在数学 和信号处理 中，解析信号 （英語：analytic signal 负频率 [ 1] 希爾伯特轉換 相关联的实值函数。
实值 函数的解析表示 是解析信号 ，包含原始函数和它的希尔伯特变换。这种表示促进了许多数学变换的发展。基本的想法是，由于频谱的埃尔米特对称 ，实值函数的傅里叶变换 （或频谱 ）的负频率成分是多余的。若是不介意处理复值函数的话，这些负频率分量可以丢弃而不损失信息。这使得函数的特定属性更易理解，并促进了调制和解调技术的衍生，如单边带。只要操作的函数没有负频率分量（也就是它仍是“解析函数”），从复数转换回实数就只需要丢弃虚部。解析表示是向量 概念的一个推广：[ 2]
创建一个解析信号的传递函数 若
s
(
t
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{\displaystyle s(t)}
实值 函数，其傅里叶变换为
S
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f
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{\displaystyle S(f)}
S
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f
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{\displaystyle S(f)}
f
=
0
{\displaystyle f=0}
埃尔米特 对称之函數：
S
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− − -->
f
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=
S
(
f
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∗ ∗ -->
,
{\displaystyle S(-f)=S(f)^{*},}
S
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f
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∗ ∗ -->
{\displaystyle S(f)^{*}}
S
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f
)
{\displaystyle S(f)}
复共轭 。函数：
S
a
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=
d
e
f
{
2
S
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,
for
f
>
0
,
S
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,
for
f
=
0
,
0
,
for
f
<
0
=
2
u
-->
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f
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⏟ ⏟ -->
1
+
sgn
-->
(
f
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S
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f
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=
S
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f
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+
sgn
-->
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S
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,
{\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}}}
其中：
u
-->
(
f
)
{\displaystyle \operatorname {u} (f)}
单位阶跃函数 ，
sgn
-->
(
f
)
{\displaystyle \operatorname {sgn}(f)}
符号函数 ，
仅包含
S
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f
)
{\displaystyle S(f)}
非负频率 分量。而且由于
S
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f
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{\displaystyle S(f)}
S
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f
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=
{
1
2
S
a
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f
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,
for
f
>
0
,
S
a
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,
for
f
=
0
,
1
2
S
a
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− − -->
f
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∗ ∗ -->
,
for
f
<
0
(Hermitian symmetry)
=
1
2
[
S
a
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+
S
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− − -->
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∗ ∗ -->
]
.
{\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}}}
s
(
t
)
{\displaystyle s(t)}
解析信号 是
S
a
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f
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{\displaystyle S_{\mathrm {a} }(f)}
s
a
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=
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F
− − -->
1
[
S
a
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]
=
F
− − -->
1
[
S
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f
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+
sgn
-->
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⋅ ⋅ -->
S
(
f
)
]
=
F
− − -->
1
{
S
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f
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}
⏟ ⏟ -->
s
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t
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+
F
− − -->
1
{
sgn
-->
(
f
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}
⏟ ⏟ -->
j
1
π π -->
t
∗ ∗ -->
F
− − -->
1
{
S
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f
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}
⏟ ⏟ -->
s
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t
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⏞ ⏞ -->
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o
n
v
o
l
u
t
i
o
n
=
s
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+
j
[
1
π π -->
t
∗ ∗ -->
s
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]
⏟ ⏟ -->
H
-->
[
s
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]
=
s
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+
j
s
^ ^ -->
(
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,
{\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}}}
其中
s
^ ^ -->
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t
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=
d
e
f
H
-->
[
s
(
t
)
]
{\displaystyle {\hat {s}}(t){\stackrel {\mathrm {def} }{{}={}}}\operatorname {\mathcal {H}} [s(t)]}
s
(
t
)
{\displaystyle s(t)}
希爾伯特轉換 ；
∗ ∗ -->
{\displaystyle *}
卷积 符号；
j
{\displaystyle j}
虛數單位 。
s
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t
)
=
cos
-->
(
ω ω -->
t
)
,
{\displaystyle s(t)=\cos(\omega t),}
ω ω -->
>
0.
