[usth/ICT2.9] Remove jupiter output and add more maff
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\documentclass[a4paper,12pt]{article}
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\usepackage[english,vietnamese]{babel}
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\usepackage{amsmath}
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\usepackage{enumerate}
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\usepackage{lmodern}
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\title{System Cascade Connection}
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\author{{\selectlanguage{vietnamese}Nguyễn Gia Phong}}
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\begin{document}
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\selectlanguage{english}\maketitle
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Given two discrete-time systems $A$ and $B$ connected in cascade to form
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a new system $C = x \mapsto B(A(x))$.
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\section{Linearity}
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If $A$ and $B$ are linear, i.e. for all signals $x_i$ and scalars $a_i$,
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\begin{align*}
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A\left(n \mapsto \sum_i a_i x_i[n]\right) &= n \mapsto \sum_i a_i A(x_i)[n]\\
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B\left(n \mapsto \sum_i a_i x_i[n]\right) &= n \mapsto \sum_i a_i B(x_i)[n]
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\end{align*}
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then $C$ is also linear
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\begin{align*}
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C\left(n \mapsto \sum_i a_i x_i[n]\right)
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&= B\left(A\left(n \mapsto \sum_i a_i x_i[n]\right)\right)\\
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&= B\left(n \mapsto \sum_i a_i A(x_i)[n]\right)\\
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&= n \mapsto \sum_i a_i B(A(x_i))[n]\\
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&= n \mapsto \sum_i a_i C(x_i)[n]
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\end{align*}
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\section{Time Invariance}
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If $A$ and $B$ are time invariant, i.e. for all signals $x$ and integers $k$,
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\begin{align*}
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A(n \mapsto x[n - k]) &= n \mapsto A(x)[n - k]\\
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B(n \mapsto x[n - k]) &= n \mapsto B(x)[n - k]\\
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\end{align*}
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then $C$ is also time invariant
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\begin{align*}
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C(n \mapsto x[n - k])
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&= B(A(n \mapsto x[n - k]))\\
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&= B(n \mapsto A(x)[n - k])\\
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&= n \mapsto B(A(x))[n - k]\\
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&= n \mapsto C(x)[n - k]
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\end{align*}
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\section{LTI Ordering}
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If $A$ and $B$ are linear and time-invariant, there exists signals $g$ and $h$
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such that for all signals $x$, $A = x \mapsto x * g$ and $B = x \mapsto x * h$,
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thus \[B(A(x)) = B(x * g) = x * g * h = x * h * g = A(x * h) = A(B(x))\]
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or interchanging $A$ and $B$ order does not change $C$.
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\section{Causality}
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If $A$ and $B$ are causal, i.e. for all signals $x$, $y$ and integers $k$,
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\begin{multline*}
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x[n] = y[n]\quad\forall n < k
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\Longrightarrow\begin{cases}
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A(x)[n] = A(y)[n]\quad\forall n < k\\
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B(x)[n] = B(y)[n]\quad\forall n < k
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\end{cases}\\
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\Longrightarrow B(A(x))[n] = B(A(y))[n]\quad\forall n < k
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\iff C(x)[n] = C(y)[n]\quad\forall n < k
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\end{multline*}
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then $C$ is also causal.
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\section{BIBO Stability}
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If $A$ and $B$ are stable, i.e. there exists a signal $x$
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and scalars $a$, $b$ that
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\begin{align*}
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|x[n]| < a\quad\forall n \in \mathbb Z
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&\Longrightarrow |A(x)[n]| < b\quad\forall n \in \mathbb Z\\
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|x[n]| < a\quad\forall n \in \mathbb Z
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&\Longrightarrow |B(x)[n]| < b\quad\forall n \in \mathbb Z
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\end{align*}
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then $C$ is also stable, i.e. there exists a signal $x$
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and scalars $a$, $b$, $c$ that
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\begin{align*}
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|x[n]| < a\;\forall n \in \mathbb Z
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&\Longrightarrow |A(x)[n]| < b\;\forall n \in \mathbb Z\\
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&\Longrightarrow |B(A(x))[n]| < c\;\forall n \in \mathbb Z
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\iff |C(x)[n]| < c\;\forall n \in \mathbb Z
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\end{align*}
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\end{document}
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