ssrs 2d barcode CHAP. 81 in Software

Drawer Code 128 Code Set C in Software CHAP. 81

A simple way to write this recursion is in terms of vectors as follows:

EXAMPLE 8.5.1 For a second-order FIR lattice tiller with reflection coefficient< r I = t and relating x(n) to f l(n) is I 2A'(:) = An(:) + r l ; ' ~ , , () := 1 + i -

The recurrence formula in Eq. (8.5) provides an algorithm to find the system function A , ( z ) from the reflection coefficients r k ,k = 1,2. . . . . p. To find the reflection coefficients r r for a given system function A ,(z), we use the s t e p d o w n recu~siotz, which is given by

The reflection coefficients are then found from the polynomials A k ( z )by setting T L = u k ( k ) .

corresponding to the second-order FIR filter A z ( z ) =

Tz = n 2 ( 2 )=

= -l2 '

So far, we have only considered the system function relating the input .u(n) to the output f,,(n). A similar set of equations relate the input x ( n ) to the output R , , ( I I ) With

the relationship between the system function A p ( z )and A',,(z)is as follows:

An important property of the lattice filter is that the roots of A,(z) will lie inside the unit circle if and only if the reflection coefficients are bounded by I in magnitude:

This property is the basis for the Schur-Cohn stability test for digital filters. Specifically, a causal filter with a system function

will be stable if and only if the reflection coefficients associated with A(z) are bounded by 1 in magnitude.

The structure for an all-pole lattice filter is shown in Fig. 8-14. As with the FIR lattice, a pth-order all-pole filter is a cascade of p stages, with each stage being a two-port network that is parameterized by its reflection coefficient rk.The two inputs, f k ( n ) and gk-1 ( n ) ,are related to the two outputs f L F ( n )and g k ( n )by a pair of I coupled difference equations:

fk-l(n) = fh(n) - r k ~ k - l ( n 1) gk(n) = ~ k - l ( n 1 ) -

+ rkfk(n)

where A,(z) is the polynomial that is generated by the recursion given in Eq. (8.5). In addition, note that the given in Eq. (8.7). system function relating x ( n ) to w ( n ) is an allpass filter with a system function Hap(z)

Fig. 8-14. A pth-order all-pole lattice filter. (a)The two-port network for the kth stage of the all-pole lattice filter. (b) Cascade of p lattice stages.

If H ( z ) is an IIR filter with p poles and q zeros,

with q 5 p, a lattice filter implementation of H ( z ) consists of two components. The first is an all-pole lattice with reflection coefficients r l ,rz,. . , F, that implements I / A p ( z ) . The second is a tapped delay line with . coefficients cq(k). The structure is illustrated in Fig. 8-15 for the case in which p = 4. The relationship between the lattice filter coefficients cq(k)and the direct form coefficients bq(k) is given by

This recursion is initialized with c q ( q )= bq(9).

A third-order low-pass elliptic filter with a cutoff frequency of w,. = 0 . 5 ~ a system function has

To implement this filter using a lattice filter structure, we first transform the denominator coefficients into reflection coefficients. Using the step-down recursion, we find