{\displaystyle \omega >0.}
于是：
s
^ ^ -->
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=
cos
-->
(
ω ω -->
t
− − -->
π π -->
/
2
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=
sin
-->
(
ω ω -->
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,
{\displaystyle {\hat {s}}(t)=\cos(\omega t-\pi /2)=\sin(\omega t),}
s
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=
s
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+
j
s
^ ^ -->
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=
cos
-->
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ω ω -->
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+
j
sin
-->
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ω ω -->
t
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=
e
j
ω ω -->
t
.
{\displaystyle s_{\mathrm {a} }(t)=s(t)+j{\hat {s}}(t)=\cos(\omega t)+j\sin(\omega t)=e^{j\omega t}.}
欧拉公式 。欧拉公式 的一个推论是
cos
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=
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j
ω ω -->
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− − -->
ω ω -->
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.
{\displaystyle \cos(\omega t)={\tfrac {1}{2}}(e^{j\omega t}+e^{j(-\omega )t}).}
负频率
这里我们使用欧拉公式来识别并丢弃负频率。
s
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=
cos
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ω ω -->
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θ θ -->
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=
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2
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e
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ω ω -->
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θ θ -->
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+
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− − -->
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ω ω -->
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θ θ -->
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)
{\displaystyle s(t)=\cos(\omega t+\theta )={\tfrac {1}{2}}(e^{j(\omega t+\theta )}+e^{-j(\omega t+\theta )})}
于是：
s
a
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=
{
e
j
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ω ω -->
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+
θ θ -->
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=
e
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|
ω ω -->
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⋅ ⋅ -->
e
j
θ θ -->
,
if
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>
0
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e
− − -->
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θ θ -->
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=
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|
ω ω -->
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⋅ ⋅ -->
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− − -->
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θ θ -->
,
if
ω ω -->
<
0.
{\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}}}
这是使用希尔伯特变换方法去除负频率分量的另一个例子。我们注意到，对于复值函数
s
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t
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{\displaystyle s(t)}
s
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{\displaystyle s_{\mathrm {a} }(t)}
s
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t
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{\displaystyle s(t)}
s
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t
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=
e
− − -->
j
ω ω -->
t
{\displaystyle s(t)=e^{-j\omega t}}
ω ω -->
>
0
{\displaystyle \omega >0}
于是：
s
^ ^ -->
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=
j
e
− − -->
j
ω ω -->
t
,
{\displaystyle {\hat {s}}(t)=je^{-j\omega t},}
s
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=
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− − -->
j
ω ω -->
t
+
j
2
e
− − -->
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ω ω -->
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=
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− − -->
j
ω ω -->
t
− − -->
e
− − -->
j
ω ω -->
t
=
0.
{\displaystyle s_{\mathrm {a} }(t)=e^{-j\omega t}+j^{2}e^{-j\omega t}=e^{-j\omega t}-e^{-j\omega t}=0.}
由于
s
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=
Re
-->
[
s
a
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)
]
{\displaystyle s(t)=\operatorname {Re} [s_{\mathrm {a} }(t)]}
Im
-->
[
s
a
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]
{\displaystyle \operatorname {Im} [s_{\mathrm {a} }(t)]}
s
a
∗ ∗ -->
(
t
)
{\displaystyle s_{\mathrm {a} }^{*}(t)}
仅 由负频率分量构成。因此
Re
-->
[
s
a
∗ ∗ -->
(
t
)
]
{\displaystyle \operatorname {Re} [s_{\mathrm {a} }^{*}(t)]}
一个函数（蓝色）和它的解析表示的模（红），显示出包络现象。 解析信号也可以表示在其随时间变化的幅度和相位（极坐标 ）：
s
a
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=
s
m
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e
j
ϕ ϕ -->
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,
{\displaystyle s_{\mathrm {a} }(t)=s_{\mathrm {m} }(t)e^{j\phi (t)},}
其中：
s
m
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=
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e
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|
s
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{\displaystyle s_{\mathrm {m} }(t){\stackrel {\mathrm {def} }{{}={}}}|s_{\mathrm {a} }(t)|}
瞬时振幅 或包络
ϕ ϕ -->
(
t
)
=
d
e
f
arg
[
s
a
(
t
)
]
{\displaystyle \phi (t){\stackrel {\mathrm {def} }{{}={}}}\arg \!\left[s_{\mathrm {a} }(t)\right]}
瞬时相位 在附图中，蓝色曲线描绘
s
(
t
)
{\displaystyle s(t)}
s
m
(
t
)
{\displaystyle s_{\mathrm {m} }(t)}
解缠的 瞬时相位的时间导数的单位为rad/s，称作瞬时角频率 ：
ω ω -->
(
t
)
=
d
e
f
d
ϕ ϕ -->
d
t
(
t
)
.
