response of the loudspeaker 10 is simulated by an extended Kalman filter algorithm (estimator of the block 16) using as input the signals 18 collected on the loudspeaker 10, these signals comprising the voltage Umes applied to the terminals of the loudspeaker by the amplifier 14 and the current i flowing in the voice coil of the loudspeaker.

More specifically, the operation of the extended Kalman filter 16 with reference to FIG. Figure 3 where block 20 schematizes the estimator of the Kalman filter based on the modeling of the response of the loudspeaker, block 22 the function h of the measurement equation and block 24 the comparison between estimated state and measured state, to derive an error signal for updating the dynamic model.

The parameters of the model to be estimated form at time n the state vector X _n (the parameter M _ms of the model being assumed to be known and invariant): $${\mathrm{X}}_{\mathrm{not}}={\left[{\mathrm{bl}}_{\mathrm{0}}{\mathrm{K}}_{\mathrm{eq}\mathrm{0}}{\mathrm{The}}_{\mathrm{0}}{\mathrm{R}}_{\mathrm{eq}}{\mathrm{R}}_{\mathrm{e}}{\mathrm{bl}}_{\mathrm{1}}{\mathrm{K}}_{\mathrm{eq}\mathrm{1}}{\mathrm{The}}_{\mathrm{1}}{\mathrm{bl}}_{\mathrm{2}}{\mathrm{K}}_{\mathrm{eq}\mathrm{2}}{\mathrm{The}}_{\mathrm{e}\mathrm{2}}{\mathrm{The}}_{\mathrm{e}\mathrm{3}}{\mathrm{The}}_{\mathrm{e}\mathrm{4}}\right]}^{\mathrm{T}}$$

The model of the loudspeaker response is considered to be invariant for the time required for the estimation. For example, if we use a fraction of T = 10 seconds of signal for the estimation, we will assume that the model remains the same during this duration T, with a noise of evolution.

Therefore, the equation of evolution of the state is simply summarized in: $${\mathrm{X}}_{\mathrm{not}+\mathrm{1}}={\mathrm{X}}_{\mathrm{not}}$$

X_n étant ici une variable cachée du déplacement, calculée récursivement à l'aide des Équations (1) et (2).The measurement of the voltage across the loudspeaker constitutes the only component of the observation vector Umes _n-1 . This measurement is compared at the estimated voltage U is _n = h (X _n ) obtained with the estimates of the parameters of the instant n and the measured current i: $${\mathrm{uest}}_{\mathrm{not}}={\mathrm{R}}_{\mathrm{e}}*{\mathrm{i}}_{\mathrm{not}}+{\mathrm{The}}_{\mathrm{e}}\mathrm{\text{'}}\left({\mathrm{x}}_{\mathrm{not}}\right)*{\mathrm{v}}_{\mathrm{not}}*{\mathrm{i}}_{\mathrm{not}}+{\mathrm{The}}_{\mathrm{e}}\left({\mathrm{x}}_{\mathrm{not}}\right)*{\mathrm{j}}_{\mathrm{not}}+\mathrm{bl}\left({\mathrm{x}}_{\mathrm{not}}\right)*{\mathrm{v}}_{\mathrm{not}}$$

X _n being here a hidden variable of the displacement, computed recursively using Equations (1) and (2).

The algorithm then calculates the derivative of the function h with respect to each of the components of the vector X: dh (X) / dBl0, dh (X) / dKeq0, ... which corresponds to the partial derivative of the estimated voltage, relative to each of the parameters of the model.

