ssrs 2d barcode Fig. 8-8. Direct form 11 realization of an IIR filter with p = q . in Software

Drawer Code128 in Software Fig. 8-8. Direct form 11 realization of an IIR filter with p = q .

Fig. 8-8. Direct form 11 realization of an IIR filter with p = q .
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8.4.2 Cascade Structure
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The cascade structure is derived by factoring the numerator and denominator polynomials of H(z):
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This factorization corresponds to a cascade of first-order filters, each having one pole and one zero. In general. the coefficients n and Bk will be complex. However, if h ( n ) is real. the roots of H(z) will occur in complex k
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CHAP. 81
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IMPLEMENTATION OF DISCRETE-TIME SYSTEMS
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conjugate pairs, and these complex conjugate factors may be combined to form second-order factors with real coefficients:
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A sixth-order 1IR filter implemented as a cascade of three second-order systems in direct form I1 is shown in Fig. 8-9.
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Fig. 8-9. A sixth-order 11R filter implemented as a cascade of three direct form 11 second-order systems.
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There is considerable flexibility in how a system may be implemented in cascade form. For example, there are different pairings of the poles and zeros and different ways in which the sections may be ordered.
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8.4.3 Parallel Structure
An alternative to factoring H ( z ) is to expand the system function using a partial fraction expansion. For example, with
if p > q and ai # c r k (the roots of the denominator polynomial are distinct), H ( z ) may be expanded as a sum of p first-order factors as follows:
where the coefficients At and a k are, in general, complex. This expansion corresponds to a sum of p first-order system functions and may be realized by connecting these systems in parallel. If h ( n ) is real, the poles of H ( z ) will occur in complex conjugate pairs, and these complex roots in the partial fraction expansion may be combined to form second-order systems with real coefficients:
Shown in Fig. 8-10 is a sixth-order filter implemented as a parallel connection of three second-order direct form I1 systems. If p 5 q, the partial fraction expansion will also contain a term of the form
which is an FIR filter that is placed in parallel with the other terms in the expansion of H(z).
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[CHAP. 8
Fig. 8-10. A sixth-order IIR filter implemented as a parallel connection of three second-order direct form 11 structures.
8.4.4 Transposed Slruclures
The transposition theorem states that the input-output properties of' a network remain unchanged after the following sequence of network operations: 1. Reverse the direction of all branches. 2. Change branch points into summing nodes and summing nodes into branch points. 3. Interchange the input and output. Applying these manipulations to a network results in what is referred to as the transposed form. Shown in Fig. 8-1 1 are second-order transposed direct form I and direct form I1 filter structures.
8.4.5 Allpass Filters
An allpass filter has a frequency response with a constant magnitude:
IH,,(&")J=~
allw
If the system function of an allpass filter is a rational function of z, it has the form
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VJ) Fig. 8-11. Transposed direct form ti lter structures. [ u )Transposed direct form I. (b) Transposed direct form 11.
If h ( n ) is real-valued, the complex roots occur in conjugate pairs, and these pairs may be combined to form second-order factors with real coefficients:
A direct form I1 implementation for one of these sections is shown in Fig. 8- 12. Because each section only has two distinct coefficients, ak and Bk, it is possible to implement these sections using as few as two multiplies.
Fig. 8-12. A second-order section of an allpass ti lter imple mented in direct form 11.
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[CHAP. 8
8.5 LATTICE FILTERS
Lattice filters have a number of interesting and important properties that make them popular in a number of different applications. These properties include modularity, low sensitivity to parameter quantization effects, and a simple criterion for ensuring filter stability. In the following sections, we present the lattice filter structure for FIR ti lters, all-pole filters, and filters that have both poles and zeros.
8.5.1 FIR Lattice Filters
An FIR lattice filter is a cascade of two-port networks as shown in Fig. 8-13. Each two-port network is defined The two inputs, fk-, ( n ) and g k - ~ ( n )are related to the outputs fk(n) , by the value of its reflection coeficient, r k . and g k ( n )by a pair of coupled difference equations
with the input to the first section being fo(n) = g o @ ) = x ( n ) .
Fig. 8-13. A pth-order FIR lattice filter. (a) The two-port network for each lattice filter module. (b)A cascade of p lattice filter modules.
With A k ( z )the system function relating the input x ( n ) to the intermediate output f k ( n ) ,
these difference equations may be solved by induction to yield the following recurrence formula for A k ( z ) :
which is called the step-up recucrion. The recursion is initialized by setting A o ( z ) = 1. This recurrence formula also defines a recurrence relation for the coefficients a k ( i )of A k ( z ) ,which is
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