System Properties
Introduction System Properties

Memoryless Systems and Systems with Memory

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A system is said to have memory if $y(t_0)$ depends on any $x(t_1)$ ,where $t_1\neq t_0$

Bounded Input Bounded Output (BIBO) Stability

A signal $x(t)$ is said to be bounded if

(1)
\begin{align} |x(t)|\leq M_x < \infty \hspace{1cm} \forall \hspace{1mm} t \end{align}
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Linearity

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I. Additivity or Superposition

$x_1(t) \longrightarrow y_1(t)$
$x_2(t) \longrightarrow y_2(t)$
$x_3(t)=x_1(t)+x_2(t) \longrightarrow y_3(t)=y_1(t)+y_2(t)$


II. Homogeneity

$x_1(t) \longrightarrow y_1(t)$
$ax_1(t) \longrightarrow ay_1(t) \hspace{0.2cm} \forall \hspace{0.2cm} a \hspace{0.2cm} \text{and} \hspace{0.2cm} x_1(t)$

A system is linear if it is both Additive and Homogeneous.

$a_1x_1(t) \longrightarrow a_1y_1(t)$
$a_2x_2(t) \longrightarrow a_2y_2(t)$
.
.
.
$a_n x_n(t) \longrightarrow a_n y_n(t)$


Thus,

$x(t)=\sum_{i=1}^{N} a_i x_i(t) \longrightarrow y(t)=\sum_{i=1}^{N} a_i y_i(t)$


If $x(t)=\sum a_i x_i(t)$ then,(2)
\begin{align} y(t)=H[x(t)]=H\big[\sum_{i=1}^{N} a_i x_i(t)\big]=\sum a_i H[x_i(t)] \end{align}
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Time Invariance

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A system is Time Invariant if its parameters do not change with time.

$y_1(t)=k.x_1(t)$ $\longleftarrow$ Time Invariant$
$y_1(t)=k(t).x_1(t)=\frac{1}{t}.x_1(t)$

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Thus,

(3)
\begin{equation} HS^{t_0}[x_1(t)]=S^{t_0}H[x_1(t)] \end{equation}

Causality

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A system is Non-causal(or anticipative) if the output at time $n_0$ ,i.e $y[n_0]$ depends on any future value of the input, i.e $x[k]$ for any $k>n_0$.
System 1: $y[n]=\frac{1}{3}(x[n]+x[n-1]+x[n-2])$$\longleftarrow$ Causal
System 2: $y[n]=\frac{1}{3}(x[n-1]+x[n]+x[n+1])$$\longleftarrow$ Non-Causal


Why do we care about non-causal systems?

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