et en dérivant et réarrangeant l'Équation (2) par rapport à p : $$\begin{array}{ll}\mathrm{d}\left({\mathrm{v}}_{\mathrm{n}}\right)/\mathrm{dp}=& \left(\mathrm{1}-{\mathrm{T}}_{\mathrm{s}}*{\mathrm{R}}_{\mathrm{eq}}/{\mathrm{M}}_{\mathrm{ms}}\right)*\mathrm{d}\left({\mathrm{v}}_{\mathrm{n}-\mathrm{1}}\right)/\mathrm{dp}+\\ & {\mathrm{T}}_{\mathrm{s}}/{\mathrm{M}}_{\mathrm{ms}}*\left({\mathrm{L}}_{\mathrm{e}}\mathrm{ʺ}\left({\mathrm{x}}_{\mathrm{n}-\mathrm{1}}\right)*{{\mathrm{i}}^{\mathrm{2}}}_{\mathrm{n}-\mathrm{1}}+\mathrm{Blʹ}\left({\mathrm{x}}_{\mathrm{n}-\mathrm{1}}\right)*{\mathrm{i}}_{\mathrm{n}-\mathrm{1}}-{\mathrm{K}}_{\mathrm{eq}}\mathrm{ʹ}\left({\mathrm{x}}_{\mathrm{n}-\mathrm{1}}\right){\mathrm{x}}_{\mathrm{n}-\mathrm{1}}\right)\\ & -{\mathrm{K}}_{\mathrm{eq}}\left({\mathrm{x}}_{\mathrm{n}-\mathrm{1}}\right))*\mathrm{d}\left({\mathrm{x}}_{\mathrm{n}-\mathrm{1}}\right)/\mathrm{dp}+{\mathrm{T}}_{\mathrm{s}}/{\mathrm{M}}_{\mathrm{ms}}*(\mathrm{dBl}\left({\mathrm{x}}_{\mathrm{n}-\mathrm{1}}\right)/\mathrm{dp}*{\mathrm{i}}_{\mathrm{n}-\mathrm{1}}\\ & +{\mathrm{dL}}_{\mathrm{e}}\mathrm{ʹ}\left({\mathrm{x}}_{\mathrm{n}-\mathrm{1}}\right)/\mathrm{dp}*{{\mathrm{i}}^{\mathrm{2}}}_{\mathrm{n}-\mathrm{1}}-{\mathrm{dK}}_{\mathrm{eq}}\left({\mathrm{x}}_{\mathrm{n}-\mathrm{1}}\right)/\mathrm{dp}*{\mathrm{x}}_{\mathrm{n}-\mathrm{1}}-{\mathrm{dR}}_{\mathrm{eq}}\left(\mathrm{n}-\mathrm{1}\right)/\mathrm{dp}*{\mathrm{v}}_{\mathrm{n}-\mathrm{1}})\end{array}$$

et : $$\mathrm{d}\left({\mathrm{x}}_{\mathrm{n}}\right)/\mathrm{dp}=\mathrm{d}\left({\mathrm{x}}_{\mathrm{n}-\mathrm{1}}\right)/\mathrm{dp}+{\mathrm{T}}_{\mathrm{s}}*\mathrm{d}\left({\mathrm{v}}_{\mathrm{n}-\mathrm{1}}\right)/\mathrm{dp}$$

If in general we denote by p one of these parameters, we derive Equation (1) with respect to p: $$\mathrm{d}\left({\mathrm{uest}}_{\mathrm{not}}\right)/\mathrm{dp}=\left({\mathrm{The}}_{\mathrm{e}}\mathrm{"}\left({\mathrm{x}}_{\mathrm{not}}\right){\mathrm{v}}_{\mathrm{not}}{\mathrm{i}}_{\mathrm{not}}+\mathrm{The}\left({\mathrm{x}}_{\mathrm{not}}\right){\mathrm{j}}_{\mathrm{not}}+\mathrm{Bl\text{'}}\left({\mathrm{x}}_{\mathrm{not}}\right){\mathrm{v}}_{\mathrm{not}}\right)*{\mathrm{dx}}_{\mathrm{not}}/\mathrm{dp}+\left({\mathrm{The}}_{\mathrm{e}}\mathrm{\text{'}}\left({\mathrm{x}}_{\mathrm{not}}\right){\mathrm{i}}_{\mathrm{not}}+\mathrm{bl}\left({\mathrm{x}}_{\mathrm{not}}\right)\right)*{\mathrm{dv}}_{\mathrm{not}}/\mathrm{dp}+\mathrm{DBL}\left({\mathrm{x}}_{\mathrm{not}}\mathrm{p}\right)/\mathrm{dp}*{\mathrm{v}}_{\mathrm{not}}+{\mathrm{dL}}_{\mathrm{e}}\left({\mathrm{x}}_{\mathrm{not}}\mathrm{p}\right)/\mathrm{dp}*{\mathrm{j}}_{\mathrm{not}}+{\mathrm{dL}}_{\mathrm{e}}\mathrm{\text{'}}\left({\mathrm{x}}_{\mathrm{not}}\mathrm{p}\right)/\mathrm{dp}*{\mathrm{v}}_{\mathrm{not}}*{\mathrm{i}}_{\mathrm{not}}$$

