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
In discrete time, integration and differentiation can be accomplished using digital filters . 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 :
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.