# 解析信号

## 定义

${\displaystyle s(t)}$ 是一个实值函数，其傅里叶变换为 ${\displaystyle S(f)}$${\displaystyle S(f)}$為一於 ${\displaystyle f=0}$ 埃尔米特对称之函數：

${\displaystyle S(-f)=S(f)^{*},}$   其中，${\displaystyle S(f)^{*}}$${\displaystyle S(f)}$复共轭

{\displaystyle {\begin{aligned}S_{\mathrm {a} }(f)&{\stackrel {\mathrm {def} }{{}={}}}{\begin{cases}2S(f),&{\text{for}}\ f>0,\\S(f),&{\text{for}}\ f=0,\\0,&{\text{for}}\ f<0\end{cases}}\\&=\underbrace {2\operatorname {u} (f)} _{1+\operatorname {sgn}(f)}S(f)=S(f)+\operatorname {sgn}(f)S(f),\end{aligned}}}

• ${\displaystyle \operatorname {u} (f)}$单位阶跃函数
• ${\displaystyle \operatorname {sgn}(f)}$符号函数

{\displaystyle {\begin{aligned}S(f)&={\begin{cases}{\frac {1}{2}}S_{\mathrm {a} }(f),&{\text{for}}\ f>0,\\S_{\mathrm {a} }(f),&{\text{for}}\ f=0,\\{\frac {1}{2}}S_{\mathrm {a} }(-f)^{*},&{\text{for}}\ f<0\ {\text{(Hermitian symmetry)}}\end{cases}}\\&={\frac {1}{2}}[S_{\mathrm {a} }(f)+S_{\mathrm {a} }(-f)^{*}].\end{aligned}}}

${\displaystyle s(t)}$解析信号${\displaystyle S_{\mathrm {a} }(f)}$ 的傅里叶逆变换：

{\displaystyle {\begin{aligned}s_{\mathrm {a} }(t)&{\stackrel {\mathrm {def} }{{}={}}}{\mathcal {F}}^{-1}[S_{\mathrm {a} }(f)]\\&={\mathcal {F}}^{-1}[S(f)+\operatorname {sgn}(f)\cdot S(f)]\\&=\underbrace {{\mathcal {F}}^{-1}\{S(f)\}} _{s(t)}+\overbrace {\underbrace {{\mathcal {F}}^{-1}\{\operatorname {sgn}(f)\}} _{j{\frac {1}{\pi t}}}*\underbrace {{\mathcal {F}}^{-1}\{S(f)\}} _{s(t)}} ^{convolution}\\&=s(t)+j\underbrace {\left[{1 \over \pi t}*s(t)\right]} _{\operatorname {\mathcal {H}} [s(t)]}\\&=s(t)+j{\hat {s}}(t),\end{aligned}}}

• ${\displaystyle {\hat {s}}(t){\stackrel {\mathrm {def} }{{}={}}}\operatorname {\mathcal {H}} [s(t)]}$${\displaystyle s(t)}$希爾伯特轉換
• ${\displaystyle *}$卷积符号；
• ${\displaystyle j}$虛數單位

## 例子

### 例1

${\displaystyle s(t)=\cos(\omega t),}$   其中  ${\displaystyle \omega >0.}$

${\displaystyle {\hat {s}}(t)=\cos(\omega t-\pi /2)=\sin(\omega t),}$
${\displaystyle s_{\mathrm {a} }(t)=s(t)+j{\hat {s}}(t)=\cos(\omega t)+j\sin(\omega t)=e^{j\omega t}.}$  第三个等式为欧拉公式

### 例2

${\displaystyle s(t)=\cos(\omega t+\theta )={\tfrac {1}{2}}(e^{j(\omega t+\theta )}+e^{-j(\omega t+\theta )})}$

${\displaystyle s_{\mathrm {a} }(t)={\begin{cases}e^{j(\omega t+\theta )}\ \ =\ e^{j|\omega |t}\cdot e^{j\theta },&{\text{if}}\ \omega >0,\\e^{-j(\omega t+\theta )}=\ e^{j|\omega |t}\cdot e^{-j\theta },&{\text{if}}\ \omega <0.\end{cases}}}$

### 例3

${\displaystyle s(t)=e^{-j\omega t}}$, 其中 ${\displaystyle \omega >0}$.

