Multirate Noble Identities

Figure 10.14 shows the two so-called ``noble identities'' for commuting downsamplers/upsamplers with ``sparse transfer functions'' which can be expressed a function of $z^{-N}$. Note that downsamplers and upsamplers are linear, time-varying operators. Therefore, operation order is very important. It is also important to note that adders or multipliers (any memoryless operators) can commute across downsamplers and upsamplers, as shown in Fig.10.15.

Figure 10.15: Commuting of downsampler with adder and gains.