Alternative Wave Variables

We have thus far considered discrete-time simulation of transverse
*displacement*
in the ideal string. It is equally valid to
choose *velocity*
, *acceleration*
, *slope*
, 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.C.10. Since integration and
differentiation are *linear* operators, and since the traveling
wave arguments are in units of time, the conversion formulas relating
,
, and
hold also for the traveling wave *components*
.

Differentiation and integration have a simple form in the
frequency domain. Denoting the *Laplace Transform* of
by

(C.36) |

where `` '' in the time argument means ``for all time,'' we have, according to the

(C.37) |

Similarly, , and so on. Thus, in the frequency domain, the conversions between displacement, velocity, and acceleration appear as shown in Fig.C.11.

In discrete time, integration and differentiation can be accomplished using digital filters [365]. Commonly used first-order approximations are shown in Fig.C.12.

If discrete-time acceleration
is defined as the sampled version of
continuous-time acceleration, *i.e.*,
, (for some fixed continuous position
which we
suppress for simplicity of notation), then the
frequency-domain form is given by the *
transform*
[488]:

(C.38) |

In the frequency domain for discrete-time systems, the first-order approximate conversions appear as shown in Fig.C.13.

The transform plays the role of the Laplace transform for discrete-time systems. Setting , it can be seen as a sampled Laplace transform (divided by ), where the sampling is carried out by halting the limit of the rectangle width at in the definition of a Reimann integral for the Laplace transform. An important difference between the two is that the frequency axis in the Laplace transform is the imaginary axis (the `` axis''), while the frequency axis in the plane is on the unit circle . As one would expect, the frequency axis for discrete-time systems has unique information only between frequencies and while the continuous-time frequency axis extends to plus and minus infinity.

These first-order approximations are accurate (though scaled by
)
at low frequencies relative to half the sampling rate, but they are
not ``best'' approximations in any sense other than being most like
the definitions of integration and differentiation in continuous time.
Much better approximations can be obtained by approaching the problem
from a *digital filter design* viewpoint, as discussed in §8.6.

- Spatial Derivatives
- Force Waves
- Wave Impedance
- State Conversions
- Power Waves
- Energy Density Waves
- Root-Power Waves
- Total Energy in a Rigidly Terminated String

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