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:

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

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.