The Impulse Response
Why one measurement describes a system completely — built up from impulses rather than asserted.
Skip to the animationHit an LTI system once with an impulse and record the output, and that single recording predicts the response to every input there will ever be — because any input is a sum of scaled, shifted impulses, and linearity and time invariance handle the scaling and the shifting.
The claim, and why it is not an overstatement
- 1Any input can be written as a continuum of scaled, shifted impulses — that is the sifting property read as a decomposition.
- 2Time invariance says the response to an impulse delayed by τ is
h(t − τ)— you already know it, for every τ, from one measurement. - 3Linearity says each response may be scaled and all of them added.
- 4Therefore
y(t) = ∫ x(τ)·h(t − τ) dτ— the convolution integral.
Both properties are needed. Time invariance shifts the responses; linearity combines them. Remove either and the claim is simply false.
What h(t) tells you at a glance
| Property | Condition on h(t) |
|---|---|
| Causal | h(t) = 0 for t < 0 |
| Stable (BIBO) | ∫|h(t)| dt < ∞ |
| Memoryless | h(t) = k·δ(t) |
| Step response | The integral of h(t) |
The step response is the integral because an LTI system commutes with integration — so measuring one gives the other for free.
It is a real measurement
Clap once in a room and record the echo: you have measured the room's impulse response. Convolution reverb convolves dry audio with a recorded space's h(t) so the audio inherits that space; room correction inverts the same measurement to cancel a room's colouration.
Cathedral impulse responses are sold as audio products. That is this topic taken literally and commercially.
The discrete case
The integral becomes y[n] = Σ x[k]·h[n − k]. An FIR filter's coefficients are literally `h[n]`, one per tap — so designing the filter means choosing an impulse response directly. An IIR filter has feedback, so its impulse response never terminates, which is what its name records.
The numbers you will be asked for
- Convolution integral
y(t) = ∫ x(τ)·h(t − τ) dτ
Valid only for LTI systems.
- Convolution sum
y[n] = Σ x[k]·h[n − k]
The discrete counterpart.
- BIBO stability
∫ |h(t)| dt < ∞
Absolutely integrable.
- Step response
s(t) = ∫ h(τ) dτ
Because an LTI system commutes with integration.
Advantages and disadvantages
Advantages
- One measurement completely characterises the system.
- Causality, stability and memory are all readable from it.
- It is directly measurable — a clap and a microphone will do.
- For an FIR filter it *is* the design.
Disadvantages
- An ideal impulse cannot be generated, so the measurement is always approximate.
- A short impulse carries little energy, so the measurement is noisy — hence sine-sweep methods.
- Only meaningful for LTI systems.
- Direct convolution is O(N²), so it is rarely computed this way.
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.