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216 Chapter 6 IMPLEMENTATION OF DISCRETE-TIME FILTERS
6.2.2 TRANSPOSED STRUCTURE
An equivalent structure to the direct form can be obtained using a procedure
called transposition. Inthis operation three steps are performed:
1. All path arrow directions are reversed.
2. All branch nodes are replaced by adder nodes, and all adder nodes are
replaced by branch nodes.
3. The input and output nodes are interchanged.
The resulting structure is called the transposed direct form structure. The
transposed direct form II structure is shown in Figure 6.3b. Problem P6.3
explains this equivalent structure.
6.2.3 MATLAB IMPLEMENTATION
In MATLAB the direct form structure is described by two row vectors;
b containing the {b n } coefficients and a containing the {a n } coefficients.
The filter function, which is discussed in Chapter 2, implements the
transposed direct form II structure.
6.2.4 CASCADE FORM
In this form the system function H(z) iswritten as a product of 2nd-order
sections with real coefficients. This is done by factoring the numerator and
denominator polynomials into their respective roots and then combining
either a complex conjugate root pair or any two real roots into 2nd-order
polynomials. In the remainder of this chapter, we assume that N is an
even integer. Then
H(z) = b 0 + b 1 z −1 + ···+ b N z −N
1+a 1 z −1 + ···+ a N z −N
= b 0
1+ b1
b 0
z −1 + ···+ b N
b0
z −N
1+a 1 z −1 + ···+ a N z −N
= b 0
K
∏
k=1
1+B k,1 z −1 + B k,2 z −2
1+A k,1 z −1 + A k,2 z −2 (6.3)
where K is equal to N 2 , and B k,1, B k,2 , A k,1 , and A k,2 are real numbers
representing the coefficients of 2nd-order sections. The 2nd-order section
H k (z) = Y k+1(z)
Y k (z)
= 1+B k,1z −1 + B k,2 z −2
1+A k,1 z −1 ; k =1,...,K
+ A k,2 z−2 with
Y 1 (z) =b 0 X(z);
Y K+1 (z) =Y (z)
is called the kth biquad section. The input to the kth biquad section is
the output from the (k − 1)th biquad section, and the output from the
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