Treść książki
Przejdź do opcji czytnikaPrzejdź do nawigacjiPrzejdź do informacjiPrzejdź do stopki
Time-frequencyresponsesofsecondorderparametricsections
15
thetimeinwhichthestationaryvalue(
σ
0or
ω
0
2)isreachedandisthesamefor
bothparametricfunctions.
2
σ
(
t
)
=
σ
0
+
De
−
γ
t
,
D
∈
[
−
σ
0
,
∞
),
γ
>
0
,
σ
0
>
0
,
(6)
ω
0
2
(
t
)
=
ω
0
2
+
Ce
−
γ
t
,
C
∈
[
−
ω
0
2
,
∞
)
,
γ
>
0
,
ω
0
2
>
0
.
(7)
a)
b)
Fig.1.ParametricfunctionsofsecondorderLTVsection:a).waveformofvariableparameterω0
2(t),
b).waveformofvariableparameter2σ(t)
Assumedvariabilityofthecoefficientsmakesbothparametricfunctionsstrictly
positive.Thiswayoffunctionsselectionallowstoobtainanalyticsolutionto
parametricsecondorderdifferentialequationandwhatfollowsthesectionresponse
toanyexcitationinaclosedform.
Theimpulseresponsesofanon-stationarysystemcanbeobtainedbysolving
equation(5),fortheexactionx(t)=δ(t-
ξ)
,śśwhere
ζ
istimewhenexcitationδ(t)is
applied.Solutiontodifferentialequationwithzeroinitialconditionsresultsin
impulseresponseh(t-
τ
)ofnon-stationarysystem[2].Ifthecoefficientsofequation
arevarying,theresponseswillbedifferentfordifferenttimemomentst=
ζ
llin
whichexactionδ(t-
τ
)isappliedtotheinputofsystem.Mentionedeffectistypical
fornon-stationarysystemsanddoesnotexistinLLSsystems.
2.3.ImpulseresponseandfrequencyresponseofthesecondorderLTVsection
Amethodofdeterminationoftime-frequencyresponsesofLTVsectionsis
describedbellow[1],[7].Assuming,thatimpulseresponseofLTVsystemisgiven
andfurtherdenotedash(t-
ζ,ζ
),theresponseforanyexaction[4]isexpressedby:
y
(
t
)
=
∫
0
t
h
(
t
−
ζ
,
ζ
)
x
(
ζ
)
d
ζ
.
(8)
Forcomplexmonoharmonicsignalx(t)=e
j
ω
tformula(8)canbepresentedas:
