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Window Design Techniques 323
where
⎧
⎪⎨ some symmetric function with respect to
w(n) = α over 0 ≤ n ≤ M − 1
⎪⎩
0, otherwise
Depending on how we define w(n), we obtain different window designs.
For example, in (7.20)
{
1, 0 ≤ n ≤ M − 1
w(n) =
= R M (n)
0, otherwise
which is the rectangular window defined earlier.
In the frequency domain the causal FIR filter response H(e jω )isgiven
by the periodic convolution of H d (e jω ) and the window response W (e jω );
that is,
H(e jω )=H d (e jω ) ∗○ W (e jω )= 1
2π
∫ π
−π
W ( e jλ) H d
(e j(ω−λ)) dλ
(7.22)
This is shown pictorially in Figure 7.8 for a typical window response, from
which we have the following observations:
1. Since the window w(n) has a finite length equal to M, its response has
a peaky main lobe whose width is proportional to 1/M , and has side
lobes of smaller heights.
H d (e jω )
−π
0
π
ω
Ripples
H(e jω )
Transition
Bandwidth
−π
0
π
−ω c ω c
−π
W (e jω )
Max Side-lobe
Height
Main Lobe
Width
Periodic
Convolution
ω
−ω c
0 ω c
Minimum
Stopband
Attenuation
π
ω
FIGURE 7.8
Windowing operation in the frequency domain
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