System properties: linearity, time invariance, causality, stability

Why it matters

Almost every analysis tool in this subject — convolution, Fourier, Laplace and z-transforms, transfer functions — is valid only for systems that are linear and time-invariant (LTI). Causality tells you whether a system can run in real time on a live sensor signal, and stability tells you whether its output stays bounded. Testing these four properties is therefore the first step before you apply any formula.

Key ideas

A system maps an input x(t) (or x[n]) to an output y(t) (or y[n]), written y = T{x}. Each property is a statement that must hold for every allowed input; one counterexample is enough to disprove it.

Linearity = additivity + homogeneity, combined as superposition:

• If x1 → y1 and x2 → y2, then a·x1 + b·x2 → a·y1 + b·y2 for all constants a, b.
• Quick test: a linear system must give zero output for zero input. So y = 3x + 2 is not linear (it is "incrementally linear" — linear plus a constant).
• Any operation that acts on the input non-linearly (squares, |x|, log x, e^x, sgn x, saturation) makes the system non-linear. Multiplying by a known function of time, such as y(t) = t·x(t), is still linear.

Time invariance

• Shift the input by t0, and the output must be the same response shifted by t0: if x(t) → y(t) then x(t − t0) → y(t − t0).
• Test: compute the response to x(t − t0), call it y1(t); compute y(t − t0) by replacing every t in the output expression; compare.
• Time-varying signs: a coefficient that depends on t (e.g. t·x(t), x(t)·cos ω0t), time scaling x(2t) or x[2n], time reversal x(−t), and integration limits fixed in absolute time such as ∫ from 0 to t.
• y(t) = x(t − 2) + 3 is time-invariant: the delay and the constant do not depend on t.

Causality

• The output at any time depends only on present and past inputs. y(t) = x(t − 1) is causal; y(t) = x(t + 1) is not.
• Check every argument of x: the system is causal only if each argument is ≤ t for every t. y(t) = x(−t) is non-causal, because at t = −1 it needs x(1). y[n] = x[2n] is non-causal (needs x[2] at n = 1).
• Memoryless (static) systems depend only on the present input, e.g. y = x². Every memoryless system is causal.
• For an LTI system: causal ⇔ h(t) = 0 for t < 0 (or h[n] = 0 for n < 0).

BIBO stability

• Every bounded input (|x| ≤ Mx < ∞) must produce a bounded output. One bounded input that gives an unbounded output proves instability; y(t) = t·x(t) with x = 1 gives y = t, unbounded.
• For an LTI system: stable ⇔ h is absolutely integrable (CT) or absolutely summable (DT).
• In transform terms (covered later): a causal CT LTI system is stable when all poles of H(s) have negative real parts; a causal DT system when all poles of H(z) lie inside the unit circle. More generally, stable ⇔ the ROC includes the jω-axis (CT) or the unit circle (DT).
• An ideal integrator y(t) = ∫ x(τ)dτ (from −∞ to t) is unstable: a step input gives a ramp.

Invertibility (also asked): distinct inputs give distinct outputs. y = 2x is invertible; y = x² is not (x and −x give the same output).

Formulas

• T{a·x1 + b·x2} = a·T{x1} + b·T{x2} — linearity (superposition), for all a, b, x1, x2.
• x(t − t0) → y(t − t0) for all t0 — time invariance (same with n, n0 for DT).
• h(t) = 0 for t < 0 / h[n] = 0 for n < 0 — causality of an LTI system.
• ∫ |h(t)| dt < ∞ (−∞ to ∞) — BIBO stability of a CT LTI system.
• Σ |h[n]| < ∞ (all n) — BIBO stability of a DT LTI system.
• |y| ≤ Mx · ∫|h(τ)|dτ — bound on the output of a stable LTI system.

Symbols: x, y input and output (any physical unit); h impulse response (unit of y per unit of x, per second for CT); t, t0 time and shift (s); n, n0 sample index and shift (integer); a, b arbitrary constants; Mx bound on |x|.

Worked examples

Example 1 (standard). Test y(t) = t·x(t) and y[n] = x[2n] for linearity, time invariance, causality and stability.

1. y = t·x(t). Linearity: t·(a·x1 + b·x2) = a·t·x1 + b·t·x2 ✓ linear.
2. Time invariance: response to x(t − t0) is t·x(t − t0); shifted output is (t − t0)·x(t − t0). They differ, so time-varying.
3. Causality: only x(t) is used — memoryless, so causal.
4. Stability: x(t) = 1 is bounded but y(t) = t grows without limit — unstable.
5. y[n] = x[2n]. Linear ✓ (no operation on amplitude). Response to x[n − n0] is x[2n − n0]; shifted output is x[2(n − n0)] = x[2n − 2n0] — time-varying. At n = 1 it needs x[2] — non-causal. |y[n]| ≤ max|x| — stable.

Example 2 (GATE level). Decide causality and stability of the LTI systems (a) h[n] = (0.8)ⁿ u[n] and (b) h(t) = e^(2t) u(−t).

1. (a) h[n] = 0 for n < 0 → causal.
2. Σ |h[n]| = Σ from 0 to ∞ of (0.8)ⁿ = 1/(1 − 0.8) = 5 — finite, so stable, with output bound |y| ≤ 5·Mx.
3. (b) h(t) is non-zero for t < 0 → non-causal.
4. ∫ |h(t)| dt = ∫ from −∞ to 0 of e^(2t) dt = [e^(2t)/2] from −∞ to 0 = 1/2 — finite, so stable.
5. Lesson: a growing exponential does not automatically mean instability; here e^(2t) is used only for negative t, where it decays. (a) causal and stable (Σ|h| = 5); (b) non-causal and stable (∫|h| = 0.5).

Common mistakes

• Calling y = 3x + 2 linear because its graph is a straight line. Zero input must give zero output.
• Calling y(t) = x(t − 2) time-varying "because there is a shift". A fixed delay is time-invariant.
• Testing time invariance by shifting only the input argument in the output formula, and forgetting to shift the t in coefficients such as t·x(t).
• Thinking y(t) = x(t) + x(t − 1) is non-causal — t − 1 is in the past, so it is causal.
• Concluding stability from one decaying input-output pair. Stability is about every bounded input; for LTI systems use ∫|h| or Σ|h|.
• Treating y = x(−t) as causal because it has no explicit future term — check negative t.

For GATE IN

Expect a one- or two-mark question giving 3–4 input-output relations and asking which property each has (often MSQ), and questions that give h(t), h[n] or a transfer function and ask for causality and stability. Practise the four tests in a fixed order on systems such as x(t)cos ωt, x(2t), x(−t), x², ∫ from −∞ to t, y[n] = n·x[n], and on impulse responses that are two-sided or growing.

Quick check

1. Is y(t) = x²(t) linear? Time-invariant?
2. Is y[n] = x[n] + x[n + 1] causal?
3. Is the LTI system h[n] = u[n] stable?
4. Is y(t) = x(t)·cos(100t) time-invariant?
5. Is y(t) = x(t − 2) + 3 linear? Answers: 1. Non-linear; time-invariant. 2. No — it needs a future sample. 3. No — Σ u[n] diverges (it is an accumulator). 4. No — the coefficient depends on t. 5. No — the constant 3 makes it non-linear (though it is time-invariant, causal and stable).