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Complemente de Electronica Mihai P. Dinca
1Lectia 2
Cap. 2
Functia de transfer Fourier
Cap. 1
Sisteme si semnale
Cap. 3
Functia de transfer LaplaceCap. 4
Raspunsul la semnal treapta. Sisteme de ordinul 1Cap. 5
Sisteme de ordin superiorCap. 6
Reactia negativaCap. 7
Amplificatoare operationaleCap. 8
Aplicatii liniare ale AO
Functia pondere (raspunsul la impuls Dirac unitar)
Imaginea Fourier a semnalelor
Functia de transfer Fourier
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Complemente de Electronica Mihai P. Dinca
2Lectia 2
Functia pondere (raspunsul la impuls Dirac unitar)
Sistem liniar
Sistem liniar ?
Impulse response
Orice semnal de intrare
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Complemente de Electronica Mihai P. Dinca
3Lectia 2
Raspunsul la impuls Dirac – masurare experimentala
Amplitudine infinita ?Durata nula ?
0 t
Nu este nevoie de pulsuri exagerat de inaltePutem masura, de exemplu, raspunsul la 0.001(t)
Nu este nevoie de pulsuri exagerat de scurte.E suficient ca durata pulsului << timpul de raspuns al sistemului (=RC).
Ocolirea dificultatilor: excitatie cu semnal treapta si derivarea raspunsului
derivareSistem liniar
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Complemente de Electronica Mihai P. Dinca
4Lectia 2
Sisteme discrete
Convolutia vazuta de la intrare
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matematica mai simpla
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Complemente de Electronica Mihai P. Dinca
5Lectia 2
Convolutia vazuta de la iesire
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Complemente de Electronica Mihai P. Dinca
6Lectia 2
Orice semnal de intrare
C – operatorul sistemului
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Sistem continuu - calculul raspunsului in domeniul timp
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Complemente de Electronica Mihai P. Dinca
7Lectia 2
Calculul raspunsului in domeniul timp
Alegem t = t’ variabila de integrare
y() - un numar
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Complemente de Electronica Mihai P. Dinca
8Lectia 2
Raspunsul la impuls – functia de memorie a sistemului
Sistem fara memorie y(t)=Ax(t)h(t)=A(t)
h(t)=A(t-) Intirziere pura (delay) y(t)=Ax(t-)
Sistem la limita stabilitatii
Sistem instabil
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Complemente de Electronica Mihai P. Dinca
9Lectia 2
Program de calcul in domeniul timp
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Complemente de Electronica Mihai P. Dinca
10Lectia 2
Descrierea sistemului prin functia pondere – puncte slabe
funcţia pondere nu este direct calculabilă din topologia circuitului
nu putem lega simplu amplificarea în curent continuu de funcţia pondere
funcţia pondere a unui sistem, obţinut din legarea în cascadă a două subsisteme, nu se poate obţine simplu din funcţiile pondere ale celor două subsisteme
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Complemente de Electronica Mihai P. Dinca
11Lectia 2
Functia pondere – avantaje
In cazul sistemelor discrete, prin alegerea unei secvente de numere drept functie h[n] putem “inventa” sisteme liniare cu proprietati convenabile (filtre digitale).
Volumul mare de calcule implicat de operatie de convolutie nu mai este o problema pentru calculatoarele moderne
Permite calculul raspunsului pentru orice semnal de intrare utilizind acelasi algoritm.
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Complemente de Electronica Mihai P. Dinca
12Lectia 2
Imaginea Fourier a semnalelor
Orice semnal periodic de perioada T=2/0
C0 componenta continua (media pe o perioada)
Componentele cu m>1 - armonice de ordin superior
Componenta cu m=1 - fundamentala
Seria Fourier
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Complemente de Electronica Mihai P. Dinca
13Lectia 2
Sinteza Fourier a semnalelor periodice
Aproximativ 9% din amplitudine !
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Complemente de Electronica Mihai P. Dinca
14Lectia 2
Analizatorul de spectru
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Complemente de Electronica Mihai P. Dinca
15Lectia 2
Semnale de energie finita – transformarea Fourier
Frecvente negative ! Cum x(t) este real,
Informatia este “repetata” la frecventele negative
este real)0()0()0( * XXX
)()( * XX
(aria totala de sub curba x(t)
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Complemente de Electronica Mihai P. Dinca
16Lectia 2
Exemple
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Complemente de Electronica Mihai P. Dinca
17Lectia 2
Exemple
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Complemente de Electronica Mihai P. Dinca
18Lectia 2
Exemple
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Complemente de Electronica Mihai P. Dinca
19Lectia 2
Proprietati ale transformatei Fourier
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Complemente de Electronica Mihai P. Dinca
20Lectia 2
Functia de transfer Fourier
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Complemente de Electronica Mihai P. Dinca
21Lectia 2
Semnal sinusoidal extindere infinita
Pentru valori pozitive ale frecvenţei Fourier, funcţia de transfer Fourier este identică cu răspunsul în frecvenţă al sistemului, aşa cum a fost el definit anterior
Functia de transfer Fourier - semnificatie
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Complemente de Electronica Mihai P. Dinca
22Lectia 2
Avantajele caracterizarii sistemelor liniare prin functia de transfer Fourier
Functia H() are semnificatie fizica directa si poate fi usor determinata experimental.
Formalismul este elegant si compact, in locul unui set de ecuatii diferentiale (eventual a uneia de grad superior) avem nevoie doar de o functie complexa de variabila reala H().
Pentru circuitele electrice calculul lui H() se face direct din topologia circuitului fara scrierea ecuattilor diferentiale.
Legarea in cascada a sistemelor se traduce simplu prin inmultirea celor doua raspunsuri in frecventa, permitind astfel reprezentarea sistemelor complexe prin scheme bloc.
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Complemente de Electronica Mihai P. Dinca
23Lectia 2
321 HHHTTT rdout
4HTT outr
4321
321
1 HHHHHHH
TT
d
out
H1 H2 H3
H4Tr
_
Scheme bloc si functia de transfer Fourier