{\displaystyle \omega (t){\stackrel {\mathrm {def} }{{}={}}}{\frac {d\phi }{dt}}(t).}
因此，瞬時頻率 赫兹 ）为：
f
(
t
)
=
d
e
f
1
2
π π -->
ω ω -->
(
t
)
.
{\displaystyle f(t){\stackrel {\mathrm {def} }{{}={}}}{\frac {1}{2\pi }}\omega (t).}
[ 3] 瞬时振幅、瞬时相位与频率在一些应用中用于测量和检测的信号的局部特征。信号的解析表示的另一个应用与调制信号 的解调有关。极坐标方便将振幅調變 和相位（或频率）调制的影响分开，对解调某些种类的信号很有效。
解析信号通常都会在频率上移位（下转换）到 0 Hz，可能会产生[非对称]负频率分量：
s
a
_ _ -->
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=
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s
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− − -->
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ω ω -->
0
t
=
s
m
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e
j
(
ϕ ϕ -->
(
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− − -->
ω ω -->
0
t
)
,
{\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)},}
其中
ω ω -->
0
{\displaystyle \omega _{0}}
[ 2]
这个函数有不同的名称，如复包络 和复基带 。复包络不是唯一的；它是由
ω ω -->
0
{\displaystyle \omega _{0}}
带通信号
s
(
t
)
{\displaystyle s(t)}
ω ω -->
0
{\displaystyle \omega _{0}}
载波频率
在其他情况下，
ω ω -->
0
{\displaystyle \omega _{0}}
低通滤波器 就可以去除感兴趣的部分。另一个动机是减少最高频率，从而降低最小的无混叠采样率。频移不加大复信号表示的数学处理难度。因此从这个意义上说，下转换的信号仍然是解析信号 。但是恢复实值表示不再是简简单单提取实部的问题了。为了避免混疊 可能需要上转换，若信号已被（离散时间）采样 ，还可能需要插值 （升採樣 ）。
若选取的
ω ω -->
0
{\displaystyle \omega _{0}}
s
a
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t
)
{\displaystyle s_{\mathrm {a} }(t)}
s
a
_ _ -->
(
t
)
{\displaystyle {\underline {s_{\mathrm {a} }}}(t)}
下边带 的单边带 信号。
参考频率的其他选择： 有时
ω ω -->
0
{\displaystyle \omega _{0}}
∫ ∫ -->
0
+
∞ ∞ -->
(
ω ω -->
− − -->
ω ω -->
0
)
2
|
S
a
(
ω ω -->
)
|
2
d
ω ω -->
.
