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The z-transform of a sequence x[n] is given by X[z] = 0.5/(1 - 2z⁻¹). It is given that the region of convergence of X[z] includes the unit circle. The value of x[0] is

1. -0.5
2. 0
3. 0.25
4. 0.5

Correct answer: 0

Solution

X(z) = 0.5/(1 - 2 z^-1) has a pole at z = 2. Since the ROC must include the unit circle, the ROC is |z| < 2, making x[n] left-sided: x[n] = -0.5*2^n u[-n-1]. That sequence is zero for n >= 0, so x[0] = 0.

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