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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 [362]. In practice, the half-sample delay must be compensated elsewhere in the filtered delay loop, such as in the delay-line interpolation filter [207]. 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 [454] for related discussion from a software implementation perspective.
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