Digital Signal Processing Reference
In-Depth Information
•
Kaiser window
⎧
⎡
⎤
⎧
2
⎫
⎛
⎜
⎞
⎟
⎪
⎪
n
−
α
⎪
⎪
⎪
⎪
⎢
⎢
⎢
⎥
⎥
⎥
I
β
1
−
0
α
wn
()
=
⎣
⎦
(5.14d)
⎨
,
0
≤≤
nM
⎪
⎪
⎪
I
(()
β
0
0
,
otherwise
⎩
where
= M/
2 and
I
0
is the modified Bessel function first kind and order zero.
The shape factor,
α
, can be adjusted to optimize the window properties,
for certain desired properties of the filter frequency response. For a given
maximum frequency response ripple,
β
δ
, and maximum transition bandwidth
∆ω
, the shape factor is given by the following equation:
⎧
0 1101
.
(
A
−
8 7
. ),
A
>
50
⎪
04
.
β=
0 5842
.
(
A
−
21
)
+
0 07886
.
(
A
−
21
),
21
≤≤
A
50
(5.15)
⎨
⎪
⎪
0
,
A
<
21
⎩
.
Additionally, the order of the Kaiser window
M
is given by the following
empirical equation:
where
A
= -20log
10
δ
A
−
8
M
=
(5.16)
2 285
.
∆ω
Finally the impulse response of the windowed causal, finite filter is given by
⎧
⎡
⎤
⎧
2
⎫
⎛
⎜
⎞
⎟
⎪
⎪
n
−
α
⎪
⎪
⎪
⎪
⎢
⎢
⎢
⎥
⎥
⎥
I
β
1
−
0
α
hn
()
=
⎣
⎦
(5.17
)
⎨
hn
(
−
α
)
,
0
≤≤
nM
d
⎪
⎪
⎪
I
()
β
0
0
,
otherwise
⎩
Note:
All w
indow functions are symmetric about the point
M/
2. This implies
the following condition:
⎧
wM n,
(
−
)
0
≤
n
≤
M
⎪
⎪
wn
()
=
(5.18)
0
,
otherwise
