基础扩展时频分析

在傅里叶级数，

${\displaystyle x(t)\approx \sum _{m=1}^{M}a_{m}\cdot e^{j2\pi mt/T}}$ 

${\displaystyle a_{m}=\int _{0}^{T}x(t)e^{-j2\pi mt/T}dt={\frac {\langle x(t),\varphi _{m}^{*}(t)\rangle }{\langle \varphi _{m}(t),\varphi _{m}^{*}(t)\rangle }}}$ 

在部分的时频分析也是意图要将讯号表示成如下的型态

${\displaystyle x(t)\approx \sum _{m=1}^{M}a_{m}\cdot \varphi _{m}(t)}$ 

其中，${\displaystyle \varphi _{m}(t)=e^{j2\pi f_{m}t}}$ ，${\displaystyle f_{m}=m/T}$ 

并且要求在M固定的情况下，将会最小化近似误差

${\displaystyle error=\int _{-\infty }^{\infty }|x(t)-\sum _{m=1}^{M}a_{m}\cdot \varphi _{m}(t)|^{2}dt}$ 

将${\displaystyle \varphi _{m}(t)}$  一般化，不同的 basis 之间不只是有频率的差异

Three-parameter atoms

${\displaystyle x(t)\approx \sum a_{t_{0},f_{0},\sigma }\cdot \varphi _{t_{0},f_{0},\sigma }(t)}$ 

${\displaystyle \varphi _{t_{0},f_{0},\sigma }(t)={\frac {2^{0.25}}{\sigma ^{0.5}}}e^{j2\pi f_{0}t-\pi (t-t_{0})^{2}/\sigma ^{2}}}$ 

当  ${\displaystyle \varphi _{t_{0},f_{0},\sigma }}$  不是正交，${\displaystyle a_{t_{0},f_{0},\sigma }}$  应该被匹配追求过程来决定

三个参数如下：

${\displaystyle t_{0}}$  控制时间轴上的中心位置

${\displaystyle f_{0}}$  控制频率轴上的中心位置

${\displaystyle \sigma }$  控制缩放的因子

Four-parameter atoms (线调频小波 chirplet)

${\displaystyle x(t)\approx \sum a_{t_{0},f_{0},\sigma ,\eta }\cdot \varphi _{t_{0},f_{0},\sigma ,\eta }(t)}$ 

${\displaystyle \varphi _{t_{0},f_{0},\sigma ,\eta }(t)={\frac {2^{0.25}}{\sigma ^{0.5}}}e^{j2\pi (f_{0}t+\eta /2t^{2})-\pi (t-t_{0})^{2}/\sigma ^{2}}}$ 

四个参数如下：

${\displaystyle t_{0}}$  控制时间轴上的中心位置

${\displaystyle f_{0}}$  控制频率轴上的中心位置

${\displaystyle \sigma }$  控制缩放的因子

${\displaystyle \eta }$  控制啁啾率

在不同的基础的短时距傅立叶转换

长球波函数(PSWF)

${\displaystyle x(t)\approxeq \sum _{n,T,\Omega ,t_{0},f_{0}}a_{n,T,\Omega ,t_{0},f_{0}}\psi _{n,T,\Omega }(t-t_{0})e^{j2\pi f_{0}t}}$ 

• S. G. Mallat and Z. Zhang, “Matching pursuits with time-frequency dictionaries,” IEEE Trans. Signal Processing, vol. 41, no. 12, pp. 3397–3415, Dec. 1993.
• A. Bultan, “A four-parameter atomic decomposition of chirplets,” IEEE Trans. Signal Processing, vol. 47, no. 3, pp. 731–745, Mar. 1999.
• C. Capus, and K. Brown. "Short-time fractional Fourier methods for the time-frequency representation of chirp signals," J. Acoust. Soc. Am. vol. 113, issue 6, pp. 3253–3263, 2003.
• Jian-Jiun Ding, Time frequency analysis and wavelet transform class note, Department of Electrical Engineering, National Taiwan University (NTU), Taipei, Taiwan, 2016.
• Jian-Jiun Ding, Time frequency analysis and wavelet transform class notes, Graduate Institute of Communication Engineering, National Taiwan University (NTU), Taipei, Taiwan, 2017.