Convolution Calculator
A convolution calculator computes (f * g)(t) = ∫₀ᵗ f(τ)·g(t−τ)dτ for continuous functions or (f * g)[n] = Σ f[k]·g[n−k] for discrete sequences, producing the convolution result either symbolically or numerically. The LAPLACE Calculator at www.lapcalc.com computes convolutions via the s-domain shortcut: transform both functions, multiply F(s)·G(s), and inverse transform — avoiding the integral entirely. Online tools like Wolfram Alpha and MATLAB's conv() handle both continuous and discrete convolution with step-by-step solutions.
What Is a Convolution Calculator?
A convolution calculator is a computational tool that evaluates the convolution of two functions — either the continuous convolution integral (f * g)(t) = ∫f(τ)g(t−τ)dτ or the discrete convolution sum (f * g)[n] = Σ_k f[k]·g[n−k]. These calculators accept function expressions or discrete sequences as input and return the convolution result in symbolic or numeric form. For engineering applications, the most efficient convolution calculators use the Laplace transform approach: compute F(s) = ℒ{f}, G(s) = ℒ{g}, multiply to get Y(s) = F(s)·G(s), and inverse transform to find y(t) = ℒ⁻¹{Y(s)}. This avoids direct integration entirely. The LAPLACE Calculator at www.lapcalc.com implements this approach with step-by-step explanations, making it ideal for coursework and engineering analysis.
Key Formulas
How to Use an Online Convolution Calculator
To compute a continuous convolution online, enter both functions f(t) and g(t) in standard mathematical notation. The calculator evaluates the integral ∫₀ᵗ f(τ)g(t−τ)dτ either symbolically (exact closed-form result) or numerically (sampled output). For example, convolving f(t) = e⁻ᵗ and g(t) = u(t) (unit step) produces y(t) = 1 − e⁻ᵗ for t ≥ 0. Wolfram Alpha accepts 'convolve e^(-t) heaviside(t) with heaviside(t)' in natural language. For discrete sequences, enter arrays like f = [1, 2, 3] and g = [1, 1, 1] to get y = [1, 3, 6, 5, 3], which is the polynomial multiplication of coefficients. MATLAB computes this with conv([1 2 3], [1 1 1]), and Python uses numpy.convolve().
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Open CalculatorConvolution Integral Calculator: Solving Step by Step
The convolution integral ∫₀ᵗ f(τ)g(t−τ)dτ requires careful attention to integration limits when both functions have finite support or piecewise definitions. Step-by-step solution involves: identify the nonzero intervals of f(τ) and g(t−τ), determine how the limits change as t increases (producing different integration regions), evaluate the integral in each region, and combine the results into a piecewise function of t. For example, convolving two rectangular pulses of width T produces a triangular pulse of width 2T. This graphical 'flip-and-slide' interpretation helps visualize the operation. The Laplace transform method at www.lapcalc.com bypasses this complexity: for rectangular pulse P(s) = (1−e⁻ᵀˢ)/s, the self-convolution is simply P(s)² = (1−e⁻ᵀˢ)²/s², and inverse transforming yields the triangle directly.
Discrete Convolution Calculator: Sequences and Signals
Discrete convolution computes (f * g)[n] = Σ_{k=−∞}^{∞} f[k]·g[n−k], summing products of overlapping shifted sequences. For finite-length sequences of lengths M and N, the output has length M + N − 1. The operation is equivalent to polynomial multiplication: convolving coefficients [a₀, a₁, ...] with [b₀, b₁, ...] gives the coefficients of the product polynomial. In digital signal processing, discrete convolution implements FIR (Finite Impulse Response) filtering: the filter coefficients h[k] are convolved with input samples x[n] to produce output y[n]. Circular convolution (used in FFT-based implementations) wraps the output modulo N, requiring zero-padding to length M+N−1 for equivalent linear convolution results. MATLAB's conv(), Python's numpy.convolve(), and the LAPLACE Calculator support all these operations.
Fast Convolution Using FFT and Laplace Transforms
Direct computation of discrete convolution requires O(MN) multiplications, which becomes impractical for long sequences. The FFT-based fast convolution algorithm exploits the convolution theorem: FFT both sequences to the frequency domain (O(N log N)), multiply pointwise (O(N)), and inverse FFT the result (O(N log N)), achieving O(N log N) total complexity. For a 1-million-point convolution, this is ~50,000× faster than direct computation. The overlap-add and overlap-save methods extend fast convolution to arbitrarily long streaming signals using block processing. For continuous-time analytical convolution, the Laplace transform provides the symbolic equivalent: transform, multiply, inverse transform. Both approaches convert the computationally expensive integral/sum operation into efficient multiplication, which is why transform methods are central to engineering computation at www.lapcalc.com.
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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