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One-Zero Loop Filter

If we relax the constraint that $ N_{\hat g}$ be odd, then the simplest case becomes the one-zero digital filter:

$\displaystyle {\hat G}(z) = {\hat g}(0) + {\hat g}(1) z^{-1}
$

When $ {\hat g}(0)={\hat g}(1)$ , the filter is linear phase, and its phase delay and group delay are equal to $ 1/2$ sample [365]. In practice, the half-sample delay must be compensated elsewhere in the filtered delay loop, such as in the delay-line interpolation filter [208]. Normalizing the dc gain to unity removes the last degree of freedom so that $ {\hat g}(0) = {\hat g}(1) = 1/2$ , and $ {\hat G}(e^{j\omega T}) = \cos\left({\omega T/ 2}\right),\,\left\vert\omega\right\vert\leq \pi f_s$ .

See [456] for related discussion from a software implementation perspective.


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``Physical Audio Signal Processing'', by Julius O. Smith III, W3K Publishing, 2010, ISBN 978-0-9745607-2-4.
Copyright © 2014-03-23 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University
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