Commutative, associative, distributive properties

LTI system

A linear time-invariant (LTI) system obeys superposition (linearity) and its behaviour does not change with time (a shifted input gives an equally shifted output). It is fully described by its impulse response $h[n]$, and

$$y[n]=x[n]*h[n]=\sum_{k=-\infty}^{\infty}x[k]\,h[n-k]$$

1. Commutative property

$$x[n]*h[n]=h[n]*x[n]$$

Proof: in $\sum_k x[k]h[n-k]$ put $m=n-k$: it becomes $\sum_m h[m]x[n-m]=h[n]*x[n]$. Meaning: the roles of input and impulse response can be swapped without changing the output.

2. Associative property

$$\{x[n]*h_1[n]\}*h_2[n]=x[n]*\{h_1[n]*h_2[n]\}$$

Meaning: two LTI systems in cascade are equivalent to a single system with $h[n]=h_1[n]*h_2[n]$. With the commutative property, the order of the cascade can also be changed.

 x -->[h1]-->[h2]--> y
   ==  x -->[h1*h2]--> y
   ==  x -->[h2]-->[h1]--> y

3. Distributive property

$$x[n]*\{h_1[n]+h_2[n]\}=x[n]*h_1[n]+x[n]*h_2[n]$$

Proof: $\sum_k x[k](h_1[n-k]+h_2[n-k])=\sum_k x[k]h_1[n-k]+\sum_k x[k]h_2[n-k]$. Meaning: two LTI systems in parallel (same input, outputs added) are equivalent to one system with $h=h_1+h_2$.

       +-->[h1]--+
 x ----+         (+)--> y   ==  x -->[h1+h2]--> y
       +-->[h2]--+

Example

$h_1=\delta[n]+\delta[n-1]$, $h_2=\delta[n]-\delta[n-1]$:

• cascade: $h_1*h_2=\delta[n]-\delta[n-2]$;
• parallel: $h_1+h_2=2\delta[n]$.

These properties let a block diagram of many LTI systems be reduced to a single equivalent $h[n]$.