What is convolution in signal processing?

Convolution is a mathematical operation that combines two signals to produce a third signal. It represents the amount of overlap between one signal and a time-reversed, shifted version of another signal. In signal processing, convolution is used for filtering, system analysis, and feature detection.

What's the difference between discrete and continuous convolution?

Discrete convolution operates on sequences (arrays of numbers) using summation: (f*g)[n] = Σf[k]g[n-k]. Continuous convolution operates on functions using integration: (f*g)(t) = ∫f(τ)g(t-τ)dτ. Discrete convolution is used in digital signal processing, while continuous convolution is used in analog systems and theoretical analysis.

How do you calculate discrete convolution step by step?

To calculate discrete convolution: 1) Flip one sequence horizontally, 2) Shift it by n positions, 3) Multiply corresponding elements, 4) Sum all products to get output[n], 5) Repeat for all n values. The output length is len(f) + len(g) - 1.

What are the properties of convolution?

Key convolution properties include: Commutative (f*g = g*f), Associative ((f*g)*h = f*(g*h)), Distributive (f*(g+h) = f*g + f*h), and Identity (f*δ = f, where δ is the impulse function). These properties make convolution a powerful mathematical tool.

What is circular convolution?

Circular convolution assumes the input sequences are periodic and performs convolution with wraparound indexing. Unlike linear convolution, the output length equals the input length. It's equivalent to linear convolution followed by aliasing and is commonly used in FFT-based processing.

What are real-world applications of convolution?

Convolution is used in: image processing (blurring, edge detection), audio processing (reverb, filtering), computer vision (feature detection), neural networks (convolutional layers), radar/sonar systems, and communication systems for channel modeling and equalization.

Signal Processing Tool

Convolution Calculator – Linear, Circular & Discrete Time

Free online convolution calculator with steps. Enter two sequences or functions and compute linear convolution, circular convolution, discrete time convolution and continuous convolution. The tool shows the convolution sum and explains each step, making it useful for signal processing, DSP, control systems and exam preparation.

You can treat it as a discrete convolution calculator,signal convolution calculator or a quick convolution of two functions calculator for continuous-time formulas.

Last updated: February 2, 2026

This is an online convolution calculator that runs in your browser—no software installation required.

Discrete and continuous convolution

Circular convolution support

Properties analysis and validation

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Convolution Calculator

Calculate discrete and continuous convolution with step-by-step analysis

Convolution Type

Sequence f[n]

Enter sequence in brackets: [1,2,3,4]

Sequence g[n]

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

Discrete Convolution:
(f * g)[n] = Σ f[k]g[n-k]

Continuous Convolution:
(f * g)(t) = ∫ f(τ)g(t-τ) dτ

Circular Convolution:
y[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.

📊 Convolution Process Diagram

Shows signal overlap, shifting, and multiplication process

Convolution Integral & Laplace Transform

For continuous-time signals, the operation becomes the convolution integral:

(f * g)(t) = ∫_−∞^∞ f(τ) g(t − τ) dτ

This makes the calculator useful as a simple convolution of two functions calculator: you specify f(t) and g(t), and the tool shows the symbolic form of the convolution integral.

In the Laplace transform domain, convolution becomes multiplication:

L{(f * g)(t)} = F(s) · G(s)

This relationship is the basis of many Laplace transform convolution calculators: instead of computing the convolution integral directly, you transform both functions, multiply in the s-domain, and then take the inverse Laplace transform to get the time-domain result.

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 Laplace transform calculator.

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Linear Convolution vs Circular Convolution

This page can be used as both a linear convolution calculator and a circular convolution calculator. The difference is how we treat the signal lengths and the indexing.

Linear Convolution Calculator

Linear convolution is the standard convolution used in most DSP and signals-and-systems courses.

Formula: (f * g)[n] = Σ f[k] g[n − k]
Output length: N + M − 1 for inputs of length N and M
Used for LTI system response and filtering
Matches the convolution sum taught in discrete-time systems

Circular Convolution Calculator

Circular convolution assumes signals are periodic and uses modulo indexing.

Formula: y[n] = Σ f[k] g[(n − k) mod N]
Output length: N (same as the period / sequence length)
Used with DFT / FFT and periodic signals
Effectively linear convolution plus aliasing (wrap-around)

Discrete Time Convolution with Steps

As a discrete time convolution calculator, the tool follows the standard step-by-step procedure students learn in class:

Write f[n] and g[n] as sequences (for example: f[n] = [1, 2, 1], g[n] = [1, 1, 1]).
Flip one sequence: g[−k] or g[−n] (time reversal).
Shift it by n samples to get g[n − k].
Multiply overlapping samples f[k] · g[n − k].
Sum all products to obtain y[n] (the convolution sum).
Repeat for all n to get the full output sequence.

This is exactly what students are looking for when they search "convolution calculator with steps" or "convolution sum calculator"—a clear sequence-by-sequence explanation of how the output is generated.

Example Analysis

Audio Filtering Application

Discrete convolution for audio signal filtering using a simple averaging filter

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.