Convolution, Graphically
Slide one shape past another and watch the overlap area rise, peak and fall — that area is the output.
Skip to the animationConvolution flips one signal, slides it past the other, and integrates the overlap — and the value it produces at each shift is simply the area of that overlap, which is why the operation spreads signals in time and why cascaded systems combine into one.
The recipe, and the step it never justifies
- 1Flip
h(τ)to geth(−τ). - 2Shift it by
tto geth(t − τ). - 3Multiply by
x(τ)point by point. - 4Integrate. That is
y(t)— for one value of t.
Why the flip? An impulse arriving at τ has been running for t − τ seconds when you observe the output at t, so you need h evaluated at that age. And t − τ decreases as τ increases, which is exactly what the flip draws. The minus sign is bookkeeping about age, not a manipulation.
What the sliding shows
- Before the two functions touch, the product is zero everywhere and so is the output.
- As they begin to overlap, the integral grows with the overlap.
- At maximum overlap the output peaks.
- As they separate, it falls back to zero.
The output at time t is the area of the overlap at that shift. Two rectangular pulses therefore convolve to a triangle. And the output lasts as long as the two inputs combined, which is why a filter with a long impulse response smears sharp edges.
The properties, which are block-diagram algebra
| Property | Meaning for systems |
|---|---|
| Commutative: x ∗ h = h ∗ x | It does not matter which you slide |
| Associative: (x ∗ h₁) ∗ h₂ = x ∗ (h₁ ∗ h₂) | Cascaded systems combine into one, with h₁ ∗ h₂ |
| Distributive: x ∗ (h₁ + h₂) | Parallel systems add |
| Identity: x ∗ δ = x | Convolving with an impulse changes nothing |
The middle two rows are block diagram reduction, which is why a chain of filters can be collapsed into a single one before any signal is processed.
Why nobody computes it this way
Direct convolution is O(N²). Transforming both signals, multiplying pointwise and transforming back is O(N log N) — a factor of about fifty thousand at a million samples.
The convolution theorem is arguably the most commercially important result in the subject: every large audio and image convolution is done in the frequency domain because of it. The graphical method is for understanding; the FFT is for doing.
The numbers you will be asked for
- Convolution
y(t) = ∫ x(τ)·h(t − τ) dτ
One integral per output point.
- Convolution theorem
x(t) ∗ h(t) ⟷ X(f)·H(f)
Convolution in time is multiplication in frequency.
- Output duration
T_y = T_x + T_h
N + M − 1 samples in discrete time.
- Cascade
h_total = h₁ ∗ h₂
Two systems in series are one system.
Advantages and disadvantages
Advantages
- Gives the exact output of any LTI system from its impulse response.
- Its properties are block diagram reduction rules.
- The graphical view makes the result predictable before any integration.
- Becomes multiplication under a transform, which is what makes it fast.
Disadvantages
- Direct computation is O(N²).
- The flip is unintuitive until it is derived rather than stated.
- Only valid for LTI systems.
- It lengthens signals, so block processing needs overlap-add or overlap-save.
Watch it work
Check yourself
question 1 / 4
One question at a time. Pick an answer to see why it is right or wrong, then move on — there is no score to keep and nothing is saved.