鑒相器特性函數 （phase detector characteristic）是相位差的函數，可以描述鉴相器 的輸出。
在鑒相器的分析時，常需要考慮時域以及相域-時域的特性[ 1]
鑒相器的特性和其實現方式以及其使用的信號種類有關。考量鑒相器特性時，允許針對高頻振盪使用平均法，也允許從時域下從相位同步系統非自治模型的分析和仿真，改為在相域-時域自治模型的分析和仿真[ 2]
考慮用類比乘法器和低通濾波器組成的鑒相器。
相域—頻域下的鑒相器 此處
f
1
(
θ
1
(
t
)
)
{\displaystyle f^{1}(\theta ^{1}(t))}
f
2
(
θ
2
(
t
)
)
{\displaystyle f^{2}(\theta ^{2}(t))}
分段 可微函數
f
1
(
θ
)
{\displaystyle f^{1}(\theta )}
f
2
(
θ
)
{\displaystyle f^{2}(\theta )}
波形 ，
θ
1
,
2
(
t
)
{\displaystyle \theta ^{1,2}(t)}
g
(
t
)
{\displaystyle g(t)}
若
f
1
,
2
(
θ
)
{\displaystyle f^{1,2}(\theta )}
θ
1
,
2
(
t
)
{\displaystyle \theta ^{1,2}(t)}
[ 3] [ 4]
ϕ
(
θ
)
{\displaystyle \phi (\theta )}
g
(
t
)
=
∫
0
t
f
1
(
θ
1
(
t
)
)
f
2
(
θ
2
(
t
)
)
d
t
{\displaystyle g(t)=\int \limits _{0}^{t}f^{1}(\theta ^{1}(t))f^{2}(\theta ^{2}(t))dt}
和相域—頻域模型的濾波輸出
G
(
t
)
=
∫
0
t
φ
(
θ
1
(
t
)
−
θ
2
(
t
)
)
d
t
{\displaystyle G(t)=\int \limits _{0}^{t}\varphi (\theta ^{1}(t)-\theta ^{2}(t))dt}
幾乎相等
g
(
t
)
−
G
(
t
)
≈
0
{\displaystyle g(t)-G(t)\approx 0}
考慮簡單的弦波輸入
f
1
(
θ
)
=
sin
(
θ
)
,
{\displaystyle f^{1}(\theta )=\sin(\theta ),}
f
2
(
θ
)
=
cos
(
θ
)
{\displaystyle f^{2}(\theta )=\cos(\theta )}
sin
(
θ
1
(
t
)
)
cos
(
θ
2
(
t
)
)
=
1
2
sin
(
θ
1
(
t
)
+
θ
2
(
t
)
)
+
1
2
sin
(
θ
1
(
t
)
−
θ
2
(
t
)
)
{\displaystyle \sin(\theta ^{1}(t))\cos(\theta ^{2}(t))={\frac {1}{2}}\sin(\theta ^{1}(t)+\theta ^{2}(t))+{\frac {1}{2}}\sin(\theta ^{1}(t)-\theta ^{2}(t))}
標準的工程假設會假設濾波器會去除高頻訊號
sin
(
θ
1
(
t
)
+
θ
2
(
t
)
)
{\displaystyle \sin(\theta ^{1}(t)+\theta ^{2}(t))}
sin
(
θ
1
(
t
)
−
θ
2
(
t
)
)
{\displaystyle \sin(\theta ^{1}(t)-\theta ^{2}(t))}
因此，其弦波訊號的鑒相器特性為
φ
(
θ
)
=
1
2
sin
(
θ
)
.
{\displaystyle \varphi (\theta )={\frac {1}{2}}\sin(\theta ).}
考慮高頻方波信號
f
1
(
t
)
=
sgn
(
sin
(
θ
1
(
t
)
)
)
{\displaystyle f^{1}(t)=\operatorname {sgn}(\sin(\theta ^{1}(t)))}
f
2
(
t
)
=
sgn
(
cos
(
θ
2
(
t
)
)
)
{\displaystyle f^{2}(t)=\operatorname {sgn}(\cos(\theta ^{2}(t)))}
[ 5]
φ
(
θ
)
=
{
1
+
2
θ
π
,
if
θ
∈
[
−
π
,
0
]
,
1
−
2
θ
π
,
if
θ
∈
[
0
,
π
]
.
{\displaystyle \varphi (\theta )={\begin{cases}1+{\frac {2\theta }{\pi }},&{\text{if }}\theta \in [-\pi ,0],\\1-{\frac {2\theta }{\pi }},&{\text{if }}\theta \in [0,\pi ].\\\end{cases}}}
考慮一般情形的片段連續輸入信號
f
1
(
θ
)
{\displaystyle f^{1}(\theta )}
f
2
(
θ
)
{\displaystyle f^{2}(\theta )}
輸入信號可以展開為傅立葉級數，
f
1
(
θ
)
{\displaystyle f^{1}(\theta )}
f
2
(
θ
)
{\displaystyle f^{2}(\theta )}
a
i
p
=
1
π
∫
−
π
π
f
p
(
x
)
sin
(
i
x
)
d
x
,
{\displaystyle a_{i}^{p}={\frac {1}{\pi }}\int \limits _{-\pi }^{\pi }f^{p}(x)\sin(ix)dx,}
b
i
p
=
1
π
∫
−
π
π
f
p
(
x
)
cos
(
i
x
)
d
x
,
{\displaystyle b_{i}^{p}={\frac {1}{\pi }}\int \limits _{-\pi }^{\pi }f^{p}(x)\cos(ix)dx,}
c
i
p
=
1
π
∫
−
π
π
f
p
(
x
)
d
x
,
p
=
1
,
2
{\displaystyle c_{i}^{p}={\frac {1}{\pi }}\int \limits _{-\pi }^{\pi }f^{p}(x)dx,p=1,2}
鑒相器特性為
[ 2]
φ
(
θ
)
=
c
1
c
2
+
1
2
∑
l
=
1
∞
(
(
a
l
1
a
l
2
+
b
l
1
b
l
2
)
cos
(
l
θ
)
+
(
a
l
1
b
l
2
−
b
l
1
a
l
2
)
sin
(
l
θ
)
)
.
{\displaystyle \varphi (\theta )=c^{1}c^{2}+{\frac {1}{2}}\sum \limits _{l=1}^{\infty }{\bigg (}(a_{l}^{1}a_{l}^{2}+b_{l}^{1}b_{l}^{2})\cos(l\theta )+(a_{l}^{1}b_{l}^{2}-b_{l}^{1}a_{l}^{2})\sin(l\theta ){\bigg )}.}
顯然，鑒相器特性
φ
(
θ
)
{\displaystyle \varphi (\theta )}
R
{\displaystyle \mathbb {R} }
有些專利是有關此分析方式的結果[ 6]
![{\displaystyle \varphi (\theta )={\begin{cases}1+{\frac {2\theta }{\pi }},&{\text{if }}\theta \in [-\pi ,0],\\1-{\frac {2\theta }{\pi }},&{\text{if }}\theta \in [0,\pi ].\\\end{cases}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9dceaa675d8ebf387974ede984742ec90da97beb)
