Direct Implementation
X
(
t
,
f
)
=
∫
−
∞
∞
w
(
t
−
τ
)
x
(
τ
)
e
−
j
2
π
f
τ
d
τ
{\displaystyle X(t,f)=\int _{-\infty }^{\infty }w(t-\tau )x(\tau )e^{-j2\pi f\tau }d\tau }
w
(
t
)
=
e
−
π
σ
t
2
{\displaystyle w(t)=e^{-\pi \sigma t^{2}}}
令
t
=
n
Δ
t
,
f
=
m
Δ
f
,
τ
=
p
Δ
t
{\displaystyle t=n\Delta _{t},f=m\Delta _{f},\tau =p\Delta _{t}}
X
(
n
Δ
t
,
m
Δ
f
)
=
∑
p
=
−
∞
∞
w
(
(
n
−
p
)
Δ
t
)
x
(
p
Δ
t
)
e
−
j
2
π
m
p
Δ
t
Δ
f
Δ
t
{\displaystyle {X}\left({n{\Delta _{t}},m{\Delta _{f}}}\right)=\sum \limits _{p=-\infty }^{\infty }{w\left({(n-p){\Delta _{t}}}\right){x}\left({p{\Delta _{t}}}\right)}{e^{-j2\pi \,mp{\Delta _{t}}{\Delta _{f}}}}{\Delta _{t}}}
w
(
t
)
≅
0
f
o
r
|
t
|
>
B
,
B
Δ
t
=
Q
{\displaystyle w(t)\cong 0\qquad for\left|t\right|>B,{\frac {B}{\Delta _{t}}}=Q}
w
(
(
n
−
p
)
Δ
t
)
≅
0
{\displaystyle w((n-p)\Delta _{t})\cong 0\qquad }
f
o
r
|
n
−
p
|
>
B
Δ
t
{\displaystyle for\left|n-p\right|>{\frac {B}{\Delta _{t}}}}
|
p
−
n
|
>
Q
{\displaystyle \left|p-n\right|>Q}
−
Q
<
p
−
n
<
Q
{\displaystyle -Q<p-n<Q}
w
(
(
n
−
p
)
Δ
t
)
{\displaystyle w((n-p)\Delta _{t})}
X
(
n
Δ
t
,
m
Δ
f
)
=
∑
p
=
n
−
Q
n
+
Q
w
(
(
n
−
p
)
Δ
t
)
x
(
p
Δ
t
)
e
−
j
2
π
m
p
Δ
t
Δ
f
Δ
t
{\displaystyle {X}\left({n{\Delta _{t}},m{\Delta _{f}}}\right)=\sum \limits _{p=n-Q}^{n+Q}{w\left({(n-p){\Delta _{t}}}\right){x}\left({p{\Delta _{t}}}\right)}{e^{-j2\pi \,mp{\Delta _{t}}{\Delta _{f}}}}{\Delta _{t}}}
e
−
π
σ
a
2
<
0.00001
{\displaystyle e^{-\pi \sigma a^{2}}<0.00001}
w
h
e
n
|
a
|
>
1.9143
{\displaystyle when\left|a\right|>1.9143}
Q
=
1.9143
σ
Δ
t
{\displaystyle Q={\frac {1.9143}{{\sqrt {\sigma }}\Delta t}}}
B
=
1.9143
σ
{\displaystyle B={\frac {1.9143}{\sqrt {\sigma }}}}
(1)
Δ
t
<
1
2
Ω
Ω
=
Ω
x
+
Ω
w
{\displaystyle {\Delta _{t}}<{\frac {1}{2\Omega }}\qquad {\Omega }={{\Omega _{x}}+{\Omega _{w}}}}
O(TFQ) T:时间取样点数 F:频率取样点数 Q:
Q
=
1.9143
σ
Δ
t
{\displaystyle Q={\frac {1.9143}{{\sqrt {\sigma }}\Delta t}}}
优点:简单实现,限制条件少
缺点:时间复杂度高
由Direct Implementation可得下式
X
(
n
Δ
t
,
m
Δ
f
)
=
∑
p
=
n
−
Q
n
+
Q
w
(
(
n
−
p
)
Δ
t
)
x
(
p
Δ
t
)
e
−
j
2
π
m
p
Δ
t
Δ
f
Δ
t
{\displaystyle {X}\left({n{\Delta _{t}},m{\Delta _{f}}}\right)=\sum \limits _{p=n-Q}^{n+Q}{w\left({(n-p){\Delta _{t}}}\right){x}\left({p{\Delta _{t}}}\right)}{e^{-j2\pi \,mp{\Delta _{t}}{\Delta _{f}}}}{\Delta _{t}}}
