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FIR Filter Structures 229
FIGURE 6.10
Direct form FIR structure
6.3.1 DIRECT FORM
The difference equation (6.7) is implemented as a tapped delay line since
there are no feedback paths. Let M =5(i.e., a 4th-order FIR filter); then
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)
The direct form structure is given in Figure 6.10. Note that since the
denominator is equal to unity, there is only one direct form structure.
6.3.2 MATLAB IMPLEMENTATION
In MATLAB the direct form FIR structure is described by the row vector
b containing the {b n } coefficients. The structure is implemented by the
filter function, in which the vector a is set to the scalar value 1, as
discussed in Chapter 2.
6.3.3 CASCADE FORM
This form is similar to that of the IIR form. The system function H(z) is
converted into products of 2nd-order sections with real coefficients. These
sections are implemented in direct form and the entire filter as a cascade
of 2nd-order sections. From (6.5)
H(z) =b 0 + b 1 z −1 + ···+ b M−1 z −M+1 (6.8)
(
= b 0 1+ b 1
z −1 + ···+ b )
M−1
z −M+1
b 0 b 0
∏ K
(
= b 0 1+Bk,1 z −1 + B k,2 z −2)
k=1
where K is equal to ⌊ M 2 ⌋, and B k,1 and B k,2 are real numbers representing
the coefficients of 2nd-order sections. For M =7the cascade form is shown
in Figure 6.11.
FIGURE 6.11
Cascade form FIR structure
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