- 2079 Baisakh · 5 marks
Check LTI: y(t) = 2x(t) + 3
Short answer
The system y(t)=2x(t)+3 is not LTI. It is non-linear because the constant offset 3 breaks superposition (zero input gives 3). It is time-invariant: delaying input yields same delayed output.
A system's output is given as y(t) = 2x(t) + 3, where x(t) is its input. Determine whether the system is LTI or not.
A system is LTI only if it is both linear and time-invariant.
Linearity
Let x₁ give y₁ = 2x₁ + 3 and x₂ give y₂ = 2x₂ + 3.
Response to a·x₁ + b·x₂: y = 2(a·x₁ + b·x₂) + 3
Combination of outputs: a·y₁ + b·y₂ = 2(a·x₁ + b·x₂) + 3(a + b)
These are equal only when a + b = 1, so superposition fails in general. (Quick test: x(t) = 0 gives y(t) = 3 ≠ 0.) The system is non-linear. It is only incrementally (affine) linear.
Time invariance
Delay the input: x(t - t₀) gives y₁ = 2x(t - t₀) + 3. Delay the output: y(t - t₀) = 2x(t - t₀) + 3. They are equal, so the system is time-invariant.
Answer: the system is not LTI, because it is non-linear, although it is time-invariant.
✓ Answer checked by Claude Opus 5.5 (high effort)
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