令
q
=
p
−
(
n
−
Q
)
→
p
=
(
n
−
Q
)
+
q
{\displaystyle q=p-(n-Q)\to p=(n-Q)+q}
Y
[
m
]
=
∑
n
=
0
N
−
1
y
[
n
]
e
−
j
2
π
m
n
N
{\displaystyle Y[m]=\sum \limits _{n=0}^{N-1}y[n]e^{-j{\frac {2\pi mn}{N}}}}
X
(
n
Δ
t
,
m
Δ
f
)
=
Δ
t
e
j
2
π
(
Q
−
n
)
m
N
∑
q
=
0
N
−
1
x
1
(
q
)
e
−
j
2
π
q
m
N
{\displaystyle {X}\left({n{\Delta _{t}},m{\Delta _{f}}}\right)={\Delta _{t}}{e^{j{\textstyle {{2\pi \,(Q-n)m} \over N}}}}\sum \limits _{q=0}^{N-1}{x_{1}\left({q}\right){e^{-j{\textstyle {{2\pi \,qm} \over N}}}}}}
x
1
{\displaystyle {x_{1}}}
其中
x
1
(
q
)
=
w
(
(
Q
−
q
)
Δ
t
)
x
(
(
n
−
Q
+
q
)
Δ
t
)
{\displaystyle {x_{1}}\left(q\right)=w\left({(Q-q){\Delta _{t}}}\right)x\left({(n-Q+q){\Delta _{t}}}\right)}
0
≤
q
≤
2
Q
{\displaystyle 0\leq q\leq 2Q}
w
(
t
)
=
e
−
π
σ
t
2
{\displaystyle w(t)=e^{-\pi \sigma t^{2}}}
x
1
(
q
)
=
0
,
2
Q
<
q
≤
N
{\displaystyle {x_{1}}\left(q\right)=0,2Q<q\leq N}
Q
=
1.9143
σ
Δ
t
{\displaystyle Q={\frac {1.9143}{{\sqrt {\sigma }}\Delta t}}}
B
=
1.9143
σ
{\displaystyle B={\frac {1.9143}{\sqrt {\sigma }}}}
Matlab及python 皆可呼叫fft函式完成
Y
[
m
]
=
∑
n
=
0
N
−
1
y
[
n
]
e
−
j
2
π
m
n
N
{\displaystyle Y[m]=\sum \limits _{n=0}^{N-1}y[n]e^{-j{\frac {2\pi mn}{N}}}}
假设
t
=
n
0
Δ
t
,
(
n
0
+
1
)
Δ
t
,
⋯
⋯
,
(
n
0
+
T
−
1
)
Δ
t
{\displaystyle t=n_{0}\Delta _{t},(n_{0}+1)\Delta _{t},\cdots \cdots ,(n_{0}+T-1)\Delta _{t}}
f
=
m
0
Δ
f
,
(
m
0
+
1
)
Δ
f
,
⋯
⋯
,
(
m
0
+
F
−
1
)
Δ
f
{\displaystyle \,f=m_{0}\Delta _{f},(m_{0}+1)\Delta _{f},\cdots \cdots ,(m_{0}+F-1)\Delta _{f}}
step 1：计算
n
0
,
m
0
,
T
,
F
,
N
,
Q
{\displaystyle n_{0},m_{0},T,F,N,Q}
step 2：
n
=
n
0
{\displaystyle n=n_{0}}
step 3：决定
x
1
(
q
)
{\displaystyle x_{1}(q)}
step 4：
X
1
(
m
)
=
F
F
T
[
x
1
(
q
)
]
{\displaystyle X_{1}(m)=FFT[x_{1}(q)]}
step 5：转换
X
1
(
m
)
{\displaystyle X_{1}(m)}
X
(
n
Δ
t
,
m
Δ
f
)
{\displaystyle X(n\Delta _{t},m\Delta _{f})}
step 6：设
n
=
n
+
1
{\displaystyle n=n+1}
n
=
n
0
+
T
+
1
{\displaystyle n=n_{0}+T+1}
(1)
Δ
t
<
1
2
Ω
Ω
=
Ω
x
+
Ω
w
{\displaystyle {\Delta _{t}}<{\frac {1}{2\Omega }}\qquad {\Omega }={{\Omega _{x}}+{\Omega _{w}}}}
(2)
Δ
t
Δ
f
=
1
N
{\displaystyle {\Delta _{t}}{\Delta _{f}}={\textstyle {1 \over {N}}}}
(3)
N
=
1
/
Δ
t
Δ
f
≥
2
Q
+
1
{\displaystyle N=1/{\Delta _{t}}{\Delta _{f}}\geq 2Q+1}
O
(
T
N
log
2
N
)
{\displaystyle O(TN{\log _{2}}N)}
