GATE Electrical Engineering - Complete Master Library

 

📘 System Properties – Linearity, Time Invariance & Stability

A system is a mathematical model that transforms an input signal into an output signal. Understanding system properties is fundamental for LTI system analysis.

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🔹 1. Linearity

A system is linear if it satisfies:

Superposition Principle: If x₁ → y₁ and x₂ → y₂ Then a x₁ + b x₂ → a y₁ + b y₂

Test Method:

Replace input by ax₁ + bx₂ Check if output becomes a y₁ + b y₂.

Example:

y(t) = 3x(t)

Linear ✔

y(t) = x²(t)

Non-linear ✖

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🔹 2. Time Invariance

A system is time invariant if time shift in input causes same shift in output.

Test Method:

Step 1: Replace x(t) by x(t − t₀) Step 2: Compare with y(t − t₀)

Example:

y(t) = x(t − 2)

Time invariant ✔

y(t) = t x(t)

Time varying ✖

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🔹 3. Causality

System is causal if output depends only on present and past inputs.

Example:

y(t) = x(t − 1)

Causal ✔

y(t) = x(t + 1)

Non-causal ✖

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🔹 4. Stability (BIBO Stability)

Bounded Input → Bounded Output.

For LTI system:

∫ |h(t)| dt < ∞

If impulse response absolutely integrable → Stable.

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🔹 5. Memory / Memoryless

Memoryless: Output depends only on present input.

y(t) = 5x(t)

Memoryless ✔

y(t) = x(t − 1)

Has memory ✖

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🔹 6. LTI System

LTI = Linear + Time Invariant. Most analysis tools (Fourier, Laplace, Convolution) apply only to LTI systems.

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🔹 7. Worked Example

Check system:

y(t) = t x(t − 1)

Linearity → Yes Time invariance → No (depends on t) Causality → Yes Memory → Yes

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🎯 GATE Important Points

• Linearity + Time invariance together → LTI
• Impulse response determines stability
• Most questions conceptual
• Practice test method carefully

Understanding LTI = Gateway to Convolution & Fourier