The general IIR lattice system in Figure 11.21 is restricted to all-pole systems. Howeve...

The general IIR lattice system in Figure 11.21 is restricted to all-pole systems. However, both poles and zeros can be implemented by the system of Figure P11.27-1 (Gray and Markel, 1973, 1975). Each of the sections in Figure P11.27-1 is described by the flow graph of Figure P11.27-2. In other words, Figure 11.21 is embedded in Figure P11.27-1 with the output formed as a linear combination of the backward prediction error sequences.

(a) Show that the system function between the input X(z) = E^(p)(z) and is

(b) Show that is an all-pass system. (This result is not needed for the rest of the problem.)

(c) The overall system function from X(z) to Y(z) is

Show that the numerator Q(z) in Eq. (P11.27-2) is a p^th-order polynomial of the form

where the coefficients c_m in Figure P11.27 are given by the equation

(d) Give a procedure for computing all the parameters needed to implement a system function such as Eq. (P11.27-2) using the lattice structure of Figure P11.27.

(e) Using the procedure described in (c), draw the complete flow graph of the lattice implementation of the system