优点:时间复杂度低
缺点:限制条件较直接实现法多
可改写为:
由Direct Implementation可得下式
X
(
n
Δ
t
,
m
Δ
f
)
=
∑
p
=
n
−
Q
n
+
Q
w
(
(
n
−
p
)
Δ
t
)
x
(
p
Δ
t
)
e
−
j
2
π
m
p
Δ
t
Δ
f
Δ
t
{\displaystyle {X}\left({n{\Delta _{t}},m{\Delta _{f}}}\right)=\sum \limits _{p=n-Q}^{n+Q}{w\left({(n-p){\Delta _{t}}}\right){x}\left({p{\Delta _{t}}}\right)}{e^{-j2\pi \,mp{\Delta _{t}}{\Delta _{f}}}}{\Delta _{t}}}
e
−
π
σ
a
2
<
0.00001
{\displaystyle e^{-\pi \sigma a^{2}}<0.00001}
w
h
e
n
|
a
|
>
1.9143
{\displaystyle when\left|a\right|>1.9143}
Q
=
1.9143
σ
Δ
t
{\displaystyle Q={\frac {1.9143}{{\sqrt {\sigma }}\Delta t}}}
B
=
1.9143
σ
{\displaystyle B={\frac {1.9143}{\sqrt {\sigma }}}}
令
e
x
p
(
−
j
2
π
m
p
Δ
t
Δ
f
)
=
e
x
p
(
−
j
π
p
2
Δ
t
Δ
f
)
e
x
p
(
j
π
(
p
−
m
)
2
Δ
t
Δ
f
)
e
x
p
(
−
j
π
m
2
Δ
t
Δ
f
)
{\displaystyle exp(-j2\pi \,mp{\Delta _{t}}{\Delta _{f}})=exp(-j\pi \,p^{2}{\Delta _{t}}{\Delta _{f}})exp(j\pi \,{(p-m)}^{2}{\Delta _{t}}{\Delta _{f}})exp(-j\pi \,m^{2}{\Delta _{t}}{\Delta _{f}})}
X
(
n
Δ
t
,
m
Δ
f
)
=
Δ
t
∑
p
=
n
−
Q
n
+
Q
w
(
(
n
−
p
)
Δ
t
)
x
(
p
Δ
t
)
e
−
j
2
π
m
p
Δ
t
Δ
f
→
X
(
n
Δ
t
,
m
Δ
f
)
=
Δ
t
e
−
j
π
m
2
Δ
t
Δ
f
∑
p
=
n
−
Q
n
+
Q
w
(
(
n
−
p
)
Δ
t
)
x
(
p
Δ
t
)
e
−
j
π
p
2
Δ
t
Δ
f
e
j
π
(
p
−
m
)
2
Δ
t
Δ
f
{\displaystyle {X}\left({n{\Delta _{t}},m{\Delta _{f}}}\right)={\Delta _{t}}\sum \limits _{p=n-Q}^{n+Q}{w\left({(n-p){\Delta _{t}}}\right){x}\left({p{\Delta _{t}}}\right)}{e^{-j2\pi \,mp{\Delta _{t}}{\Delta _{f}}}}\to {X}\left({n{\Delta _{t}},m{\Delta _{f}}}\right)={\Delta _{t}}{e^{-j\pi \,m^{2}{\Delta _{t}}{\Delta _{f}}}}\sum \limits _{p=n-Q}^{n+Q}{w\left({(n-p){\Delta _{t}}}\right){x}\left({p{\Delta _{t}}}\right)}{e^{-j\pi \,p^{2}{\Delta _{t}}{\Delta _{f}}}}{e^{j\pi \,{(p-m)}^{2}{\Delta _{t}}{\Delta _{f}}}}}
Step1:
x
1
[
p
]
=
w
(
(
n
−
p
)
Δ
t
)
x
(
p
Δ
t
)
e
−
j
π
p
2
Δ
t
Δ
f
{\displaystyle x_{1}[p]=w((n-p)\Delta _{t})x(p\Delta _{t})e^{-j\pi p^{2}\Delta _{t}\Delta _{f}}}
n
−
Q
≤
p
≤
n
+
Q
{\displaystyle \quad \quad n-Q\leq p\leq n+Q}
Step2:
X
2
[
n
,
m
]
=
∑
p
=
n
−
Q
n
+
Q
x
1
[
p
]
c
[
m
−
p
]
c
[
m
]
=
e
j
π
m
2
Δ
t
Δ
f
{\displaystyle X_{2}[n,m]=\sum _{p=n-Q}^{n+Q}x_{1}[p]c[m-p]\quad \quad c[m]=e^{j\pi m^{2}\Delta _{t}\Delta _{f}}}
Step3:
X
(
n
Δ
t
,
m
Δ
f
)
=
Δ
t
e
−
j
π
m
2
Δ
t
Δ
f
X
2
[
m
,
n
]
