Given the transfer function for a causal discrete-time LTI system: $H(z) = \frac{2z^2+3}{(z+\frac{1}{3})(z-\frac{1}{3})} = \frac{2z^2+3}{z^2 - \frac{1}{9}}$

System Stability Analysis

A causal discrete-time LTI system is BIBO stable if all its poles lie strictly inside the unit circle ($|z| < 1$).

• The poles are the roots of the denominator $z^2 - \frac{1}{9} = 0$.
• The poles are $z_1 = \frac{1}{3}$ and $z_2 = -\frac{1}{3}$.
• Checking magnitudes: $|z_1| = \frac{1}{3} < 1$ and $|z_2| = \frac{1}{3} < 1$.

Since both poles are inside the unit circle, the system is stable.

Minimum Phase System Check

A causal system is minimum phase if all its poles and zeros lie strictly inside the unit circle.

• Poles: $z = \pm \frac{1}{3}$ (inside the unit circle).
• Zeros are roots of $2z^2 + 3 = 0 \implies z^2 = -\frac{3}{2} \implies z = \pm j\sqrt{\frac{3}{2}}$.
• Magnitude of zeros: $| \pm j\sqrt{\frac{3}{2}} | = \sqrt{\frac{3}{2}} \approx 1.22 > 1$.

Since there are zeros outside the unit circle, the system is not minimum phase.

Initial Value of Impulse Response

For a causal system, the initial value of the impulse response, $h[0]$, is found by evaluating $H(z)$ as $z \to \infty$.

• Calculate the limit: $\lim_{z \to \infty} H(z) = \lim_{z \to \infty} \frac{2z^2+3}{z^2 - \frac{1}{9}}$
• Divide by the highest power of $z$ ($z^2$): $\lim_{z \to \infty} \frac{2 + \frac{3}{z^2}}{1 - \frac{1}{9z^2}} = \frac{2+0}{1-0} = 2$.

The initial value of the impulse response, $h[0]$, is 2.

Final Value of Impulse Response

The final value, $h[\infty]$, is determined using the final value theorem. This theorem applies if all poles of $z H(z)$ lie strictly inside the unit circle.

• Consider $z H(z) = z \frac{2z^2+3}{z^2 - \frac{1}{9}} = \frac{2z^3+3z}{z^2 - \frac{1}{9}}$.
• The poles of $z H(z)$ are the same as the poles of $H(z)$, which are $z = \pm \frac{1}{3}$.
• Since $| \pm \frac{1}{3} | < 1$, all poles of $z H(z)$ are strictly inside the unit circle.

Therefore, the final value of the impulse response, $h[\infty]$, is 0.

Conclusion

Based on the analysis:

• Statement 1 (The system is stable) is true.
• Statement 2 (The system is a minimum phase system) is false.
• Statement 3 (The initial value of the impulse response is 2) is true.
• Statement 4 (The final value of the impulse response is 0) is true.

The correct statements are 1, 3, and 4.