Determine the zeros of the following FIR systems and identify whether the following system is minimum phase, maximum phase, mixed phase. Also comment on stability.
H2 (z) =1-z-1-6z-2                                      

To solve this problem, we will determine the zeros of the given FIR (Finite Impulse Response) system, \( H_2(z) \), and then classify the system as minimum phase, maximum phase, mixed phase, and comment on its stability.

### Step 1: Express the System in a Standard Form
The given FIR system is:

\[
H_2(z) = 1 - z^{-1} - 6z^{-2}
\]

To find the zeros, we need to solve for \( z \) in the equation \( H_2(z) = 0 \).

### Step 2: Find the Zeros
Set \( H_2(z) = 0 \):

\[
1 - z^{-1} - 6z^{-2} = 0
\]

Multiply the entire equation by \( z^2 \) to eliminate negative powers of \( z \):

\[
z^2 - z - 6 = 0
\]

Now, solve the quadratic equation:

\[
z^2 - z - 6 = 0
\]

This can be factored as:

\[
(z - 3)(z + 2) = 0
\]

So, the zeros are:

\[
z = 3 \quad \text{and} \quad z = -2
\]

### Step 3: Determine 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**: Some zeros are inside the unit circle, and some are outside.

For the zeros \( z = 3 \) and \( z = -2 \):

- \( |3| > 1 \)
- \( |-2| > 1 \)

Both zeros are outside the unit circle, which means the system is a **Maximum Phase** system.

### Step 4: Comment on Stability
For stability, all poles of the system must lie inside the unit circle. Since this is an FIR system (which inherently has all poles at the origin, \( z = 0 \)), it is **always stable** regardless of the zeros' location.

### Conclusion:
- **Zeros**: \( z = 3 \) and \( z = -2 \)
- **Phase Type**: Maximum Phase
- **Stability**: The system is stable.

Would you like a detailed explanation on any of these steps or have further questions? Here are some related topics you might find interesting:

1. How to determine the poles of a system and their effect on stability.
2. Difference between FIR and IIR systems.
3. The significance of minimum phase systems in signal processing.
4. Methods to find zeros of higher-order polynomials in digital signal processing.
5. The impact of zero location on the frequency response of an FIR filter.

**Tip:** FIR filters are often preferred in digital signal processing because they can be designed to be exactly linear phase, ensuring no phase distortion.

Learn how to determine the zeros of the FIR system H2(z) = 1 - z^{-1} - 6z^{-2}, classify its phase type (maximum phase), and comment on its stability. This problem explores key concepts in digital signal processing, FIR filter design, and stability analysis.

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