{\displaystyle X(n\Delta _{t},m\Delta _{f})=\Delta _{t}e^{-j\pi m^{2}\Delta _{t}\Delta _{f}}X_{2}[m,n]}
(1)
Δ
t
<
1
2
Ω
Ω
=
Ω
x
+
Ω
w
{\displaystyle {\Delta _{t}}<{\frac {1}{2\Omega }}\qquad {\Omega }={{\Omega _{x}}+{\Omega _{w}}}}
O
(
T
N
log
2
N
)
{\displaystyle O(TN{\log _{2}}N)}
优点:限制条件与Direct Implementation法一样基本上没有限制
缺点:时间复杂度与FFT-Based Method(快速傅立叶转换)一样 但由于加伯变换无法使用Recursive Method(递回法)所以此不能算是缺点 
![{\displaystyle Gx_{2}(t,f)={\sqrt[{4}]{2}}\int _{-\infty }^{\infty }e^{-\pi (\tau -t)^{2}}e^{-j2\pi f\tau }x(\tau )\,d\tau }](https://wikimedia.org/api/rest_v1/media/math/render/svg/5f24e0746f4181926a2bca6889b7575897735b4d)
![{\displaystyle G_{x}(t,f)={\sqrt[{4}]{\sigma }}\int _{-\infty }^{\infty }e^{-\sigma \pi (\tau -t)^{2}}e^{-j2\pi f\tau }x(\tau )d\tau }](https://wikimedia.org/api/rest_v1/media/math/render/svg/0bc988812ad5b07cb2aec7d92467f8e2836cdaef)
![{\displaystyle G_{x}(t,f)\simeq {\sqrt[{4}]{\sigma }}\int _{t-{\frac {1.9143}{\sqrt {\sigma }}}}^{t+{\frac {1.9143}{\sqrt {\sigma }}}}e^{-\sigma \pi (\tau -t)^{2}}e^{-j2\pi f\tau }x(\tau )d\tau }](https://wikimedia.org/api/rest_v1/media/math/render/svg/f929ad8b72be6144a350a43ef76d4fcaeba260cf)
![{\displaystyle G_{x}(t,f)={\sqrt[{4}]{\sigma (t)}}\int _{-\infty }^{\infty }e^{-\sigma (t)\pi (\tau -t)^{2}}e^{-j2\pi f\tau }x(\tau )d\tau }](https://wikimedia.org/api/rest_v1/media/math/render/svg/5e3159615aeb5177ba10391dd062990bd843430d)
![{\displaystyle Y[m]=\sum \limits _{n=0}^{N-1}y[n]e^{-j{\frac {2\pi mn}{N}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cc331a613689cf802fd4bceea34e000e320d4788)
![{\displaystyle X_{1}(m)=FFT[x_{1}(q)]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/dbf390598152c42280900168151ea7b5b0f94acb)
![{\displaystyle x_{1}[p]=w((n-p)\Delta _{t})x(p\Delta _{t})e^{-j\pi p^{2}\Delta _{t}\Delta _{f}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c6d196e6e3da81f669dda7d669f5160b5e732b91)
![{\displaystyle X_{2}[n,m]=\sum _{p=n-Q}^{n+Q}x_{1}[p]c[m-p]\quad \quad c[m]=e^{j\pi m^{2}\Delta _{t}\Delta _{f}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/35e701ab4fcf0fa1edb16e73bf56af9db288e41e)
![{\displaystyle X(n\Delta _{t},m\Delta _{f})=\Delta _{t}e^{-j\pi m^{2}\Delta _{t}\Delta _{f}}X_{2}[m,n]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/14a4dd812be8493640853ad78e9dbabf9729a329)
