1. The problem is to understand and apply the Z-transform, which is a powerful tool in signal processing and discrete-time systems. 2. The Z-transform of a discrete-time signal $x[n]$ is defined as: $$X(z) = \sum_{n=-\infty}^{\infty} x[n] z^{-n}$$ where $z$ is a complex variable. 3. Important rules: - Linearity: $Z\{a x[n] + b y[n]\} = a X(z) + b Y(z)$ - Time shifting: $Z\{x[n-k]\} = z^{-k} X(z)$ - Convolution in time domain corresponds to multiplication in Z-domain. 4. To apply the Z-transform, identify the sequence $x[n]$, substitute into the formula, and simplify the sum. 5. Example: For $x[n] = a^n u[n]$ where $u[n]$ is the unit step, the Z-transform is: $$X(z) = \sum_{n=0}^{\infty} a^n z^{-n} = \sum_{n=0}^{\infty} (a z^{-1})^n = \frac{1}{1 - a z^{-1}}, \quad |z| > |a|$$ 6. This shows how the Z-transform converts a sequence into a function of $z$, useful for analyzing system behavior.