Digital Signal Processing Reference
In-Depth Information
An average FIR filter
K
a
(
p
) should isolate the constant (˙(
t
)=
X
˙
1
at
ω
1
=
ω
0
) or low-frequency
component (˙(
t
)=
X
˙
1
(
t
) at
ω
1
≠
ω
0
). The filters should suppress higher harmonics and a damped
oscillatory component with the complex frequency
p
=−
β
0
+
jω
0
.
The considered filters should have low sensitivity to a change of damping coefficient
β
0
in the
range from 10÷200 sec
-1
and the frequency
ω
1
=2
π
(
50±5
)
rad/sec. The acceptable static error of
signal processing should not be more than 0,5%, and the acceptable dynamic error at
t
≥
T
1
should not be higher than 3%.
FIR filter analysis is performed at input signal of a device as a set of semi-infinite or finite
damped oscillatory components according to the algebraic expressions from the Table 7 and
the Table 8.
An example of FIR filter analysis using Mathcad at compound input signals as a set of
sequentially adjacent finite signals is given on the fig.6.
ë
X1 e
p
×
(
t
-
t1
)
X2 e
p
×
(
t
-
t2
)
û
Function
fx X1 X2
,
(
p, t1
,
,
t2
,
t
)
:=
×
×
F t
(
-
t1
)
-
×
×
F t
(
-
t2
)
p r
¾
( )
t
¾
p r
¾
-
ë
û
G
p r
-
ë
-
(
p r
-
) t
û
G
-
(
)
W G r, p, t
,
:=
×
1
-
e
+
×
1
-
e
ORIGIN
:=
1
j
:=
-
w1
:=
2p 50
×
w0
:=
2p 50
×
b0
:=
20
r1
:=
j w1
×
r2
:=
-
b0
p0
:=
j w0
×
b1
:=
5
dw j × p2
:=
t1
:=
0
t2
:=
0.2
t3
:=
0.6
t4
:=
0.7
t5
:=
1.8
t6
:=
3.6
(
)
T
0.3 e
- 0.5
×
p
- 0.5
×
p
- 0.5
×
p
- 0.5
×
p
- 0.5
×
p
- 0.5
×
p
- 0.5
×
p
Input signal device
Z
:=
×
-
0.1
0.05 e
×
-
2.5
2.5
0
2 e
×
- e
×
1 e
×
0.4 e
×
0.4 e
×
(
)
T
(
)
T
r
:=
r1
3 r1
×
5 r1
×
r1
r2
0
r1 b1
-
+
j w1
×
r1
r1
+
dw
r1
-
dw
t1
:=
t1 t1 t1 t2 t2 t3 t4 t4 t5 t5 t5
)
T
(
t2
:=
t2 t2 t2 t3 t3 t4 t5 t5 t6 t6 t6
K
:=
length Z
(
)
k
:=
1
..
K
¾
( )
k
K
(
)
ë
(
)
ë
û
û
r
t2
k
-
t1
k
1
2
é
ë
r
k
t2
k
-
t1
k
ù
û
fx
( )
k
( )
k
e
( )
k
=
z t
(
)
:=
×
fx Z
k
Z
k
e
,
×
,
r
k
,
t1
k
,
t2
k
,
t
+
Z
,
Z
×
,
r
,
t1
k
,
t2
k
,
t
k
1
( )
k
X
k
:=
2 Z
k
×
p1
k
:=
r
k
-
p0
p2
k
:=
r
-
p0
Input signal filter
K
(
)
(
)
é
ë
é
ë
ù
û
fx
é
ë
( )
k
( )
k
e
p2
k
ù
û
ù
û
1
2
p1
k
t2
k
-
t1
k
t2
k
-
t1
k
=
x t
(
)
:=
×
fx X
k
X
k
e
,
×
,
p1
k
,
t1
k
,
t2
k
,
t
+
X
,
X
×
,
p2
k
,
t1
k
,
t2
k
,
t
k
1
(
)
T
80.4832e
j 4.2732
×
37.932 e
j 0.5887
×
)
T
)
T
FIR filter
G
:=
×
×
r
:=
(
-
22.9881
+
j 62.3049
×
-
23.2599
+
j 186.8944
×
T1
:=
0.051
TF
:=
(
T1
T1
r T1
×
G1
:=
diag G
(
) e
×
M
:=
length G
(
)
m
:=
1 M
..
M
M
1
2
1
2
(
)
(
)
=
=
K1 p
(
,
t
)
:=
W G
m
r
m
,
p, t
,
K2 p
(
,
t
)
:=
W G1
m
r
m
,
p, t
,
-
T1
m
1
m
1
(
)
(
)
p1
k
t2
k
-
t1
k
( )
k
e
p2
k
t2
k
-
t1
k
X1
k
:=
X
k
e
×
X2
k
:=
X
×
K
(
(
(
)
(
)
)
fx X
k
K2 p1
k
(
(
)
(
)
)
)
=
y1 t
(
)
:=
fx X
k
K1 p1
k
×
,
t
-
t1
k
,
X1
k
K1 p1
k
×
,
t
-
t2
k
,
p1
k
,
t1
k
,
t2
k
,
t
-
×
,
t
-
t1
k
,
X1
k
K2 p1
k
×
,
t
-
t2
k
,
p1
k
,
t1
k
+
T1
,
t2
k
+
T1
,
t
k
1
K
( )
k
K1 p2
k
(
)
(
)
( )
k
K2 p2
k
(
)
(
)
=
ë
ë
û
fx
ë
û
û
y2 t
(
)
:=
fx
X
×
,
t
-
t1
k
,
X2
k
K1 p2
k
×
,
t
-
t2
k
,
p2
k
,
t1
k
,
t2
k
,
t
-
X
×
,
t
-
t1
k
,
X2
k
K2 p2
k
×
,
t
-
t2
k
,
p2
k
,
t1
k
+
T1
,
t2
k
+
T1
,
t
k
1
y t
(
)
:=
0.5
×
(
y1 t
(
)
+
y2 t
(
)
)
Tz
:=
0.03
zz t
(
)
:=
if
(
t
>
Tz
,
z t
(
-
Tz
)
,
0
)
4
2
zz
(
t
)
y
(
t
)
-
y
(
t
)
0
2
0
0.5
1
1.5
2
2.5
3
3.5
t
Figure 5.
FIR filter analysis using Mathcad software
