# 传递函数

## 解释

${\displaystyle Y(s)=H(s)\,X(s)}$

${\displaystyle H(s)={\frac {Y(s)}{X(s)}}={\frac {{\mathcal {L}}\left\{y(t)\right\}}{{\mathcal {L}}\left\{x(t)\right\}}}}$

${\displaystyle H(z)={\frac {Y(z)}{X(z)}}}$

### 从微分方程直接推导

${\displaystyle L[u]={\frac {d^{n}u}{dt^{n}}}+a_{1}{\frac {d^{n-1}u}{dt^{n-1}}}+\dotsb +a_{n-1}{\frac {du}{dt}}+a_{n}u=r(t)}$

${\displaystyle p_{L}(\lambda )=\lambda ^{n}+a_{1}\lambda ^{n-1}+\dotsb +a_{n-1}\lambda +a_{n}\,}$

${\displaystyle H(s)={\frac {1}{p_{L}(s)}},\qquad p_{L}(s)\neq 0.}$

## 信号处理

${\displaystyle X(s)={\mathcal {L}}\left\{x(t)\right\}\equiv \int _{0}^{\infty }x(t)e^{-st}\,dt}$
${\displaystyle Y(s)={\mathcal {L}}\left\{y(t)\right\}\equiv \int _{0}^{\infty }y(t)e^{-st}\,dt}$ .

${\displaystyle Y(s)=H(s)X(s)\,}$

${\displaystyle H(s)={\frac {Y(s)}{X(s)}}}$

### 频响函数

${\displaystyle x(t)=Xe^{j\omega t}=|X|e^{j(\omega t+\arg(X))}}$

${\displaystyle y(t)=Ye^{j\omega t}=|Y|e^{j(\omega t+\arg(Y))}}$
${\displaystyle Y=|Y|e^{j\arg(Y)}}$ .

${\displaystyle \tau _{\phi }(\omega )=-{\frac {\phi (\omega )}{\omega }}}$

${\displaystyle \tau _{g}(\omega )=-{\frac {d\phi (\omega )}{d\omega }}}$ .

${\displaystyle G(\omega )={\frac {|Y|}{|X|}}=|H(j\omega )|\ }$
${\displaystyle \phi (\omega )=\arg(Y)-\arg(X)=\arg(H(j\omega ))}$ .

${\displaystyle H(j\omega )=\operatorname {Re} (\omega )+j\operatorname {Im} (\omega )}$

## 光学

${\displaystyle \mathrm {MTF} (f)={\frac {M(\mathrm {image} )}{M(\mathrm {source} )}}}$

${\displaystyle M={\frac {(L_{\mathrm {max} }-L_{\mathrm {min} })}{(L_{\mathrm {max} }+L_{\mathrm {min} })}}}$

## 参考文献

1. ^ Bernd Girod, Rudolf Rabenstein, Alexander Stenger, Signals and systems, 2nd ed., Wiley, 2001, ISBN 978-0-471-98800-7 p. 50
2. ^ The Oxford Dictionary of English, 3rd ed., "Transfer function"
3. ^ M. A. Laughton; D.F. Warne. Electrical Engineer's Reference Book 16. Newnes. : 14/9–14/10. ISBN 978-0-08-052354-5.
4. ^ E. A. Parr. Logic Designer's Handbook: Circuits and Systems 2nd. Newness. 1993: 65–66. ISBN 978-1-4832-9280-9.
5. ^ Ian Sinclair; John Dunton. Electronic and Electrical Servicing: Consumer and Commercial Electronics. Routledge. 2007: 172. ISBN 978-0-7506-6988-7.
6. ^ Birkhoff, Garrett; Rota, Gian-Carlo. Ordinary differential equations. New York: John Wiley & Sons. 1978. ISBN 0-471-05224-8.[页码请求]
7. 郁道银; 谈恒英. 工程光学（第4版）. 机械工业. 2016-02-01. ISBN 9787111519621.
8. ^ Valentijn De Smedt, Georges Gielen and Wim Dehaene. Temperature- and Supply Voltage-Independent Time References for Wireless Sensor Networks. Springer. 2015: 47. ISBN 978-3-319-09003-0.