Convolution Calculator
Calculate the discrete convolution of two sequences with step-by-step visualization
Discrete Convolution: (x * h)[n] = Σ x[k] · h[n - k]
Enter comma-separated values for each sequence. Output length = len(x) + len(h) - 1
Enter numbers separated by commas
Enter numbers separated by commas
Quick Examples
Convolution Result: x * h
y[n] =
Output Length:
Input Sequence x[n]
Length:
Kernel h[n]
Length:
Step-by-Step Calculation
Output Sequence Values
| Index (n) | y[n] |
|---|---|
Convolution Properties
| Property | Formula |
|---|---|
| Commutativity | x * h = h * x |
| Associativity | (x * h) * g = x * (h * g) |
| Distributivity | x * (h + g) = x * h + x * g |
| Identity | x * δ = x |
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About Convolution Calculator
What is Convolution?
Convolution is a mathematical operation that combines two sequences (or signals) to produce a third sequence. It's fundamental in signal processing, image processing, and probability theory.
The Convolution Formula
The discrete convolution of two sequences x[n] and h[n] is defined as:
(x * h)[n] = Σ x[k] · h[n-k]
Where:
- x[n] is the input sequence
- h[n] is the impulse response (kernel)
- y[n] is the output sequence
- The sum is computed over all valid indices k
How Convolution Works
- Flip one sequence (usually h[k])
- Shift the flipped sequence by n positions
- Multiply corresponding elements
- Sum all the products
- Repeat for each output position
Properties of Convolution
Commutativity
x * h = h * x
Associativity
(x * h) * g = x * (h * g)
Distributivity
x * (h + g) = x * h + x * g
Identity
x * δ = x (where δ is the unit impulse)
Output Length
For two sequences of length M and N, the convolution result has length: Length = M + N - 1
Applications
- Signal Processing: Filtering, system analysis
- Image Processing: Edge detection, blurring, sharpening
- Probability: Sum of independent random variables
- Neural Networks: Convolutional layers in CNNs
- Audio Processing: Reverb, echo effects
Example
For x = [1, 2, 3] and h = [1, 0, 1]:
- y[0] = 1×1 = 1
- y[1] = 1×0 + 2×1 = 2
- y[2] = 1×1 + 2×0 + 3×1 = 4
- y[3] = 2×1 + 3×0 = 2
- y[4] = 3×1 = 3
Result: y = [1, 2, 4, 2, 3]