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CHAP. 81
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IMPLEMENTATION O F DISCRETE-TIME SYSTEMS
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A simple way to write this recursion is in terms of vectors as follows:
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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 -
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r2= i, the system function
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and the second-order system function relating . r ( ~ r ) fz(n) is to
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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
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In terms of the coefficients a k ( i ) ,this recursion is
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The reflection coefficients are then found from the polynomials A k ( z )by setting T L = u k ( k ) .
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EXAMPLE 8.5.2 To find the reflection coefticients T I and 1': 1 - f z - 2 , we begin by setting
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corresponding to the second-order FIR filter A z ( z ) =
Tz = n 2 ( 2 )=
Next, we find A 1 ( z ) us~ng step-down recursion. the
Because cl,(l) = 0. r l = 0. Therefore, the reflection coefficients are
= 0 and
= -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:
IMPLEMENTATION OF DISCRETE-TIME SYSTEMS
[CHAP. 8
Thus, f,(n) and g p ( n )are related by an allpass filter, F,(z) = Hu,f(z)G,(z), where
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.
8.5.2 All-Pole Lattice Filters
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)
The system function relating the input x ( n ) to the output y ( n ) is
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.
CHAP. 81
IMPLEMENTATION OF DISCRETE-TIME SYSTEMS
8.5.3 IZR Lattice Filters
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
Similarly, a recursion that generates the coefficients cq(k)from the coefficients h q ( k )is
This recursion is initialized with c q ( q )= bq(9).
EXAMPLE 8.5.3
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
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