Convolution Calculator

About Convolution Calculator

Welcome to the Convolution Calculator, a comprehensive free online tool for computing discrete and continuous convolution with detailed step-by-step solutions and interactive visualizations. Whether you are a student learning signal processing, an engineer analyzing linear systems, or a researcher working with mathematical operations, this calculator provides everything you need to understand and compute convolutions accurately.

What is Convolution?

Convolution is a fundamental mathematical operation that combines two functions (or signals) to produce a third function. It describes how the shape of one function is modified by another. Convolution is denoted by the asterisk symbol (*) and is essential in signal processing, image processing, probability theory, and many engineering applications.

In signal processing, convolution determines the output of a Linear Time-Invariant (LTI) system when given an input signal and the system's impulse response. This makes it one of the most important operations in understanding how systems transform signals.

Discrete Convolution

For discrete-time signals, the convolution of sequences x[n] and h[n] is defined as:

For finite-length sequences of length N and M, the output has length N + M - 1.

Circular Convolution

Circular (or cyclic) convolution is used when signals are periodic or when working with the Discrete Fourier Transform (DFT). For N-point circular convolution:

The modulo operation causes the indices to wrap around, making circular convolution suitable for periodic signal analysis.

Continuous Convolution

For continuous-time functions, the convolution integral is defined as:

For causal signals (signals that are zero for t less than 0), the limits become 0 to t.

Features of This Convolution Calculator

• Multiple Convolution Types: Supports discrete linear convolution, discrete circular convolution, and continuous convolution (integral form).
• Step-by-Step Solutions: Provides detailed mathematical breakdown showing each step of the convolution process, helping you understand the calculations.
• Interactive Visualizations: Generates Chart.js graphs showing input signals and the convolution output for visual understanding.
• Flexible Input Formats: Enter sequences with or without brackets (1, 2, 3 or [1, 2, 3]) and functions using standard mathematical notation.
• Quick Examples: Pre-set example buttons let you explore different convolution scenarios instantly.
• MathJax Rendering: Beautiful mathematical formulas rendered with professional typesetting.

How to Use This Calculator

1. Select convolution type: Choose between Discrete Linear Convolution (for standard signal processing), Discrete Circular Convolution (for DFT applications), or Continuous Convolution (for mathematical functions).
2. Enter input signals or functions: For discrete convolution, enter comma-separated values (e.g., 1, 2, 3, 4). For continuous convolution, enter mathematical expressions (e.g., t, sin(t), exp(-t)).
3. Use examples: Click the example buttons to quickly load common convolution scenarios and see how different inputs produce different results.
4. Calculate and analyze: Click "Calculate Convolution" to see the result with complete step-by-step solutions, calculation tables, and interactive visualizations.

Properties of Convolution

Convolution has several important mathematical properties that are useful in signal processing and analysis:

Applications of Convolution

Signal Processing

Convolution is fundamental to signal filtering. When you convolve an input signal with a filter's impulse response, you get the filtered output. This is how low-pass filters, high-pass filters, and band-pass filters process signals.

Image Processing

In image processing, 2D convolution is used for operations like blurring, sharpening, edge detection, and embossing. Convolutional kernels (small matrices) slide across images to produce various effects.

Audio Processing

Convolution reverb simulates acoustic spaces by convolving dry audio with the impulse response of a room or hall. This creates realistic reverb effects that capture the unique characteristics of physical spaces.

Neural Networks

Convolutional Neural Networks (CNNs) use convolution as their core operation. Learnable convolution kernels extract features from images, making CNNs extremely effective for image recognition and computer vision tasks.

System Analysis

For any Linear Time-Invariant (LTI) system, the output y(t) equals the convolution of the input x(t) with the system's impulse response h(t). This relationship is foundational to control systems and communication systems analysis.

Probability Theory

The probability density function of the sum of two independent random variables equals the convolution of their individual PDFs. This is used extensively in statistics and stochastic processes.

