Signals & Systems Lab · Continuous Time

Convolution, in continuous time

Same idea as before, but now the signals are smooth functions and the sum becomes an integral — an area building up as one signal slides across the other.

For continuous-time signals, convolution flips one signal in time, slides it across the other, multiplies the overlap, and instead of adding discrete samples, it integrates — accumulating the area under the product curve.

y(t) = ∫_−∞^∞ x(τ) · h(t − τ) dτ

Drag the slider to move t and watch h(t − τ) (cyan) slide across x(τ) (amber). The thin green bars are a Riemann approximation — the little rectangular slices this tool actually sums up to estimate the integral. As you advance t, each resulting area gets added to the output curve below, one point at a time.

Signals are windowed to τ = −6…6 so the integral stays finite and fast to compute in the browser.

Signal A — x(t)

Signal B — h(t)

shift t = -12.0

Overlap — x(τ) and h(t−τ) (flipped & shifted) τ axis

Output — y(t) = ∫ x(τ)h(t−τ) dτ t axis

y(−12.0) ≈ 0 (no overlap yet)

x(τ) — Signal A, fixed in place h(t−τ) — Signal B, flipped & slid to t Riemann slices — pieces integrated into y(t)