Step 1: Define the transfer function. The transfer function \(H(z)\) expresses the relationship between the Z-transform of the output, \(Y(z)\), and the Z-transform of the input, \(X(z)\):\[ H(z) = \frac{Y(z)}{X(z)} \]Given \( H(z) = \frac{1}{1 - \frac{1}{4}z^{-1}} \), we can express this relationship as:\[ \frac{Y(z)}{X(z)} = \frac{1}{1 - \frac{1}{4}z^{-1}} \]
Step 2: Rearrange the equation to isolate \(Y(z)\) and \(X(z)\) terms. Cross-multiplication eliminates the fraction:\[ Y(z) \left( 1 - \frac{1}{4}z^{-1} \right) = X(z) \]Distribute \(Y(z)\) on the left-hand side:\[ Y(z) - \frac{1}{4}z^{-1}Y(z) = X(z) \]
Step 3: Apply the inverse Z-transform to derive the difference equation. Utilizing the time-shifting property of the Z-transform, \( \mathcal{Z}^{-1}\{z^{-k}F(z)\} = f[n-k] \), we take the inverse Z-transform of each term:\[ \mathcal{Z}^{-1}\{Y(z)\} \rightarrow y[n] \]\[ \mathcal{Z}^{-1}\{\frac{1}{4}z^{-1}Y(z)\} \rightarrow \frac{1}{4}y[n-1] \]\[ \mathcal{Z}^{-1}\{X(z)\} \rightarrow x[n] \]Combining these results in the following difference equation:\[ y[n] - \frac{1}{4}y[n-1] = x[n] \]