To determine the zeros of the FIR (Finite Impulse Response) system and identify whether the system is minimum phase, maximum phase, or mixed phase, we need to analyze the polynomial $H_1(z) = 6 + z^{-1} + 6z^{-2}$.

Step 1: Rewrite the Transfer Function

Let's rewrite the transfer function $H_1(z)$ in terms of positive powers of $z$:

$H_1(z) = 6 + z^{-1} + 6z^{-2} = 6z^2 + z + 6$

Step 2: Find the Zeros of the System

To find the zeros, set $H_1(z) = 0$:

$6z^2 + z + 6 = 0$

This is a quadratic equation of the form $az^2 + bz + c = 0$, where $a = 6$, $b = 1$, and $c = 6$.

The roots (zeros) of this quadratic equation can be found using the quadratic formula:

$z = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

Substituting the values of $a$, $b$, and $c$:

$z = \frac{-1 \pm \sqrt{1^2 - 4 \cdot 6 \cdot 6}}{2 \cdot 6}$

$z = \frac{-1 \pm \sqrt{1 - 144}}{12}$

$z = \frac{-1 \pm \sqrt{-143}}{12}$

$z = \frac{-1 \pm j\sqrt{143}}{12}$

Here, $j$ is the imaginary unit. Thus, the zeros of the system are:

$z_1 = \frac{-1 + j\sqrt{143}}{12}, \quad z_2 = \frac{-1 - j\sqrt{143}}{12}$

Step 3: Identify the Phase Type

• Minimum Phase: All zeros are inside the unit circle $|z| < 1$.
• Maximum Phase: All zeros are outside the unit circle $|z| > 1$.
• Mixed Phase: Zeros are both inside and outside the unit circle.

To determine whether the system is minimum, maximum, or mixed phase, we need to check the magnitudes of the zeros:

$|z_1| = \left|\frac{-1 + j\sqrt{143}}{12}\right|, \quad |z_2| = \left|\frac{-1 - j\sqrt{143}}{12}\right|$

Calculating the magnitude:

$|z_1| = |z_2| = \frac{\sqrt{(-1)^2 + (\sqrt{143})^2}}{12} = \frac{\sqrt{1 + 143}}{12} = \frac{\sqrt{144}}{12} = \frac{12}{12} = 1$

Both zeros lie on the unit circle $|z| = 1$.

Conclusion on Phase Type

Since the zeros lie exactly on the unit circle, the system is neither minimum phase nor maximum phase but can be considered mixed phase because it doesn't strictly fall into either category.

Step 4: Comment on Stability

For FIR systems, the stability is inherently guaranteed because the impulse response is finite in length. Therefore, regardless of the zeros, the system will always be stable.

Summary

• Zeros of the system: $z_1 = \frac{-1 + j\sqrt{143}}{12}$, $z_2 = \frac{-1 - j\sqrt{143}}{12}$
• Phase Type: Mixed phase (since the zeros lie on the unit circle)
• Stability: Stable (since it is an FIR system)

Would you like more details or have any questions?

Related Questions:

1. How do you find the poles of a system and what do they signify?
2. What is the difference between minimum phase and maximum phase systems?
3. How does the location of zeros affect the frequency response of an FIR filter?
4. Can an FIR filter be unstable? If so, under what conditions?
5. How are FIR systems implemented in digital signal processing?

Tip:

To determine the phase type of an FIR system, always check the location of its zeros relative to the unit circle in the complex plane.