无论是傅里叶级数 ，小波分析 ，或是其他的基础，在单个基础上的线性扩展都是不足的。傅立叶基础在时域上提供了很弱的表现，小波不能很好地适应于具有窄高频的傅立叶转换。在这两种情况下，都很难从其扩展系数中侦测和识别出信号的模样，因为讯号在整个基础上是被稀释。因此，我们必须用大量的傅立叶基础或是小波来表示具有些微误差的整个讯号。在参考文献中提出了一些匹配追踪算法，已在给定基础量时使误差最小化。
在傅里叶级数 ，
x
(
t
)
≈
∑
m
=
1
M
a
m
⋅
e
j
2
π
m
t
/
T
{\displaystyle x(t)\approx \sum _{m=1}^{M}a_{m}\cdot e^{j2\pi mt/T}}
a
m
=
∫
0
T
x
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t
)
e
−
j
2
π
m
t
/
T
d
t
=
⟨
x
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)
,
φ
m
∗
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t
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⟩
⟨
φ
m
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t
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,
φ
m
∗
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t
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⟩
{\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
x
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t
)
≈
∑
m
=
1
M
a
m
⋅
φ
m
(
t
)
{\displaystyle x(t)\approx \sum _{m=1}^{M}a_{m}\cdot \varphi _{m}(t)}
其中，
φ
m
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t
)
=
e
j
2
π
f
m
t
{\displaystyle \varphi _{m}(t)=e^{j2\pi f_{m}t}}
f
m
=
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
e
r
r
o
r
=
∫
−
∞
∞
|
x
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t
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−
∑
m
=
1
M
a
m
⋅
φ
m
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t
)
|
2
d
t
{\displaystyle error=\int _{-\infty }^{\infty }|x(t)-\sum _{m=1}^{M}a_{m}\cdot \varphi _{m}(t)|^{2}dt}
将
φ
m
(
t
)
{\displaystyle \varphi _{m}(t)}
Three-parameter atoms [ 编辑 ]
x
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t
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≈
∑
a
t
0
,
f
0
,
σ
⋅
φ
t
0
,
f
0
,
σ
(
t
)
{\displaystyle x(t)\approx \sum a_{t_{0},f_{0},\sigma }\cdot \varphi _{t_{0},f_{0},\sigma }(t)}
φ
t
0
,
f
0
,
σ
(
t
)
=
2
0.25
σ
0.5
e
j
2
π
f
0
t
−
π
(
t
−
t
0
)
2
/
σ
2
{\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
φ
t
0
,
f
0
,
σ
{\displaystyle \varphi _{t_{0},f_{0},\sigma }}
a
t
0
,
f
0
,
σ
{\displaystyle a_{t_{0},f_{0},\sigma }}
匹配追求过程
三个参数如下：
t
0
{\displaystyle t_{0}}
f
0
{\displaystyle f_{0}}
σ
{\displaystyle \sigma }
[ 编辑 ]
x
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t
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≈
∑
a
t
0
,
f
0
,
σ
,
η
⋅
φ
t
0
,
f
0
,
σ
,
η
(
t
)
{\displaystyle x(t)\approx \sum a_{t_{0},f_{0},\sigma ,\eta }\cdot \varphi _{t_{0},f_{0},\sigma ,\eta }(t)}
φ
t
0
,
f
0
,
σ
,
η
(
t
)
=
2
0.25
σ
0.5
e
j
2
π
(
f
0
t
+
η
/
2
t
2
)
−
π
(
t
−
t
0
)
2
/
σ
2
{\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：
t
0
{\displaystyle t_{0}}
f
0
{\displaystyle f_{0}}
σ
{\displaystyle \sigma }
η
{\displaystyle \eta }
在不同的基础的短时距傅立叶转换
x
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t
)
≊
∑
n
,
T
,
Ω
,
t
0
,
f
0
a
n
,
T
,
Ω
,
t
0
,
f
0
ψ
n
,
T
,
Ω
(
t
−
t
0
)
e
j
2
π
f
0
t
{\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. 
