Circular Convolution
Theory
Circular convolution of two sequences x(n) and h(n) of length N is defined as:
y(n) = ฮฃ x(k) · h((n − k) mod N)
- Sequences are treated as periodic
- Uses modulo operation (wrap-around)
- Output length is N
Example
x(n) = [1, 2, 3]
h(n) = [4, 5, 6]
Step-by-Step Solution
y(0)
y(0) = 1×4 + 2×6 + 3×5 = 4 + 12 + 15 = 31
y(1)
y(1) = 1×5 + 2×4 + 3×6 = 5 + 8 + 18 = 31
y(2)
y(2) = 1×6 + 2×5 + 3×4 = 6 + 10 + 12 = 28
Final Answer
y(n) = [31, 31, 28]
Alternative Method (Table)
| Shift | Shifted h(n) | Calculation | Result |
|---|---|---|---|
| n = 0 | [4, 5, 6] | 1×4 + 2×6 + 3×5 | 31 |
| n = 1 | [6, 4, 5] | 1×5 + 2×4 + 3×6 | 31 |
| n = 2 | [5, 6, 4] | 1×6 + 2×5 + 3×4 | 28 |
Key Concept
Circular convolution corresponds to multiplication in the frequency domain using DFT.
- Time domain: Circular convolution
- Frequency domain: Multiplication
Conclusion
Normal convolution slides without wrapping, while circular convolution wraps around like a loop.