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Multirate Noble Identities

Figure 11.13 illustrates the so-called noble identities for commuting downsamplers/upsamplers with ``sparse transfer functions'' that can be expressed a function of $ z^{-N}$ . 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.


\begin{psfrags}
% latex2html id marker 30272\psfrag{nd}{ $N\downarrow$\ }\psfrag{hz}{ $H(z)$\ }\psfrag{hzn}{ $H(z^N)$\ }\psfrag{equal}{ $\equiv$\ }\begin{figure}[htbp]
\includegraphics[width=0.9\twidth]{eps/noble}
\caption{Multirate noble identities}
\end{figure} %
\end{psfrags}

Figure 11.14: Commuting of downsampler with adder and gains.
\includegraphics[width=0.9\twidth]{eps/noble_commute}


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``Spectral Audio Signal Processing'', by Julius O. Smith III, W3K Publishing, 2011, ISBN 978-0-9745607-3-1.
Copyright © 2016-07-18 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University
CCRMA