Elementary recirculating filter

The elementary recirculating filter is the recirculating comb filter of Figure 7.7 with a complex-valued feedback gain $P$ as shown in Figure 8.11 (part a). By the same analysis as before, feeding this network a sinusoid whose $n$th sample is $Z^n$ gives an output of:

$${{1} \over {1 - P {Z^{-1}}}} {Z^n}$$

so the transfer function is

$$H(Z) = {{1} \over {1 - P {Z^{-1}}}}$$

The recirculating filter is stable when $\vert P\vert < 1$; when, instead, $\vert P\vert> 1$ the output grows exponentially as the delayed sample recirculates.

Figure 8.11: The elementary recirculating filter: (a) block diagram; (b) frequency response.

The transfer function is thus just the inverse of that of the non-recirculating filter (first form). If you put the two in series with $P=Q$, the output theoretically equals the input. (This analysis only demonstrates it for sinusoidal inputs; that it follows for other signals as well can be verified by working out the impulse response of the combined network).