Consider relaxed LTI system. A relaxed system means if input x (n) is zero then output y (n) = 0 is zero. Let us say unit impulse 𝛿 (n) is applied to this system then its output is denoted by h (n). h (n) we called as the impulse response of the system.
Step 1 :
T
𝛿 (n) → y (n) = h (n)
𝛿 (n) → y (n) = h (n)
Step 2 :
T
𝛿 (n-k) → y (n) = h (n-k)
𝛿 (n-k) → y (n) = h (n-k)
Step 3 :
T
x (k) 𝛿 (n-k) → y (n) = x (k) h(n-k)
Step 4 :
∞ T ∞
∑ = x (k) 𝛿 (n-k) = y (n) → ∑ x (k) h(n-k)
k= -∞ k= -∞
y (n) =
∞
∑ = x (k) h (n-k)
k= -∞
∞
x (n) * h (n) = ∑ x (k) h(n-k)
k= -∞