Let's go through each part of this problem systematically.

Given:

The linear difference equation is: $y(n) - y(n-1) + \frac{1}{4} y(n-2) = x(n) - \frac{1}{4} x(n-1)$

Part (a): Deduce the Unit Sample Response of the System

1. Find the system's transfer function $H(z)$: To determine the unit sample response (impulse response $h(n)$), we need to find the transfer function $H(z) = \frac{Y(z)}{X(z)}$.

   Taking the $z$-transform of both sides (assuming zero initial conditions), we get: $Y(z) - z^{-1} Y(z) + \frac{1}{4} z^{-2} Y(z) = X(z) - \frac{1}{4} z^{-1} X(z)$

   Factoring out $Y(z)$ and $X(z)$: $Y(z) \left(1 - z^{-1} + \frac{1}{4} z^{-2}\right) = X(z) \left(1 - \frac{1}{4} z^{-1}\right)$

   So the transfer function $H(z)$ is: $H(z) = \frac{Y(z)}{X(z)} = \frac{1 - \frac{1}{4} z^{-1}}{1 - z^{-1} + \frac{1}{4} z^{-2}}$

2. Find the inverse $z$-transform of $H(z)$: To determine $h(n)$, we perform the partial fraction expansion of $H(z)$ and take the inverse $z$-transform. This will provide the unit sample response $h(n)$ in the time domain.

Part (b): Determine the Stability of the System

The system's stability can be checked by examining the poles of $H(z)$. For stability, all poles must lie inside the unit circle in the $z$-plane.

1. The denominator of $H(z)$, $1 - z^{-1} + \frac{1}{4} z^{-2}$, can be rewritten as: $z^2 - z + \frac{1}{4} = 0$
2. Solving this quadratic equation gives the poles of the system. If both poles have magnitudes less than 1, the system is stable.

Part (c): Estimate the Response of the Input Signal $x(n) = \left( \frac{1}{4} \right)^n u(n)$

To find the response $y(n)$ to the input $x(n) = \left( \frac{1}{4} \right)^n u(n)$:

1. We can use the convolution sum: $y(n) = x(n) * h(n)$ where $*$ denotes convolution.
2. Alternatively, if the transfer function $H(z)$ is known, we can determine $Y(z) = H(z) X(z)$ and then take the inverse $z$-transform to obtain $y(n)$.

Would you like further details on solving each step, including finding the specific unit sample response or stability condition?

Follow-up Questions:

1. How do you solve for the poles in a difference equation?
2. Why is it important for the poles to lie within the unit circle for stability?
3. Can you explain the significance of the unit sample response in LSI systems?
4. What is the purpose of using the convolution sum to find the system’s output?
5. How does the input $\left( \frac{1}{4} \right)^n u(n)$ affect the system’s response compared to a simple impulse input?

Tip:

When dealing with linear difference equations, converting to the $z$-domain simplifies the process by allowing algebraic manipulation instead of recursive calculations.