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Causality of y(t) = t·x(t)

Short answer

y(t)=t·x(t) is causal and memoryless; output depends solely on the present input. It is not non-causal. Anticausal only in the trivial sense of not using future inputs.

Test the given function y(t) = t·x(t) for causality, non-causality and anti-causality.

y(t) = t·x(t)

  • Causal: the output at time t depends only on the present input x(t) (multiplied by t); no future value is needed. So the system is causal.
  • Non-causal: it would need x(t + τ) with τ > 0, which is not so; hence it is not non-causal.
  • Anticausal: an anticausal system needs only present and future values. The output depends on the present value only, so it technically also satisfies the anticausal definition (every memoryless system is both causal and anticausal). It is not a strict anticausal system, since it uses no future input.

Check: at t = 2, y(2) = 2x(2); at t = -3, y(-3) = -3x(-3). Each uses only the same instant.

Answer: causal (and memoryless); not non-causal; anticausal only in the trivial sense of a memoryless system. (The system is time-varying but this does not affect causality.)

✓ Answer checked by Claude Opus 5.5 (high effort)

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