Bandpass Signals(帶通訊號，或稱窄帶信號)

帶通訊號

一個實的帶通訊號$x(t)$能夠表示爲

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

其中$r(t)$是幅度調製或包絡，$\phi_x(t)$是相位調製，$f_0$是載波頻率，$r(t)$和$\phi_x(t)$的變化比$f_0$要小得多。頻率調製表示爲

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

瞬時頻率

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

若是信號帶寬B遠小於中心頻率$f_0$，則信號$x(t)$稱爲帶通訊號。

帶通訊號也能夠由兩個互爲正交的低通訊號(的調製)來表示，即

\[x(t) = {x_I}(t)\cos 2\pi {f_0}t - {x_Q}(t)\sin 2\pi {f_0}t\]

其中

\[\begin{array}{l}
{x_I}(t) = r(t)\cos {\phi _x}(t)\\
{x_Q}(t) = r(t)\sin{\phi _x}(t)
\end{array}\]

解析信號(Analytic Signal)或預包絡(Pre-Envelope)

對於給定的實信號$x(t)$，其Hilbert變換爲

\[\hat x(t) = x(t)*\frac{1}{{\pi t}}\]

定義解析信號

\[\psi (t) = x(t) + j\hat x(t)\]

解析信號本質上是原信號的正頻譜部分，是實信號的一種「簡練」形式，常稱爲$x(t)$的預包絡，由於$x(t)$的包絡能夠經過對$\psi (t)$簡單求模獲得。

帶通訊號的預包絡與復包絡

帶通訊號$x(t)$的Hilbert變換爲

\[\hat x(t) = {x_I}(t)\sin 2\pi {f_0}t + {x_Q}(t)\cos2\pi {f_0}t\]

對應的解析信號爲

\[\psi (t) = x(t) + j\hat x(t) = \left[ {{x_I}(t) + j{x_Q}(t)} \right]{e^{j2\pi {f_0}t}} = \tilde x(t){e^{j2\pi {f_0}t}}\]

信號$\tilde x(t) = {x_I}(t) + j{x_Q}(t) $是$x(t)$的復包絡。所以，包絡信號及其對應的相位爲

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

所以，實帶通訊號$x(t)$、解析信號$\phi(t)$及復包絡$\tilde x(t)$之間的關係以下：

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