### Half-Rate Ladder Waveguide Filters

The delays preceding the two inputs to each scattering junction can be ``pushed'' into the junction and then ``pulled'' out to the outputs and combine with the delays there. (This is easy to show using the Kelly-Lochbaum scattering junction derived in §C.8.4.) By performing this operation on every other section in the DWF chain, the*half-rate ladder waveguide filter*shown in Fig.C.25 is obtained [432].

figure[htbp] This ``half-rate'' ladder DWF is so-called because its sampling rate can be cut in half due to each delay being two-samples long. It has advantages worth considering, which will become clear after we have derived conventional ladder digital filters below. Note that now

*pairs*of scattering junctions can be computed in parallel, and an exact physical interpretation remains. That is, it can still be used as a general purpose physical modeling building block in this more efficient (half-rate) form. Note that when the sampling rate is halved, the physical wave variables (computed by summing two traveling-wave components) are at the same time only every other spatial sample. In particular, the physical transverse forces on the right side of scattering junctions and in Fig.C.25 are

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