System Properties: Linearity, Time-Invariance, Causality, and Stability

The four defining properties that classify systems for analysis

Signals & SystemsElectrical Engineering Year 2Free preview

⏱️ About 16 min

Why can we predict the output of an audio equalizer to any imaginable input — without ever testing every possible signal? The answer lies in four properties that make systems mathematically tractable.

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The big idea: When a system is both linear and time-invariant, its entire behavior is captured by a single function: the impulse response. Everything else — convolution, frequency response, filtering — follows from this.

🎯 By the end, you'll be able to

Define linearity in terms of additivity and homogeneity, and test whether a given system satisfies superposition
Explain time-invariance and determine whether a system's output shifts faithfully with a delayed input
Identify causal versus non-causal systems by checking whether the output depends on future input values
Apply the absolute-summability criterion to determine BIBO stability of a discrete-time LTI system

What Is a System?

A system, in the signals-and-systems sense, is any process that transforms an input signal into an output signal. Mathematically, we write y(t) = T{x(t)} for a continuous-time system, or y[n] = T{x[n]} for a discrete-time system, where T{·} denotes the transformation operator. A circuit that filters noise, a microphone that converts sound pressure to voltage, and a digital equalizer are all systems.

Before we can analyze a system — predict its output, understand its behavior, or design it to meet specifications — we need to classify it. Four properties do most of the heavy lifting in that classification: linearity, time-invariance, causality, and stability. When a system is both linear and time-invariant (LTI), an extraordinary set of analytical tools becomes available, including convolution, frequency-domain analysis, and transform methods. This lesson defines each property precisely, with worked tests and counter-examples.

Linearity: Additivity and Homogeneity

Linearity is really two properties bundled together. Additivity says that if input x₁ produces output y₁ and input x₂ produces output y₂, then the combined input x₁ + x₂ must produce y₁ + y₂. Homogeneity (also called scaling) says that if x produces y, then a·x must produce a·y for any scalar a. Together, these give us the superposition principle: T{a·x₁ + b·x₂} = a·y₁ + b·y₂ for any inputs x₁, x₂ and any scalars a, b.

Superposition is the single most powerful property in linear systems theory. It means we can decompose a complicated input into simpler pieces, find the response to each piece individually, and sum the results. Without linearity, the response to a sum is not the sum of the responses, and this decomposition strategy collapses. Even a system that looks almost linear — such as y(t) = 2x(t) + 3, which is just a scaled copy of the input plus a constant — fails superposition because of that additive offset. The constant breaks both additivity and homogeneity: doubling the input does not double the output, and the response to a sum is not the sum of the individual responses.

\[ T\{a\,x_1(t) + b\,x_2(t)\} = a\,T\{x_1(t)\} + b\,T\{x_2(t)\} = a\,y_1(t) + b\,y_2(t) \]

The superposition principle: linearity requires additivity and homogeneity simultaneously.

Time-Invariance: Shifts Propagate Faithfully

Time-invariance means the system's behavior does not change over time. Formally, if x(t) produces y(t), then a delayed version x(t − t₀) must produce y(t − t₀) — the same output, just shifted by the same amount. The system's internal parameters are fixed; it does not learn or age or change its response depending on when you apply the input.

A resistor with constant resistance is time-invariant; a resistor whose resistance drifts with temperature over the course of an experiment is not. Time-invariance, combined with linearity, is what makes the impulse response a complete characterization of the system. If the system were time-varying, the response to an impulse at t = 0 could differ from the response to an impulse at t = 5, and a single h[n] would not suffice to predict every output.

\[ x(t) \to y(t) \;\Longrightarrow\; x(t - t_0) \to y(t - t_0) \]

Time-invariance: a shifted input produces an identically shifted output.

Causality: No Output Before Input

A causal system is one where the output at any time t depends only on the input at times up to and including t — never on future input values. This is a physical necessity for real-time systems: you cannot respond to a signal you have not yet received. A system defined by y[n] = x[n] − x[n−1] is causal; a system defined by y[n] = x[n+1] − x[n] is not, because computing y[n] requires knowing x[n+1], which has not arrived yet.

