Laplace Convolution Calculator

Calculator

Select models for f(t) and g(t), set parameters, and choose numerical settings. For best Laplace agreement, use a larger t_max and enough steps.

f(t) model *

Pick the function family for f(t).

f amplitude A

Used by all models.

f exponent B

Used by exponential/damped sine.

f power N

Used by polynomial.

f frequency W

Used by sine/cosine/damped sine.

f shift T0

Used by shifted step.

g(t) model *

Pick the function family for g(t).

g amplitude A

Used by all models.

g exponent B

Used by exponential/damped sine.

g power N

Used by polynomial.

g frequency W

Used by sine/cosine/damped sine.

g shift T0

Used by shifted step.

t_max

Upper time bound for tables and Laplace truncation.

Grid points

More points give smoother series.

t_eval

Single time point highlighted in results.

Convolution algorithm

Choose accuracy vs speed for h(t).

Integration method

Used for integral mode and Laplace estimates.

int_steps

Substeps for each convolution integral (integral mode).

s value

Laplace evaluation point s ≥ 0.

lap_steps

Substeps for Laplace integrals on [0, t_max].

Result appears above after submission.

Example data table

Case	f(t)	g(t)	Suggested t_max	Suggested s	Why it’s useful
A	e^−t	sin(3t)	8	1.2	Stable decay helps truncation at t_max.
B	t²	e^−2t	10	1.0	Polynomial growth needs larger bounds for checks.
C	H(t−2)	cos(2t)	12	1.5	Shifted step demonstrates causal offsets clearly.

Try matching these with the model dropdowns and parameters (A=1, B and W as shown, N for powers, T0 for shifts).

Formula used

The time-domain convolution of two causal functions is:

(f * g)(t) = ∫₀^t f(τ) · g(t − τ) dτ

The Laplace transform of a function x(t) is:

X(s) = ∫₀^∞ x(t) e^{−s t} dt

The convolution theorem states: L{f*g} = F(s)·G(s). This calculator estimates all integrals numerically on [0, t_max], so agreement depends on truncation and step settings.

How to use this calculator

Choose models for f(t) and g(t).
Enter parameters (A, B, N, W, T0). Only relevant fields affect each model.
Set t_max and grid points to control the time series resolution.
Pick an algorithm: integral mode for accuracy, discrete mode for speed.
Pick the integration method and steps. Higher steps improve precision.
Click Calculate. Download CSV/PDF for reporting.

Accuracy tip

If you see large Laplace errors, increase t_max and lap_steps. For oscillatory signals, increase points and steps as well.

Convolution as a system response tool

Convolution links two causal signals by accumulating overlap over time. The output h(t) equals the integral of f(τ) times g(t−τ). If f is an input and g an impulse response, the curve shows how delayed contributions build. Use t_eval to highlight a time and read h(t) directly. Download CSV to audit values, and PDF to record settings for reports and coursework without retyping parameters later.

Numerical grid and stability choices

A practical computation needs a finite grid. This tool samples 0 to t_max with dt = t_max/(points−1). Increasing points improves smoothness, while larger t_max captures more tail energy. For decaying exponentials, t_max around 6–12 often balances speed and accuracy. For growing polynomials, choose a larger bound and more steps.

Integral mode versus discrete mode

Integral mode runs a quadrature for every time sample. It is slower, but it handles sharp features and shifted steps reliably when int_steps is high. Discrete mode uses a dt-weighted sum, which is fast for exploratory runs. With oscillations, prefer more points to reduce phase and amplitude drift.

Laplace verification and truncation effects

The calculator estimates F(s), G(s), and L{h}(s) on a finite interval. The theorem predicts L{f*g} = F(s)G(s), but truncation replaces infinity with t_max. Improve agreement by increasing lap_steps and t_max, especially when s is small. The e^{−s t} column helps you see the weighting applied.

Interpreting error metrics in practice

Absolute error reports |L{h}(s)−F(s)G(s)|; relative error scales by |F(s)G(s)|. When the product is near zero, relative error can jump, so compare both. If errors stay large, raise lap_steps first, then int_steps in integral mode. Expand t_max to reduce tail truncation for lightly damped signals.

Recommended settings for common signals

For negative B exponentials, Simpson’s rule with 400–1200 lap_steps is usually steady. For sines and cosines, raise points to 151–201 and set t_max to several periods, such as 8–16 when W is 2–4. For delayed steps, set T0 and increase int_steps near the jump.

FAQs

Q1. Why does the convolution start at zero time?

Convolution here assumes causal signals, so integration runs from 0 to t. At t=0 the interval length is zero, giving h(0)=0. For shifted steps, h(t) begins changing after the delay.

Q2. When should I use discrete mode?

Use discrete mode for fast exploration, large point counts, or when you mainly need a smooth curve. For sharp discontinuities or high accuracy at specific t values, integral mode with higher int_steps is better.

Q3. Why is L{h}(s) not equal to F(s)G(s)?

The Laplace integrals are truncated at t_max, so long tails are omitted. Small s values decay slowly and amplify truncation. Increase t_max and lap_steps, and consider more points for oscillatory functions.

Q4. What does t_max control?

t_max sets the plotted time range and the upper limit for all numerical Laplace estimates. A larger t_max captures more energy from slowly decaying signals, but increases runtime because more grid and integration work is required.

Q5. How do I represent a delayed step?

Choose the shifted step model and set T0 to the delay. The function becomes A·H(t−T0). For better resolution around the jump, increase points and int_steps, then verify the curve near t=T0.

Q6. Which settings help oscillatory signals?

Increase points to reduce aliasing and choose t_max to include several periods. Raise lap_steps for Laplace checks and int_steps for integral mode. Simpson’s rule often performs well when the waveform is smooth.