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IIR Filter Structures 221
6.2.6 PARALLEL FORM
In this form the system function H(z) iswritten as a sum of 2nd-order
sections using partial fraction expansion.
H(z) = B(z)
A(z) = b 0 + b 1 z −1 + ···+ b M z −M
1+a 1 z −1 + ···+ a N z −N
= ˆb 0 + ˆb 1 z −1 + ···+ ˆb N−1 z 1−N M−N
1+a 1 z −1 + ···+ a N z −N + ∑
C k z −k
0
} {{ }
only if M≥N
=
K∑ B k,0 + B k,1 z −1 M−N
1+A k,1 z −1 + A k,2 z −2 + ∑
C k z −k
k=1
0
} {{ }
only if M≥N
(6.4)
where K is equal to N 2 , and B k,0, B k,1 , 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)
=
B k,0 + B k,1 z −1
1+A k,1 z −1 ; k =1,...,K
+ A k,2 z−2 with
Y k (z) =H k (z)X(z), Y(z) = ∑ Y k (z), M < N
is the kth proper rational biquad section. The filter input is available to
all biquad sections as well as to the polynomial section if M ≥ N (which
is an FIR part). The output from these sections is summed to form the
filter output. Now each biquad section H k (z) can be implemented in direct
form II. Due to the summation of subsections, a parallel structure can be
built to realize H(z). As an example, consider M = N =4.Figure 6.7
shows a parallel-form structure for this 4th-order IIR filter.
6.2.7 MATLAB IMPLEMENTATION
The following function dir2par converts the direct-form coefficients {b n }
and {a n } into parallel form coefficients {B k,i } and {A k,i }.
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