1_Introduction

1.Signals

1.1Classification

DT/CT

Real/Complex

Periodic/Aperiodic 周期/非周期

Casual/Anti-casual 因果/非因果

Right-/Left-sided

Bounded/Unbounded

Even/Odd

Energy and Power

For CT signals:

\[ \begin{align} E_{\infty}&=\lim_{T\to \infty}\int_{-T}^T|x(t)|^2dt=\int_{-\infty}^\infty|x(t)|^2dt \\ P_{\infty}&=\lim_{T\to \infty}\frac{1}{2T}\int_{-T}^T|x(t)|^2dt \end{align} \]

For DT signals:

\[ \begin{align} E_{\infty}&=\lim_{N\to \infty}\sum_{n=-N}^{+N}|x[n]|^2dt=\sum_{n=-\infty}^{+\infty}|x[n]|^2 \\ P_{\infty}&=\lim_{N\to \infty}\frac{1}{2N+1}\sum_{n=-N}^N|x[n]|^2 \end{align} \]

1.2Building-Block Signals

For DT signal

\[ \delta[n]= \begin{cases} 1 \quad if \ n=0\\ 0 \quad otherwise \end{cases} \]

For CT signal

Unit-Impulse signal

Unit-Step signal

\[ u(t)=\int_{-\infty}^t\delta(\lambda)d\lambda= \begin{cases} 1 \quad if \ t\geq 0\\ 0 \quad otherwise \end{cases} \]

1.3Transformations of time

Time shift/reversal/scaling

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2.Systems

2.1Classification

with/without memory

Stable/Non-stable

Causal/Noncausal

A system is casual if the output at time \(t_0\) depends only on the input for \(t\leq t_0\) 判断系统是否是因果的，只需要判断x(·)的因变量是否永远小于等于y(·)

Time-invariant时不变

判据：t在x()里，且只能是t

Linear

判据：所有项都有x，且都是一次