Time-varying Systems

Time-varying distributed systems have not been examined in any detail in the scattering simulation literature, though time-varying WDFs [177] and DWNs [166] have both been proposed, with a focus on vocal tract modelling. Though it is true that time-variations in material parameters generally render a system non-passive, we will show here how passive network representations may be developed for an important class of systems.

Consider a system of the form

$$\frac{\partial}{\partial t}\left( {\bf P}({\bf x},t){\bf w}\right... ...bf w}}{\partial x_{j}}+{\bf B}({\bf x},t){\bf w} + {\bf f}({\bf x}, t) = {\bf0}$$ (6.8)

which is a simple generalization of the $(n+1)$D symmetric hyperbolic form (3.1) to the case where ${\bf P}$ and ${\bf B}$ depend on both the spatial coordinates ${\bf x}$ and time $t$; ${\bf P}$ is assumed to be positive definite for all values of these coordinates and smoothly-varying. The matrices ${\bf A}_{j}$ are again assumed to be constant and symmetric, and ${\bf B}$ is not required to have any particular structure. It is easy to show that in this form, it is not possible to arrive immediately at an energy condition such as (3.5). In order to put system (6.8) into more useful form, note that we can factor ${\bf P}$ as ${\bf P}$ = ${\bf P}^{\frac{T}{2}}{\bf P}^{\frac{1}{2}}$ where ${\bf P}^{\frac{T}{2}}$ is some left matrix square root of ${\bf P}$. We can then rewrite (6.8) as

$${\bf P}^{\frac{T}{2}}\frac{\partial {\bf P}^{\frac{1}{2}}{\bf w}}... ...A}_{j}\frac{\partial {\bf w}}{\partial x_{j}}+{\bf B}{\bf w} + {\bf f} = {\bf0}$$

Now introduce a new dependent variable ${\bf z}$ defined by ${\bf w} = e^{\kappa}{\bf z}$, where $\kappa = \kappa(t)$ and is assumed differentiable. Then, in terms of the new variable ${\bf z}$, we have

$${\bf P}^{\frac{T}{2}}\frac{\partial {\bf P}^{\frac{1}{2}}{\bf z}}... ...n}{\bf A}_{j}\frac{\partial {\bf z}}{\partial x_{j}} + \tilde{{\bf f}} = {\bf0}$$

with $\tilde{{\bf f}} = e^{-\kappa}{\bf f}$. Assuming that this source term is zero, we can then take the inner product of this expression with ${\bf z}^{T}$ to get

$$\frac{\partial}{\partial t}\left(\frac{1}{2}{\bf z}^{T}{\bf Pz}\r... ...partial x_{j}}\left({\bf z}^{T}{\bf A}_{j}{\bf z}\right) = -{\bf z}^{T}{\bf Qz}$$

where

$${\bf Q} = \frac{\partial \kappa}{\partial t}{\bf P}+\frac{1}{2}\frac{\partial {\bf P}}{\partial t}+\frac{1}{2}\left({\bf B}+{\bf B}^{T}\right)$$

If ${\bf Q}$ is positive semi-definite, then integrating over $\mathbb{R}^{n}$ gives the energy condition

$$\frac{d}{dt}\int_{\mathbb{R}^{n}}\frac{1}{2}{\bf z}^{T}{\bf Pz}d{\bf x} \leq 0$$

which is identical to the condition derived in §3.2, under the replacement of ${\bf w}$ with ${\bf z}$. As long as ${\bf B}$ and the time derivative of ${\bf P}$ are bounded, it is always possible to make a choice of $\kappa$ such that ${\bf Q}$ is positive semi-definite. For instance, we can choose $\kappa = \kappa_{0}t$, with

$$\kappa_{0} \geq -\frac{1}{2}\min_{{\bf x}\in\mathbb{R}^{n}, t\geq... ...partial{\bf P}}{\partial t}+{\bf B}+{\bf B}^{T}\right)}{\lambda_{min}({\bf P})}$$

where $\lambda_{min}(\cdot)$ signifies ``minimum eigenvalue of.'' Here, we essentially have a passivity condition in an exponentially-weighted norm.

