The preceding derivation generalizes immediately to frequency-dependent losses. First imagine each in Fig.C.7 to be replaced by , where for passivity we require
In the time domain, we interpret as the impulse response corresponding to . We may now derive the frequency-dependent counterpart of Eq. (C.31) as follows:
where denotes convolution (in the time dimension only). Define filtered node variables by
Then the frequency-dependent FDTD scheme is simply
We see that generalizing the FDTD scheme to frequency-dependent losses requires a simple filtering of each node variable by the per-sample propagation filter . For computational efficiency, two spatial lines should be stored in memory at time : and , for all . These state variables enable computation of , after which each sample of ( ) is filtered by to produce for the next iteration, and is filtered by to produce for the next iteration.
The frequency-dependent generalization of the FDTD scheme described in this section extends readily to the digital waveguide mesh. See §C.14.5 for the outline of the derivation.