フィルタの特性関数
特性関数（characteristic function）\(K(s)\)を，
\begin{gather}
\frac{P_{max}}{P_2}= 1+|K(s)|^2
\end{gather}
で定義し（\(s = j\omega\)），伝達関数（transfer function, transducer function）\(H(s)\)も次のように定義する．
\begin{gather}
H(s)= \sqrt{\frac{P_{max}}{P_2}}
= \frac{1}{2} \sqrt{\frac{R_2}{R_1}} \frac{E}{V_2}
\end{gather}
よって，
\begin{gather}
|H(s)|^2= 1+|K(s)|^2
\end{gather}
次に，反射係数との関係を導出する．
\begin{eqnarray}
1-\frac{P_2}{P_{max}}
&=& 1-4 \frac{R_1}{R_2} \left| \frac{V_2}{E} \right|^2
= 1-4 \frac{R_1}{R_2} \cdot \frac{R_2 \Re (Z_{in,1})}{|R_1+Z_{in,1}|^2}
\nonumber \\
&=& \frac{(R_1+Z_{in,1})(R_1+Z_{in,1}^*)-4R_1 \Re (Z_{in,1})}{|R_1+Z_{in,1}|^2}
\nonumber \\
&=& \frac{R_1^2+R_1(Z_{in,1}+Z_{in,1}^*)+|Z_{in,1}|^2-2R_1(Z_{in,1}+Z_{in,1}^*)}{|R_1+Z_{in,1}|^2}
\nonumber \\
&=& \frac{R_1^2 - R_1(Z_{in,1}+Z_{in,1}^*) + |Z_{in,1}|^2}{|R_1+Z_{in,1}|^2}
\nonumber \\
&=& \frac{(R_1 - Z_{in,1})(R_1-Z_{in,1}^*)}{|R_1+Z_{in,1}|^2}
\nonumber \\
&=& \left| \frac{Z_{in,1}-R_1}{Z_{in,1}+R_1} \right|^2
= |\Gamma_1|^2
\end{eqnarray}
つまり，
\begin{eqnarray}
|\Gamma_1|^2
&=& 1-\frac{1}{|H|^2}
= \frac{|H|^2-1}{|H|^2}
\nonumber \\
&=& \frac{|K|^2}{|H|^2}
= \frac{|K|^2}{1-|K|^2}
\end{eqnarray}
これより，特性関数\(K\)に関して次式が成り立つ．
\begin{gather}
|K|^2 = |\Gamma_1|^2 |H|^2
\\
K K^* = \Gamma_1 \Gamma_1^* H H^*
\end{gather}