A Reformulation

• For reasons to be discussed in the next section, Van Duyne reformulates the second-order filter of Eq. 1 with $r_{k} = 1$ as
    $$\frac{1}{1 + z^{-1} H_{k}(z)},$$ (2)
  where $H_{k}$ is a first order allpass filter of the form
    $$H_{k}(z) = \frac{a_k + z^{-1}}{1 + a_k z^{-1}}$$ (3)
  and $a_k = -\cos(2 \pi f_{k} T_{s})$.

• With $r_{k} = 1$, the resonator does not decay but rather behaves like an oscillator with frequency $f_{k}$.

• Eq. 2 has an additional zero at $-a_k$ that is not present in Eq. 1, though this has little affect on the resonating behavior of the filter.

• A block diagram of the resulting filter structure is shown in Fig. 13.

  Figure 13: The CMS resonator filter structure.