SC · LTI and convolution

The Impulse Response

Know an LTI system's response to a single impulse and you know everything about it. See for yourself that this response h(t) is the system's fingerprint, and that when the impulse is scaled up or moved, the response follows exactly.

Lay the output onto the target response

The red bar is the impulse fed into the system: the bar's height is the impulse's size a, and the place where it stands is its position t₀. The output it produces is the blue curve, which starts swinging the instant the impulse arrives and then dies away. Move the size a and the position t₀ to lay the output a·h(t − t₀) exactly over the gold dashed target, and the output turns gold as well.

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 only two things need matching: the size a and the position t₀.

Far off

Strike the system with one impulse

Feed an impulse δ(t), with width 0 and area 1, into a system once, and the output that comes out is called the impulse response h(t). Just as striking a bell once draws out that bell's own ring, h(t) is a reaction unique to that system, which is why it is called the system's fingerprint. The h(t) on this screen is a signal that swings and dies away, 1.3e^(−1.1t) sin 2.6t, and it is 0 before the impulse arrives, so the system is causal.

Amplitude and position follow along

Because the system is linear, scaling the impulse by a scales the response by a, and because it is time-invariant, delaying the impulse by t₀ delays the response by t₀ with its shape unchanged. Put 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, chopped finely enough, can be seen as a line of impulses of different sizes, one at each instant, and the sifting property guarantees that this view is valid. If the system is LTI, each impulse leaves behind a copy of h in its own place and at its own size, and the total 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 needed to lay the output over the target response were two, the size a and the position t₀. The shape of the output could not be touched, because that shape is the system's fixed fingerprint h. Scaling the impulse enlarges the response and moving it moves the response, but the shape is always h. Know this one h and you know everything the LTI system will do; next comes convolution, which splits any input into impulses and adds up these responses.

The impulse response h(t) is the output an LTI system produces when an impulse δ(t) goes in. Thanks to linearity and time invariance, a·δ(t − t₀) → a·h(t − t₀), so the output is a scaled and shifted copy of h. That is why the single h determines the LTI system completely. The output for any input is the superposition of the copies of h left by the impulses that make up the input, and this leads to convolution.

On to the next unit

The impulse response is a single signal that holds everything about an LTI system. In the next unit, the intuition of convolution, any input is seen as a line of impulses, and the copies of h left by each impulse are overlapped and slid and added to build the output. The single a·h(t − t₀) copy you just saw is one piece of that sum. Convolution, in the end, carries out this unit's superposition continuously.