廣義多項式韋格納頻譜圖 （generalized polynomial Wigner spectrogram），是一種用於時頻分析 的方法，屬於信號處理 的範疇。一個好的時頻分析講求在頻譜圖上要有高的解析度，並且不能有相交項(cross term)，才能得到準確的瞬時頻率 ，但這兩點之間常須進行取捨。韋格納分佈 雖然解析度較高，但在許多情況下會有相交項，例如瞬時頻率為高階指數函數時或多組件時；在瞬時頻率為高階指數函數時多項式韋格納分佈 加伯轉換 沒有相交項，但解析度較低，廣義頻譜圖 雖然強化了加伯轉換的解析度，但仍比韋格納分佈來得模糊。
廣義多項式韋格納頻譜圖透過結合廣義頻譜圖 與多項式韋格納分佈
韋格納分佈
编辑
W
x
(
t
,
f
)
=
∫
−
∞
∞
x
(
t
+
τ
/
2
)
x
∗
(
t
−
τ
/
2
)
e
−
j
2
π
τ
f
d
τ
{\displaystyle W_{x}(t,f)=\int _{-\infty }^{\infty }x(t+\tau /2)x^{*}(t-\tau /2)e^{-j2\pi \tau \,f}d\tau }
=
∫
−
∞
∞
X
(
f
+
η
/
2
)
⋅
X
∗
(
f
−
η
/
2
)
e
j
2
π
t
η
⋅
d
η
{\displaystyle =\int _{-\infty }^{\infty }X(f+\eta /2)\cdot X^{*}(f-\eta /2)e^{j2\pi t\eta }\cdot d\eta }
多項式韋格納分佈
编辑
在
x
(
t
)
=
exp
(
j
2
π
∑
n
=
1
q
2
+
1
n
a
n
t
n
)
{\displaystyle x(t)=\exp(j2\pi \sum _{n=1}^{{\frac {q}{2}}+1}na_{n}t^{n})}
P
W
V
D
x
(
t
,
f
)
=
∫
−
∞
∞
exp
(
j
2
π
∑
n
=
1
q
2
+
1
n
a
n
t
n
−
1
τ
)
e
−
j
2
π
τ
f
d
τ
=
∫
−
∞
∞
[
∏
ℓ
=
1
q
/
2
x
(
t
+
d
ℓ
τ
)
x
∗
(
t
−
d
−
ℓ
τ
)
]
e
−
j
2
π
τ
f
d
τ
{\displaystyle PWVD_{x}(t,f)=\int _{-\infty }^{\infty }\exp(j2\pi \sum _{n=1}^{{\frac {q}{2}}+1}na_{n}t^{n-1}\tau )\ e^{-j2\pi \tau f}d\tau =\int _{-\infty }^{\infty }[\prod _{\ell =1}^{q/2}x(t+d_{\ell }\tau )x^{*}(t-d_{-\ell }\tau )]e^{-j2\pi \tau f}d\tau }
透過設定
d
ℓ
{\displaystyle d_{\ell }}
exp
(
j
2
π
∑
n
=
1
q
2
+
1
n
a
n
t
n
−
1
τ
)
=
∏
ℓ
=
1
q
/
2
x
(
t
+
d
ℓ
τ
)
x
∗
(
t
−
d
−
ℓ
τ
)
{\displaystyle \exp(j2\pi \sum _{n=1}^{{\frac {q}{2}}+1}na_{n}t^{n-1}\tau )=\prod _{\ell =1}^{q/2}x(t+d_{\ell }\tau )x^{*}(t-d_{-\ell }\tau )}
即可得到，
P
W
V
D
x
(
t
,
f
)
=
∫
−
∞
∞
exp
(
j
2
π
∑
n
=
1
q
2
+
1
n
a
n
t
n
−
1
τ
)
d
τ
≅
δ
(
f
−
∑
n
=
1
q
2
+
1
n
a
n
t
n
−
1
τ
)
{\displaystyle PWVD_{x}(t,f)=\int _{-\infty }^{\infty }\exp(j2\pi \sum _{n=1}^{{\frac {q}{2}}+1}na_{n}t^{n-1}\tau )d\tau \cong \delta (f-\sum _{n=1}^{{\tfrac {q}{2}}+1}na_{n}t^{n-1}\tau )}
亦即
x
(
t
)
{\displaystyle x(t)}