Causality matters less for offline processing, where the entire signal is available in memory, but it is essential for understanding real-time filters, feedback control, and any system that must operate sample-by-sample as data streams in.

🔑 Why LTI Is Special

When a system is both linear and time-invariant, its entire behavior is captured by a single function: the impulse response h[n]. This means one experiment — feeding in a unit impulse — reveals everything needed to predict the output for any input. The next lesson shows exactly how.

BIBO Stability: Bounded In, Bounded Out

A system is BIBO (Bounded-Input Bounded-Output) stable if every bounded input produces a bounded output. If there exists some bounded input that drives the output to infinity, the system is unstable. Stability is a practical concern: an unstable filter can saturate, overflow, or produce meaningless results.

For a discrete-time LTI system, there is an elegant necessary and sufficient condition: the system is BIBO stable if and only if its impulse response is absolutely summable, that is, the sum of |h[n]| over all n converges to a finite value. This connects stability directly to the impulse response — a preview of just how central h[n] will be in the next lesson.

\[ \sum_{n=-\infty}^{\infty} |h[n]| < \infty \quad \Longleftrightarrow \quad \text{BIBO stable} \]

A discrete-time LTI system is BIBO stable if and only if its impulse response is absolutely summable.

📝 Worked example: Test whether the system y(t) = 2x(t) + 3 is linear by checking superposition with x₁ = 1 and x₂ = 2.

Apply x₁ = 1: the output is y₁ = 2(1) + 3 = 5.
Apply x₂ = 2: the output is y₂ = 2(2) + 3 = 7.
If the system were linear, the combined input x₁ + x₂ = 3 should produce y₁ + y₂ = 5 + 7 = 12.
But the actual output for input 3 is y(3) = 2(3) + 3 = 9.
Since 9 ≠ 12, superposition fails — the additive constant breaks linearity.

✓ NOT linear — 9 ≠ 12, superposition fails because of the additive constant.

✏️ Practice: A discrete-time LTI system has impulse response h[n] = (0.5)^n for n ≥ 0 (zero otherwise). Compute S = Σ|h[n]| for n = 0 to ∞. Report the value of S (a finite S means the system is BIBO stable).

Solution

The impulse response is h[n] = (0.5)^n for n ≥ 0, so |h[n]| = (0.5)^n since all terms are non-negative.
Compute S = Σ_{n=0}^{∞} (0.5)^n. This is a geometric series with first term a = 1 and common ratio r = 0.5.
Using the geometric series formula S = 1/(1 − r) = 1/(1 − 0.5) = 1/0.5 = 2.
Since S = 2 is finite, the impulse response is absolutely summable (Σ|h[n]| < ∞) and the system is BIBO stable.

Check your understanding

1. Which property requires that T{a·x₁ + b·x₂} = a·y₁ + b·y₂ for any scalars a, b?

This is the superposition principle, which combines additivity and homogeneity — the defining test for linearity.

2. A discrete-time LTI system has h[n] = (0.8)^n for n ≥ 0. The sum Σ|h[n]| = 1/(1−0.8) = 5. Is the system BIBO stable?

BIBO stability for a discrete-time LTI system requires Σ|h[n]| < ∞. The sum equals 5, which is finite, so the system is stable despite having infinitely many nonzero terms.

✅ Key takeaways

A system transforms an input signal into an output signal; classifying systems by four properties — linearity, time-invariance, causality, and stability — determines which analytical tools apply.
Linearity requires both additivity and homogeneity (superposition). A system like y(t) = 2x(t) + 3 fails because the additive constant breaks superposition.
Time-invariance means a shifted input produces an identically shifted output; the system's behavior does not change over time.
A discrete-time LTI system is BIBO stable if and only if its impulse response is absolutely summable: Σ|h[n]| < ∞.

➡️ We now have the vocabulary to classify systems. But how do we actually compute the output of an LTI system for an arbitrary input? The answer — convolution — is the subject of the next lesson, and it rests entirely on the impulse response h[n].

Want to test yourself on this? Try the Electrical Aptitude test →