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Window Design Techniques 329
Hamming Window : M=45
Amplitude Response in dB
1
0
w(n)
Decibels
43
0
23.84
−22 0 22
n
Amplitude Response
60
−1 0 1
frequency in π units
0
Accumulated Amplitude Response
Width=(6.6)*pi/M
Wr
Decibels
0
0 22 45
frequency in π units
54
70
−1 1
frequency in π units
FIGURE 7.13 Hamming window: M =45
7.3.5 BLACKMAN WINDOW
This window is also similar to the previous two but contains a second
harmonic term and is given by
⎧
( ) (
⎨
2πn
4πn
0.42 − 0.5 cos
M−1
+0.08 cos
M−1
w(n) =
⎩
0, otherwise
)
, 0 ≤ n ≤ M − 1
(7.28)
This window and its frequency-domain responses are shown in Figure 7.14.
In Table 7.1 we provide a summary of fixed window function characteristics
in terms of their transition widths (as a function of M) and their
minimum stopband attenuations in dB. Both the approximate as well as
the exact transition bandwidths are given. Note that the transition widths
and the stopband attenuations increase as we go down the table. The
Hamming window appears to be the best choice for many applications.
7.3.6 KAISER WINDOW
This is an adjustable window function that is widely used in practice. The
window function is due to J. F. Kaiser and is given by
√
]
I 0
[β 1 − (1 − 2n
M−1 )2
w(n) =
, 0 ≤ n ≤ M − 1 (7.29)
I 0 [β]
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