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 xi, for i = 1 to 8, and image is denoted by ci, for i = 0 to 3. From (12.74), each xi(n), for i = 3 to 8, has unit power. In terms of the state covariance matrix, Kii = 1, for i = 3 to 8, where Kii is the i-th diagonal element of K. Then, to scale the filter, we only need to compute the powers of x1(n) and x2(n), which can be computed from (11.25). In the case of the normalized lattice filter, K11 and K22 can be computed more efficiently using the orthonormality of the Schur polynomials. From Fig 12.17, the polynomial at the state x1 is given by:

image

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

image

Therefore, using the orthonormality of the Schur polynomials,

image

By the same way,

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

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