Téori sistem LTI nyaéta téori dina widang téknik listrik , hususna dina perkara sirkuit, pamrosésan sinyal , jeung téori kadali, anu naliti réspon hiji sistem waktu invarian liniér kana sambarang sinyal asupan.
Réspon impuls
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Réspon kana impuls, h(t), tina hiji sistem LTI nyaéta réspon sistem kasebut mangsa diasupan sinyal
δ
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t
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{\displaystyle \delta (t)}
h
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δ
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{\displaystyle h(t)={\mathcal {T}}[\delta (t)]}
Fungsi h(t) téh sambarang, sarta ajénna teu kudu enol pikeun t<0.
Upama h(t) = 0 pikeun t<0 mangka sistem kasebut disebut kausal.
Réspon kana asupan sambarang
édit
Réspon y(t) ti sistem LTI kana sambarang asupan x(t) bisa dinyatakeun sabagé konvolusi tina x(t) jeung réspon impuls h(t) sistem dimaksud, nyaéta:
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=
x
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∗
h
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∫
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∞
∞
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h
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t
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d
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{\displaystyle y(t)=x(t)*h(t)=\int _{-\infty }^{\infty }x(\tau )h(t-\tau )\,d\tau }
Lantaran konvolusi téh sifatna komutatif mangka urang ogé bisa nganyatakeun kaluaran salaku:
y
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=
h
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∗
x
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∞
∞
h
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x
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{\displaystyle y(t)=h(t)*x(t)=\int _{-\infty }^{\infty }h(\tau )x(t-\tau )\,d\tau .}
Réspon ti sistem kausal
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Ti persamaan-persamaan di luhur, réspon y(t) tina sistem LTI dinyatakeun ku:
y
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=
∫
−
∞
t
x
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h
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d
τ
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0
∞
x
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h
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d
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{\displaystyle y(t)=\int _{-\infty }^{t}x(\tau )h(t-\tau )\,d\tau =\int _{0}^{\infty }x(t-\tau )h(\tau )\,d\tau }
Hiji sinyal x(t) disebut kausal upama inyana mibanda harga enol pikeun t<0. Jadi lamun input x(t) ogé kausal mangka:
y
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=
∫
0
t
x
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h
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t
−
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d
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h
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{\displaystyle y(t)=\int _{0}^{t}x(\tau )h(t-\tau )\,d\tau =\int _{0}^{t}x(t-\tau )h(\tau )\,d\tau }
Réspon frékuénsi
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Ku cara nerapkeun teorema konvolusi kana persamaan y(t) = x(t) * h(t) mangka urang meunangkeun persamaan:
Y
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ω
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=
X
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H
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{\displaystyle Y(\omega )=X(\omega )H(\omega )}
di mana
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=
F
{
x
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t
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}
{\displaystyle X(\omega )={\mathcal {F}}\{x(t)\}}
Y
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=
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{
y
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}
{\displaystyle Y(\omega )={\mathcal {F}}\{y(t)\}}
H
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=
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{
h
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{\displaystyle H(\omega )={\mathcal {F}}\{h(t)\}}
H
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{
h
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}
{\displaystyle H(\omega )={\mathcal {F}}\{h(t)\}}
H
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=
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{
h
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=
Y
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X
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{\displaystyle H(\omega )={\mathcal {F}}\{h(t)\}={\frac {Y(\omega )}{X(\omega )}}}
Hubungan antara asupan jeung kaluaran dina sistem LTI Ku cara nerapkeun kabalikan transformasi Fourier kana persamaan
Y
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ω
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=
X
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ω
)
H
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ω
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{\displaystyle Y(\omega )=X(\omega )H(\omega )}
y
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1
2
π
∫
−
∞
∞
X
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ω
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H
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e
j
ω
t
d
ω
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{\displaystyle y(t)={\frac {1}{2\pi }}\int _{-\infty }^{\infty }X(\omega )H(\omega )\ e^{j\omega t}\,d\omega ,}
Hsu, Hwei P., Schaum's Outline of Théory and Problems of Analog and Digital Communications, McGraw Hill, 1993 
![{\displaystyle h(t)={\mathcal {T}}[\delta (t)]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/12f14f1b15c6e47809469aa5d35886fceba1012a)
