Digital Signal Processing Reference
In-Depth Information
sions for calculating forced and free components of a filter reaction by the first method (items
4,5), components of a filter reaction by the second method (item6) are given in the Table 1.
№ Name
Expression
Remark
x
(
t
)=Re
(
˙
T
e
p
t
)
,
X
(
p
)=Re
(
˙
T
˙
=
˙
n
N
=
Xm
n
e
−
jφ
n
N
,
p
=
p
n N
= −
β
n
+
jω
n N
p
−
p
n N
)
1. Input signal
1
g
(
t
)=Re
(
˙
T
e
q
t
)
,
K
(
p
)=Re
(
˙
T
˙
=
˙
m
M
=
k
m
e
−
j
ϕ
m
M
,
q
=
ρ
m M
= −
α
m
+
jw
m M
p
−
ρ
m M
)
2. Filter
1
K
(
p
,
t
)=Re
(
˙
T
1−
e
−
(
p
−
ρ
m
)
t
)
K
(
p
,
t
)=
∫
0
t
3.
Time dependent
transfer function
g
(
τ
)
e
−
pτ
dτ
p
−
ρ
m
M
y
1
(
t
)=Re
(
˙
T
e
p
t
)
˙
=diag(
˙
)
K
(
p
)
4. Forced components
y
2
(
t
)=Re
(
˙
T
e
q
t
)
˙
=diag(
˙
)
X
(
q
)
5. Free components
y
(
t
)=Re
(
˙
(
t
)
T
e
p
t
)
˙
(
t
)=diag(
˙
)
K
(
p
,
t
)
6. Filter reaction
Table 1.
IIR filters analysis
The operation of the real part extraction on the right side of the expression in the items 1,2,3
for
X
(
p
),
K
(
p
),
K
(
p
,
t
) is solved in terms of the complex coefficients
˙
m
and
ρ
m
with no
relevance to the complex variable
p
.
The first method is a complex amplitude method generalization for definition of forced and
free components for filter reaction at semi-infinite or finite input signals [6]. The advantages
of this method are related to simple algebraic operations, which are used for determining the
parameters of linear system reaction (filter, linear circuit) components to an input action
described by a set of semi-infinite or finite damped oscillatory components. To analyze a filter
it is needed to use simple algebraic operations and operate a set of complex amplitudes and
frequencies of forced and free filter reaction components. In this case, there are simple relations
between complex amplitudes of output signal forced components and complex amplitudes of
an input signal (item 4 Table 1), between complex amplitudes of output signal free components
and complex amplitudes of a filter impulse function (item5).
The time-and-frequency approach in the second analysis method applies to a filter transfer
function, i.e. time dependent transfer function of the filter is used [6,8]. In that case, instead of
two sets of filter reaction components only one of them may be used.
Analysis methods given in the Table 1 enable to reduce effectively the computational costs
when performing a filter analysis by using simple algebraic operations to determine the forced
and free components of a filter reaction to an input action as a set of damped oscillatory
components. Therefore, the considered analysis methods for linear systems (filters) can be
effectively applied for performance analysis of signal processing by frequency filters.
