信号

信号是关于时间的函数，可分为连续时间信号和离散时间信号两类。

所有信号都可以由一些基础信号组合构建。

基础信号

constant signal 或者 DC signal

\(
u(t) = k
\)
unit step signal

\(
u(t) =
\begin{cases}
0,t<0
\newline
1,t\geq 0
\end{cases}
\)
unit ramp signal

\(
u(t) =
\begin{cases}
0,t<0
\newline
t,t\geq 0
\end{cases}
\)
sinusoidal signal

\(
\begin{split}
u(t) = &A\mathrm{cos}(wt \pm \phi)
\newline
\mbox{or}\quad &A\mathrm{sin}(wt \pm \phi)
\end{split}
\)
exponential signal

\(
u(t) = e^{at}
\)

脉冲信号

Dirac’s delta function 或者说 impulse δ 是一种理想化的信号，满足：

在靠近 t = 0 时特别大
在远离 t = 0 时特别小
函数曲线下面积为 1
函数的具体形状是无所谓的
ϵ 很小

在图中 δ 被表示为一个箭头：

The δ function is defined with the following property:

\(
\int_a^b f(t) \delta (t) \mathrm{d} t = f(0)
\), provided \(a<0\), \(b>0\), and \(f\) is continous at \(t = 0\).

Scaled impulses & Sifting property

\(\alpha \delta (t - T)\) is an impulse at time \(T\), with magnitude \(\alpha\)

\(\int_a^b \alpha \delta (t - T)f(t)\mathrm{d}t = \alpha f(T)\)

for \(a < T < b\) and \(f\) is continuous at \(T\).

Physical interpretation

Impulse function are used to model physical signals

that act over short time intervals
whose effect depends on integral of signal

系统

A system can be viewed as a process in which input signals are transformed by the system or cause the system to respond in some way, resulting in other signals as output.