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214 Chapter 6 IMPLEMENTATION OF DISCRETE-TIME FILTERS
6.2 IIR FILTER STRUCTURES
The system function of an IIR filter is given by
H(z) = B(z) ∑ M
A(z) = n=0 b nz −n
∑ N
n=0 a nz = b 0 + b 1 z −1 + ···+ b M z −M
−n 1+a 1 z −1 + ···+ a N z −N ; a 0 =1
(6.1)
where b n and a n are the coefficients of the filter. We have assumed without
loss of generality that a 0 =1. The order of such an IIR filter is called N if
a N ̸=0.The difference equation representation of an IIR filter is expressed
as
M∑
N∑
y(n) = b m x(n − m) − a m y(n − m) (6.2)
m=0
m=1
Three different structures can be used to implement an IIR filter:
1. Direct form: In this form the difference equation (6.2) is implemented
directly as given. There are two parts to this filter, namely the moving
average part and the recursive part (or equivalently, the numerator
and denominator parts). Therefore this implementation leads to two
versions: direct form I and direct form II structures.
2. Cascade form: In this form the system function H(z) inequation
(6.1) is factored into smaller 2nd-order sections, called biquads. The
system function is then represented as a product of these biquads. Each
biquad is implemented in a direct form, and the entire system function
is implemented as a cascade of biquad sections.
3. Parallel form: This is similar to the cascade form, but after factorization,
a partial fraction expansion is used to represent H(z) asasum
of smaller 2nd-order sections. Each section is again implemented in a
direct form, and the entire system function is implemented as a parallel
network of sections.
We will briefly discuss these forms in this section. IIR filters are generally
described using the rational form version (or the direct form structure)
of the system function. Hence we will provide MATLAB functions for
converting direct form structures to cascade and parallel form structures.
6.2.1 DIRECT FORM
As the name suggests, the difference equation (6.2) is implemented as
given using delays, multipliers, and adders. For the purpose of illustration,
let M = N =4.Then the difference equation is
y(n) =b 0 x(n)+b 1 x(n − 1) + b 2 x(n − 2) + b 3 x(n − 3) + b 4 x(n − 4)
− a 1 y(n − 1) − a 2 y(n − 2) − a 3 y(n − 3) − a 4 y(n − 4)
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