Digital Signal Processing Reference
In-Depth Information
off between transition band roll-off and attenuation in the stopband. One of the
window functions that developed because of this fact was the Kaiser window
function, as shown in Figure 7.10. The expression for the window coefficients as
given in (7.28) is based on the modified Bessel function of the first kind
I
o
. The
value β generally ranges from 3 to 9 and can be used to control the trade-off
between the transition band and stopband characteristics.
Figure 7.10
Kaiser windows with various β values.
⎡
⎤
2
⎛ ⋅
2
n
⎞
⎢
⎣
⎥
⎦
I
β
⋅
1
−
⎝
⎠
o
M
w
(
n
)
=
w
(
−
n
)
=
,
kais
kais
()
I
β
o
n
=
0
1
…
,
M
(7.28)
A reasonable estimate of β in the equation above has been determined
empirically by Kaiser, as shown in (7.29):
⎧
0
.
1102
⋅
(
A
−
8
.
7
)
,
for
A
>
50
⎨
0
.
4
β
=
0
.
5842
⋅
(
A
−
21
)
+
0
.
07886
⋅
(
A
−
21
),
for
21
≤
A
≤
50
(7.29)
⎩
0
0
for
A
<
21
In (7.29), the variable
A
represents the larger of the band errors (δ
p
or δ
s
)
expressed as attenuation, as shown in (7.30):
