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121 lines
2.7 KiB
Markdown
121 lines
2.7 KiB
Markdown
---
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title: Chirp - 啁啾
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tags:
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- basic
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- signal
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date: 2023-06-30
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---
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啁啾（Chirp）是指频率随时间而改变（增加或减少）的信号。其名称来源于这种信号听起来类似鸟鸣的啾声。
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Chirp常常被用在sonar, radar, laser systems里。其中，为了能够测量长距离又保留时间的分辨率，雷达需要短时间的派冲波但是又要持续的发射信号，啁啾信号可以同时保留连续信号和脉冲的特性，因此被应用在雷达和声纳探测上。
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# Definition
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## 瞬时频率 (instantaneous angular frequency)
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有一信号，$x(t)=A\sin{(\phi(t))}$，其瞬时角频率为
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$$
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\omega(t)=\frac{d\phi(t)}{dt}
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$$
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经适当归一化后得到瞬时频率
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$$
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f(t)=\frac{1}{2\pi}\frac{d\phi(t)}{dt}
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$$
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## 啁啾度
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对前两式再求导，得到瞬时角频率的变化速率为**瞬时角啁啾度**(instantaneous angular chirpyness)
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$$
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\gamma(t)=\frac{d^2\phi(t)}{dt^2}
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$$
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类似有**瞬时（普通）啁啾度**(instantaneous ordinary chirpyness)
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$$
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c(t)=\frac{1}{2\pi}\gamma(t)=\frac{1}{2\pi}\frac{d^2\phi(t)}{dt^2}
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$$
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# Types
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## Linear
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啁啾的瞬时频率$f(t)$呈线性变化
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$$f(t)=f_0 + ct$$
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$$
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c = \frac{f_1-f_0}{T}
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$$
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c是一个常值
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Also，
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$$
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\phi(t)=\phi_0 + 2\pi \int_{0}^t f(\tau)d\tau =\phi_0 = 2\pi(\frac{c}{2}t^2 + f_0 t)
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$$
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相位为t的二次函数，从而可以继续推导出信号在time domain：
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$$
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x(t)=A \cos{(\phi_0 + 2\pi (\frac{c}{2}t^2 + f_0 t))}
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$$
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这种Linear Chirp信号也被称为二次相位讯号(**quadratic-phase signal**)
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## Exponential
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Exponential chirp，也叫geometric chirp，瞬时频率以指数变化，即$f(t_2)/f(t_1)$会是常数
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signal frequency:
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$$
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f(t)=f_0 k^t
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$$
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$$
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k = (\frac{f(T)}{f_0})^{\frac{1}{T}} = \text{constant}
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$$
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相位:
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$$
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\phi(t)=\phi_0 + 2\pi\int_0^t f(\tau)d\tau = \phi_0 + 2\pi f_0 (\frac{k^t - 1}{\ln(k)})
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$$
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time-domain:
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$$
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x(t) = \sin{[\phi_0 + 2\pi f_0(\frac{k^t - 1}{\ln(k)})]}
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$$
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## Hyperbolic
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双曲线线性调频用于雷达应用，因为它们在被多普勒效应([Doppler Effect](physics/wave/doppler_effect.md))扭曲后显示出最大的匹配滤波器([Matched filter](https://en.wikipedia.org/wiki/Matched_filter))响应。
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signal frequency:
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$$
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f(t) = \frac{f_0 f_1 T}{(f_0 - f_1)t + f_1T}
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$$
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Phase:
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$$
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\phi(t) = \phi_0 + 2\pi \int_0^t f(\tau)d\tau = \phi_0 + 2\pi \frac{-f_0f_1 T}{f_1 - f_0}\ln(1 - \frac{f_1-f_0}{f_1 T}t)
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$$
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time-domain:
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$$
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x(t) = \sin{[\phi_0 + 2\pi \frac{-f_0f_1 T}{f_1 - f_0}\ln(1 - \frac{f_1-f_0}{f_1 T}t)]}
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$$
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