[신호 및 시스템] Lec 03 - Properties of Systems

Precaution

본 게시글은 서울대학교 박성준 교수님의 신호 및 시스템 (26 Spring) 강의록입니다.

Linearity

• For two valid inputs $x_1(t), x_2(t)$ and their respective output $y_1(t) = H{x_1(t)}$, $y_2(t) = H{x_2(t)}$, a linear system satisfy below for arnitary $\alpha, \beta$

\[H\{\alpha x_1(t) + \beta x_2(t)\} = \alpha y_1(t) + \beta y_2(t)\]

Time-invariance

• A system is time invarient if it satisfies equation for any time shift $t_0 (n_0)$

\[y(t-t_0) = H\{x(t-t_0)\}\]

• $y(t-t_0)$ : substitute $t \leftarrow t-t_0 $ for all $t$ in $y(t)$
• $H{x(t-t_0)}$ : only put $x(t-t_0)$ for system’s input
• The system is time-invarient if the system is commutative with time delay

Causality

• A system is causal if the output at any time depends only on values of the input at the present time and in the past
• If the output $y(t)$ uses $x(t’) \ t’>t$ for any instant of time $t$, the system is not causal

BIBO stability

• Bounded input bounded output stable
• A system is BIBO stable if the input to the system is bounded, then the output is also bounded.
• for any input $x(t)$ that satisfies $\vert x(t) \vert < B$ for all time $t$ with a positive $B$, then there exists a constant $B_y>0$ s.t. $\vert y(t)\vert < B_y$ for all $t$

Memoryless

• A system is memoryless if its output for each value of the independent time variable is dependent only on the input at that same time
• If the output $y(t) $ uses only $x(t)$ for any time $t$, the system is memoryless