+++
rss = "Properties of cascade connected systems analyzed via anonymous functions"
date = Date(2020, 4, 15)
tags = ["fun", "math"]
+++

# System Cascade Connection

Given two discrete-time systems $A$ and $B$ connected in cascade to form
a new system $C = x \mapsto B(A(x))$, we examine the following properties:

\toc

## Linearity

If $A$ and $B$ are linear, i.e. for all signals $x_i$ and scalars $a_i$,

\[\begin{aligned}
  A\left(n \mapsto \sum_i a_i x_i[n]\right) = n \mapsto \sum_i a_i A(x_i)[n]\\
  B\left(n \mapsto \sum_i a_i x_i[n]\right) = n \mapsto \sum_i a_i B(x_i)[n]
\end{aligned}\]

then $C$ is also linear

\[\begin{aligned}
  C\left(n \mapsto \sum_i a_i x_i[n]\right)
  &= B\left(A\left(n \mapsto \sum_i a_i x_i[n]\right)\right)\\
  &= B\left(n \mapsto \sum_i a_i A(x_i)[n]\right)\\
  &= n \mapsto \sum_i a_i B(A(x_i))[n]\\
  &= n \mapsto \sum_i a_i C(x_i)[n]
\end{aligned}\]

## Time Invariance

If $A$ and $B$ are time invariant,
i.e. for all signals $x$ and integers $k$,

\[\begin{aligned}
  A(n \mapsto x[n - k]) &= n \mapsto A(x)[n - k]\\
  B(n \mapsto x[n - k]) &= n \mapsto B(x)[n - k]
\end{aligned}\]

then $C$ is also time invariant

\[\begin{aligned}
  C(n \mapsto x[n - k])
  &= B(A(n \mapsto x[n - k]))\\
  &= B(n \mapsto A(x)[n - k])\\
  &= n \mapsto B(A(x))[n - k]\\
  &= n \mapsto C(x)[n - k]
\end{aligned}\]

## LTI Ordering

If $A$ and $B$ are linear and time-invariant, there exists
signals $g$ and $h$ such that for all signals $x$,
$A = x \mapsto x * g$ and $B = x \mapsto x * h$, thus 

\[B(A(x)) = B(x * g) = x * g * h = x * h * g = A(x * h) = A(B(x))\]

or interchanging $A$ and $B$ order does not change $C$.

## Causality

If $A$ and $B$ are causal,
i.e. for all signals $x$, $y$ and any choise of integer $k$,

\[\begin{aligned}
  \forall n < k, x[n] = y[n]\quad
  \Longrightarrow &\;\begin{cases}
  \forall n < k, A(x)[n] = A(y)[n]\\
  \forall n < k, B(x)[n] = B(y)[n]
  \end{cases}\\
  \Longrightarrow &\;\forall n < k, B(A(x))[n] = B(A(y))[n]\\
  \Longleftrightarrow &\;\forall n < k, C(x)[n] = C(y)[n]
\end{aligned}\]

then $C$ is also causal.

## BIBO Stability

If $A$ and $B$ are stable, i.e. there exists a signal $x$
and scalars $a$ and $b$ that for all integers $n$,

\[\begin{aligned}
  |x[n]| < a &\Longrightarrow |A(x)[n]| < b\\
  |x[n]| < a &\Longrightarrow |B(x)[n]| < b
\end{aligned}\]

then $C$ is also stable, i.e. there exists a signal $x$
and scalars $a$, $b$ and $c$ that for all integers $n$,

\[\begin{aligned}
  |x[n]| < a\quad
  \Longrightarrow &\;|A(x)[n]| < b\\
  \Longrightarrow &\;|B(A(x))[n]| < c\\
  \Longleftrightarrow &\;|C(x)[n]| < c
\end{aligned}\]