Systems & the LTI idea

A system is anything that takes a signal in and puts a signal out: an amplifier, an echo chamber, a blur filter, a car's suspension. Two innocent-looking properties — linearity and time-invariance — turn the zoo of all systems into something fully understandable.

Build the intuition

Linearity: scaling and stacking survive

Linear means: double the input, double the output; feed in a sum, get the sum of the individual outputs. You've met this exact idea — it's what made matrices predictable in linear algebra. Linearity lets you decompose a complicated input into easy pieces, push each through separately, and add the results. Divide and conquer, certified.

T(a x_1 + b x_2) = a\,T(x_1) + b\,T(x_2)

Time-invariance: the system doesn't care what time it is

Play a chord now or play it tomorrow — a concert hall colors it identically. A system is time-invariant when delaying the input only delays the output, changing nothing else. Together with linearity, this means the system's behavior is one fixed personality, not a moving target.

Why LTI is the master key

Here's the payoff chain: any input can be written as a stack of shifted, scaled impulses. Linearity says the output is the same stack of shifted, scaled impulse-responses. So one measurement — kick the system once, record what comes out — determines everything. That stacking operation has a name: convolution, next lesson.

See it move

InteractiveA matrix is a verb

Rotate 45°: The whole plane turns together. Every game camera move is a cousin of this matrix. Matrix: [0.71, -0.71; 0.71, 0.71]

Linearity, met before: a matrix transforms every vector by one fixed rule — scaling and sums survive. LTI systems are this same discipline, applied to signals over time.

A worked example

Test a system for linearity

System A: output = 3 × input. Double the input → output doubles. Sum two inputs → outputs add. Linear ✓
System B: output = input². Feed x = 2: out 4. Feed x = 4: out 16 — doubling the input quadrupled the output. Not linear ✗
System C: output = input + 5. Feed 0: out 5. But linearity demands zero in → zero out. Not linear ✗ (close, though — engineers call it affine).
The scaling test catches most impostors in one line.

Out in the world

Concert halls are LTI systems

Clap once on an empty stage and record the reverberation — that's the hall's impulse response. Convolve any dry studio recording with it and you hear that recording as if performed in the hall. “Convolution reverb” plugins used on most film scores are literally this lesson, productized.

Common confusion, cleared

“Linear means the graph is a straight line through anywhere.”

It must pass through zero: zero input must give zero output. y = 2x qualifies; y = 2x + 1 doesn't. The scaling-and-stacking tests are the real definition.

“Real systems are too messy for LTI math to matter.”

Many are beautifully LTI in their operating range (circuits, acoustics, optics) — and engineers deliberately design systems to be LTI precisely so this math applies. The theory shapes the hardware, not just describes it.

Recap

A system maps input signals to output signals.
Linear: scaling and sums pass through. Time-invariant: behavior doesn't drift.
LTI ⇒ one impulse response determines the response to everything.

Progress saves in this browser.