The impulse response of a causal, linear, time-invariant,

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The impulse response of a causal, linear, time-invariant, continuous-time system is h(t). The output y(t) of the same system to an input x(t), where x(t) = 0 for t < – 2, is—

1. ∫^t₀ h(τ) x(t – τ )dτ
2. ∫^t_{– 2} h(τ) x(t – τ )dτ
3. ∫^{t – 2}_{+ 2} h(τ) x(t – τ )dτ
4. ∫^{t + 2}₀ h(τ) x(t – τ )dτ

Correct Option: B

In the given question all the alternative are linear and time invariant, but not causal all, for t < – 2. Hence, for the given system to be causal lower limit should be – 2 and upper limit should be t.
i.e., y(t) = ∫^t_{– 2} h(τ) x(t – τ )dt
Hence, alternative (B) is the correct choice.