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 kana impuls, h(t), tina hiji sistem LTI nyaéta réspon sistem kasebut mangsa diasupan sinyal
δ
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{\displaystyle \delta (t)}
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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.
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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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h
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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:
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{\displaystyle y(t)=h(t)*x(t)=\int _{-\infty }^{\infty }h(\tau )x(t-\tau )\,d\tau .}
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Ti persamaan-persamaan di luhur, réspon y(t) tina sistem LTI dinyatakeun ku:
y
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∫
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∞
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h
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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
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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 }
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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=
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H
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{\displaystyle Y(\omega )=X(\omega )H(\omega )}
di mana
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{\displaystyle X(\omega )={\mathcal {F}}\{x(t)\}}
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{\displaystyle Y(\omega )={\mathcal {F}}\{y(t)\}}
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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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{\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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{\displaystyle Y(\omega )=X(\omega )H(\omega )}
y
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2
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∞
∞
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H
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e
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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)
