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228 Chapter 6 IMPLEMENTATION OF DISCRETE-TIME FILTERS
This example shows that by using the MATLAB functions developed in this
section, we can probe and construct a wide variety of structures. □
6.3 FIR FILTER STRUCTURES
A finite-duration impulse response filter has a system function of the form
H(z) =b 0 + b 1 z −1 + ···+ b M−1 z 1−M =
Hence the impulse response h(n) is
M−1
∑
n=0
b n z −n (6.5)
{
bn , 0 ≤ n ≤ M − 1
h(n) =
0, else
and the difference equation representation is
(6.6)
y(n) =b 0 x(n)+b 1 x(n − 1) + ···+ b M−1 x(n − M +1) (6.7)
which is a linear convolution of finite support.
The order of the filter is M − 1, and the length of the filter (which
is equal to the number of coefficients) is M. The FIR filter structures are
always stable, and they are relatively simple compared to IIR structures.
Furthermore, FIR filters can be designed to have a linear-phase response,
which is desirable in some applications.
We will consider the following four structures:
1. Direct form: In this form the difference equation (6.7) is implemented
directly as given.
2. Cascade form: In this form the system function H(z) in(6.5) is factored
into 2nd-order factors, which are then implemented in a cascade
connection.
3. Linear-phase form: When an FIR filter has a linear-phase response,
its impulse response exhibits certain symmetry conditions. In this form
we exploit these symmetry relations to reduce multiplications by about
half.
4. Frequency-sampling form: This structure is based on the DFT of
the impulse response h(n) and leads to a parallel structure. It is also
suitable for a design technique based on the sampling of frequency
response H(e jω ).
We will briefly describe these four forms along with some examples.
The MATLAB function dir2cas developed in the previous section is also
applicable for the cascade form.
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