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18
JanuszWalczak,AnnaRomanowska
Extractionofimpulseresponseh(t-
ζ,ζ
)fromequation(13)gives:
h
(
t
−
ζ
,
t
)
=
A
[
q
2
(
t
)
q
1
(
ζ
)
−
q
1
(
t
)
q
2
(
ζ
)
]
e
−
a
(
ζ
)
e
a
(
t
)
.
(30)
Takingintoaccount
τ
=−
t
ζ
andh(
τ
,t)=h(
τ
,t-
τ
)resultsin:
h
(
τ
,
t
)
=
A
[
q
2
(
t
)
q
1
(
t
−
τ
)
−
q
1
(
t
)
q
2
(
t
−
τ
)
]
e
a
(
τ
)
.
(31)
Therefore,onthebaseofthemethodformchapter2.3onecanobtainthefrequency
responses:
∞
H
(
j
ω
,
t
)
=
∫
A
[
q
2
(
t
)
q
1
(
t
−
τ
)
−
q
1
(
t
)
q
2
(
t
−
τ
)
]
e
a
(
τ
)
e
−
j
ωτ
d
τ
.
(32)
0
Finally,thesystemfrequencyresponseisexpressedbyformula(32).
Precisedefinitionofvaluesandparametersformagnitudeandphase
characteristicH(j
ω
,t)fromsolutionstoLTVdifferentialequationscanbepossible
onlyforspecialclassesofparametricfunctions.Aboverelationhasbeenillustrated
byexample(Fig.2),inwhichthemagnitudeofthesystemresponseforthesecond
orderLTVsectionwithexponentiallyvaryingparameter
ω
0
2(t)hasbeenplotted.
a)
b)
x(t)
IIorderLTVsection
CaseI
ω
o
2(t)
y(t)
c)
Fig.2.FrequencyresponsesofthesecondorderLTVsectionwithvaryingparameterω0
2(t):a)block
diagramforthesecondorderLTVsection,b)waveformofthefunctionω0
2(t),c)magnitudeofLTV
sectionresponse
