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= -∞