Side-by-side comparison
| Parameter | Time Invariant | Time Variant System |
|---|---|---|
| Definition | Shift in input by t0 causes same shift in output: y(t−t0) | A time shift in input does NOT produce a proportional shift in output |
| System parameters | Resistances, capacitances, inductances are constant | At least one parameter changes with time, e.g., L(t) or C(t) |
| Impulse response | h(t, τ) depends only on (t − τ); written as h(t − τ) | h(t, τ) depends on both t and τ separately |
| Convolution validity | y(t) = x(t) * h(t) is valid | Convolution integral changes form; not simple * operation |
| Transfer function | H(s) or H(jω) is well-defined and constant | No fixed H(s); must use time-varying state equations |
| Real example | Fixed Butterworth LPF using TL072, fc = 1 kHz | FM modulator — carrier frequency varies with message signal |
| Test method | Apply x(t − t0), check if output is y(t − t0) | Output differs from shifted version for at least one input |
| Typical LTI tool | Laplace Transform, Fourier Transform, Z-Transform | Floquet theory, time-varying state space, numerical ODE |
| Common GATE trap | y(t) = x(t)·cos(2π·1000·t) — this is TIME-VARIANT | y(t) = x(t − 2) — this IS time-invariant (just a shift) |
| Application | Most communication receivers, fixed digital filters | Radar Doppler processing, kalman filter with varying noise |
Key differences
Time invariance means the system's behaviour does not change with when you apply the input. The formal test: replace x(t) with x(t − t0) and check if output becomes y(t − t0). The classic GATE trap is y(t) = x(t) · cos(2000πt): replacing t with (t − t0) gives x(t−t0)·cos(2000π(t−t0)), which is NOT the same as y(t−t0) = x(t−t0)·cos(2000πt0). So it is time-variant. A fixed RC circuit is LTI; an FM modulator whose effective gain changes with the message is time-variant.
When to use Time Invariant
Use a time-invariant model when the circuit components are fixed and the operating conditions do not change — for example, analysing the step response of a second-order RLC circuit with R = 100 Ω, L = 10 mH, C = 1 µF.
When to use Time Variant System
Model the system as time-variant when a component's value or the system's gain changes with time — for example, in an amplitude modulator where y(t) = x(t)·cos(2π·100000·t), the multiplying carrier makes the effective gain time-dependent.
Recommendation
For GATE, treat every system as time-invariant until the test fails — then call it time-variant. In interviews, choose the LTI framework for any fixed-hardware problem and flag time-variance only when you see an explicit time-dependent coefficient like a multiplier, compressor, or Doppler shift.
Exam tip: The GATE examiner's favourite test case is y(t) = x(t)·cos(ω0·t) — always perform the formal delay-and-compare test and write out each step; stating the answer without working earns no marks.
Interview tip: An interviewer at a telecom company will ask you to classify y[n] = n·x[n] — point out that the coefficient n changes with time, so the system is time-variant, and cannot be described by a single H(z).