SC · Signal basics

Properties of a System

A system takes a signal and turns it into another. We classify that transformation by independent properties: linear, time-invariant, causal. You will see firsthand that the most important one, linearity, is exactly superposition.

When do the two paths agree

Compare passing the sum of two inputs through the system once (gold) against passing each separately and adding afterward (blue dashed). Lower the nonlinearity ε and find when the two paths overlap exactly.

combine first: S{x₁+x₂}pass separately: S{x₁}+S{x₂}

nonlinearity ε0.45

System and superposition status

S{x} = x + 0.45 x²

Where the two inputs overlap, the two paths diverge sharply. The order of combining changes the result, so superposition is broken. This system is nonlinear.

Nonlinear

Linearity is exactly superposition

A linear system keeps two promises. Scaling the input by a scales the output by a (homogeneity), and adding two inputs adds their two outputs (additivity). Bundled into one line: S{a·x₁ + b·x₂} = a·S{x₁} + b·S{x₂}. So however you write an input as a sum of pieces, the output is the same combination of their outputs. A nonlinear term (say x²) creates a cross term that breaks this equality.

ObserveS{a x₁ + b x₂} = a S{x₁} + b S{x₂}

Linear = scaling and sums pass through intact.

Choosex(t − t₀) → ?

Time-invariant = delayed input, delayed output.

Fill iny(t): x(τ), τ ?

Causal = looks only at past and present.

On your owny = x ?

An LTI output is a convolution.

Time invariance: delay in, delay out

A time-invariant system does not change its behavior over time. So delaying the input by t₀ merely delays the output by the same t₀, shape unchanged. In symbols, if x(t) → y(t), then x(t − t₀) → y(t − t₀). It gives the same response whether you feed it today or tomorrow.

Causality and LTI

A causal system has an output that depends only on present and past inputs, never the future. That is, y(t) is fixed by x(τ) for τ ≤ t. Every system that runs in real time is causal. A system that is both linear and time-invariant is called LTI, and an LTI system is fixed completely by a single impulse response, with its output given by convolution y = x ∗ h. The entire next chapter stands on this LTI.

Back to the first screen

Only when the nonlinearity dropped to 0 did the two paths coincide exactly. That agreement — combine first or combine last, same result — is superposition, and that is linearity. Linear, time-invariant, causal are three independent questions you put to a system: are sums and scalings preserved, does a delay just delay the output, does it avoid looking at the future? Linear and time-invariant together make the system an LTI, fully told by a single impulse response.

A system is classified by three independent properties. Linear means superposition holds: S{a·x₁ + b·x₂} = a·S{x₁} + b·S{x₂} (homogeneity + additivity). Time-invariant means if x(t) → y(t) then x(t − t₀) → y(t − t₀); behavior does not change over time. Causal means y(t) depends only on x(τ) for τ ≤ t, never the future. Linear and time-invariant together is LTI, where a single impulse response h fixes the output as y = x ∗ h.

On to the next chapter

When one system has both linearity and time-invariance it becomes LTI, and from here the real power of signals and systems emerges. The next chapter, LTI and convolution, captures an LTI completely by a single signal — its impulse response — and computes the output for any input by convolution. The superposition you just saw is the very key that makes this work: convolution breaks any input into a sum of impulses and adds their impulse responses back by superposition.