## 12.6 DERIVATION OF SCALED-NORMALIZED LATTICE FILTER

By applying the slow-down and retiming/pipelining (SRP) transformation (see Section 11.7) with *M =* 2 to a 3rd-order normalized lattice filter, we obtain the filter structure in Fig. 12.17. In Fig. 12.17, each delay element is represented by *x*_{i}, for *i* = 1 to 8, and is denoted by *c*_{i}, for *i* = 0 to 3. From (12.74), each *x*_{i}(*n*), for *i* = 3 to 8, has unit power. In terms of the state covariance matrix, *K*_{ii} = 1, for *i* = 3 to 8, where **K**_{ii} is the *i*-th diagonal element of **K**. Then, to scale the filter, we only need to compute the powers of *x*_{1}(*n*) and *x*_{2}(*n*), which can be computed from (11.25). In the case of the normalized lattice filter, *K*_{11} and *K*_{22} can be computed more efficiently using the orthonormality of the Schur polynomials. From Fig 12.17, the polynomial at the state *x*_{1} is given by:

*Fig. 12.17* A 3rd-order normalized lattice filter after upsampling and retiming with *M* = 2.

Therefore, using the orthonormality of the Schur polynomials,

By the same way,

In general, for the summing node on the top line of module *i*, the power is . To have ...