# 訊號與系統/第三章 連續時間系統之時域分析/3.4 零狀態響應－LTI系統之重疊積分

## LTI系統之重疊積分(Superposition Integral)

    (1)利用單位脈衝函數的篩選特性(sifting property) ，任意輸入


    (2)經由LTI系統作用後的輸出${\displaystyle \mathbf {y} (t)}$ 為


${\displaystyle \mathbf {T} [x(t)]=T[\int _{-\infty }^{\infty }x(\tau )\delta (t-\tau )\,d\tau ]}$

    (3)假設〝積分〞和LTI系統〝T〞的作用順序可對調，則


${\displaystyle \mathbf {y} (t)=\int _{-\infty }^{\infty }x(\tau )T[\delta (t-\tau )]\,d\tau }$

    (4)由於非時變的特性可知，${\displaystyle \mathbf {T} [\delta (t-\tau )]=h(t-\tau )}$


${\displaystyle \mathbf {y} (t)=\int _{-\infty }^{\infty }x(\tau )h(t-\tau )\,d\tau }$

(1)由下圖可知，LTI系統的任意輸入訊號${\displaystyle \mathbf {x} (t)}$可用單位脈波函數${\displaystyle \mathbf {r} ect(t)}$所形成之階梯函數(stairstep function)來近似


${\displaystyle \mathbf {\hat {x}} (t)=\sum _{n=-\infty }^{\infty }x(n\triangle \lambda )rect({t-n\triangle \lambda \over \triangle \lambda })}$

${\displaystyle =\mathbf {\sum } _{n=-\infty }^{\infty }x(n\triangle \lambda ){1 \over \triangle \lambda }rect({t-n\triangle \lambda \over \triangle \lambda })\triangle \lambda }$

© Rodger E. Ziemer, William H. Tranter, D. Ronald Fannin, Signals & Systems: Continuous and Discrete, 4th ed., Prentice Hall International, 1998.

(2)明顯地，${\displaystyle \mathbf {\hat {x}} (t)}$會趨近${\displaystyle \mathbf {x} (t)}$當${\displaystyle \mathbf {\Delta } \lambda }$趨近為 0

(3)${\displaystyle \mathbf {\lim } _{\Delta \lambda \to 0}{rect[(t-n\Delta \lambda )/\Delta \lambda ] \over \Delta \lambda }=\delta (t-\lambda )}$


(4) 假設${\displaystyle \mathbf {\hat {h}} (t)=T[{rect(t/\Delta \lambda ) \over \Delta \lambda }]}$當${\displaystyle \mathbf {\Delta } \lambda \to 0}$


${\displaystyle \mathbf {\lim } _{\Delta \lambda \to 0}{\hat {h}}(t)=\lim _{\Delta \lambda \to 0}T[{rect(t/\Delta \lambda ) \over \Delta \lambda }]}$

${\displaystyle \mathbf {=} T[\lim _{\Delta \lambda \to 0}{rect(t/\Delta \lambda ) \over \Delta \lambda }]}$

${\displaystyle \mathbf {=} T[\delta (t)]}$

${\displaystyle \mathbf {=} h(t)}$

(5)考慮LTI系統對${\displaystyle \mathbf {\hat {x}} (t)}$的響應 : 假設${\displaystyle \mathbf {\hat {h}} (t)=T[{rect(t/\Delta \lambda ) \over \Delta \lambda }]}$


${\displaystyle \mathbf {\hat {y}} (t)=T[{\hat {x}}(t)]}$

${\displaystyle \mathbf {=} T[\sum _{n=-\infty }^{\infty }x(n\Delta \lambda ){1 \over \Delta \lambda }rect({t-n\Delta \lambda \over \Delta \lambda })\Delta \lambda ]}$

(線性系統滿足重疊定理)

${\displaystyle \mathbf {=} \sum _{n=-\infty }^{\infty }x(n\Delta \lambda )T[{1 \over \Delta \lambda }rect({t-n\Delta \lambda \over \Delta \lambda })]\Delta \lambda }$

(非時變特性)

${\displaystyle \mathbf {\sum } _{n=-\infty }^{\infty }x(n\Delta \lambda ){\hat {h}}(t-n\Delta \lambda )\Delta \lambda }$

© Rodger E. Ziemer, William H. Tranter, D. Ronald Fannin, Signals & Systems: Continuous and Discrete, 4th ed., Prentice Hall International, 1998.

(6)取${\displaystyle \mathbf {\Delta } \lambda \to 0}$


${\displaystyle \mathbf {\Rightarrow } n\Delta \lambda \to \lambda ,{\hat {x}}(t)\to x(t),{\hat {h}}\to h(t)}$

${\displaystyle \mathbf {\lim } _{\Delta \lambda \to 0}{\hat {y}}(t)=\lim _{\Delta \lambda \to 0}T[{\hat {x}}(t)]}$

${\displaystyle \mathbf {=} T[\lim _{\Delta \lambda \to 0}{\hat {x}}(t)]}$

${\displaystyle \mathbf {=} T[x(t)]}$

${\displaystyle \mathbf {=} y(t)}$

## 範例3.9

© Rodger E. Ziemer, William H. Tranter, D. Ronald Fannin, Signals & Systems: Continuous and Discrete, 4th ed., Prentice Hall International, 1998.

## 範例3.10

考慮一系統的單位脈衝響應為${\displaystyle \mathbf {h} (t)={1 \over RC}e^{-t/RC}}$，令輸入訊號為單位步階函數${\displaystyle \mathbf {x} (t)=u(t)}$，試求系統的輸出${\displaystyle \mathbf {y} (t)}$。

【解】


${\displaystyle y(t)\mathbf {=} \int _{-\infty }^{\infty }u(t-\tau )h(\tau )\,d\tau }$

${\displaystyle \mathbf {=} \int _{-\infty }^{t}h(\tau )\,d\tau }$

${\displaystyle \mathbf {=} \int _{-\infty }^{t}{1 \over RC}e^{-\tau /RC}u(\tau )\,d\tau }$

${\displaystyle \mathbf {\int } _{0}^{t}{1 \over RC}e^{-\tau /RC}\,d\tau }$

${\displaystyle \mathbf {=} 1-e^{-t/RC}}$

注意：此一輸出也稱為系統的單位步階響應(unit step response)


${\displaystyle \mathbf {y} (t)=\int _{-\infty }^{t}h(\tau )\,d\tau }$

## 線性時變系統之重疊積分

●非時變系統之單位脈衝響應：


${\displaystyle \mathbf {\delta } (t)\to T\to h(t)}$

${\displaystyle \mathbf {\delta } (t-\lambda )\to T\to h(t-\lambda )}$

●時變系統之單位脈衝響應：


${\displaystyle \mathbf {\delta } (t)\to T\to h(t,0)}$

${\displaystyle \mathbf {\delta } (t-\lambda )\to T\to h(t,\lambda )}$

(假設在${\displaystyle \mathbf {\delta } (\lambda )}$輸入前，系統是rest)

●任意輸入訊號${\displaystyle \mathbf {x} (t)}$，則系統的響應(輸出)為：


${\displaystyle \mathbf {y} (t)=\int _{-\infty }^{\infty }x(\lambda )h(t,\lambda )\,d\lambda }$