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Length Three FIR Loop Filter

The simplest nondegenerate example of the loop filters of §6.8 is the three-tap FIR case ( $ N_{\hat g}=3$). The symmetry constraint leaves two degrees of freedom in the frequency response:10.1

$\displaystyle {\hat G}(e^{j\omega T}) = {\hat g}(0) + 2{\hat g}(1) \cos(\omega T) $

If the dc gain is normalized to unity, then $ {\hat g}(0)+2{\hat g}(1)=1$, and there is only one remaining degree of freedom which can be interpreted as a damping control. As damping is increased, the duration of free vibration is reduced at all nonzero frequencies, and the decay of higher frequencies is accelerated relative to lower frequencies, provided

$\displaystyle {\hat g}(0) \ge 2{\hat g}(1) > 0. $

In this coefficient range, the string-loop amplitude response can be described as a ``raised cosine'' having a unit-magnitude peak at dc, and minimum gains $ {\hat g}(0)-2{\hat g}(1)\ge 0$ at plus and minus half the sampling rate ( $ \omega T=\pm\pi$).
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Phasing with 2nd-Order Allpass Filters