${\displaystyle {\hat {s}}(t)=je^{-j\omega t},}$
${\displaystyle s_{\mathrm {a} }(t)=e^{-j\omega t}+j^{2}e^{-j\omega t}=e^{-j\omega t}-e^{-j\omega t}=0.}$

## 应用

### 包络和瞬时相位

${\displaystyle s_{\mathrm {a} }(t)=s_{\mathrm {m} }(t)e^{j\phi (t)},}$

• ${\displaystyle s_{\mathrm {m} }(t){\stackrel {\mathrm {def} }{{}={}}}|s_{\mathrm {a} }(t)|}$ 称作瞬时振幅包络英语envelope (waves)
• ${\displaystyle \phi (t){\stackrel {\mathrm {def} }{{}={}}}\arg \!\left[s_{\mathrm {a} }(t)\right]}$ 称作瞬时相位

${\displaystyle \omega (t){\stackrel {\mathrm {def} }{{}={}}}{\frac {d\phi }{dt}}(t).}$

${\displaystyle f(t){\stackrel {\mathrm {def} }{{}={}}}{\frac {1}{2\pi }}\omega (t).}$  [3]

### 复包络/基带

${\displaystyle {\underline {s_{\mathrm {a} }}}(t){\stackrel {\mathrm {def} }{{}={}}}s_{\mathrm {a} }(t)e^{-j\omega _{0}t}=s_{\mathrm {m} }(t)e^{j(\phi (t)-\omega _{0}t)},}$

${\displaystyle \int _{0}^{+\infty }(\omega -\omega _{0})^{2}|S_{\mathrm {a} }(\omega )|^{2}\,d\omega .}$

${\displaystyle \int _{-\infty }^{+\infty }[\omega (t)-\omega _{0}]^{2}|s_{\mathrm {a} }(t)|^{2}\,dt}$

${\displaystyle \int _{-\infty }^{+\infty }[\phi (t)-(\omega _{0}t+\theta )]^{2}\,dt.}$

## 注释

1. ^ "the complex envelope (or complex amplitude)"[6]
2. ^ "the complex envelope (or complex amplitude)", p.586 [7]
3. ^ "Complex envelope is an extended interpretation of complex amplitude as a function of time." p.85[8]

## 参考文献

1. ^ Mathematics of the Discrete Fourier Transform (DFT), with Audio Applications --- Second Edition, by Julius O. Smith III, W3K Publishing, 2007, ISBN 978-0-9745607-4-8. Copyright © 2014-04-21 by Julius O. Smith III Center for Computer Research in Music and Acoustics (CCRMA), Stanford University, https://ccrma.stanford.edu/~jos/r320/Analytic_Signals_Hilbert_Transform.html[7/16/2014 1:07:57 PM]
2. Bracewell, Ron. The Fourier Transform and Its Applications. McGraw-Hill, 1965. p269
3. ^ B. Boashash, "Estimating and Interpreting the Instantaneous Frequency of a Signal-Part I: Fundamentals", Proceedings of the IEEE, Vol. 80, No. 4, pp. 519-538, April 1992
4. ^ Justice, J. Analytic signal processing in music computation. IEEE Transactions on Acoustics, Speech, and Signal Processing. 1979-12-01, 27 (6): 670–684. ISSN 0096-3518. doi:10.1109/TASSP.1979.1163321.
5. ^ B. Boashash, “Notes on the use of the Wigner distribution for time frequency signal analysis”, IEEE Trans. on Acoustics, Speech, and Signal Processing , vol. 26, no. 9, 1987
6. ^ Hlawatsch, Franz; Auger, François. Time-Frequency Analysis. John Wiley & Sons. 2013-03-01. ISBN 9781118623831 （英语）.
7. ^ Driggers, Ronald G. Encyclopedia of Optical Engineering: Abe-Las, pages 1-1024. CRC Press. 2003-01-01. ISBN 9780824742508 （英语）.
8. ^ Okamoto, Kenʼichi. Global Environment Remote Sensing. IOS Press. 2001-01-01. ISBN 9781586031015 （英语）.

## 延伸阅读

• Leon Cohen, Time-frequency analysis, Prentice Hall, Upper Saddle River, 1995.
• Frederick W. King, Hilbert Transforms, vol. II, Cambridge University Press, Cambridge, 2009.
• B. Boashash, Time-Frequency Signal Analysis and Processing: A Comprehensive Reference, Elsevier Science, Oxford, 2003.