# 传递函数

## 解释

$Y(s) = H(s) \, X(s)$

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

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

### 从微分方程直接推导

$L[u] = \frac{d^nu}{dt^n} + a_1\frac{d^{n-1}u}{dt^{n-1}} + \dotsb + a_{n-1}\frac{du}{dt} + a_nu = r(t)$

$p_L(\lambda) = \lambda^n + a_1\lambda^{n-1} + \dotsb + a_{n-1}\lambda + a_n\,$

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

## 信号处理

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

$Y(s) = H(s) X(s) \,$

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

### 频响函数

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

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

$\tau_{\phi}(\omega) = -\frac{\phi(\omega)}{\omega}$

$\tau_{g}(\omega) = -\frac{d\phi(\omega)}{d\omega}$.

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

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

## 光学

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

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

## 参考文献

1. ^ Bernd Girod, Rudolf Rabenstein, Alexander Stenger, Signals and systems, 2nd ed., Wiley, 2001, ISBN 0-471-98800-6 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. ^ 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.