Consider a generalization of the source-free (1+1)D transmission line system,

$$\begin{eqnarray}\frac{\partial (li)}{\partial t} + \frac{\partial... ...rtial (cu)}{\partial t} + \frac{\partial i}{\partial x}+gu &=& 0 \end{eqnarray}$$ (6.9a)

where $l$, $c$, $r$ and $g$, are all smooth positive functions of $x$ and $t$. Introducing the variables

$$i_{1} = i e^{-\kappa_{0}t}\hspace{1.5in}i_{2} = ue^{-\kappa_{0}t}/r_{0}$$

where $r_{0}$ is a positive constant as well as the scaled time variable $t' = v_{0}t$, and transformed coordinates as per (3.18), we can rewrite this system as

$$\sqrt{L_{1}}\frac{\partial}{\partial t'}\left(\sqrt{L_{1}}i_{1}\r... ...)+\frac{r_{0}}{\sqrt{2}}\frac{\partial}{\partial t_{2}}\left(i_{1}-i_{2}\right) =$$ 0

$$\sqrt{L_{2}}\frac{\partial}{\partial t'}\left(\sqrt{L_{2}}i_{2}\r... ...)+\frac{r_{0}}{\sqrt{2}}\frac{\partial}{\partial t_{2}}\left(i_{1}-i_{2}\right) =$$ 0

with

$$L_{1} = v_{0}l-r_{0}\hspace{1.0in}L_{2} = v_{0}cr_{0}^{2}-r_{0}$$

and

$$r_{1} = \kappa_{0}l+\frac{1}{2}\frac{\partial l}{\partial t}+r\hs... ... = r_{0}^{2}\left(\kappa_{0}c+\frac{1}{2}\frac{\partial c}{\partial t}+g\right)$$

Under the choices

$$r_{0} = \sqrt{l_{min}/c_{min}}\hspace{0.5in} v_{0}\geq 1/\sqrt{l_{min}c_{min}}$$

where now we have

$$l_{min} = \min_{x\in\mathbb{R}, t\geq 0}l\hspace{1.0in}c_{min} = \min_{x\in\mathbb{R}, t\geq 0}c$$

then $L_{1}$ and $L_{2}$ are non-negative, and the terms involving them can be interpreted as voltages across passive inductors, if power-normalized waves are employed (see §3.5.1 for more information on this definition of inductors). If we also choose

$$\kappa_{0}\geq -\frac{1}{2}\min\left(\min_{x\in\mathbb{R}, t\geq ... ...athbb{R}, t\geq 0}\big(\frac{1}{c}\frac{\partial c}{\partial t}+g/c\big)\right)$$ (6.10)

then $r_{1}$ and $r_{2}$ are non-negative and can be interpreted as passive resistances. The resulting MDKC is shown in Figure 6.4; an MDWD network can be immediately obtained through the methods discussed in Chapter 3, or network manipulations and alternative integration rules may be employed to get a DWN. A balanced form (see §3.12) is also possible, and gives a less strict bound on $v_{0}$, but the bound on $\kappa_{0}$ remains unchanged.

Figure 6.4: MDKC for time-varying (1+1)D transmission line system (6.9). The exponential weighting of the current variables can be viewed (formally) as a time-varying transformer coupling.

A direct application of this MDKC to an important music synthesis problem would be the simulation of acoustic wave propagation in the vocal tract, under time-varying conditions. Such a system of PDEs is mentioned in [145], and has the exact form of (6.9), with $r=g=0$, and under the replacements

$$l\rightarrow \frac{\rho}{A}\hspace{0.3in}c\rightarrow \frac{A}{\r... ...mma^{2}}\hspace{0.3in}i\rightarrow v\hspace{0.3in}u\rightarrow p+\rho\gamma^{2}$$

where $\rho$ is the air density, $\gamma$ is the speed of sound, $A(x,t)$ is the surface area of the tube, $v(x,t)$ is the volume velocity and $p(x,t)$ is the pressure variation. The condition (6.10) then reduces to

$$\kappa_{0}\geq \max_{{\bf x}\in\mathbb{R}, t\geq 0}\left\vert\frac{\partial \ln(\sqrt{A})}{\partial t}\right\vert$$

If the time variation in $A$ is slow, then $\kappa_{0}$ will be close to zero, and the exponential weighting will not be overly severe. The problems, for real-time synthesis applications, are that we will need to have an a priori estimate of the maximal time variation of the vocal tract area, and that we will apply an exponential weighting to the signal output from the scattering simulation. This exponential weighting may be viewed as a passive operation involving time-varying transformers (as shown in Figure 6.4).