McSinyx

System Cascade Connection

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

  1. Linearity
  2. Time Invariance
  3. LTI Ordering
  4. Causality
  5. BIBO Stability

Linearity

If AA and BB are linear, i.e. for all signals xix_i and scalars aia_i,

A(niaixi[n])=niaiA(xi)[n]B(niaixi[n])=niaiB(xi)[n]\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 CC is also linear

C(niaixi[n])=B(A(niaixi[n]))=B(niaiA(xi)[n])=niaiB(A(xi))[n]=niaiC(xi)[n]\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 AA and BB are time invariant, i.e. for all signals xx and integers kk,

A(nx[nk])=nA(x)[nk]B(nx[nk])=nB(x)[nk]\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 CC is also time invariant

C(nx[nk])=B(A(nx[nk]))=B(nA(x)[nk])=nB(A(x))[nk]=nC(x)[nk]\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 AA and BB are linear and time-invariant, there exists signals gg and hh such that for all signals xx, A=xxgA = x \mapsto x * g and B=xxhB = x \mapsto x * h, thus

B(A(x))=B(xg)=xgh=xhg=A(xh)=A(B(x))B(A(x)) = B(x * g) = x * g * h = x * h * g = A(x * h) = A(B(x))

or interchanging AA and BB order does not change CC.

Causality

If AA and BB are causal, i.e. for all signals xx, yy and any choise of integer kk,

n<k,x[n]=y[n] {n<k,A(x)[n]=A(y)[n]n<k,B(x)[n]=B(y)[n] n<k,B(A(x))[n]=B(A(y))[n] n<k,C(x)[n]=C(y)[n]\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 CC is also causal.

BIBO Stability

If AA and BB are stable, i.e. there exists a signal xx and scalars aa and bb that for all integers nn,

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

then CC is also stable, i.e. there exists a signal xx and scalars aa, bb and cc that for all integers nn,

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