the application of an input signal. A well-designed filter should have a fast response, a
small rise time, a small settling time, and a small overshoot.
In theory, both the steady-state and transient performance should be considered in
the design of a digital filter. However, it is difficult to consider these two specifications
simultaneously. In practice, we first design a filter to meet the magnitude specifications.
Once this filter is obtained, we check its phase response and transient performance. If
they are satisfactory, the design is completed. Otherwise, we must repeat the design
process. Once the transfer function has been determined, we can obtain a realization of
the filter. This will be discussed later.
5.2 FIR Filtering
The signal-flow diagram of the FIR filter is shown in Figure 3.6. As discussed in
Chapter 3, the general I/O difference equation of FIR filter is expressed as
yn b
0
xn b
1
xn À 1 Á Á Á b
LÀ1
xn À L 1
LÀ1
l0
b
l
xn À l,
5:2:1
where b
l
are the impulse response coefficients of the FIR filter. This equation describes
the output of the FIR filter as a convolution sum of the input with the impulse response
of the system. The transfer function of the FIR filter defined in (5.2.1) is given by
Hz b
0
b
1
z
À1
Á Á Á b
LÀ1
z
ÀLÀ1
LÀ1
l0
b
l
z
Àl
:
5:2:2
5.2.1 Linear Convolution
As discussed in Section 3.2.2, the output of the linear system defined by the impulse
response h(n) for an input signal x(n) can be expressed as
yn xn à hn
I
lÀI
hlxn À l:
5:2:3
Thus the output of the LTI system at any given time is the sum of the input samples
convoluted by the impulse response coefficients of the system. The output at time n
0
is
given as
yn
0
I
lÀI
hlxn
0
À l:
5:2:4
Assuming that n
0
is positive, the process of computing the linear convolution involves
the following four steps:
FIR FILTERING
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