# 啁啾

## 定義

### 瞬時頻率

${\displaystyle \omega (t)={\frac {\mathrm {d} \phi (t)}{\mathrm {d} t}},}$

${\displaystyle f(t)={\frac {1}{2\pi }}{\frac {\mathrm {d} \phi (t)}{\mathrm {d} t}}.}$

### 啁啾度

${\displaystyle \gamma (t)={\frac {\mathrm {d} ^{2}\phi (t)}{\mathrm {d} t^{2}}},}$

${\displaystyle c(t)={\frac {1}{2\pi }}\gamma (t)={\frac {1}{2\pi }}{\frac {\mathrm {d} ^{2}\phi (t)}{\mathrm {d} t^{2}}}.}$ [1]

## 分類

### 線性

 線性啁啾 線性啁啾的波形；頻率線性遞增，波長越來越短的正弦波（五次） 播放此文件有问题？请参见媒體幫助。

${\displaystyle c={\frac {f_{1}-f_{0}}{T}}}$

{\displaystyle {\begin{aligned}\phi (t)&=\phi _{0}+2\pi \int _{0}^{t}f(\tau )\,\mathrm {d} \tau \\&=\phi _{0}+2\pi \int _{0}^{t}\left(c\tau +f_{0}\right)\,\mathrm {d} \tau \\&=\phi _{0}+2\pi \left({\frac {c}{2}}t^{2}+f_{0}t\right),\end{aligned}}}

${\displaystyle x(t)=A\cos \left(\phi _{0}+2\pi \left({\frac {c}{2}}t^{2}+f_{0}t\right)\right).}$

FFT 啁啾

## 參考資料

1. ^ Mann, Steve and Haykin, Simon; The Chirplet Transform: A generalization of Gabor's Logon Transform; Vision Interface '91.[1]页面存档备份，存于互联网档案馆
2. ^ Easton, R.L. Fourier Methods in Imaging. Wiley. 2010: 703 [2014-12-03]. ISBN 9781119991861. （原始内容存档于2021-11-28）.