##

State Space Formulation

In this section, we will summarize and extend the above discussion by
means of a *state space analysis* [220].

### FDTD State Space Model

Let
denote the FDTD state for one of the two subgrids at time
, as defined by Eq.(E.10). The other subgrid is handled
identically and will not be considered explicitly. In fact, the other
subgrid can be dropped altogether to obtain a *half-rate,
staggered grid* scheme [55,147]. However, boundary
conditions and input signals will couple the subgrids, in general. To
land on the same subgrid after a state update, it is necessary to
advance time by two samples instead of one. The state-space model for
one subgrid of the FDTD model of the ideal string may then be written
as

To avoid the issue of boundary conditions for now, we will continue working with the infinitely long string. As a result, the state vector denotes a point in a space of countably infinite dimensionality. A proper treatment of this case would be in terms of operator theory [325]. However, matrix notation is also clear and will be used below. Boundary conditions are taken up in §E.4.3.

When there is a general input signal vector , it is necessary to augment the input matrix to accomodate contributions over both time steps. This is because inputs to positions at time affect position at time . Henceforth, we assume and have been augmented in this way. Thus, if there are input signals , , driving the full string state through weights , , the vector is of dimension :

forms the output signal as an arbitrary linear combination of states. To obtain the usual displacement output for the subgrid, is the matrix formed from the identity matrix by deleting every other row, thereby retaining all displacement samples at time and discarding all displacement samples at time in the state vector :

The intra-grid state update for even is then given by

For odd , the update in Eq.(E.25) is used. Thus, every other row of , for time , consists of the vector preceded and followed by zeros. Successive rows for time are shifted right two places. The rows for time consist of the vector aligned similarly:

### DW State Space Model

As discussed in §E.2, the traveling-wave decomposition Eq.(E.4) defines a linear transformation Eq.(E.10) from the DW state to the FDTD state:

Since is invertible, it qualifies as a linear transformation for performing a

*change of coordinates*for the state space. Substituting Eq.(E.27) into the FDTD state space model Eq.(E.24) gives

Multiplying through Eq.(E.28) by gives a new state-space representation of the same dynamic system which we will show is in fact the DW model of Fig.E.2:

(E.30) |

where

To verify that the DW model derived in this manner is the computation diagrammed in Fig.E.2, we may write down the state transition matrix for one subgrid from the figure to obtain the permutation matrix ,

and displacement output matrix :

#### DW Displacement Inputs

We define general DW inputs as follows:

(E.33) | |||

(E.34) |

The th block of the input matrix driving state components and multiplying is then given by

Typically, input signals are injected equally to the left and right along the string, in which case

To show that the directly obtained FDTD and DW state-space models correspond to the same dynamic system, it remains to verify that . It is somewhat easier to show that

A straightforward calculation verifies that the above identity holds, as expected. One can similarly verify , as expected. The relation provides a recipe for translating any choice of input signals for the FDTD model to equivalent inputs for the DW model, or vice versa. For example, in the scalar input case (), the DW input-weights become FDTD input-weights according to

Finally, when and for all , we obtain the result familiar from Eq.(E.23):

####

DW Non-Displacement Inputs

Since a displacement input at position corresponds to
symmetrically exciting the right- and left-going traveling-wave
components and , it is of interest to understand what
it means to excite these components *antisymmetrically*. As
discussed in §E.3.3, an antisymmetric excitation of
traveling-wave components can be interpreted as a *velocity*
excitation. It was noted that localized velocity excitations in the
FDTD generally correspond to non-localized velocity excitations in the
DW, and that velocity in the DW is proportional to the *spatial
derivative* of the difference between the left-going and right-going
traveling displacement-wave components (see Eq.(E.13)). More
generally, the antisymmetric component of displacement-wave excitation
can be expressed in terms of any wave variable which is linearly
independent relative to displacement, such as acceleration, slope,
force, momentum, and so on. Since the state space of a vibrating
string (and other mechanical systems) is traditionally taken to be
position and velocity, it is perhaps most natural to relate the
antisymmetric excitation component to velocity.

In practice, the simplest way to handle a velocity input in a DW simulation is to first pass it through a first-order integrator of the form

to convert it to a displacement input. By the equivalence of the DW and FDTD models, this works equally well for the FDTD model. However, in view of §E.3.3, this approach does not take full advantage of the ability of the FDTD scheme to provide localized velocity inputs for applications such as simulating a piano hammer strike. The FDTD provides such velocity inputs for ``free'' while the DW requires the external integrator Eq.(E.37).

Note, by the way, that these ``integrals'' (both that done internally by the FDTD and that done by Eq.(E.37)) are merely sums over discrete time--not true integrals. As a result, they are exact only at dc (and also trivially at , where the output amplitude is zero). Discrete sums can also be considered exact integrators for impulse-train inputs--a point of view sometimes useful when interpreting simulation results. For normal bandlimited signals, discrete sums most accurately approximate integrals in a neighborhood of dc. The KW-converter filter has analogous properties.

#### Input Locality

The DW state-space model is given in terms of the FDTD state-space
model by Eq.(E.31). The similarity transformation matrix
is
bidiagonal, so that
and
are both approximately
diagonal when the output is string displacement for all . However,
since
given in Eq.(E.11) is upper triangular, the input matrix
can replace sparse input matrices
with only
half-sparse
, unless successive columns of
are equally
weighted, as discussed in §E.3. We can say that local
K-variable excitations may correspond to *non-local* W-variable
excitations. From Eq.(E.35) and Eq.(E.36), we see that
*displacement inputs are always local in both systems*.
Therefore, local FDTD and non-local DW excitations can only occur when
a variable dual to displacement is being excited, such as velocity.
If the external integrator Eq.(E.37) is used, all inputs are
ultimately displacement inputs, and the distinction disappears.

