- 2074 Asoj · 3 marks
Check y(t) = x(t-2) + x(2-t)
Short answer
The system y(t)=x(t-2)+x(2-t) is linear but time-variant due to the time reversal in x(2-t). It is non-causal because x(2-t) depends on future inputs (e.g., at t=0 it uses x(2)).
Check whether the following system is linear, time invariant and causal or not: y(t) = x(t-2) + x(2-t)
y(t) = x(t - 2) + x(2 - t)
Linearity
For x = a·x₁ + b·x₂, y = a[x₁(t-2) + x₁(2-t)] + b[x₂(t-2) + x₂(2-t)] = a·y₁ + b·y₂. The system is linear.
Time invariance
Delay the input by t₀: x₁(t) = x(t - t₀). Then
y₁(t) = x₁(t - 2) + x₁(2 - t) = x(t - t₀ - 2) + x(2 - t - t₀)
Delay the output by t₀: y(t - t₀) = x(t - t₀ - 2) + x(2 - (t - t₀)) = x(t - t₀ - 2) + x(2 - t + t₀)
The second terms differ (x(2 - t - t₀) against x(2 - t + t₀)), so the system is time-variant. The time reversal in x(2 - t) causes this.
Causality
x(t - 2) is a past value (fine), but x(2 - t) at t = 0 is x(2), a future value; for any t < 1 it uses a future input. So the system is non-causal.
Answer: linear, time-variant, non-causal.
✓ Answer checked by Claude Opus 5.5 (high effort)
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