Find the response of the linear time invariant system h(k) to the input x(n), where n is time index and k is index of filter coefficient.

*Solution.*

The response y(n) of an *LTI* system to the input signal x(n) is the sum of delayed inputs x(n-k) weighted by the impulse response h(k):

n | x(n) | x(n-1) | x(n-2) | y(n) |

0 | 1 | 0 | 0 | 0.5 |

1 | 1 | 1 | 0 | 1.5 |

2 | 0 | 1 | 1 | 1.5 |

3 | 0 | 0 | 1 | 0.5 |

4 | 0 | 0 | 0 | 0 |