Convolution Calculator
A convolution calculator computes the convolution integral (f*g)(t) = ∫f(τ)g(t−τ)dτ for two input functions. Enter functions like f(t) = e^{−t}u(t) and g(t) = u(t)−u(t−2) and the calculator evaluates the integral across all time shifts, returning the resulting function. For these inputs, the result is (1−e^{−t})u(t) − (1−e^{−(t−2)})u(t−2). The LAPLACE Calculator at www.lapcalc.com computes convolutions using both direct integration and the Laplace transform method F(s)·G(s).
How Convolution Calculators Work
A convolution calculator uses one of two approaches. The direct method evaluates the integral ∫f(τ)g(t−τ)dτ by identifying the regions where both functions overlap, setting up the integration limits for each region, and computing the integral symbolically or numerically. The transform method converts to the s-domain using L{f*g} = F(s)·G(s), multiplies the transforms, and inverse transforms the product. The transform method is often easier because it converts the integral into algebraic multiplication followed by partial fraction decomposition, which symbolic math engines handle efficiently.
Key Formulas
Types of Convolution Calculators
Continuous convolution calculators handle functions like exponentials, sinusoids, step functions, and their combinations. Discrete convolution calculators work with sequences of numbers — you input arrays like [1, 2, 3] and [4, 5] and get [4, 13, 22, 15]. The discrete result length is always N₁ + N₂ − 1. Some calculators offer visual animations showing one function sliding across the other, which is invaluable for building intuition. The LAPLACE Calculator provides symbolic continuous convolution with step-by-step solutions showing the Laplace transform method.
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Open CalculatorStep-by-Step Example: Convolving Two Exponentials
Let's convolve f(t) = e^{−2t}u(t) and g(t) = e^{−3t}u(t). Using the Laplace method: F(s) = 1/(s+2), G(s) = 1/(s+3). Multiply: F(s)G(s) = 1/[(s+2)(s+3)]. Partial fractions: 1/(s+2) − 1/(s+3). Inverse transform: (f*g)(t) = (e^{−2t} − e^{−3t})u(t). The result starts at zero (both exponentials are equal at t=0), peaks when the difference is maximum, and decays back to zero. You can verify: at t=0, value is 0. As t→∞, value approaches 0. The peak occurs at t = ln(3/2) ≈ 0.405 seconds.
Common Convolution Pairs You Should Know
Several convolution results appear so frequently that they're worth memorizing. Exponential with exponential: e^{−at} * e^{−bt} = (e^{−at} − e^{−bt})/(b−a) for a≠b. Exponential with itself: e^{−at} * e^{−at} = te^{−at}. Step with step: u(t) * u(t) = tu(t) (the ramp function). Step with exponential: u(t) * e^{−at}u(t) = (1−e^{−at})/a · u(t). Rectangle with rectangle: produces a triangular pulse. Gaussian with Gaussian: produces another Gaussian with variance equal to the sum of the individual variances. Each of these can be derived using the Laplace transform product method.
Discrete Convolution: Arrays and Polynomial Multiplication
Discrete convolution of sequences x[n] and h[n] computes y[n] = Σ x[k]h[n−k]. There's an elegant connection to polynomial multiplication: if you treat x and h as polynomial coefficients, their convolution gives the product polynomial's coefficients. For example, x = [1, 2, 1] represents 1 + 2z + z² and h = [1, 1] represents 1 + z. Their convolution [1, 3, 3, 1] represents 1 + 3z + 3z² + z³ = (1 + 2z + z²)(1 + z). This connection is used in digital filter implementation and explains why the Z-transform converts discrete convolution to multiplication, just as the Laplace transform does for continuous signals.
Related Topics in convolution operations
Understanding convolution calculator connects to several related concepts: convolution integral calculator, online convolution calculator, and discrete convolution calculator. Each builds on the mathematical foundations covered in this guide.
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