In this section, we will summarize and extend the above discussion by
means of a state space analysis .
State Space Formulation
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 . 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 :
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:
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:
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 :
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
Finally, when and for all , we obtain the result familiar from Eq.(E.23):
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
DW Non-Displacement Inputs
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.
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.
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
The simplest choice of state transformation matrix is obtained by cropping it to size :
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 .
reflectance . The condition for passivity of the reflectance is simply that its gain be bounded by 1 at all frequencies :
A very simple case, used, for example, in the Karplus-Strong plucked-string algorithm, is the two-point-average filter:
Kelly-Lochbaum scattering junction [297,447] can be introduced into the string at the fourth sample by the following perturbation
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.
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.