Properties of linear convolution

Linear convolution or proof of LTI system is completely characterized by unit impulse response h(n).

These properties are :

• Commutative property
• Associative property
• Distributive property

1.  Commutative property :

x (n) * h (n) = h (n) * x (n)

2. Distributive property :

x (n) * [ h₁ (n) +  h₂ (n) ] =  [ x (n) * h₁ (n) ]  + [   x (n) * h₂ (n) ]

3. Associative property :

[ x (n) * h₁ (n) ] *  h₂ (n)  =  x (n) * [ h₁ (n) *  h₂ (n) ] 

Linear convolution or proof of LTI system is completely characterized by unit impulse response h(n).

These properties are :

• Commutative property
• Associative property
• Distributive property

1.  Commutative property :

x (n) * h (n) = h (n) * x (n)

2. Distributive property :

x (n) * [ h₁ (n) +  h₂ (n) ] =  [ x (n) * h₁ (n) ]  + [   x (n) * h₂ (n) ]

3. Associative property :

[ x (n) * h₁ (n) ] *  h₂ (n)  =  x (n) * [ h₁ (n) *  h₂ (n) ]