Discretization in the Spectral Domain

and for an exponential state at frequencies , we have

where and . The ``impedance'' is here and clearly satisfies MD positive realness criterion given in (3.31) (and furthermore is MD-lossless) if . As in the lumped case, the trapezoid rule, now applied in the direction, can be interpreted as a spectral mapping

where is some arbitrary step-size in the direction. For notational purposes, we have used

to represent the frequency domain equivalent of a unit shift in the direction. In complete analogy with the lumped case, (3.43) implies that

This shift can of course also be written in terms of delays and shifts in the coordinates. For example, consider the coordinate transformation defined in (3.18). In this case we have, in the frequency domain,

where and are the frequency variables conjugate to and respectively. (We assume that our spatial domain is of infinite extent, so that corresponds to an imaginary Fourier transform variable.) Suppose we have also chosen the step-sizes in the two coordinates such that the grids overlap, that is, , where is the shift in the pure time direction. Then, for a shift of in the direction, we can write

or

where represents a delay of duration in the time direction, and corresponds to a shift over distance in the positive space direction. Similarly, we can write

For a more complex example, consider again the transformation defined by

which maps coordinates to a five-dimensional coordinates . A shift of in direction corresponds to a transmittance of the form

where represents a unit shift (of length ) in the -direction, and as before, corresponds to a unit delay of . The other shifts can be written as

where represents a unit shift (of length ) in the direction. At a given grid point in the old coordinates, the unit delays , interpreted as directional shifts, refer to points on the grid at the previous time step, and located one grid point away in the , , and directions, respectively. The unit delay is simply a unit time delay.

It is important to note the manner in which the special character of the coordinate transformation manifests itself here. Due to the positivity requirement on the elements of the last column of
, a unit delay in *any* of the directions will always include some delay in the pure time direction. By means of this requirement, and the introduction of wave variables, MDWD networks can, in the same way as their lumped counterparts, be designed in which delay-free loops do not appear. Such networks, when used for simulation, will give rise, in general, to *explicit* numerical schemes [176].