附录G 式(9.5-7)的推导¶
由于条件概率密度 \(f_{0}(\boldsymbol{r}/\varphi_{0})\) 和 \(f_{1}(\boldsymbol{r}/\varphi_{1})\) 是在 \(\varphi_{0}\) 给 \(\varphi_{1}\) 给定条件下的概率密度，所以它们只决定于噪声的统计特性。故参照式 (9.1-12) 和式 (9.1-13) 可以写出：
\[
f _ {0} (\textbf {r} / \varphi_ {0}) = \frac {1}{(\sqrt {2 \pi} \sigma_ {\mathrm{n}}) ^ {k}} \exp \Bigl \{- \frac {1}{n _ {0}} \int_ {0} ^ {T _ {*}} [ r (t) - s _ {0} (t, \varphi_ {0}) ] ^ {2} \mathrm{d} t \Bigr \} \quad (\text {附G-1})
\]
\[
f _ {1} (\boldsymbol {r} / \varphi_ {1}) = \frac {1}{(\sqrt {2 \pi} \sigma_ {\mathrm{n}}) ^ {k}} \exp \left\{- \frac {1}{n _ {0}} \int_ {0} ^ {T _ {\mathrm{s}}} [ r (t) - s _ {1} (t, \varphi_ {1}) ] ^ {2} \mathrm{d} t \right\} \quad (\text {附G-2})
\]
将式 \((9.5-2)\) 、式 \((9.5-3)\) 、式 (附 G-1)、式 (附 G-2) 代入式 \((9.5-5)\) 和式 \((9.5-6)\) ，得到
\[
f _ {0} (\textbf {r}) = \int_ {0} ^ {2 \pi} \Bigl (\frac {1}{2 \pi} \Bigr) \frac {1}{(\sqrt {2 \pi} \sigma_ {\mathrm{n}}) ^ {k}} \exp \Bigl \{- \frac {1}{n _ {0}} \int_ {0} ^ {T _ {\mathrm{s}}} [ r (t) - s _ {0} (t, \varphi_ {0}) ] ^ {2} \mathrm{d} t \Bigr \} \mathrm{d} \varphi_ {0} \quad (\text {附G-3})
\]
\[
f _ {1} (\boldsymbol {r}) = \int_ {0} ^ {2 \pi} \left(\frac {1}{2 \pi}\right) \frac {1}{(\sqrt {2 \pi} \sigma_ {\mathrm{n}}) ^ {k}} \exp \left\{- \frac {1}{n _ {0}} \int_ {0} ^ {T _ {\mathrm{s}}} [ r (t) - s _ {1} (t, \varphi_ {1}) ] ^ {2} \mathrm{d} t \right\} \mathrm{d} \varphi_ {1}
\]
(附 G - 4)
将式 (9.5-1a) 和式 (9.5-1b) 代入式 (附 G-3) 和式 (附 G-4)，得到
\[
f _ {0} (\textbf {r}) = K \int_ {0} ^ {2 \pi} \Bigl (\frac {1}{2 \pi} \Bigr) \exp \Bigl \{\frac {2}{n _ {0}} \int_ {0} ^ {T _ {*}} r (t) V \cos (\omega_ {0} t + \varphi_ {0}) \mathrm{d} t \Bigr \} \mathrm{d} \varphi_ {0} \qquad (\text {附G-5})
\]
\[
f _ {1} (\textbf {r}) = K \int_ {0} ^ {2 \pi} \Bigl (\frac {1}{2 \pi} \Bigr) \exp \Bigl \{\frac {2}{n _ {0}} \int_ {0} ^ {T _ {\mathrm{s}}} r (t) V \cos (\omega_ {1} t + \varphi_ {1}) \mathrm{d} t \Bigr \} \mathrm{d} \varphi_ {1} \qquad (\text {附G-6})
\]
式中：
\[
K = \exp (- E _ {1} / n _ {0}) \exp \left[ (- 1 / n _ {0}) \int_ {0} ^ {T _ {\mathrm{s}}} r ^ {2} (t) \mathrm{d} t \right] / (\sqrt {2 \pi} \sigma_ {\mathrm{n}}) ^ {k} \tag {附G-7}
\]
现在将式 (附 G-5) 右端大括弧中的积分化简如下:
\[
\begin{array}{l} \frac {2}{n _ {0}} \int_ {0} ^ {T _ {\mathrm{s}}} r (t) V \cos (\omega_ {0} t + \varphi_ {0}) \mathrm{d} t = \frac {2 V}{n _ {0}} \int_ {0} ^ {T _ {\mathrm{s}}} r (t) (\cos \omega_ {0} t \cos \varphi_ {0} - \sin \omega_ {0} t \sin \varphi_ {0}) \mathrm{d} t \\ = \frac {2 V}{n _ {0}} (X _ {0} \cos \varphi_ {0} - Y _ {0} \sin \varphi_ {0}) = \frac {2 V}{n _ {0}} \sqrt {X _ {0} ^ {2} + Y _ {0} ^ {2}} \cos (\varphi_ {0} + \varphi) \\ \end{array}