and by deriving and rearranging Equation (2) with respect to p: $$\begin{array}{ll}\mathrm{d}\left({\mathrm{v}}_{\mathrm{not}}\right)/\mathrm{dp}=& \left(\mathrm{1}-{\mathrm{T}}_{\mathrm{s}}*{\mathrm{R}}_{\mathrm{eq}}/{\mathrm{M}}_{\mathrm{ms}}\right)*\mathrm{d}\left({\mathrm{v}}_{\mathrm{not}-\mathrm{1}}\right)/\mathrm{dp}+\\ & {\mathrm{T}}_{\mathrm{s}}/{\mathrm{M}}_{\mathrm{ms}}*\left({\mathrm{The}}_{\mathrm{e}}\mathrm{"}\left({\mathrm{x}}_{\mathrm{not}-\mathrm{1}}\right)*{{\mathrm{i}}^{\mathrm{2}}}_{\mathrm{not}-\mathrm{1}}+\mathrm{Bl\text{'}}\left({\mathrm{x}}_{\mathrm{not}-\mathrm{1}}\right)*{\mathrm{i}}_{\mathrm{not}-\mathrm{1}}-{\mathrm{K}}_{\mathrm{eq}}\mathrm{\text{'}}\left({\mathrm{x}}_{\mathrm{not}-\mathrm{1}}\right){\mathrm{x}}_{\mathrm{not}-\mathrm{1}}\right)\\ & -{\mathrm{K}}_{\mathrm{eq}}\left({\mathrm{x}}_{\mathrm{not}-\mathrm{1}}\right))*\mathrm{d}\left({\mathrm{x}}_{\mathrm{not}-\mathrm{1}}\right)/\mathrm{dp}+{\mathrm{T}}_{\mathrm{s}}/{\mathrm{M}}_{\mathrm{ms}}*(\mathrm{DBL}\left({\mathrm{x}}_{\mathrm{not}-\mathrm{1}}\right)/\mathrm{dp}*{\mathrm{i}}_{\mathrm{not}-\mathrm{1}}\\ & +{\mathrm{dL}}_{\mathrm{e}}\mathrm{\text{'}}\left({\mathrm{x}}_{\mathrm{not}-\mathrm{1}}\right)/\mathrm{dp}*{{\mathrm{i}}^{\mathrm{2}}}_{\mathrm{not}-\mathrm{1}}-{\mathrm{dK}}_{\mathrm{eq}}\left({\mathrm{x}}_{\mathrm{not}-\mathrm{1}}\right)/\mathrm{dp}*{\mathrm{x}}_{\mathrm{not}-\mathrm{1}}-{\mathrm{dR}}_{\mathrm{eq}}\left(\mathrm{not}-\mathrm{1}\right)/\mathrm{dp}*{\mathrm{v}}_{\mathrm{not}-\mathrm{1}})\end{array}$$

and $$\mathrm{d}\left({\mathrm{x}}_{\mathrm{not}}\right)/\mathrm{dp}=\mathrm{d}\left({\mathrm{x}}_{\mathrm{not}-\mathrm{1}}\right)/\mathrm{dp}+{\mathrm{T}}_{\mathrm{s}}*\mathrm{d}\left({\mathrm{v}}_{\mathrm{not}-\mathrm{1}}\right)/\mathrm{dp}$$

These equations allow recursively calculating the Jacobian matrix (which, in this case, is a simple vector): $$\mathrm{H}=\left[\mathrm{dUest}/{\mathrm{DBL}}_{\mathrm{0}},\mathrm{dUest}/{\mathrm{dK}}_{\mathrm{eq}\mathrm{0}},...,\mathrm{dUest}/{\mathrm{dL}}_{\mathrm{e}\mathrm{4}}\right]$$

