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H(z) is a transfer function of a real system. When a signal x[n] = (1 + j)ⁿ is the input to such a system, the output is zero. Further, the Region Of Convergence (ROC) of (1 - 1/2 z⁻¹) H(z) is the entire Z-plane (except z = 0). It can then be inferred that H(z) can have a minimum of

1. one pole and one zero
2. one pole and two zeros
3. two poles and one zero
4. two poles and two zeros

Correct answer: one pole and two zeros

Solution

The output being zero for the input signal indicates that the system has a specific behavior with respect to the poles and zeros. Given that the ROC is the entire Z-plane except at z = 0, it suggests that H(z) must have at least one pole to ensure stability, and having two zeros allows for the cancellation of the input signal, leading to a zero output.

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