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

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

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

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

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 ( $\omega T=\pm\pi$ ).