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14
JanuszWalczak,AnnaRomanowska
Relationsbetweeninputandoutputsignalsinthefrequencydomainaredescribed
bythefollowingequation:
Y
(
j
ω
)
=
H
(
j
ω
)
X
(
j
ω
)
.
(4)
Thisequationcharacterizesmodificationofinputsignalspectrumdependingon
propertiesofmagnitude|H(j
ω
)|andphase
ϕ
(
ω
)frequencycharacteristics.
Basicdynamiccharacteristicsoflinearstationarysystems,suchasstep-
response,impulse-responseandfrequency-response,canbegeneralizedfornon-
stationaryLTVsystems[1],[2].Methodsoffrequencyresponsedeterminationare
usedinsynthesisofLTVdigital[4]andanalog[1]filters.
Thepurposeofthispaperconsistsinpresentationofadefinitionoftime-
frequencyresponsesoftheLTVsection,descriptionofthemethodofdetermining
time-frequencyresponsesanditsapplicationtothesecondorderLTVsection.
Detailedanalysisofthissectionhadbeencarriedinworks:[10],[11],[12].
2.TIME–FREQUENCYRESPONSESOFTHESECONDORDER
SYSTEMS
2.1.LTVsectionmodelanditsinterpretation
Ananalogcircuitwithtime-varyingparameterscanbeobtainedbyvarying
parametersinstationarysystem[3],[8].Theequationofthegeneralizedsecond-
orderparametricsectionobtainedbyvaryingparametersindifferentialequationof
LLSsection(1)andisexpressedby:
y
"
(
t
)
+
2
ω
0
σ
(
t
)
y
'
(
t
)
+
ω
0
2
(
t
)
y
(
t
)
=
k
(
t
)
x
(
t
)
,
(5)
Inordertoanalyzedynamicpropertiesofthesecond-orderparametricsectionone
oughttoconsiderthefollowingcasesofstatement(5):
I.
LTVsectionwithvaryingcoefficient
ω
0t
2
(
)
andconstantcoefficient
σ
(t),
II.
LTVsectionwithvaryingcoefficient
σ
(t)andconstantcoefficient
ω
0t
2
(
)
,
III.LTVsectionwithsimultaneousvariationofbothcoefficients
ω
0t
2
(
)
and
σ
(t).
Forallcasesconstantcomponentkhasbeenassumedasconstantcoefficient.
2.2.Variabilityofparametricfunctions
Processesinparametricsystemsaredescribedbydifferentialequationswhere
coefficientsdependontime.Solutionstotheseequationsarenotalwaysknown
thereforeexactanalysisispossibleonlyforspecialcasesoffunctionvariabilityof
parameters.InthispaperthreementionedearliercasesofequationofLTVsection
withexponentiallyvaryingparametershavebeenanalyzed.Variabilityof
parametricfunctiondescribedbyequations(6)and(7)hasbeenconsidered.The
coefficientsDandCarefunctionsvaluesfort=0,whereascoefficient
γ
describes
