Biomedical Engineering Reference
In-Depth Information
x
K
(
n
)
x
0
(
n
)
x
1
(
n
)
x
2
(
n
)
…
G
(
z
)
G
(
z
)
G
(
z
)
w
0
w
1
w
K
y
(
n
)
…
FIgURE 4.2:
An overall diagram of a generalized feedforward filter [
19
].
x
0
(
n
) is an instantaneous
input and
y
(
n
) is a filter output.
depth from the filter order [
19
]. Memory depth in the present context is defined as the center of
mass of the impulse response of the last tap of the GFF filter [
19
].
The gamma filter is a special case of the GFF with
G
(
z
) =
μ
/(
z
- (1 -
μ
)), where
μ
is a feedback
parameter. The impulse response of the transfer function from an input to the
k
th tap, denoted as
g
k
(
n
), is given by
æ
ö
k
æ
è
ö
ø
æ
è
ö
ø
÷
÷
÷
÷
÷
=
-
n
1
1
m
ç
ç
ç
ç
ç
÷
÷
÷
÷
÷
÷
÷
÷
ç
ç
ç
ç
ç
ç
ç
ç
)
(4.21)
-
1
-
1
k
n
-
k
g
( )
n
=
Z
(
G
( ))
z
=
Z
m m
(
1
-
)
u n
(
-
k
k
k
z
- -
(
1
m
)
k
-
è
ø
where
Z
−1
(∙) indicates the inverse
z
transform and
u
(
n
) the step function. When
μ
= 1, the gamma
filter becomes an FIR filter. The stability of the gamma filter in adaptation is guaranteed when 0 <
μ
< 2 because of a local feedback structure.
The memory depth
D
with a feedback parameter
μ
in the
K
th-order gamma filter is given
by
=
m
for
μ <
1, or
D
K
=
2
m
for
μ >
1.
D
(4.22)
If we defined the resolution
R
≡
μ
, the property of the gamma delay line can be described as
K
= ´
for
μ <
1
,
or
K
D
R
for
μ > 1.
(4.23)
= ´ -
D
(
2
R
)
