How it works
Linearity requires both additivity and homogeneity: if x₁(t)→y₁(t) and x₂(t)→y₂(t), then ax₁(t)+bx₂(t)→ay₁(t)+by₂(t). Time-invariance requires that x(t−t₀) produces y(t−t₀) for any delay t₀. The system y(t) = tx(t) is linear but not time-invariant; y(t) = x(2t) is LTI? No — it is linear but time-varying because compressing time changes when features appear. Causality requires h(t) = 0 for t < 0, meaning the system does not respond before the input arrives. BIBO stability requires ∫|h(t)|dt < ∞.
Key points to remember
BIBO (Bounded Input Bounded Output) stability: a system is stable if and only if its impulse response is absolutely integrable for continuous-time, or absolutely summable for discrete-time. For an IIR filter with h[n] = aⁿu[n], stability requires |a| < 1 — the same unit-circle pole condition from Z-transform analysis. Memoryless systems have y(t) depending only on x(t) at the same instant; all memory systems depend on past (or future) inputs. An invertible system has a unique inverse that recovers x from y. The ideal delay y(t) = x(t − 2) is LTI, causal (for delay ≥ 0), stable, and has memory.
Exam tip
The examiner always asks you to check whether a given system like y[n] = nx[n] or y(t) = x(t²) satisfies linearity and time-invariance — test each property separately with a formal proof, never by inspection.