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GATE 2015 EE – Question 45

Signals and Systems · Applications of Fourier Transform for continuous and discrete time signals, Laplace Transform and Z transform · 2 marks · Multiple choice

Consider a discrete time signal given by

$$x[n] = (-0.25)^n u[n] + (0.5)^n u[-n-1]$$

The region of convergence of its Z-transform would be

  1. the region inside the circle of radius 0.5 and centered at origin
  2. the region outside the circle of radius 0.25 and centered at origin
  3. the annular region between the two circles, both centered at origin and having radii 0.25 and 0.5
  4. the entire Z plane.

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Correct answer: (C) the annular region between the two circles, both centered at origin and having radii 0.25 and 0.5

Explanation

The first term is right-sided, so it converges for $|z| > 0.25$. The second term is left-sided, so it converges for $|z| < 0.5$. The signal converges where both do, which is the ring $0.25 < |z| < 0.5$.