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Causality, stability; energy and power signals

Short answer

A causal system's output depends only on present/past inputs. BIBO stability means bounded input yields bounded output; for LTI, ∫|h|dt<∞. An energy signal has finite energy E=∫|x|²dt, zero power (e.g., e^{-t}u(t)). A power signal has finite power P=lim(1/2T)∫|x|²dt, infinite energy (e.g., A cos ω₀t).

Define causality and stability properties of systems. What are energy and power signals. Explain with example.

Causality

A system is causal if the output at time t depends only on the present and past values of the input (not the future). Example: y(t) = x(t) + x(t - 1) is causal; y(t) = x(t + 1) is non-causal. For an LTI system, h(t) = 0 for t < 0.

Stability

A system is BIBO stable if every bounded input gives a bounded output. For an LTI system, ∫|h(t)| dt < ∞. Example: h(t) = e^(-2t)u(t) is stable (∫ = 1/2); h(t) = u(t) (integrator) is unstable.

Energy and power signals

  • Energy signal: finite energy 0 < E = ∫|x(t)|² dt < ∞, and zero average power. Example: x(t) = e^(-t)u(t) has E = 1/2.
  • Power signal: finite average power 0 < P = lim (1/2T) ∫_{-T}^{T}|x(t)|² dt < ∞, with infinite energy. Example: x(t) = A cos(ω₀t) has P = A²/2.

Periodic signals are power signals; non-periodic decaying (finite-duration) signals are energy signals. No signal is both.

✓ Answer checked by Claude Opus 5.5 (high effort)

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