Discrete Convolution Visualization & Animation
Reverse/shift \(h[k-n]\) → Multiply \(x[n]h[k-n]\) → Sum \(y[k]\)
Slide \(k\) to see how the overlap between \(x[n]\) and \(h[k-n]\) produces the convolved output $$ y[k]=\sum_{n=-\infty}^{\infty}x[n]h[k-n] $$
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\(k=0,\;y[k]=0\)
Sum of \(x[n]h[k-n]\) equals \(y[k]\).
Convolution: definition & properties
Definition
\(y[k]=\sum_{n=-\infty}^{\infty} x[n]h[k-n]\)
Graphically: reverse/shift \(h\), multiply sample-by-sample with \(x\), then sum all products to get one output sample \(y[k]\).
Changing \(k\) moves the overlap window and traces the output sequence.
Key properties
- Commutative: \(x*h=h*x\) — swapping inputs gives the same sequence.
- Shift (delta): \(x[n]*\delta[n-k_0]=x[n-k_0]\) — a shifted impulse shifts the signal.
- Identity: \(x[n]*\delta[n]=x[n]\) — the impulse is the neutral element.
- Length hint: \(\mathrm{len}(y)=\mathrm{len}(x)+\mathrm{len}(h)-1\) for finite-length sequences.