Convolution Calculator
Calculate discrete and continuous convolution operations with step-by-step analysis. Our signal processing calculator supports linear, circular convolution and convolution properties for educational and engineering applications.
Last updated: December 15, 2024
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Enter sequence in brackets: [1,2,3,4]
Second sequence for convolution
Convolution Results
Convolution Result:
[1.00, 3.00, 4.00, 3.00, 1.00]
Output Length:
5 samples
Analysis:
Discrete convolution of sequences f[n] = [1, 2, 1] and g[n] = [1, 1, 1] produces a sequence of length 5. Each output sample is computed as the sum of products of overlapping input samples.
Calculation Steps:
- Input sequences: f[n] = [1, 2, 1], g[n] = [1, 1, 1]
- Convolution formula: (f * g)[n] = Σ f[k]g[n-k]
- Output length: 3 + 3 - 1 = 5
- Sample calculations:
- y[0] = 1.00
- y[1] = 3.00
- y[2] = 4.00
Convolution Properties:
- • Commutative: f * g = g * f
- • Associative: (f * g) * h = f * (g * h)
- • Distributive: f * (g + h) = f * g + f * h
- • Identity: f * δ = f (delta function is identity)
Quick Example Result
For sequences f = [1, 2, 1] and g = [1, 1, 1]:
f * g = [1, 3, 5, 3, 1]
Output length: 3 + 3 - 1 = 5 samples
How This Calculator Works
Our convolution calculator applies fundamental signal processing principles to analyze the interaction between two signals or functions. The calculator uses mathematical convolutionoperations to compute linear, circular, and continuous convolution with detailed step-by-step analysis.
Convolution Formulas
(f * g)[n] = Σ f[k]g[n-k](f * g)(t) = ∫ f(τ)g(t-τ) dτy[n] = Σ f[k]g[(n-k) mod N]These formulas represent different types of convolution operations. Discrete convolution uses summation over sequences, continuous convolution uses integration over functions, and circular convolution uses modular arithmetic for periodic signals.
Shows signal overlap, shifting, and multiplication process
Signal Processing Foundation
Convolution is a fundamental operation in signal processing, systems analysis, and mathematical modeling. It describes the output of a linear time-invariant system when given an input signal, and it's essential for understanding filtering, system response, and signal transformation. The operation represents how each input sample influences multiple output samples.
- Linear convolution produces output length: N + M - 1 for inputs of length N and M
- Circular convolution produces output length equal to input length (periodic assumption)
- Convolution is commutative, associative, and distributive
- The impulse response of a system determines its convolution behavior
Sources & References
- Digital Signal Processing: Principles, Algorithms, and Applications - John G. Proakis, Dimitris G. Manolakis (4th Edition)Comprehensive treatment of convolution in digital signal processing
- IEEE Signal Processing Society - Signal Processing Education ResourcesProfessional standards for signal processing education and applications
- MIT OpenCourseWare - Signals and SystemsAcademic materials on convolution theory and applications
Need help with other signal processing calculations? Check out our Fourier transform calculator and Z-transform calculator.
Get Custom Calculator for Your PlatformExample Analysis
Given Signals:
- Input signal: [1, 4, 2, 3, 1]
- Filter kernel: [0.33, 0.33, 0.33]
- Operation: Linear convolution
Convolution Steps:
- y[0] = 1×0.33 = 0.33
- y[1] = 1×0.33 + 4×0.33 = 1.65
- y[2] = 1×0.33 + 4×0.33 + 2×0.33 = 2.31
- Continue for all positions...
Result: [0.33, 1.65, 2.31, 2.97, 1.98, 1.32, 0.33]
The convolution applies a 3-point averaging filter to smooth the input signal. Each output sample is the weighted average of 3 consecutive input samples, demonstrating how convolution implements digital filtering.
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