Excitation Examples
Localized Displacement Excitations
Whenever two adjacent components of



will initialize only
, a solitary left-going pulse
of amplitude 1 at time
, as can be seen from Eq.
(E.11) by adding the leftmost columns
explicitly written for
. Similarly, the initialization

gives rise to an isolated right-going pulse
, corresponding
to the leftmost column of
plus the first column on the left not
explicitly written in Eq.
(E.11). The superposition of these two
examples corresponds to a physical impulsive excitation at time 0 and
position
:
Thus, the impulse starts out with amplitude 2 at time 0 and position

In summary, we see that to excite a single sample of displacement
traveling in a single-direction, we must excite equally a pair of
adjacent colums in
. This corresponds to equally weighted
excitation of K-variable pairs the form
.
Note that these examples involved only one of the two interleaved computational grids. Shifting over an odd number of spatial samples to the left or right would involve the other grid, as would shifting time forward or backward an odd number of samples.
Localized Velocity Excitations
Initial velocity excitations are straightforward in the DW paradigm,
but can be less intuitive in the FDTD domain. It is well known that
velocity in a displacement-wave DW simulation is determined by the
difference of the right- and left-going waves
[437]. Specifically, initial velocity waves can
be computed from from initial displacement waves
by spatially
differentiating
to obtain traveling slope waves
, multiplying by minus the tension
to obtain force
waves, and finally dividing by the wave impedance
to
obtain velocity waves:
where




We can see from Eq.(E.11) that such asymmetry can be caused by
unequal weighting of
and
. For example, the
initialization

corresponds to an impulse velocity excitation at position
. In this case, both interleaved grids are excited.
More General Velocity Excitations
From Eq.(E.11), it is clear that initializing any single K variable
corresponds to the initialization of an infinite number of W
variables
and
. That is, a single K variable
corresponds to only a single column of
for only one of the
interleaved grids. For example,
referring to Eq.
(E.11),
initializing the K variable
to -1 at time
(with all other
intialized to 0)
corresponds to the W-variable initialization

with all other W variables being initialized to zero.
In view of earlier remarks, this corresponds to an impulsive velocity
excitation on only one of the two subgrids. A schematic
depiction from to
of the W variables at time
is as
follows:
![]() |
(E.14) |
Below the solid line is the sum of the left- and right-going traveling-wave components, i.e., the corresponding K variables at time





![]() |
(E.15) |
![]() |
(E.16) |
![]() |
(E.17) |
![]() |
(E.18) |
The sequence
![$ [\dots,1,0,1,0,1,\dots]$](http://www.dsprelated.com/josimages_new/pasp/img4578.png)





Due to the independent interleaved subgrids in the FDTD algorithm, it is nearly always non-physical to excite only one of them, as the above example makes clear. It is analogous to illuminating only every other pixel in a digital image. However, joint excitation of both grids may be accomplished either by exciting adjacent spatial samples at the same time, or the same spatial sample at successive times instants.
In addition to the W components being non-local, they can demand a
larger dynamic range than the K variables. For example, if the entire
semi-infinite string for is initialized with velocity
,
the initial displacement traveling-wave components look as follows:
![]() |
(E.19) |
and the variables evolve forward in time as follows:
![]() |
(E.20) |
![]() |
(E.21) |
![]() |
(E.22) |
Thus, the left semi-infinite string moves upward at a constant velocity of 2, while a ramp spreads out to the left and right of position




where denotes the set of all integers.
While the FDTD excitation is also not local, of course, it is
bounded for all
.
Since the traveling-wave components of initial velocity excitations are generally non-local in a displacement-based simulation, as illustrated in the preceding examples, it is often preferable to use velocity waves (or force waves) in the first place [447].
Another reason to prefer force or velocity waves is that displacement
inputs are inherently impulsive. To see why this is so, consider that
any physically correct driving input must effectively exert some
finite force on the string, and this force is free to change
arbitrarily over time. The ``equivalent circuit'' of the infinitely
long string at the driving point is a ``dashpot'' having real,
positive resistance
. The applied force
can be
divided by
to obtain the velocity
of the string driving
point, and this velocity is free to vary arbitrarily over time,
proportional to the applied force. However, this velocity must be
time-integrated to obtain a displacement
. Therefore,
there can be no instantaneous displacement response to a finite
driving force. In other words, any instantaneous effect of an input
driving signal on an output displacement sample is non-physical except
in the case of a massless system. Infinite force is required to move
the string instantaneously. In sampled displacement simulations, we
must interpret displacement changes as resulting from time-integration
over a sampling period. As the sampling rate increases, any
physically meaningful displacement driving signal must converge to
zero.
Additive Inputs
Instead of initial conditions, ongoing input signals can be defined
analogously. For example, feeding an input signal into the FDTD
via
corresponds to physically driving a single sample of string displacement at position


Interpretation of the Time-Domain KW Converter
As shown above, driving a single displacement sample in the
FDTD corresponds to driving a velocity input at position
on two
alternating subgrids over time. Therefore, the filter
acts as the filter
on either subgrid alone--a
first-order difference. Since displacement is being simulated, velocity
inputs must be numerically integrated. The first-order difference can
be seen as canceling this integration, thereby converting a velocity
input to a displacement input, as in Eq.
(E.23).
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State Space Formulation
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State Transformations