\]
\[
= \frac {2 V}{n _ {0}} M _ {0} \cos (\varphi_ {0} + \varphi) \tag {附G-8}
\]
式中： \(X_0 = \int_0^{T_*}r(t)\cos \omega_0t\mathrm{d}t\) （附 G-9）
\[
Y _ {0} = \int_ {0} ^ {T _ {\mathrm{s}}} r (t) \sin \omega_ {0} t \mathrm{d} t \tag {附G-10}
\]
\[
M _ {0} = \sqrt {X _ {0} ^ {2} + Y _ {0} ^ {2}}, M _ {0} \geqslant 0 \tag {附G-11}
\]
\[
\varphi = \arctan (Y _ {0} / X _ {0}) \tag {附G-12}
\]
将式 (附 G-8) 代入式 (附 G-5) 得
\[
\begin{array}{l} f _ {0} (\boldsymbol {r}) = K \frac {1}{2 \pi} \int_ {0} ^ {2 \pi} \exp \left[ \frac {2 V}{n _ {0}} M _ {0} \cos (\varphi_ {0} + \varphi) \right] \mathrm{d} \varphi_ {0} \\ = K I _ {0} \left(\frac {2 V}{n _ {0}} M _ {0}\right) \\ \end{array}
\]
式中： \(I_{0}(u)=\frac{1}{2\pi}\int_{0}^{2\pi}\exp[u\cos(\varphi_{0}+\varphi)]\mathrm{d}\varphi_{0}\) (附 G-14)
\(I_{0}(u)\) 是第一类零阶修正贝塞尔函数，它的值可以由查表得到。
同样地，可以将式 (附 G-6) 化简为
\[
f _ {1} (\textbf {r}) = K I _ {0} \Bigl (\frac {2 V}{n _ {0}} M _ {1} \Bigr) \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad
\]
式中： \(M_{1}=\sqrt{X_{1}^{2}+Y_{1}^{2}}\) (附 G-16)
\[
X _ {1} = \int_ {0} ^ {T _ {*}} r (t) \cos \omega_ {1} t \mathrm{d} t \qquad \cdot \qquad \qquad \qquad (\text {附G-17})
\]
\[
Y _ {1} = \int_ {0} ^ {T _ {*}} r (t) \sin \omega_ {1} t \mathrm{d} t \tag {附G-18}
\]
于是，式 (附 G-18) 的判决准则就变成
\[
\left\{ \begin{array}{l l} \text {若接收矢量} r \text {使} I _ {0} \Big (\frac {2 V}{n _ {0}} M _ {1} \Big) < I _ {0} \Big (\frac {2 V}{n _ {0}} M _ {0} \Big), \text {则判发送码元是“0”} \\ \text {若接收矢量} r \text {使} I _ {0} \Big (\frac {2 V}{n _ {0}} M _ {0} \Big) < I _ {0} \Big (\frac {2 V}{n _ {0}} M _ {1} \Big), \text {则判发送码元是“1”} \end{array} \right. \tag {附G-19}
\]
由于此修正贝塞尔函数是单调增函数，所以式 (附 G-19) 中的判决准则可以化简为
\[
\left\{ \begin{array}{l l} \text { 若接收矢量 } r \text { 使 } M _ {1} ^ {2} < M _ {0} ^ {2}, \text { 则判为发送码元是“0” } \\ \text { 若接收矢量 } r \text { 使 } M _ {0} ^ {2} < M _ {1} ^ {2}, \text { 则判为发送码元是“1” } \end{array} \right. \tag {附G-20}
\]
式 (附 G-20) 就是最终判决条件，其中:
\[
\begin{array}{l} M _ {0} = \sqrt {X _ {0} ^ {2} + Y _ {0} ^ {2}}, \quad M _ {1} = \sqrt {X _ {1} ^ {2} + Y _ {1} ^ {2}}, \\ X _ {0} = \int_ {0} ^ {T _ {*}} r (t) \cos \omega_ {0} t \mathrm{d} t, Y _ {0} = \int_ {0} ^ {T _ {*}} r (t) \sin \omega_ {0} t \mathrm{d} t \\ X _ {1} = \int_ {0} ^ {T _ {\mathrm{s}}} r (t) \cos \omega_ {1} t \mathrm{d} t, Y _ {1} = \int_ {0} ^ {T _ {\mathrm{s}}} r (t) \sin \omega_ {1} t \mathrm{d} t \\ \end{array}
\]
附录 G 式 (9.5-7) 的推导
