Bandpass Signals(带通信号，或称窄带信号)

带通信号

一个实的带通信号$x(t)$可以表示为

\[x(t) = r(t)\cos (2\pi f_0 t + \phi_x(t)) \]

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

\[f_m(t) = \frac{1}{2\pi} \frac{d}{dt}\phi_x(t) \]

瞬时频率

\[{f_i}(t) = \frac{1}{{2\pi }}\frac{d}{{dt}}\left( {2\pi {f_0}t + {\phi _x}(t)} \right) = {f_0} + {f_m}(t)\]

如果信号带宽B远小于中心频率$f_0$，则信号$x(t)$称为带通信号。

带通信号也可以由两个互为正交的低通信号(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，因为$x(t)$的包络可以通过对$\psi (t)$简单求模得到。

带通信号的预包络与复包络

带通信号$x(t)$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。因此，包络信号及其对应的相位为

\[\begin{array}{l}
a(t) = |{x_I}(t) + j{x_Q}(t)| = |\psi (t)|\\
\psi (t) = \arg (\tilde x(t)) = \angle \tilde x(t)
\end{array}\]

因此，实带通信号$x(t)$、解析信号$\phi(t)$及复包络$\tilde x(t)$之间的关系如下：

\[\begin{array}{l}
x(t) = r(t)\cos (2\pi {f_0}t + {\phi _x}(t))\\
x(t) = {x_I}(t)\cos 2\pi {f_0}t - {x_Q}(t)\sin 2\pi {f_0}t\\
\psi (t) = x(t) + j\hat x(t) \equiv \tilde x(t){e^{j2\pi {f_0}t}}\\
\tilde x(t) = {x_I}(t) + j{x_Q}(t)
\end{array}\]