Figure 11.13 illustrates the so-called noble identities for commuting downsamplers/upsamplers with ``sparse transfer functions'' that can be expressed a function of . Note that downsamplers and upsamplers are linear, time-varying operators. Therefore, operation order is important. Also note that adders and multipliers (any memoryless operators) may be commuted across downsamplers and upsamplers, as shown in Fig.11.14.
Two-Channel Critically Sampled Filter Banks
Filtering and Downsampling, Revisited