📘 Discrete Time Convolution – Complete Explanation
For a discrete LTI system, output is obtained using summation instead of integration. This is easier and frequently asked in GATE.
🔹 1. Convolution Formula
y[n] = x[n] * h[n] = Σ x[k] h[n − k]
Summation over all k.🔹 2. Step-by-Step Procedure
1. Flip h[k] → h[−k] 2. Shift → h[n−k] 3. Multiply 4. Sum overlapping terms🔹 3. Example 1 (Finite Length Signals)
Given:x[n] = {1, 2, 1} h[n] = {1, 1}
Compute y[n].Step 1: Length
Output length = 3 + 2 − 1 = 4Step 2: Compute Each Value
n=0:y[0] = 1×1 = 1
n=1:y[1] = (1×1) + (2×1) = 3
n=2:y[2] = (2×1) + (1×1) = 3
n=3:y[3] = 1×1 = 1
Final Answer:
y[n] = {1, 3, 3, 1}
🔹 4. Example 2
Given:x[n] = u[n] h[n] = aⁿ u[n]
Then:y[n] = Σ a^(n-k)
Result:y[n] = (1 − a^(n+1)) / (1 − a)
🔹 5. Important Properties
- Commutative
- Associative
- Distributive
🔹 6. Output Length Rule
If: Length of x = L Length of h = M Then:Output length = L + M − 1
🎯 GATE Important Tips
- Most problems finite length
- Be careful with indexing
- Output length formula important
- Write table method to avoid mistakes
Discrete Convolution = Controlled Summation of Overlap
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