What variable name must be used in function expressions?

Always use x as the variable in both f(x) and g(x) expressions.

What is the difference between causal and direct window mode?

Causal mode restricts the integration bounds to [max(τmin, 0), min(τmax, t)] assuming signals are zero for negative values, while direct window mode integrates across the entire [τmin, τmax] range.

How many output time points can be calculated at once?

You can evaluate between 1 and 8 comma-separated time values in a single calculation.

How does the number of Simpson intervals impact the result?

Higher interval counts increase numerical integration accuracy for steep or rapidly oscillating signals, with a default setting of 2000.

Which mathematical functions are supported?

Supported operations include +, -, *, /, ^, parentheses, and standard functions like exp, sin, cos, sqrt, and ln.

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Convolution Integral Calculator (Numeric)

Numeric (f*g)(t) = ∫ f(τ)g(t−τ)dτ by composite Simpson, in causal or direct-window mode, at up to 8 output points.

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2 examples

e^(−t) * e^(−2t) = e^(−t) − e^(−2t) — causal exponentials

Two causal exponentials convolve to the difference of exponentials: (f*g)(t) = e^(−t) − e^(−2t). At t = 0.5, 1, 2, 3 the values are 0.2387, 0.2325, 0.1170 and 0.0473, which the Simpson integral reproduces.

Convolution integral (composite Simpson, 2000 intervals) f(τ) = exp(-x) g(τ) = exp(-2x) Mode: causal (both signals zero for t < 0) Window: τ ∈ [0, 20] (f*g)(0.5) = 0.2387 (f*g)(1) = 0.2325 (f*g)(2) = 0.117 (f*g)(3) = 0.0473

Gaussian * Gaussian = √(π/2)·e^(−t²/2) — direct window mode

The convolution of two unit Gaussians e^(−τ²) is again a Gaussian: √(π/2)·e^(−t²/2) ≈ 1.2533·e^(−t²/2). Integrating over the window [−5, 5] gives 1.2533, 0.7602 and 0.1696 at t = 0, 1, 2.

Convolution integral (composite Simpson, 2000 intervals) f(τ) = exp(-x^2) g(τ) = exp(-x^2) Mode: direct window (full τ range) Window: τ ∈ [-5, 5] (f*g)(0) = 1.2533 (f*g)(1) = 0.7602 (f*g)(2) = 0.1696

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8 options

ContentPaste or type the main input values.3

f(τ) expression (variable x)textRequiredFirst signal, written in the variable x. Supports + - * / ^, parentheses, and functions like exp, sin, cos, sqrt, ln.g(τ) expression (variable x)textRequiredSecond signal, written in the variable x (it is evaluated at t − τ inside the integral).Output t values (comma-separated)textRequired1–8 evaluation points t at which (f*g)(t) is reported.

SettingsAdjust formats, ranges, numbers, and modes.5

ModeselectRequiredWindow start τminnumberOptionalLower integration bound.Window end τmaxnumberOptionalUpper integration bound.Simpson intervalsnumberOptionalNumber of Simpson subintervals per integral (even, 100–20000).Decimal PlacesnumberOptional

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Convolution Integral Calculator (Numeric)

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1. Evaluating Causal Exponential Decay System Response

2. Gaussian Kernel Convolution in Windowed Mode

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