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  • Question
  • Consider a stable and causal system with impulse response h(t) and system function H(S). Suppose H(S) is rational, contains a pole at S = - 2, and does not have a zero at the origin. The location of all other poles and zero is unknown for each of the following statements. Let us determine whether statement is true or false.

    1. f[h(t) e-3t] converges
    2. Consider a stable and causal system with impulse response h(t) and system function H(S). Suppose H(S
    3. h(t) has finite duration
    4. H(s) = H(- s)
    Choose correct option.


  • Options
  • A. 1 - True, 2 - False, 3 - True, 4 - False
  • B. 1 - False, 2 - False, 3 - False, 4 - True
  • C. 1 - False, 2 - False, 3 - False, 4 - False
  • D. 1 - True, 2 - can't say, 3 - True, 4 - can't say

  • Correct Answer
  • 1 - False, 2 - False, 3 - False, 4 - False 

    Explanation
    Statement 1 is false, since f{h(t)e3t} corresponds to the value of the Laplace transform of h(t) at s = 3.

    If this converges, it implies that s = - 3 is in the ROC.

    A casual and stable system must always have its ROC to the right of all its poles. However, s = - 3 is not to the right of the pole at s = - 2.

    Statement 2 is false, because it is equivalent to stating that H(0) = 0. This contradicts the fact that H(s) does not have a zero at the origin.

    Statement 3 is false. If h(t) is of finite duration, then if its Laplace transform has any points in its ROC, ROC must be the entire s-plane.

    However, this is not consistent with H(s) having a pole at s = - 2.

    Statement 4 is false. If it were true, then H(s) has a pole at s = - 2, it must also have a pole at s = 2.

    This is inconsistent with the fact that all the poles of a causal and stable system must be in the left half of the s-plane.


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