What Is Convolution?
Convolution blends two functions into a third that measures how much they overlap as one slides across the other. For causal signals (zero before t = 0), the convolution integral is
(f ∗ g)(t) = ∫0t f(τ)·g(t − τ) dτ
The recipe is flip, shift, multiply, integrate: reverse g in time, slide it right by t, multiply it point by point against f, and add up the overlapping area. The result is the response of a linear time-invariant system with impulse response g to an input f, which is why convolution is the single most important operation in signal processing and the solution formula for driven differential equations. This calculator evaluates the integral on a fine grid, graphs the convolution against both inputs, and verifies the answer with the Laplace convolution theorem.
How to Use the Convolution Calculator
Enter two functions of t (treated as zero for t < 0) and a time horizon. The steps state the definition, the flip-and-slide interpretation, the computation method, a table of sample values, the location of the convolution's peak, and a numerical check against the convolution theorem L{f ∗ g} = L{f}·L{g}. The graph shows f and g dashed with f ∗ g solid, so the smoothing effect is visible: convolving with a decaying exponential or a box blurs sharp features, exactly what a moving average or a low-pass filter does. For the transform side of the story, see the Laplace transform calculator and the inverse Laplace transform calculator.
Worked Example
Convolve f(t) = e−t with the unit step g(t) = 1. The integral is
(f ∗ g)(t) = ∫0t e−τ·1 dτ = 1 − e−t
which is the classic charging curve of an RC circuit: feed a step voltage into a first-order system whose impulse response decays exponentially, and the output rises smoothly toward its steady state. The Laplace check confirms it instantly: L{e−t} = 1/(s + 1) and L{1} = 1/s, whose product 1/[s(s + 1)] is exactly the transform of 1 − e−t. The two-boxes chip shows the other iconic case, convolving a box with a box produces a triangle, the reason repeated averaging drives any signal toward a bell shape (a discrete central limit theorem in action).
Why Convolution Is Everywhere
Convolution is the answer to a single question asked across the sciences: what does a linear system do to an arbitrary input? Any linear time-invariant system is completely characterized by its response g to a unit impulse, and its response to any input f is f ∗ g, because f can be decomposed into a continuum of scaled, shifted impulses and the system responds to each independently (superposition), then sums. This is why the impulse response and the transfer function (its Laplace transform) tell you everything. The operation is commutative (f ∗ g = g ∗ f), associative, and distributes over addition, and its crown jewel is the convolution theorem: convolution in the time domain is plain multiplication in the frequency domain. That equivalence is what makes the Fast Fourier Transform revolutionary, it turns an expensive sliding-integral into a cheap pointwise product, accelerating everything from audio filters to polynomial multiplication to large-integer arithmetic.
Common Mistakes to Avoid
- Forgetting to flip g. The integrand is g(t − τ), the time-reversed and shifted copy, not g(τ). Skipping the flip computes a correlation, a related but different operation used for pattern matching.
- Getting the limits wrong for causal signals. When both are zero before 0, the overlap runs only from τ = 0 to τ = t; extending the limits to ±∞ drags in nonexistent signal.
- Expecting (f ∗ g)(t) = f(t)·g(t). Convolution is an integral of a product over a sliding window, not a pointwise product; the two agree only in the frequency domain, never in time.
- Ignoring the widening support. Convolving a signal of length A with one of length B yields length A + B; the result spreads out, which is precisely the blurring that filters exploit and edge-detection must undo.
- Mislabeling correlation as convolution. Correlation uses g(τ − t) (no flip) and measures similarity; convolution uses the flip and models system response. Neural-network "convolution" layers are technically correlations, a persistent naming confusion.
Real-World Applications
Convolution runs the modern world. Every digital audio effect, reverb, echo, equalization, is a convolution of the sound with a filter kernel or a recorded room impulse response; convolution reverb literally convolves your track with the acoustics of a cathedral. Image processing blurs, sharpens, and detects edges by convolving pixels with small kernels, and convolutional neural networks stack thousands of such operations to recognize objects. In optics, a telescope or microscope image is the true scene convolved with the instrument's point-spread function, and deconvolution sharpens it back. Probability theory: the density of a sum of independent random variables is the convolution of their densities, which is how the normal distribution emerges as an attractor. Control engineering computes system output as input convolved with impulse response; seismology and medical imaging (CT, MRI) reconstruct signals by deconvolution. Even multiplying two polynomials or two large numbers is a convolution of their coefficient sequences, the operation FFT-based algorithms accelerate. Pair this with the second-order ODE solver, whose driven solutions are convolutions of the forcing with the impulse response.
Frequently Asked Questions
What does convolution actually compute?
The sliding overlap of two functions: at each shift t, it multiplies one signal against a flipped, shifted copy of the other and integrates. Physically it is the output of a linear time-invariant system whose impulse response is one function, driven by the other as input.
Why do you flip one of the functions?
Because a system's output at time t sums contributions from all past inputs, weighted by how long ago they arrived, and "how long ago" is t − τ. The g(t − τ) term encodes that fading memory, and the reversal is what distinguishes convolution from correlation.
What is the convolution theorem?
Convolution in time equals multiplication in the transform domain: L{f ∗ g} = L{f}·L{g} for Laplace, and the analogous product for Fourier. It converts a costly integral into a cheap product, the mathematical engine behind fast filtering and the FFT.
How is convolution different from correlation?
Correlation slides one signal against the other without flipping, measuring similarity at each lag; convolution flips first, modeling system response. They coincide when the kernel is symmetric. Deep-learning "convolution" layers actually perform correlation, a widespread but harmless misnomer.
Why does convolving two boxes give a triangle?
As one box slides across the other, the overlap area grows linearly, plateaus, then shrinks linearly, tracing a triangle. Convolving again rounds the corners, and repeated self-convolution converges to a Gaussian, a hands-on demonstration of the central limit theorem.
What are the algebraic properties of convolution?
It is commutative, associative, and distributes over addition, so it behaves like a multiplication, indeed it is multiplication of Laplace or Fourier transforms in disguise. The unit impulse (Dirac delta) is its identity element: f ∗ δ = f, leaving any signal unchanged.
How does convolution solve differential equations?
The solution of a driven linear ODE is the forcing function convolved with the system's impulse response (Green's function). Instead of solving anew for each input, you find the impulse response once and convolve, which is exactly what the Laplace-transform method automates.
Why does convolution blur signals?
Because each output value is a weighted average over a window of inputs, smoothing out rapid variation. Convolving with a wide kernel is low-pass filtering; the wider the kernel, the more high-frequency detail it removes, which the convolution theorem states precisely as multiplying by a narrow frequency response.
What is deconvolution?
The inverse operation: recovering an original signal from its convolved (blurred) version, given the kernel. It is done by dividing in the frequency domain, but noise makes it delicate, which is why deblurring photos and reconstructing CT scans require careful regularized deconvolution rather than naive division.
How does this connect to multiplying polynomials?
The coefficients of a product polynomial are the convolution of the two factor coefficient sequences. This turns fast multiplication of polynomials and large integers into an FFT-based convolution, the algorithmic reason schoolbook multiplication can be beaten for very large numbers.