加伯轉換
编辑
G
x
,
w
(
t
,
f
)
=
∫
−
∞
∞
w
(
t
−
τ
)
x
(
τ
)
e
−
j
2
π
f
τ
d
τ
{\displaystyle {G_{x,{w}}}\left({t,f}\right)=\int _{-\infty }^{\infty }{w\left({t-\tau }\right)x\left(\tau \right)\,{e^{-j2\pi \,f\,\tau }}d\tau }}
亦即使用高斯函數 做為短時距傅立葉轉換 的窗函數 。 廣義頻譜圖
编辑
S
P
x
,
w
1
,
w
2
(
t
,
f
)
=
G
x
,
w
1
(
t
,
f
)
G
x
,
w
2
∗
(
t
,
f
)
{\displaystyle S{P_{x,{w_{1}},{w_{2}}}}(t,f)=G_{x,{w_{1}}}(t,f)G_{x,{w_{2}}}^{*}(t,f)}
其中
w
1
(
t
)
,
w
2
(
t
)
{\displaystyle w_{1}(t),\ w_{2}(t)}
G
x
,
w
1
(
t
,
f
)
=
∫
−
∞
∞
w
1
(
t
−
τ
)
x
(
τ
)
e
−
j
2
π
f
τ
d
τ
{\displaystyle {G_{x,{w_{1}}}}\left({t,f}\right)=\int _{-\infty }^{\infty }{{w_{1}}\left({t-\tau }\right)x\left(\tau \right)\,{e^{-j2\pi \,f\,\tau }}d\tau }}
G
x
,
w
2
(
t
,
f
)
=
∫
−
∞
∞
w
2
(
t
−
τ
)
x
(
τ
)
e
−
j
2
π
f
τ
d
τ
{\displaystyle {G_{x,{w_{2}}}}\left({t,f}\right)=\int _{-\infty }^{\infty }{{w_{2}}\left({t-\tau }\right)x\left(\tau \right)\,{e^{-j2\pi \,f\,\tau }}d\tau }}
若
w
1
(
t
)
=
w
2
(
t
)
{\displaystyle w_{1}(t)=w_{2}(t)}
頻譜圖 。
不過根據測不準原理，較窄的窗函數，時間解析度較好，而頻率解析度較差；相反的，較寬的窗函數，頻率解析度較好，而時間解析度較差。
因此若兩個窗函數一個較窄一個較寬，加伯轉換後會得到解析度分別在時域與頻域較好的兩個頻譜圖，再透過相乘即可得到解析度在時頻兩域均好的頻譜圖。 廣義多項式韋格納頻譜圖
编辑
C
x
(
t
,
f
)
=
p
(
S
P
x
(
t
,
f
)
,
|
P
W
V
D
x
(
t
,
f
)
|
)
{\displaystyle C_{x}(t,f)=p\left(SP_{x}(t,f),\ |PWVD_{x}(t,f)|\right)}
p
(
x
,
y
)
{\displaystyle p(x,y)}
如果在
min
(
x
,
y
)
=
0
{\displaystyle \min(x,y)=0}
p
(
x
,
y
)
=
0
{\displaystyle p(x,y)=0}
例如：
p
(
x
,
y
)
=
x
y
{\displaystyle p(x,y)=xy}
C
x
(
t
,
f
)
=
S
P
x
(
t
,
f
)
|
P
W
V
D
x
(
t
,
f
)
|
{\displaystyle C_{x}(t,f)=SP_{x}(t,f)|PWVD_{x}(t,f)|}
P
W
V
D
x
(
t
,
f
)
≠
0
{\displaystyle PWVD_{x}(t,f)\neq 0}
S
P
x
(
t
,
f
)
=
0
{\displaystyle SP_{x}(t,f)=0}
S
P
x
(
t
,
f
)
≠
0
{\displaystyle SP_{x}(t,f)\neq 0}
P
W
V
D
x
(
t
,
f
)
≃
0
{\displaystyle PWVD_{x}(t,f)\simeq 0}
C
x
(
t
,
f
)
≃
0
{\displaystyle C_{x}(t,f)\simeq 0}
p
(
x
,
y
)
=
x
α
y
