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Check properties of y(t) = 2x(t+1)

Short answer

y(t)=2x(t+1) is linear, time-invariant, and stable (bounded input yields bounded output). It is non-causal because the output depends on the future input x(t+1).

Check linearity, causality, stability and time invariance characteristics of system y(t) = 2x(t+1).

y(t) = 2x(t + 1)

Linearity

Response to a·x₁ + b·x₂ is 2[a·x₁(t+1) + b·x₂(t+1)] = a·y₁ + b·y₂. Linear.

Causality

At time t the output uses x(t + 1), the input one second in the future. Non-causal.

Stability

If |x(t)| ≤ M, then |y(t)| = 2|x(t + 1)| ≤ 2M. A bounded input gives a bounded output. Stable. (h(t) = 2δ(t + 1), and ∫|h| dt = 2.)

Time invariance

Delay the input: x₁(t) = x(t - t₀) gives y₁(t) = 2x₁(t + 1) = 2x(t + 1 - t₀). Delay the output: y(t - t₀) = 2x(t - t₀ + 1). These are equal. Time-invariant.

Answer: linear, time-invariant, non-causal and stable (an LTI system that advances the signal by 1 s and doubles it).

✓ Answer checked by Claude Opus 5.5 (high effort)

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