1. **Problem Statement:** Explain the Z-transform definition and key properties including differentiation, initial and final value theorems, and convolution property. 2. **Z-Transform Definition:** The Z-transform of a sequence $x[n]$ is defined as: $$ X(z) = \mathcal{Z}\{x[n]\} = \sum_{n=-\infty}^{\infty} x[n] z^{-n} $$ This transforms a discrete-time sequence into a complex frequency domain representation. 3. **Differentiation Property (Time-Multiplication):** - Starting from the definition: $$ X(z) = \sum_n x[n] z^{-n} $$ - Differentiate w.r.t. $z$: $$ \frac{d}{dz} X(z) = \sum_n x[n] \frac{d}{dz} z^{-n} = \sum_n x[n] (-n) z^{-n-1} $$ - Multiply by $-z$: $$ -z \frac{d}{dz} X(z) = \sum_n n x[n] z^{-n} = \mathcal{Z}\{n x[n]\} $$ - More generally: $$ \mathcal{Z}\{n^k x[n]\} = \left(-z \frac{d}{dz}\right)^k X(z) $$ This property relates multiplication by $n$ in time domain to differentiation in $z$-domain. 4. **Initial Value Theorem:** - For a causal sequence ($x[n]=0$ for $n<0$), $$ x[0] = \lim_{z \to \infty} X(z) $$ - Proof: $$ X(z) = \sum_{n=0}^\infty x[n] z^{-n} = x[0] + \frac{x[1]}{z} + \frac{x[2]}{z^2} + \cdots $$ - Taking limit as $z \to \infty$: $$ \lim_{z \to \infty} X(z) = x[0] + 0 + 0 + \cdots = x[0] $$ This allows finding the initial value directly from $X(z)$. 5. **Final Value Theorem:** - For a stable system (all poles of $(1 - z^{-1})X(z)$ inside unit circle), $$ \lim_{n \to \infty} x[n] = \lim_{z \to 1} (1 - z^{-1}) X(z) = \lim_{z \to 1} \frac{z-1}{z} X(z) $$ - This gives the steady-state value of the sequence from its Z-transform. 6. **Convolution Property:** - Time domain convolution: $$ y[n] = (x * h)[n] = \sum_{k=-\infty}^\infty x[k] h[n-k] $$ - Z-transform of convolution: $$ Y(z) = \mathcal{Z}\{y[n]\} = X(z) H(z) $$ - Here, $X(z) = \mathcal{Z}\{x[n]\}$ and $H(z) = \mathcal{Z}\{h[n]\}$. - The ROC of $Y(z)$ includes the intersection of ROCs of $X(z)$ and $H(z)$. - This property simplifies convolution (complex sum) into multiplication in $z$-domain, facilitating system analysis. 7. **Checklist When Using These Properties:** - Always verify ROC and causality/stability before applying initial or final value theorems. - When differentiating $X(z)$, carefully track powers of $z^{-n}$ using $-z \frac{d}{dz}$. - For convolution, ensure sequences are absolutely summable or ROCs overlap for validity. **Summary:** The Z-transform converts discrete sequences into complex frequency domain, enabling analysis via algebraic operations. Differentiation property links time multiplication to $z$-differentiation. Initial and final value theorems extract sequence start and steady-state values from $X(z)$. Convolution in time domain becomes multiplication in $z$-domain, simplifying LTI system analysis.