• 1°) Prédiction du système (par utilisation du modèle et du bruit du modèle) : $${\mathrm{X}}_{\mathrm{n}|\mathrm{n}-\mathrm{1}}={\mathrm{X}}_{\mathrm{n}-\mathrm{1}|\mathrm{n}-\mathrm{1}}$$
  
  $${\mathrm{P}}_{\mathrm{n}|\mathrm{n}-\mathrm{1}}={\mathrm{P}}_{\mathrm{n}-\mathrm{1}|\mathrm{n}-\mathrm{1}}+{\mathrm{Q}}_{\mathrm{n}}$$
  
  
  Q_n étant la matrice de covariance du bruit de modèle
• 2°) Mise à jour du système : $${\mathrm{Uest}}_{\mathrm{n}}=\mathrm{h}\left({\mathrm{X}}_{\mathrm{n}|\mathrm{n}-\mathrm{1}}\right)$$
  
  $${\mathrm{Uerror}}_{\mathrm{n}}={\mathrm{Umes}}_{\mathrm{n}}-{\mathrm{Uest}}_{\mathrm{n}}$$
  
  $$\mathrm{Calcul\; de}{\mathrm{H}}_{\mathrm{n}}=\left[{\mathrm{dUest}}_{\mathrm{n}}/{\mathrm{dBl}}_{\mathrm{0}},{\mathrm{dUest}}_{\mathrm{n}}/{\mathrm{dK}}_{\mathrm{eq}\mathrm{0}},\dots ,{\mathrm{dUest}}_{\mathrm{n}}/{\mathrm{dL}}_{\mathrm{e}\mathrm{4}}\right]$$
  
  $${\mathrm{S}}_{\mathrm{n}}={\mathrm{H}}_{\mathrm{n}}{\mathrm{P}}_{\mathrm{n}|\mathrm{n}-\mathrm{1}}{{\mathrm{H}}_{\mathrm{n}}}^{\mathrm{T}}+{\mathrm{R}}_{\mathrm{n}}$$
  
  
  S_n étant la matrice d'erreur de la mise à jour,
  R_n étant la matrice de covariance du bruit d'observation,
  K_n étant le gain par lequel l'erreur est multipliée,
  X_nln étant le vecteur d'état à estimer, et
  P_nln étant la mise à jour de la matrice de covariance (décrivant le bruit) $${\mathrm{K}}_{\mathrm{n}}={\mathrm{P}}_{\mathrm{n}|\mathrm{n}-\mathrm{1}}{{\mathrm{H}}_{\mathrm{n}}}^{\mathrm{T}}{{\mathrm{S}}_{\mathrm{n}}}^{-1}$$
  
  $${\mathrm{X}}_{\mathrm{n}|\mathrm{n}}={\mathrm{X}}_{\mathrm{n}|\mathrm{n}-\mathrm{1}}+{\mathrm{K}}_{\mathrm{n}}*{\mathrm{Uerror}}_{\mathrm{n}}$$
  
  $${\mathrm{P}}_{\mathrm{n}|\mathrm{n}}=\left(\mathrm{l}-{\mathrm{K}}_{\mathrm{n}}{\mathrm{H}}_{\mathrm{n}}\right){\mathrm{P}}_{\mathrm{n}|\mathrm{n}-\mathrm{1}}$$
  

The different steps of the algorithm can be summarized as follows:

• 1 °) Prediction of the system (using model and model noise): $${\mathrm{X}}_{\mathrm{not}|\mathrm{not}-\mathrm{1}}={\mathrm{X}}_{\mathrm{not}-\mathrm{1}|\mathrm{not}-\mathrm{1}}$$
  
  $${\mathrm{P}}_{\mathrm{not}|\mathrm{not}-\mathrm{1}}={\mathrm{P}}_{\mathrm{not}-\mathrm{1}|\mathrm{not}-\mathrm{1}}+{\mathrm{Q}}_{\mathrm{not}}$$
  
