Signals, Systems, and Why LTI Matters

A signal is a function of one or more independent variables that carries information about a phenomenon. Speech is pressure against time; an ECG is voltage against time; a photograph is brightness against two spatial coordinates; a month of rainfall figures is millimetres against day number.

Notice what is *not* in the definition: nothing about electricity, and nothing about time specifically. The mathematics in this subject applies to any of the four examples above, which is why the same transforms turn up in image processing, geophysics and finance.

Continuous vs discrete is about the *variable*; analogue vs digital is about the *value*. A signal can be discrete-time and analogue — that is exactly what a sample is before it reaches the ADC.

A system is a rule assigning an output signal to every input signal. An amplifier, a room's acoustics, a car's suspension, a bank account, a thermostat — all systems under this definition.

Stated that broadly, almost nothing can be said about one. You cannot predict the output for a new input without applying it and seeing. The whole of the rest of the subject is a set of restrictions that buy back predictive power, and there are exactly two of them that matter.

Linearity is two conditions that are almost always used together:

1. 1Homogeneity — scaling the input scales the output by the same factor. a·x(t) → a·y(t).
2. 2Additivity — the response to a sum is the sum of the responses. x₁ + x₂ → y₁ + y₂.
3. 3Together they are superposition: a·x₁ + b·x₂ → a·y₁ + b·y₂, which is the form worth memorising because it tests both at once.

A squarer, y = x², fails both. Doubling the input quadruples the output, and (x₁+x₂)² = x₁² + 2x₁x₂ + x₂² carries a cross term. That cross term is the point: in a linear system the parts of an input never interact, and every non-linearity is some version of them interacting.

y = 2x + 3 is not linear, despite being a straight line. Zero in must give zero out. Systems of this shape are called *incrementally linear*, and are handled by analysing the deviation from the constant.

A system is time-invariant if delaying the input only delays the output, with no other change: x(t − t₀) → y(t − t₀). Informally, the box behaves the same today as tomorrow — its rule contains no explicit reference to the clock.

y(t) = t·x(t) fails, because the same input applied at t = 5 is scaled by 5 and at t = 1 by 1. So does any system whose parameters are being changed while it runs. Real components drift with temperature and age; over the seconds a measurement takes, we treat them as time-invariant anyway, and that approximation is doing real work.

Take an LTI system and hit it once with an impulse. Record the output, h(t) — the impulse response. That single recording now predicts the response to every input there will ever be.

1. 1Any input can be written as a sum of scaled, shifted impulses.
2. 2Time invariance says the response to an impulse shifted by τ is h(t − τ).
3. 3Linearity says the response to the sum is the sum of the responses.
4. 4So y(t) = ∫ x(τ)·h(t − τ) dτ — the convolution, written x(t) ∗ h(t).

There is a second consequence, and it is the one that makes the Fourier chapters possible. Feed an LTI system a sinusoid and what comes out is a sinusoid of exactly the same frequency — only the amplitude and phase differ. A complex exponential is an eigenfunction of every LTI system, and no other class of system has an input shape it cannot distort.

This is why a non-linear or time-varying system is a genuinely hard problem rather than a harder version of an easy one. None of convolution, Fourier, Laplace, transfer functions or pole-zero analysis applies to it.

Causal

The output at time t depends only on inputs at or before t. Any system operating in real time must be causal; an offline one processing a recorded file need not be.

Memoryless

The output at t depends only on the input at t. A resistor is memoryless; a capacitor is not, because its voltage records what the current has been doing.

Stable (BIBO)

Every bounded input produces a bounded output. For an LTI system this is exactly the condition that h(t) is absolutely integrable.

Invertible

Distinct inputs produce distinct outputs, so the input can be recovered. The basis of equalisation, and of undoing a channel's distortion.