{\displaystyle \int _{0}^{+\infty }(\omega -\omega _{0})^{2}|S_{\mathrm {a} }(\omega )|^{2}\,d\omega .}
另外，[ 4]
ω ω -->
0
{\displaystyle \omega _{0}}
解缠的 瞬时相位
ϕ ϕ -->
(
t
)
{\displaystyle \phi (t)}
∫ ∫ -->
− − -->
∞ ∞ -->
+
∞ ∞ -->
[
ω ω -->
(
t
)
− − -->
ω ω -->
0
]
2
|
s
a
(
t
)
|
2
d
t
{\displaystyle \int _{-\infty }^{+\infty }[\omega (t)-\omega _{0}]^{2}|s_{\mathrm {a} }(t)|^{2}\,dt}
再或者（对最佳
θ θ -->
{\displaystyle \theta }
∫ ∫ -->
− − -->
∞ ∞ -->
+
∞ ∞ -->
[
ϕ ϕ -->
(
t
)
− − -->
(
ω ω -->
0
t
+
θ θ -->
)
]
2
d
t
.
{\displaystyle \int _{-\infty }^{+\infty }[\phi (t)-(\omega _{0}t+\theta )]^{2}\,dt.}
在信号处理领域，维格纳–威利分布 定义中需要解析信号，因此该方法在实际应用中具有理想特性。[ 5]
有时复包络与复振幅 同义；[ a] [ b] [ c] 包络 振幅 。
^ "the complex envelope (or complex amplitude)"[ 6]
^ "the complex envelope (or complex amplitude)", p.586 [ 7]
^ "Complex envelope is an extended interpretation of complex amplitude as a function of time." p.85[ 8]
^ ``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 （页面存档备份 ，存于互联网档案馆 ）[7/16/2014 1:07:57 PM]
^ 2.0 2.1 Bracewell, Ron. The Fourier Transform and Its Applications . McGraw-Hill, 1965. p269
^ 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
^ Justice, J. Analytic signal processing in music computation . IEEE Transactions on Acoustics, Speech, and Signal Processing. 1979-12-01, 27 (6): 670–684 [2016-08-05 ] . ISSN 0096-3518 doi:10.1109/TASSP.1979.1163321 存档 于2014-10-20）. ^ 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
^ Hlawatsch, Franz; Auger, François. Time-Frequency Analysis . John Wiley & Sons. 2013-03-01. ISBN 9781118623831（英语） . ^ Driggers, Ronald G. Encyclopedia of Optical Engineering: Abe-Las, pages 1-1024 . CRC Press. 2003-01-01 [2016-08-05 ] . ISBN 9780824742508存档 于2014-10-21） （英语） . ^ Okamoto, Kenʼichi. Global Environment Remote Sensing . IOS Press. 2001-01-01. ISBN 9781586031015（英语） .
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.
![{\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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/641488b89157adff28b410cdd5db079397e49e42)
![{\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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/555facdeeb8a983551c93cdac29a005618604881)
![{\displaystyle {\hat {s}}(t){\stackrel {\mathrm {def} }{{}={}}}\operatorname {\mathcal {H}} [s(t)]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/eb5eb71b4428ffad1d004d4edff86f4b6553f001)
![{\displaystyle s(t)=\operatorname {Re} [s_{\mathrm {a} }(t)]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/01b2b0b048c344b45666f9bdb18eafbe3d9579fd)
![{\displaystyle \operatorname {Im} [s_{\mathrm {a} }(t)]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a46373663e75d8458d4a3c4ed17306abbe6a2062)
![{\displaystyle \operatorname {Re} [s_{\mathrm {a} }^{*}(t)]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d40821bf3a3c2ec9cea68e74480df669643cabe9)
![{\displaystyle \phi (t){\stackrel {\mathrm {def} }{{}={}}}\arg \!\left[s_{\mathrm {a} }(t)\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/17665909218975f34e9917c7312a4d051f15afbd)
![{\displaystyle \int _{-\infty }^{+\infty }[\omega (t)-\omega _{0}]^{2}|s_{\mathrm {a} }(t)|^{2}\,dt}](https://wikimedia.org/api/rest_v1/media/math/render/svg/49707ed8e6b2a70796b660e58addaa6cd63054a5)
![{\displaystyle \int _{-\infty }^{+\infty }[\phi (t)-(\omega _{0}t+\theta )]^{2}\,dt.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/260b4957a0fdd2b8444092dfd7fb331e169ff50d)