Question

For a causal discrete-time LTI system with transfer function: \[ H(z) = \frac{2z^2 + 3}{(z + \frac{1}{3})(z - \frac{1}{3})}, \] which of the following statements is/are true?

• A. The system is stable.
• B. The system is a minimum phase system.
• C. The initial value of the impulse response is 2.
• D. The final value of the impulse response is 0.

Hint:

For discrete LTI systems, stability is determined by pole locations, while phase classification depends on the location of zeros. Always apply the final value theorem carefully based on the pole positions.

Answer 1

The Correct Option is A

Step 1: Verify system stability.
A system is stable if all its poles lie inside the unit circle. For the given system, the poles are located at \(z = -\frac{1}{3}\) and \(z = \frac{1}{3}\). Both poles are within the unit circle, so the system is stable. Step 2: Determine if the system is minimum phase.
A system is classified as minimum phase if all its zeros lie inside the unit circle. In this case, the zeros of \(H(z)\) are outside the unit circle. Therefore, the system is not minimum phase. Step 3: Evaluate the initial value of the impulse response.
The initial value of the impulse response corresponds to \(H(z)\) evaluated at \(z = 1\): \[ H(1) = \frac{2(1)^2 + 3}{(1 + \frac{1}{3})(1 - \frac{1}{3})} = 2. \] Step 4: Determine the final value of the impulse response.
The final value theorem applies only when the denominator of \(H(z)\) has no pole at \(z = 1\). Since there is a pole at \(z = 1\), the final value theorem does not apply, and the impulse response converges to \(0\). Final Answer: \[ \boxed{{(1), (3), (4)}} \]