SC · LTI and convolution

The Impulse Response

An LTI system is fully known once you know its response to a single impulse. See firsthand that this response h(t) is the system’s fingerprint, and that scaling and moving the impulse scales and moves the response in step.

Lay the output onto the target response

Feed a single red impulse into the system and the blue response comes out. The dashed curve is the target response to match. Move the impulse’s amplitude a and position t₀ so the output a·h(t−t₀) lands exactly on the target.

input a·δ(t−t₀)output a·h(t−t₀)target

amplitude a0.50

position t₀-0.60

Current output

y(t) = 0.50 h(t − -0.60)

The output’s size or position differs from the target. The output is always a copy of the same shape h, so you only need to set the amplitude a and the position t₀.

Far off

Strike the system with one impulse

Feed the system one impulse δ(t) — zero width, area 1 — and the output that comes out is called the impulse response h(t). Like striking a bell once and hearing that bell’s own ring, h(t) is the reaction unique to that system. This is the system’s fingerprint.

Amplitude and position follow along

Because the system is linear, scaling the impulse by a scales the response by a; because it is time-invariant, delaying the impulse by t₀ delays the response by the same t₀ with its shape intact. Together, a·δ(t−t₀) → a·h(t−t₀). So the output is always a copy of the same h, with only its size and position changed.

Observeδ(t) → h(t)

An impulse in gives the impulse response out.

Choosea δ(t) → ?

Amplitude multiplies the response directly.

Fill inδ(t − t₀) → ?

A delayed impulse, a delayed response.

On your owna δ(t − t₀) → ?

Apply amplitude and position together.

So any input can be predicted

Any input, sliced finely, can be seen as a row of impulses of different sizes, one at each instant — the sifting property guarantees it. If the system is LTI, each impulse leaves a copy of h at its own place scaled by its size, and the whole output is the superposition of all these copies. This adding-up is exactly the convolution of the next unit.

Back to the first screen

The only handles you needed to match the target were two — amplitude a and position t₀. You could not touch the output’s shape, because that shape is the system’s fixed fingerprint h. Scaling the impulse scales the response, moving the impulse moves it, but the shape is always h. Know this one h and you know everything an LTI system will do. Next comes convolution, which slices any input into impulses and adds up these responses.

The impulse response h(t) is the output an LTI system produces when fed the impulse δ(t). By linearity and time-invariance, a·δ(t − t₀) → a·h(t − t₀): the output is a scaled, shifted copy of h. So a single h completely determines the LTI system. The output for any input is the superposition of copies of h for the impulses that make up the input — which leads to convolution.

On to the next unit

The impulse response is a single signal that holds everything about an LTI system. The next unit, the intuition of convolution, views any input as a row of impulses and builds the output by overlapping and sliding the copies of h each impulse leaves behind, adding them up. The one copy a·h(t−t₀) you just saw is a single piece of that sum. Convolution is, in the end, this unit’s superposition carried out continuously.