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324 Chapter 7 FIR FILTER DESIGN
2. The periodic convolution (7.22) produces a smeared version of the ideal
response H d (e jω ).
3. The main lobe produces a transition band in H(e jω ) whose width is
responsible for the transition width. This width is then proportional to
1/M . The wider the main lobe, the wider will be the transition width.
4. The side lobes produce ripples that have similar shapes in both the
passband and stopband.
Basic window design idea For the given filter specifications, choose
the filter length M and a window function w(n) for the narrowest main
lobe width and the smallest side lobe attenuation possible.
From observation 4, we note that the passband tolerance δ 1 and the
stopband tolerance δ 2 cannot be specified independently. We generally
take care of δ 2 alone, which results in δ 2 = δ 1 .Wenow briefly describe
various well-known window functions. We will use the rectangular window
as an example to study their performances in the frequency domain.
7.3.1 RECTANGULAR WINDOW
This is the simplest window function but provides the worst performance
from the viewpoint of stopband attenuation. It was defined earlier by
{
1, 0 ≤ n ≤ M − 1
w(n) =
0, otherwise
(7.23)
Its frequency response function is
(
sin
ωM
)]
W (e )=[ jω 2
sin ( )
ω
2
M−1
−jω
e
2 ⇒ W r (ω) = sin ( )
ωM
2
sin ( )
ω
2
which is the amplitude response. From (7.22) the actual amplitude response
H r (ω) isgiven by
H r (ω) ≃ 1
2π
∫
ω+ω c
−π
W r (λ) dλ = 1
2π
ω+ω ∫ c
−π
sin ( )
ωM
2
sin ( )
ω
dλ, M ≫ 1 (7.24)
2
This implies that the running integral of the window amplitude response
(or accumulated amplitude response) is necessary in the accurate analysis
of the transition bandwidth and the stopband attenuation. Figure 7.9
shows the rectangular window function w (n), its amplitude response
W (ω), the amplitude response in dB, and the accumulated amplitude
response (7.24) in dB. From the observation of plots in Figure 7.9, we can
make several observations.
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