Linear vs Circular Convolution

Understanding the difference between linear and circular convolution is crucial for proper signal processing:

Linear Convolution

• Output length: N + M - 1 for inputs of length N and M
• No wraparound - indices extend beyond original signal length
• Used for general signal processing and filtering
• Represents actual physical convolution of finite signals

Circular Convolution

• Output length: max(N, M) after zero-padding to equal lengths
• Uses modulo arithmetic for wraparound
• Required when using DFT for efficient computation
• Linear convolution can be obtained from circular by zero-padding to length N + M - 1

Input Format Guide

Discrete Sequences

Enter signal values separated by commas. Brackets are optional:

• 1, 2, 3, 4 - Simple comma-separated values
• [1, 2, 3, 4] - With square brackets
• 0.5, 1.5, 2.5 - Decimal values supported
• -1, 0, 1, 0, -1 - Negative values supported

Continuous Functions

Enter mathematical expressions using standard notation:

• t - Linear function
• t**2 or t^2 - Polynomial (use ** for exponents)
• sin(t), cos(t), tan(t) - Trigonometric functions
• exp(t), exp(-t) - Exponential functions
• log(t) - Natural logarithm
• 2*t + 3 - Combinations with constants

Common Convolution Examples

Moving Average Filter

A 3-point moving average filter smooths data: h[n] = [1/3, 1/3, 1/3]. Convolving with this filter averages each point with its neighbors.

Edge Detection

The difference kernel h[n] = [1, -1] detects transitions. Convolving with this finds where signal values change abruptly.

Gaussian Smoothing

Gaussian kernels like [0.25, 0.5, 0.25] provide smooth, bell-shaped averaging that reduces noise while preserving signal structure.

Differentiation

The kernel [1, -2, 1] approximates the second derivative, useful for detecting peaks and curvature in signals.

Frequently Asked Questions

What is convolution in signal processing?

Convolution is a mathematical operation that combines two signals to produce a third signal. It describes how the shape of one signal is modified by another. In signal processing, convolution is used to determine the output of a linear time-invariant (LTI) system when given an input signal and the system's impulse response.

What is the difference between linear and circular convolution?

Linear convolution produces an output of length N+M-1 where N and M are the input lengths. It's used for non-periodic signals. Circular convolution assumes periodic signals and produces an output of the same length as the inputs. The indices wrap around using modulo arithmetic, making it suitable for DFT-based computations.

How do I use the discrete convolution calculator?

Enter your signal values as comma-separated numbers (e.g., 1, 2, 3). You can optionally use brackets [1, 2, 3]. Select either Linear or Circular convolution type, then click Calculate. The calculator will show the result with step-by-step calculations and visualizations.

What functions are supported for continuous convolution?

The continuous convolution calculator supports polynomial functions (t, t**2, t**3), exponential functions (exp(t), exp(-t)), trigonometric functions (sin(t), cos(t), tan(t)), logarithmic functions (log(t)), and combinations thereof. Use ** for exponents and standard mathematical notation.

What are common applications of convolution?

Convolution is used in signal filtering (low-pass, high-pass, band-pass filters), image processing (blur, edge detection, sharpening), audio processing (reverb, echo effects), system analysis (determining system output from impulse response), neural networks (convolutional layers in CNNs), and probability (sum of random variables).

Why does my convolution result have more elements than the inputs?

For linear convolution, if input x has N elements and h has M elements, the output has N + M - 1 elements. This is because the convolution "slides" one signal across the other, and partial overlaps at the beginning and end contribute to the output length.

How is convolution related to the Fourier Transform?

According to the Convolution Theorem, convolution in the time domain equals multiplication in the frequency domain. This property allows efficient convolution computation using FFT: transform both signals, multiply, and inverse transform. This reduces complexity from O(N*M) to O(N log N).

Additional Resources

Learn more about convolution and signal processing:

• Convolution - Wikipedia
• Convolution Integrals - Khan Academy
• Signal Processing Stack Exchange

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