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Physical Audio Signal Processing
    Digital Waveguide Theory
       Alternative Wave Variables

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Alternative Wave Variables

We have thus far considered discrete-time simulation of transverse displacement $ y$ in the ideal string. It is equally valid to choose velocity $ v\isdeftext {\dot y}$, acceleration $ a\isdeftext {\ddot y}$, slope $ y'$, or perhaps some other derivative or integral of displacement with respect to time or position. Conversion between various time derivatives can be carried out by means integrators and differentiators, as depicted in Fig.H.10. Since integration and differentiation are linear operators, and since the traveling wave arguments are in units of time, the conversion formulas relating $ y$, $ v$, and $ a$ hold also for the traveling wave components $ y^\pm , v^\pm , a^\pm $.

Figure H.10: Conversions between various time derivatives of displacement: $ y = $ displacement, $ v = {\dot y}= $ velocity, $ a = {\ddot y}= $ acceleration, where $ {\dot y}$ denotes $ dy/dt$ and $ {\ddot y}$ denotes $ d^2y/dt^2$.
\includegraphics[scale=0.9]{eps/fwaveconversions}

Differentiation and integration have a simple form in the frequency domain. Denoting the Laplace Transform of $ y(t,x)$ by

$\displaystyle Y(s,x) \isdef {\cal L}_s\{y(\cdot,x)\} \isdef \int_0^\infty y(t,x) e^{-st} dt$ (H.36)

where ``$ \cdot$'' in the time argument means ``for all time,'' we have, according to the differentiation theorem for Laplace transforms [290],

$\displaystyle {\cal L}_s\{{\dot y}(\cdot,x)\} = s Y(s,x) - y(0,x)$ (H.37)

Similarly, $ {\cal L}_s\{\dot y^{+}\} = s Y^{+}(s) - y^{+}(0)$, and so on. Thus, in the frequency domain, the conversions between displacement, velocity, and acceleration appear as shown in Fig.H.11.

Figure H.11: Conversions between various time derivatives of displacement in the frequency domain.
\includegraphics[scale=0.9]{eps/ffdwaveconversions}

In discrete time, integration and differentiation can be accomplished using digital filters [371]. Commonly used first-order approximations are shown in Fig.H.12.

Figure H.12: Simple approximate conversions between time derivatives in the discrete-time case: a) The first-order difference $ {\hat v}(n) = y(n) - y(n-1)$. b) The first-order ``leaky'' integrator $ {\hat y}(n) = v(n) + g {\hat y}(n-1)$ with loss factor $ g$ (slightly less than $ 1$) used to avoid infinite DC build-