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**ICMC 95, Banff (Canada), 1995**

Copyright © Ircam - Centre Georges-Pompidou 1995

Physical models usually require the complete construction of a model for each new instrument. On the other hand our SSMs provide a methodology for modular construction of instruments [Tas94]. This construction is done automatically in our formal calculus environment. It allows to obtain different theoretical representations (state space equation, external or modal representation) and a synthesiser which simulates the instrument.

This paper focuses on the production of state space representation of hierarchical networks. Thus, we only briefly present the physical systems involved in wind instruments. We give some basics concerning the state space representation and we describe in detail the methodology that we used to combine modules together into networks of arbitrary complexity. In the third section, we apply this methodology to transmission lines, and in the last section we describe two software realizations of synthesiser builders.

In order to obtain a discrete model, the acoustic tube is spatially
discretized in *N* elementary tubes of equal length and given section.
This spatial discretisation is directly related to the sampling period
which is the exact delay for a round trip in an elementary tube.

For a lossless tube the usual description of propagating waves in terms
of superposition of ingoing and outgoing waves leads to the
Kelly-Lochbaum structure. This structure is made of two delays and is
parameterized by one reflection coefficient which represents the
discontinuity between two elementary tubes. For state space description,s
we prefer the lattice structure which is obtained by putting all the
delays in the upper or in the lower branch of the lattice and by *
normalizing* input and output waves [Mat90]. In that way we represent an
acoustic tube by a quadripole system with an *N*-dimensional state space
internal vector. As we normalize input and output waves of the acoustic
tube in the lattice representation, we have to denormalize the inputs
and the outputs of the whole system to obtain physical values.

When taking into account viscothermal losses, we prove that the acoustic
tube is still described by a *generalization* of the lattice filter.
Following the delay we add a filter which approximates the
frequency-dependent attenuation [Mat94]. The attenuation factor being proportional to (where is the frequency), we use specific techniques to approximate it by a digital filter. We get the same structure for the state space equation than in the lossless case except for some scalars which are replaced by block matrices.

A wind instrument is an assembly of interconnected tubes. Connections of tubes are modeled by junction modules which implement Kirchoff's laws. At this time, we only consider two-port and three-port junctions. As in electrical circuit systems, networks have to be oriented. Consequently, we distinguish four three-port junctions.

Furthermore, a wide range of physical systems can be considered as linear dynamical systems. Then, the formal expression of such a model becomes (1.b) where the system is now represented by four matrices (

To build a network, we proceed by recursive combinations of two sub-systems. Considering two systems

Figure 1:Serial (a), parallel (b) and feedback (c) connections of two modules.

For the serial combination the output of *S*_{1} is connected to the
input of *S*_{2} and we have = _{2} and = _{1}. For
the parallel combination, input and output are simply a concatenation :
= (_{1},_{2}) and = (_{1},_{2}). And for the
feedback combination, we have = _{1} = _{2} and = _{1} - _{2}.

Notice that these combinations have been defined in order to construct networks in a very general way. They do not directly have a physical meaning. In particular, the connection of two acoustic tubes is not a simple serial connection but is done through the connection of two quadripoles which involves 4 serial, 2 parallel and 1 feedback connections (fig. 3).

For example, in order to identify an unknown parameter by using the Kalman extended filtering techniques, we classically add this parameter as an extra state space variable to the internal state vector. Consequently, we have to derive automatically a new state space equation corresponding to the enlarged state vector.

Establishing an external representation of physical systems is very useful, for example, for performing real-time synthesis driven by physical parameters. The synthesis becomes very efficient computationally speaking. Such an external representation is obtained by evaluating :

is a transfer matrix and it reduces to a scalar expression when input and output vectors are one-dimensional. In this case is the transfer function corresponding to an ARMA filter.

Modal representation of physical systems is well adapted to the context
of sound synthesis because of the structure of the sound output. Modal
synthesisers such as Modalys (previously known as Mosaic) need a
data base of modes adapted to the structure which is to be excited.
Modes can be obtained by establishing the state space equation for
the structure and by diagonalizing the matrix *A* of dynamics. This is
performed by formal or numerical calculus.

We are interested in knowing in advance whether a network is computationally

Figure 2:(a) Simulable feedback; (b) serial q-junction of two quadripoles.

However, if the feedback loop is broken by at least one delay, as shown
in figure 2.a, we get an explicit expression for the
output ([*n*] = *g*_{1}([*n*] + g_{2}([*n-1*])))(see also [FC90]).

Figure 3:decomposition of a serial q-junction.

A serial q-junction between two quadripoles involves a feedback connection. It turns out that serial q-junctions of quadripoles can not be simulated in the general case, but can be simulated if each incriminated loop is broken by a delay.

Figure 4:decomposition of a parallel q-junction.

Basically a delay can be extracted either in the upper line (the ingoing waves) or in the lower line (the outgoing waves) (fig. 5). We call them respectively

We deduce some interesting properties: two

Figure 5:Two kinds of simulable quadripoles: the quad-up and the quad-down.

Notice that the most simple quadripole, i.e. the identity connection
( *s*^{+} = *e*^{+} and *s*^{-} = *e*^{-} ) is neither a quad-up nor a quad-down, and
is therefore not simulable for serial q-junction purposes, which could
become a drawback in the description of some transmission line networks.

If the network can be split in term of q-junctions, serial, parallel connections of oriented quad-ups and junctions

Figure 6:Network of transmission lines.

Fig. 7.a demonstrates how the network may be split into a regular pattern. Figure 7.b represents the pattern (a) as the parallel connection of different quad-ups. Thus this network is simulated by using simple basic elements (only

Figure 7:Network decomposition.

The Tube library is a library built around the statespace library which implements known linear models: cylindrical tubes with or without viscothermal losses, boundary conditions, junctions and T-junctions.

Usable objects are either Compound Modules[3], or any non-abstract class which implements a particular system. At this point we use parallel and serial connections, and q-junctions. We also implement classes which represent boundary conditions, tube models with or without viscothermal losses [Mat95], waveguide junctions [Tas94], non-linear reed models [Rod94]. This list may be completed by the implementation of other physical models.

Each non-abstract Module is described by an algorithm which computes an output from the inputs and its internal state. Some internal parameters (such as the reflection coefficient for a boundary condition) can be updated at run-time by a specific method. Others cannot (such as the length of a tube which cannot change dynamically).

We provide users with interactive interface with links, modules and vectors for designing and simulating systems and for controlling the synthesis with time-varying parameters.

Figure 8:Inheritance tree for the Module class.

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