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Window Design Techniques 325
Rectangular Window : M=45
Amplitude Response in dB
1
0
w(n)
Decibels
13
0
0 22 45
n
Amplitude Response
40
−1 0 1
frequency in π units
Accumulated Amplitude Response
45
0
Width=(1.8)*pi/M
Wr
0
Decibels
21
−1 0 1
frequency in π units
50
−1 1
frequency in π units
FIGURE 7.9 Rectangular window: M =45
1. The amplitude response W r (ω) has the first zero at ω = ω 1 , where
ω 1 M
2
= π or ω 1 = 2π
M
Hence the width of the main lobe is 2ω 1 =4π/M. Therefore the approximate
transition bandwidth is 4π/M.
2. The magnitude of the first side lobe (which is also the peak side lobe
magnitude) is approximately at ω =3π/M and is given by
(
∣ W r ω = 3π )∣ ∣∣∣ sin ( )
3π
2
=
M ∣sin ( )
3π
∣ ≃ 2M for M ≫ 1
3π
2M
Comparing this with the main lobe amplitude, which is equal to M,
the peak side lobe magnitude is
of the main lobe amplitude.
2
=21.22% ≡ 13 dB
3π
3. The accumulated amplitude response has the first side lobe magnitude
at 21 dB. This results in the minimum stopband attenuation of 21 dB
irrespective of the window length M.
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