BSA2_Curs 3
of 12
description
curs 3 bazele sistemelor
Transcript of BSA2_Curs 3
RECONSTITUIREA SEMNALULUI DE BAZ DIN SEMNALUL EANTIONAT
RECONSTITUIREA SEMNALULUI DE BAZ DIN SEMNALUL EANTIONAT
RECONSTITUIREA IDEAL
Reconstituirea ideal se face cu urmatoarea schem :
Breviar
Fie ; n condiii iniiale nule.
Se numete funcie de transfer: .
Presupunem c la intrare se aplic un impuls, deci .
Rspunsul sistemului = funcie pondere.
Funcia de transfer .
Fie i - rspuns la frecven.
Modulul se numete amplificare.
Argumentul se numete defazaj.
Daca ne referim la FTJ ideal ca sistem de refacere a semnalului de baza avem:
Aceast relaie se deduce din relaiile : i
S aplicm transformata Fourier inversa:
.
Obinem : .
pentru ca (vezi mai sus)Calculam pt FTJ ideal:
Se tie c : Deci am obtinut funcia pondere a FTJ ideal.
Rspunsul artat grafic demonstreaza c FTJ ideal este un element necauzal. Rspunsul h(t) ncepe s apar nainte de aplicarea impulsului la intrare.
;
Funciile se numesc funcii eantion.
S reinem c utiliznd funciile eantion obinem :
METODE PRACTICE DE REFACERE A SEMNALULUI DE BAZA
DIN SEMNALUL EANTIONAT. Utilizarea unui FTJ (nu ideal ci real):
Eroarea de aproximare e generat de abaterea FTJ fa de FTJ ideal.
. Utilizarea extrapolatoarelor :
Un extrapolator extrapoleaz (prezice) evoluia semnalului ntre momentul curent iT i momentul de eantionare care va urma.
1. Extrapolator de ordin 0 (extrapolator cardinal, element de reinere )
Semnalul refcut e un semnal constant pe poriuni.
Valoarea medie a semnalului refcut e decalata n urm fa de cu . Acest decalaj reprezint principala eroare introdus de extrapolatorul cardinal.
h(t) este raspunsul la impuls unitar, i.e. functia pondere (notata si w(t))
(1)
Caracteristicile de frecven ale lui EC
- rspuns la frecven
Extrapolatorul de ordinul 1
- la ieire el furnizeaz un semnal refcut liniar pe poriuni
Semnalul de baza este desenat cu linie intrerupta.
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- rspuns la frecven
(demonstrat la EC )
- caracteristica de frecven
LEGTURA DINTRE TRANSFORMATA LAPLACE
A SEMNALULUI DE BAZ I TRANSFORMATA LAPLACE
A SEMNALULUI EANTIONAT
Fie semnalul de baz .Presupunem c avem :
(1)
Astfel admite transformata Fourier
(2)De asemenea admite transformata Laplace :
(3)
Condiia de existen a transformatei Fourier este mult mai restrictiv dect a transformatei Laplace. Dac este ndeplinit condiia (1), atunci trecerea de la transformata Laplace la Fourier i invers, se face prin nlocuirea (permutarea) .
(4)
(5)
(6)
- model n domeniul s; (7)
x - pol
0 zeroRepartitia poli-zerouri pentru semnalul de baza si semnalul esantionat:
(8)
n relaia (8), pentru n variabila s se modific doar partea imaginar :
Prin eantionare, planul s se mparte n fii orizontale de nlime .
Modelul matematic al semnalului de baz se va regsi n fiecare fie. Exist o legtur ntre modelul n domeniul frecvenei i modelul n domeniul s. n primul caz se multiplic pe axa frecvenelor caracteristica spectral a semnalului de baz cu perioada de multiplicare .n al doilea caz, se multiplic n fiile planului s, distribuia poli-zerouri a semnalului de baz.
T
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FTJ ideal
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1
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H(s)
y(t)
u(t)
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3T
t
t
1
2T
2T
T
4T
3T
-1
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Evoluia median a semnalului
-T
iT
X(t)
Sampler and hold
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FTJ
iT
t
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T
EC
T
T
T
2T
T
-T
2T
t
t
t
1
t
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t
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1
t
2
1
t
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t
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T
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Intrare FTJ ideal
1
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t
1
L
0
L
-T
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T
-T
-T
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t
2T
T
t
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EC
2T
T
t
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-2
-2
t
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t
1
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t
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z
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Eantionare
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*
*
*
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Im
Re
Eantionare
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Re s
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3T
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T
2T
t
*
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i=1
i=-1
i=0
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2T
3T
-T
T
2T
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