  
  Q _n being the covariance matrix of the model noise
• 2 °) Update of the system: $${\mathrm{uest}}_{\mathrm{not}}=\mathrm{h}\left({\mathrm{X}}_{\mathrm{not}|\mathrm{not}-\mathrm{1}}\right)$$
  
  $${\mathrm{uerror}}_{\mathrm{not}}={\mathrm{Umes}}_{\mathrm{not}}-{\mathrm{uest}}_{\mathrm{not}}$$
  
  $$\mathrm{Calculation\; of}{\mathrm{H}}_{\mathrm{not}}=\left[{\mathrm{dUest}}_{\mathrm{not}}/{\mathrm{DBL}}_{\mathrm{0}},{\mathrm{dUest}}_{\mathrm{not}}/{\mathrm{dK}}_{\mathrm{eq}\mathrm{0}},...,{\mathrm{dUest}}_{\mathrm{not}}/{\mathrm{dL}}_{\mathrm{e}\mathrm{4}}\right]$$
  
  $${\mathrm{S}}_{\mathrm{not}}={\mathrm{H}}_{\mathrm{not}}{\mathrm{P}}_{\mathrm{not}|\mathrm{not}-\mathrm{1}}{{\mathrm{H}}_{\mathrm{not}}}^{\mathrm{T}}+{\mathrm{R}}_{\mathrm{not}}$$
  
  
  S _n being the error matrix of the update,
  R _n is the covariance matrix of the observation noise,
  K _n being the gain by which the error is multiplied,
  X _nln being the state vector to be estimated, and
  P _nln being the update of the covariance matrix (describing the noise) $${\mathrm{K}}_{\mathrm{not}}={\mathrm{P}}_{\mathrm{not}|\mathrm{not}-\mathrm{1}}{{\mathrm{H}}_{\mathrm{not}}}^{\mathrm{T}}{{\mathrm{S}}_{\mathrm{not}}}^{-1}$$
  
  $${\mathrm{X}}_{\mathrm{not}|\mathrm{not}}={\mathrm{X}}_{\mathrm{not}|\mathrm{not}-\mathrm{1}}+{\mathrm{K}}_{\mathrm{not}}*{\mathrm{uerror}}_{\mathrm{not}}$$
  
  $${\mathrm{P}}_{\mathrm{not}|\mathrm{not}}=\left(\mathrm{l}-{\mathrm{K}}_{\mathrm{not}}{\mathrm{H}}_{\mathrm{not}}\right){\mathrm{P}}_{\mathrm{not}|\mathrm{not}-\mathrm{1}}$$
  

The estimate of the parameters of the loudspeaker model at time n is given by the state vector X _{n | n} .

The state vector X _{n | n} thus obtained can be used for various purposes.

The knowledge of the response of the loudspeaker, and in particular of the excursion x of the membrane (hidden variable, not measured but estimated thanks to the extended Kalman filter) may notably be used as input data to a limiter stage 26 ( Figure 2 ): the instantaneous value x of the excursion is compared with a determined threshold X _max beyond which this excursion is considered too important, with the risk of damaging the loudspeaker, the appearance of distortions, etc. If the threshold is exceeded, the The limiter determines an attenuation gain, less than unity, that will be applied to the incident signal E to reduce its amplitude, so that the excursion remains within the allowed range.

Another treatment that can be applied to the audio signal is compensation for non-linearities (block 28). Indeed, insofar as the speaker response is modeled, it is possible to predict the nonlinearities of this response and to compensate for them by an appropriate inverse processing applied to the signal. Such treatment is in itself known, and for this reason it will not be described in more detail.

It will be noted that a compensation of the non-linearities is likely to add power to the signal obtained at the output. It is therefore necessary at this stage to check that the compensated signal of the non-linearities does not exceed a permissible limit of excursion of the membrane - in the contrary case a global gain of attenuation, lower than unity, will be applied to the signal to keep this excursion within the allowed range.

According to another aspect of the invention, the extended Kalman estimator operates on the fly, directly from the current audio signal reproduced by the loudspeaker, by collecting the electrical parameters on this loudspeaker (voltage, current) during the reproduction of this audio signal.

Indeed, there is no theoretical constraint on the signal exciting the speaker membrane so that the extended Kalman filter estimation method can be implemented.

The system can thus be used with a high-fidelity consumer installation, operating in a manner that is transparent to the user: there is no need to ask the user to reproduce a particular type of calibration signal (white noise, succession of tones, etc.) so that the algorithm can estimate the parameters of the loudspeaker, the latter can operate continuously while the music is played. However, in order to best estimate the linear and nonlinear parameters of the T / S model, in particular the parameters B1 (x), K _eq (x) and L _e (x) which depend on the displacement x of the membrane, It is preferable that the signal played move this membrane sufficiently so that the estimate is the best possible.

