带通讯号ui
一个实的带通讯号$x(t)$能够表示为通信
\[x(t) = r(t)\cos (2\pi f_0 t + \phi_x(t)) \]co
其中$r(t)$是幅度调制或包络，$\phi_x(t)$是相位调制，$f_0$是载波频率，$r(t)$和$\phi_x(t)$的变化比$f_0$要小得多。频率调制表示为ps
\[f_m(t) = \frac{1}{2\pi} \frac{d}{dt}\phi_x(t) \]带宽
瞬时频率
\[{f_i}(t) = \frac{1}{{2\pi }}\frac{d}{{dt}}\left( {2\pi {f_0}t + {\phi _x}(t)} \right) = {f_0} + {f_m}(t)\]
若是信号带宽B远小于中心频率$f_0$，则信号$x(t)$称为带通讯号。
带通讯号也能够由两个互为正交的低通讯号(的调制)来表示，即
\[x(t) = {x_I}(t)\cos 2\pi {f_0}t - {x_Q}(t)\sin 2\pi {f_0}t\]
其中
\[\begin{array}{l}
{x_I}(t) = r(t)\cos {\phi _x}(t)\\
{x_Q}(t) = r(t)\sin{\phi _x}(t)
\end{array}\]
解析信号(Analytic Signal)或预包络(Pre-Envelope)
对于给定的实信号$x(t)$，其Hilbert变换为
\[\hat x(t) = x(t)*\frac{1}{{\pi t}}\]
定义解析信号
\[\psi (t) = x(t) + j\hat x(t)\]
解析信号本质上是原信号的正频谱部分，是实信号的一种“简练”形式，常称为$x(t)$的预包络，由于$x(t)$的包络能够经过对$\psi (t)$简单求模获得。
带通讯号的预包络与复包络
带通讯号$x(t)$的Hilbert变换为
\[\hat x(t) = {x_I}(t)\sin 2\pi {f_0}t + {x_Q}(t)\cos2\pi {f_0}t\]
对应的解析信号为
\[\psi (t) = x(t) + j\hat x(t) = \left[ {{x_I}(t) + j{x_Q}(t)} \right]{e^{j2\pi {f_0}t}} = \tilde x(t){e^{j2\pi {f_0}t}}\]
信号$\tilde x(t) = {x_I}(t) + j{x_Q}(t) $是$x(t)$的复包络。所以，包络信号及其对应的相位为
\[\begin{array}{l}
a(t) = |{x_I}(t) + j{x_Q}(t)| = |\psi (t)|\\
\psi (t) = \arg (\tilde x(t)) = \angle \tilde x(t)
\end{array}\]
所以，实带通讯号$x(t)$、解析信号$\phi(t)$及复包络$\tilde x(t)$之间的关系以下：
\[\begin{array}{l}x(t) = r(t)\cos (2\pi {f_0}t + {\phi _x}(t))\\x(t) = {x_I}(t)\cos 2\pi {f_0}t - {x_Q}(t)\sin 2\pi {f_0}t\\\psi (t) = x(t) + j\hat x(t) \equiv \tilde x(t){e^{j2\pi {f_0}t}}\\\tilde x(t) = {x_I}(t) + j{x_Q}(t)\end{array}\]