Convolution Visualization & Animation

Flip/shift \(h(t-\tau)\) → Multiply \(x(\tau)h(t-\tau)\) → Integrate \(y(t)\)

Slide \(t\) to see how the overlap between \(x(\tau)\) and \(h(t-\tau)\) produces the convolved output $$ y(t)=\int_{-\infty}^{\infty}x(\tau)h(t-\tau)\,\mathrm d\tau $$




0.0
\(t=0.0,\;y(t)=0\)

Area under \(x(\tau)h(t-\tau)\) equals \(y(t)\).

Convolution: definition & properties

Definition of Convolutiuon

\(y(t)=\int_{-\infty}^{\infty} x(\tau)h(t-\tau)\,d\tau\)

Graphically: reverse/shift \(h\), multiply with \(x\), then integrate the overlap to get one output sample \(y(t)\).

Moving the slider changes overlap, so the output curve is traced point by point.

Key properties

  • Commutative: \(x*h=h*x\) — swapping inputs gives the same output.
  • Shift (delta): \(x*\delta(t-T)=x(t-T)\) — a shifted impulse shifts the signal.
  • Identity: \(x*\delta=x\) — the impulse acts like a neutral element.

Try it

Mini FAQ

Why do we flip/reverse h?

In convolution we evaluate \(h(t-\tau)\), so \(h\) is first time-reversed to \(h(-\tau)\) and then shifted by \(t\). This is why the overlap with \(x(\tau)\) changes as you move the slider.

What does the slider t/k represent?

The slider controls \(t\), the output time where we compute one sample \(y(t)=\int x(\tau)h(t-\tau)\,d\tau\). Moving it slides the overlap window and traces \(y(t)\) point-by-point.

Linear vs circular convolution (discrete)

Linear convolution is the full non-wrapping overlap. Circular convolution wraps indices modulo \(N\); FFT-based multiplication gives circular convolution unless you zero-pad enough to recover the linear result.

Next steps

This demo visualizes graphical convolution. Use the Calculator for arbitrary signals or Training for guided exercises.

Compute full convolution (Calculator) Need practice & examples? See Convolution Training