鑒相器特性函數

考慮簡單的弦波輸入${\displaystyle f^{1}(\theta )=\sin(\theta ),}$ ${\displaystyle f^{2}(\theta )=\cos(\theta )}$以及積分器濾波器。

${\displaystyle \sin(\theta ^{1}(t))\cos(\theta ^{2}(t))={\frac {1}{2}}\sin(\theta ^{1}(t)+\theta ^{2}(t))+{\frac {1}{2}}\sin(\theta ^{1}(t)-\theta ^{2}(t))}$

標準的工程假設會假設濾波器會去除高頻訊號${\displaystyle \sin(\theta ^{1}(t)+\theta ^{2}(t))}$，不改變其低頻訊號${\displaystyle \sin(\theta ^{1}(t)-\theta ^{2}(t))}$。

因此，其弦波訊號的鑒相器特性為

${\displaystyle \varphi (\theta )={\frac {1}{2}}\sin(\theta ).}$

考慮高頻方波信號${\displaystyle f^{1}(t)=\operatorname {sgn}(\sin(\theta ^{1}(t)))}$ and ${\displaystyle f^{2}(t)=\operatorname {sgn}(\cos(\theta ^{2}(t)))}$。 針對此訊號，已有論文研究出類似的結果[5] 。 方波訊號的鑒相器特性為

${\displaystyle \varphi (\theta )={\begin{cases}1+{\frac {2\theta }{\pi }},&{\text{if }}\theta \in [-\pi ,0],\\1-{\frac {2\theta }{\pi }},&{\text{if }}\theta \in [0,\pi ].\\\end{cases}}}$

考慮一般情形的片段連續輸入信號${\displaystyle f^{1}(\theta )}$, ${\displaystyle f^{2}(\theta )}$。

輸入信號可以展開為傅立葉級數，${\displaystyle f^{1}(\theta )}$和${\displaystyle f^{2}(\theta )}$傅立葉級數的係數如下：

${\displaystyle a_{i}^{p}={\frac {1}{\pi }}\int \limits _{-\pi }^{\pi }f^{p}(x)\sin(ix)dx,}$${\displaystyle b_{i}^{p}={\frac {1}{\pi }}\int \limits _{-\pi }^{\pi }f^{p}(x)\cos(ix)dx,}$${\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]

${\displaystyle \varphi (\theta )=c^{1}c^{2}+{\frac {1}{2}}\sum \limits _{l=1}^{\infty }{\bigg (}(a_{l}^{1}a_{l}^{2}+b_{l}^{1}b_{l}^{2})\cos(l\theta )+(a_{l}^{1}b_{l}^{2}-b_{l}^{1}a_{l}^{2})\sin(l\theta ){\bigg )}.}$

顯然，鑒相器特性${\displaystyle \varphi (\theta )}$是在${\displaystyle \mathbb {R} }$內的週期性、連續有界函數。

有些專利是有關此分析方式的結果[6]。