Figure C.3:
Digital simulation of the ideal, lossless waveguide
with observation points at
and
. (The symbol
``
'' denotes a one-sample delay.)
![\includegraphics[scale=0.9]{eps/fideal}](http://www.dsprelated.com/josimages_new/pasp/img3287.png) |

In this section, we interpret the sampled d'Alembert
traveling-wave
solution of the ideal
wave equation as a
digital filtering framework.
This is an example of what are generally known as
digital
waveguide models [
430,
431,
433,
437,
442].
The term
![$ y_r\left[(n-m)T\right]\isdef y^{+}(n-m)$](http://www.dsprelated.com/josimages_new/pasp/img3288.png)
in Eq.

(
C.16) can
be thought of as the output of an

-sample
delay line whose input is

. In general, subtracting a positive number

from a time
argument

corresponds to
delaying the waveform by

samples. Since

is the right-going component, we draw its
delay
line with input

on the left and its output

on the
right. This can be seen as the upper ``rail'' in Fig.
C.3
Similarly, the term
![$ y_l\left[(n+m)T\right]\isdeftext y^{-}(n+m)$](http://www.dsprelated.com/josimages_new/pasp/img3290.png)
can be
thought of as the
input to an

-sample delay line whose
output is

. Adding

to the time argument

produces an

-sample waveform
advance. Since

is the left-going component, it makes
sense to draw the delay line with its input

on the right
and its output

on the left.
This can be seen as the lower ``rail'' in Fig.
C.3.
Note that the position along the string,
meters,
is laid out from left to right in the diagram, giving a physical
interpretation to the horizontal direction in the diagram. Finally,
the left- and right-going
traveling waves must be summed to produce a
physical output according to the formula
 |
(C.17) |
We may compute the physical string
displacement at any spatial
sampling
point

by simply adding the upper and lower rails together at position

along the delay-line pair. In Fig.
C.3, ``
transverse
displacement outputs'' have been arbitrarily placed at

and

.
The diagram is similar to that of well known ladder and lattice digital
filter structures (§
C.9.4,[
297]),
except for the delays along the upper rail,
the absence of
scattering junctions, and the direct physical
interpretation. (A
scattering junction implements partial reflection and
partial transmission in the waveguide.) We could proceed to ladder and
lattice filters by (1) introducing a perfectly reflecting (rigid or free)
termination at the far right, and (2) commuting the delays rightward from
the upper rail down to the lower rail [
432,
434]. The absence of
scattering junctions is due to the fact that the string has a uniform
wave
impedance. In acoustic tube simulations, such as for voice
[
87,
297] or wind instruments [
195],
lossless scattering
junctions are used at changes in cross-sectional tube area and lossy
scattering junctions are used to implement tone holes. In
waveguide
bowed-string synthesis (discussed in a later section), the bow itself
creates an active, time-varying, and
nonlinear scattering junction on the
string at the bowing point.
Any ideal, one-dimensional waveguide can be simulated in this way. It
is important to note that the simulation is
exact at the
sampling instants, to within the numerical precision of the samples
themselves. To avoid
aliasing associated with sampling, we
require all waveshapes traveling along the string to be initially
bandlimited to less than half the sampling frequency. In other
words, the highest frequencies present in the
signals 
and

may not exceed half the temporal sampling frequency

; equivalently, the highest
spatial
frequencies in the shapes

and

may not exceed
half the spatial sampling frequency

.
A C program implementing a plucked/struck string model in the form of
Fig.
C.3 is available at
http://ccrma.stanford.edu/~jos/pmudw/.
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Any String State to Traveling Slope-Wave Components