To decide whether an excitation signal E can be used to update the Kalman estimator, when music is played, the last T seconds (typically T = 10 seconds) of the signal are permanently stored in a buffer 30 ( Figure 2 ).

The displacement of the membrane is calculated continuously by applying Equations (1) and (2) of the estimator (block 32), with speaker parameters which are fixed and correspond to the results of the last estimation made by the Kalman filter.

The effective value x_eff (n) of this displacement is calculated (block 32) every N samples (typically N = 24000 samples), for example by the following formula: $$\mathrm{x\_eff}\left(\mathrm{not}\right)=\mathrm{sqrt}\left(\left(\mathrm{x}{\left(\mathrm{not}\right)}^{\mathrm{2}}+\mathrm{x}{\left(\mathrm{not}-\mathrm{1}\right)}^{\mathrm{2}}+...+\mathrm{x}{\left(\mathrm{not}-\mathrm{NOT}\right)}^{\mathrm{2}}\right)/\mathrm{NOT}\right)$$

If this effective value is greater than a given threshold x_seuil (block 34) for a number of consecutive times corresponding to the time T, then it is considered that the T last seconds of signal played are valid and the filter update is activated. Kalman so that it can use these last T seconds of signal to re-estimate the parameters of the response of the speaker.

Claims (6)

A method of processing a digital audio signal intended to be reproduced by equipment comprising an electrodynamic loudspeaker whose global response as a function of the electrical signal applied to its terminals is defined by a set of electrical, mechanical and acoustic parameters,
this process comprising: a) determining an observation vector comprising only measurements of electrical parameters, with: · A measurement of the voltage (U) at the terminals of the loudspeaker, and A measurement of the current (i) passing through the loudspeaker; b) determining a state vector (X) by applying the voltage and current measurements to a predictive filter estimator incorporating a representation of a dynamic model of the loudspeaker, said predictive filter being a Kalman filter extended able to: · To make a prediction of the state vector (X), and · Recalibrate this prediction by calculating an estimate of the voltage (Uest) from the state vector and the measured current and comparing this estimate with the measurement of the voltage (U _mes ); and c) applying to the audio signal a processing function according to the state vector (X),
characterized in that the components of the state vector include: · Linear response parameter values of the loudspeaker included in the group: electrical resistance (R _e ) and mechanical resistance (R _eq ), and Polynomial coefficients of non-linear loudspeaker response parameters included in the group: force factor (Bl ₀ , Bl ₁ , Bl ₂ ), equivalent stiffness (K _eq0 , K _eq1 , K _eq2 ) and electrical inductance ( L _e0 , L _e1 , L _e2 , L _e3 , L _e4 ). The method of claim 1, wherein said processing applied to the audio signal is a compensation processing of the non-linearities of the loudspeaker response, as determined from the state vector delivered by the predictive filter estimator . The method of claim 1, wherein said processing applied to the audio signal comprises: c1) calculating a current value of excursion (x) of the loudspeaker according to i) an amplification gain of the audio signal and ii) the response of the loudspeaker as determined from the vector status issued by the predictive filter estimator; c2) comparing the current excursion value thus calculated with a maximum excursion value; and c3) calculating a possible attenuation of said amplification gain in case the current excursion value exceeds the maximum excursion value. The method of claim 1, wherein the components of the state vector (X) further comprise additional acoustic parameter values representative of the loudspeaker response associated with a back cavity provided with a decompression vent. The method of claim 1, wherein the determination of the state vector of step b) and performed on the fly from the current audio signal object of the processing of step c) and reproduced by the speaker, by collecting the electrical parameters at the terminals of the loudspeaker during the reproduction of this audio signal. The method of claim 5, comprising the steps of: storing a sequence of samples of the audio signal for a predetermined duration; analyzing the sequence for calculating an energy parameter of the memorized audio signal; if the calculated energy parameter is greater than a predetermined threshold, enabling the estimation by the predictive filter; - if not, inhibit the estimation by the predictive filter and retain the values of the state vector previously estimated.

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