SC · Signal basics

Properties of a System

A system takes in a signal and turns it into another signal. Sort that transformation by three independent properties, linear, time-invariant and causal, and see for yourself that the most important of them, linearity, is the same thing as superposition.

When do the two paths agree

The two faint humps are the test inputs x₁ and x₂, and the system is S{u} = u + εu². Compare the result of adding the two inputs first and passing them through once (gold) with the result of passing them through separately and adding afterwards (blue dashed): when ε is large, the two pull apart in the middle, where the inputs overlap. Turn the nonlinearity ε down and find when the two paths lie exactly on top of each other.

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

nonlinearity ε0.45

System and superposition status

S{x} = x + 0.45 x²

In the middle, where the two inputs overlap, the two paths pull far apart. When the adding happens changes the result, so superposition is broken and this system is nonlinear.

Nonlinear

Linearity is exactly superposition

A linear system keeps two rules. Scale the input by a and the output scales by a (homogeneity); feed in the sum of two inputs and the output is the sum of the two outputs (additivity). Tied into one line, these read S{a·x₁ + b·x₂} = a·S{x₁} + b·S{x₂}, so if any input is written as a sum of pieces, the output is the same combination of their outputs. When a u² term creeps in, as in the figure's system, a cross term appears, and the gap between the two paths is exactly S{x₁ + x₂} − S{x₁} − S{x₂} = 2εx₁x₂. That is why the paths separate only where both inputs are large together.

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 ?

The output of an LTI system is a convolution.

Time invariance: delay in, delay out

A time-invariant system behaves the same at every time. So delaying the input by t₀ simply delays the output by t₀ with its shape unchanged; in symbols, if x(t) → y(t), then x(t − t₀) → y(t − t₀). It gives the same response whether the input arrives today or tomorrow. The figure's S{u} = u + εu² has coefficients that do not change with time, so it is time-invariant whatever ε is.

Causality and LTI

In a causal system the output depends only on present and past inputs and cannot see future ones; that is, y(t) is set by x(τ) with τ ≤ t alone, and every system that works in real time is causal. The figure's system is memoryless, looking only at the current input, so it is causal as well, which shows once more that the three properties are independent. A system that is both linear and time-invariant is called LTI; an LTI system is completely determined by a single impulse response, and its output comes out as the convolution y = x ∗ h. The whole next chapter stands on this LTI foundation.

Back to the first screen

Only when the nonlinearity ε was turned down to 0 did the two paths lie exactly on top of each other. That agreement, the same result whether you add first or later, is superposition, which is linearity. Linear, time-invariant and causal are three independent questions to put to a system: is it preserved when inputs are added and scaled, is a delayed input delayed in exactly the same way, and does it avoid looking at the future? A system that is both linear and time-invariant becomes an LTI system, fully described by a single impulse response.

Systems are sorted by three independent properties. Linear means superposition holds, that is, S{a·x₁ + b·x₂} = a·S{x₁} + b·S{x₂} (homogeneity + additivity). Time-invariant means that if x(t) → y(t), then x(t − t₀) → y(t − t₀): the behavior does not change with time. Causal means y(t) depends only on x(τ) with τ ≤ t and never looks at the future. A system that is linear and time-invariant is LTI, and its output is set by a single impulse response h as y = x ∗ h.

On to the next chapter

When one system has both linearity and time invariance it becomes LTI, and from here comes the real power of signals and systems. In the next chapter, LTI systems and convolution, an LTI system is captured completely by a single signal, the impulse response, and the output for any input is computed by convolution. The superposition you just saw is the key to that calculation, because convolution means splitting any input into a sum of impulses and adding the impulse responses back together by superposition.