WDFs and the Numerical Integration of ODEs

In the closed network of Figure 1.10, we have left the two one-ports unspecified. Suppose we connect an inductor, of constant inductance $L>0$ at the left-hand port, and a capacitor of constant capacitance $C>0$ at the right-hand port, as shown in Figure 1.11(a). Then the voltage-current relations are defined by

$$v_{1} = L\frac{di_{1}}{dt}\hspace{1.5in}i_{2} = C\frac{dv_{2}}{dt}$$

When these relations are closed by Kirchoff's parallel connection rules (1.23), it is possible to write a single second-order ODE describing the time-evolution of the circuit state,

$$\frac{d^{2}w}{dt^{2}} = -\frac{1}{LC}w$$

where $w(t)$ stands for any of the voltages or currents in the network. This network thus behaves as a harmonic oscillator, of frequency $1/\sqrt{LC}$; the voltages and currents, assumed real, evolve according to

$$w(t) = A\cos(t/\sqrt{LC})+B\sin(t/\sqrt{LC})$$

for some arbitrary constants $A$ and $B$ determined by the initial voltages and currents in the network. The network is also lossless; if we define the total stored energy of this network $E(t)$ by

$$E(t)\triangleq \underbrace{\frac{1}{2}Li_{1}^{2}}_{\begin{minipag... ...ergy stored in electric field surrounding capacitor}\end{center}\end{minipage}}$$ (1.25)

then

$$\frac{dE}{dt} = Li_{1}\frac{di_{1}}{dt}+Cv_{2}\frac{dv_{2}}{dt} =... ...i_{2}v_{2} = 0\hspace{0.2in}\Longrightarrow\hspace{0.2in} E(t) = {\rm constant}$$

In other words, energy is traded back and forth between the two circuit elements, but is not dissipated.

Figure 1.11: The LC harmonic oscillator-- (a) a parallel connection of an inductor, of inductance $L$ and a capacitor of capacitance $C$, and (b) the corresponding wave digital network.

Though we have not explicitly derived the forms of the wave digital inductor and capacitor, this is a good opportunity to see what these elements look like--the wave digital network corresponding to the LC harmonic oscillator circuit is shown in Figure 1.11(b). (The reader may glance ahead to §2.3.4 for a glimpse of how these forms are arrived at.) We have a parallel adaptor, which is a digital signal processing block defined by equations (1.24), terminated on delay elements (one of which incorporates a sign inversion). Special choices of the port resistances $R_{1}$ and $R_{2}$ (marked in the figure) were chosen in order to obtain these simple signal-flow graphs. This diagram implies a recursion, which, like the digital waveguide network methods consists of a scattering step, and a delay step (possibly with sign inversion). Because it makes use of only two delay operators, it should be obvious that this simple network must behave as a two-pole resonator--the discrete-time counterpart to the continuous-time harmonic oscillator. The wave digital network thus behaves as a numerical integrator.

We can define the total discrete-time stored energy of this network by

$$E_{WD}(n) = \frac{1}{R_{1}}(a_{1}(n))^{2}+\frac{1}{R_{2}}(a_{2}(n))^{2}$$

which is simply a weighted sum of the squares of the signal values stored in the delay registers at time step $n$. Clearly, this quantity remains unchanged after undergoing delays and the scattering operation, i.e., we have

$$E_{WD}(n) =$$

It is simple to identify this quantity with the energy (1.25) of the continuous-time LC network. Although this example is very simple, the same ideas can be applied to large networks, and the result is always an explicitly recursible structure for which passivity can be simply guaranteed.