β
{\displaystyle p(x,y)=x^{\alpha }y^{\beta }}
C
x
(
t
,
f
)
=
S
P
x
α
(
t
,
f
)
|
P
W
V
D
x
β
(
t
,
f
)
|
{\displaystyle C_{x}(t,f)=SP_{x}^{\alpha }(t,f)|PWVD_{x}^{\beta }(t,f)|}
也可以加個閾，
C
x
(
t
,
f
)
=
t
h
r
[
S
P
x
α
(
t
,
f
)
]
|
P
W
V
D
x
β
(
t
,
f
)
|
{\displaystyle C_{x}(t,f)=thr[SP_{x}^{\alpha }(t,f)]|PWVD_{x}^{\beta }(t,f)|}
t
h
r
(
x
)
=
{
x
−
Δ
,
x
>
Δ
0
,
x
≤
Δ
{\displaystyle thr(x)={\begin{cases}x-\Delta &,x>\Delta \\0&,x\leq \Delta \end{cases}}}
Δ
{\displaystyle \Delta }
C
x
(
t
,
f
)
=
min
{
A
1
S
P
x
(
t
,
f
)
,
A
2
|
P
W
V
D
x
(
t
,
f
)
|
}
{\displaystyle C_{x}(t,f)=\min\{A_{1}SP_{x}(t,f),\ A_{2}|PWVD_{x}(t,f)|\}}
C
x
(
t
,
f
)
=
S
P
x
α
(
t
,
f
)
|
P
W
V
D
x
β
(
t
,
f
)
|
⋅
{
[
S
P
x
(
t
,
f
)
>
Δ
1
]
&
[
|
P
W
V
D
x
(
t
,
f
)
|
>
Δ
2
]
}
{\displaystyle C_{x}(t,f)=SP_{x}^{\alpha }(t,f)|PWVD_{x}^{\beta }(t,f)|\cdot \{[SP_{x}(t,f)>\Delta _{1}]\&[|PWVD_{x}(t,f)|>\Delta _{2}]\}}
優缺點比較
编辑
参考资料
编辑
Jian-Jiun Ding, Time frequency analysis and wavelet transform class notes, the Department of Electrical Engineering, National Taiwan University (NTU), Taipei, Taiwan, 2018. 
![{\displaystyle PWVD_{x}(t,f)=\int _{-\infty }^{\infty }\exp(j2\pi \sum _{n=1}^{{\frac {q}{2}}+1}na_{n}t^{n-1}\tau )\ e^{-j2\pi \tau f}d\tau =\int _{-\infty }^{\infty }[\prod _{\ell =1}^{q/2}x(t+d_{\ell }\tau )x^{*}(t-d_{-\ell }\tau )]e^{-j2\pi \tau f}d\tau }](https://wikimedia.org/api/rest_v1/media/math/render/svg/0fabbee7bb16dd268b16861fa452a4a2f553f164)
![{\displaystyle C_{x}(t,f)=thr[SP_{x}^{\alpha }(t,f)]|PWVD_{x}^{\beta }(t,f)|}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4a8537f2aa07b1742e060a6addcdb39cbaf83664)
![{\displaystyle C_{x}(t,f)=SP_{x}^{\alpha }(t,f)|PWVD_{x}^{\beta }(t,f)|\cdot \{[SP_{x}(t,f)>\Delta _{1}]\&[|PWVD_{x}(t,f)|>\Delta _{2}]\}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e8862a8135f50e7d0a83cacb962385a3ad2909c4)