###

Boundary Conditions

The relations of the previous section do not hold exactly when the
string length is finite. A finite-length string forces consideration
of *boundary conditions*. In this section, we will introduce
boundary conditions as perturbations of the state transition matrix.
In addition, we will use the DW-FDTD equivalence to obtain physically
well behaved boundary conditions for the FDTD method.

Consider an ideal vibrating string with spatial samples. This is a sufficiently large number to make clear most of the repeating patterns in the general case. Introducing boundary conditions is most straightforward in the DW paradigm. We therefore begin with the order 8 DW model, for which the state vector (for the 0th subgrid) will be

#### Resistive Terminations

Let's begin with simple ``resistive'' terminations at the string endpoints, resulting in the reflection coefficient at each end of the string, where corresponds to nonnegative (passive) termination resistances [447]. Inspection of Fig.E.2 makes it clear that terminating the left endpoint may be accomplished by setting

(E.38) |

The simplest choice of state transformation matrix is obtained by cropping it to size :

where
and
. We see that the left
FDTD termination is *non-local* for , while the right
termination is local (to two adjacent spatial samples) for all .
This can be viewed as a consequence of having ordered the FDTD state
variables as
instead of
. Choosing the other ordering
interchanges the endpoint behavior. Call these orderings Type I and
Type II, respectively. Then
; that is, the similarity
transformation matrix
is transposed when converting from Type I
to Type II or vice versa. By anechoically coupling a Type I FDTD
simulation on the right with a Type II simulation on the left,
general resistive terminations may be obtained on both ends which are
localized to two spatial samples.

In nearly all musical sound synthesis applications, at least one of the string endpoints is modeled as rigidly clamped at the ``nut''. Therefore, since the FDTD, as defined here, most naturally provides a clamped endpoint on the left, with more general localized terminations possible on the right, we will proceed with this case for simplicity in what follows. Thus, we set and obtain

#### Boundary Conditions as Perturbations

To study the effect of boundary conditions on the state transition matrices and , it is convenient to write the terminated transition matrix as the sum of of the ``left-clamped'' case (for which ) plus a series of one or more rank-one perturbations. For example, introducing a right termination with reflectance can be written

where is the matrix containing a 1 in its th entry, and zero elsewhere. (Following established convention, rows and columns in matrices are numbered from 1.)

In general, when is odd, adding
to
corresponds to a *connection* from left-going waves to
right-going waves, or vice versa (see Fig.E.2). When is
odd and is even, the connection flows from the right-going to the
left-going signal path, thus providing a termination (or partial
termination) on the right. Left terminations flow from the bottom to
the top rail in Fig.E.2, and in such connections is even
and is odd. The spatial sample numbers involved in the connection
are
and
, where
denotes the greatest integer less than or equal to
.

The rank-one perturbation of the DW transition matrix Eq.(E.39) corresponds to the following rank-one perturbation of the FDTD transition matrix :

In general, we have

Thus, the general rule is that transforms to a matrix which is zero in all but two rows (or all but one row when ). The nonzero rows are numbered and (or just when ), and they are identical, being zero in columns , and containing starting in column .

#### Reactive Terminations

In typical string models for virtual musical instruments, the ``nut
end'' of the string is rigidly clamped while the ``bridge end'' is
terminated in a *passive reflectance* . The condition
for passivity of the reflectance is simply that its gain be bounded
by 1 at all frequencies [447]:

A very simple case, used, for example, in the Karplus-Strong plucked-string algorithm, is the two-point-average filter:

This gives the desired filter in a half-rate, staggered grid case. In the full-rate case, the termination filter is really

Another often-used string termination filter in digital waveguide models is specified by [447]

where is an overall gain factor that affects the decay rate of all frequencies equally, while controls the relative decay rate of low-frequencies and high frequencies. An advantage of this termination filter is that the delay is always one sample, for all frequencies and for all parameter settings; as a result, the tuning of the string is invariant with respect to termination filtering. In this case, the perturbation is

where

The filtered termination examples of this section generalize immediately to arbitrary finite-impulse response (FIR) termination filters . Denote the impulse response of the termination filter by

#### Interior Scattering Junctions

A so-called *Kelly-Lochbaum scattering junction*
[297,447] can be introduced into the string at the fourth
sample by the following perturbation

A single time-varying scattering junction provides a reasonable model for plucking, striking, or bowing a string at a point. Several adjacent scattering junctions can model a distributed interaction, such as a piano hammer, finger, or finite-width bow spanning several string samples.

Note that scattering junctions separated by one spatial sample (as typical in ``digital waveguide filters'' [447]) will couple the formerly independent subgrids. If scattering junctions are confined to one subgrid, they are separated by two samples of delay instead of one, resulting in round-trip transfer functions of the form (as occurs in the digital waveguide mesh). In the context of a half-rate staggered-grid scheme, they can provide general IIR filtering in the form of a ladder digital filter [297,447].

### Lossy Vibration

The DW and FDTD state-space models are equivalent with respect to lossy traveling-wave simulation. Figure E.4 shows the flow diagram for the case of simple attenuation by per sample of wave propagation, where for a passive string.

The DW state update can be written in this case as

*e.g.*,

### State Space Summary

We have seen that the DW and FDTD schemes correspond to state-space models which are related to each other by a simple change of coordinates (similarity transformation). It is well known that such systems exhibit the same transfer functions, have the same modes, and so on. In short, they are the same linear dynamic system. Differences may exist with respect to spatial locality of input signals, initial conditions, and boundary conditions.

State-space analysis was used to translate initial conditions and boundary conditions from one case to the other. Passive terminations in the DW paradigm were translated to passive terminations for the FDTD scheme, and FDTD excitations were translated to the DW case in order to interpret them physically.

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