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Suppose x[n] is an absolutely summable discrete-time signal. Its z-transform is a rational function with two poles and two zeroes. The poles are at z = ±2j. Which one of the following statements is TRUE for the signal x[n]?

Suppose x[n] is an absolutely summable discrete-time signal. Its z-transform is a rational function with two poles and two zeroes. The poles are at z = ±2j. Which one of the following statements is TRUE for the signal x[n]? Correct Answer It is a non-causal signal.

Concept:

  • If a sequence is absolutely summable, then its DTFT will exist and the ROC will include the unit circle.
  • If ROC includes the unit circle then the system will be a non-causal system, because causal systems are right-sided signals with ROC extending to the right side of z-plane.

Analysis:

  • Given sequence is summable, so its ROC will include the unit circle, which implies that the DTFT exists.
  • Poles location shown below:

    [ alt="F1 S.B Madhu 18.11.19 D 7" src="//storage.googleapis.com/tb-img/production/19/11/F1_S.B_Madhu_18.11.19_D%207.png" style="width: 175px; height: 174px;">
     
  • Here ROC includes the unit circle as can be easily seen from the above graph which means that the ROC is left sided.
  • Hence it is a non-causal system

  • So Option 3 is Correct.

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