# Z-Transform & ROC for Given Sequences Let's solve each sequence step by step. --- ## (i) \( x_1[n] = \delta[n+2] + 2\delta[n] - 3\delta[n-1] \) ###Step 1 Z-TransformRecall the Z-transform of \(\[n-k]\) is \(z^{-k}\): \[ \begin{align*} X_1(z) &= \mathcal{Z} \{ \delta[n+2] \} + 2\mathcal{Z} \{ \delta[n] \} - 3\mathcal{Z} \{ \delta[n-1] \} \\ &= z^{2} + 2 \cdot 1 - 3z^{-1} \\ &= z^2 + 2 - 3z^{-1} \end{align*} \] Or, all terms as powers of \(z^{-1}\): \[ X_1(z) = z^2 + 2 - 3z^{-1} \] ### **Step 2: Region of Convergence (ROC)** For finite-length sequences (impulse responses), the ROC is **all \(z\) except possibly \(z = \) and \(z = \infty\)**. - **ROC:** All \(z\) except \(z=,\, \infty\) --- ## (ii) \( x_2[n] = (.7)^{n-3}u[n-2] \) ### **Step 1: Simplify the sequence** Let \( k = n-2 \), so \( n = k+2 \): \[ x_2[n] = (.7)^{n-3}u[n-2] = (.7)^{(n-2)-1} u[n-2] = (.7)^{n-2} \cdot (.7)^{-1} u[n-2] = (.7)^{-1} (.7)^{n-2} u[n-2] \] ### **Step 2: Z-Transform** Recall: \(\mathcal{Z}\{ a^n u[n] \} = \frac{1}{1 - az^{-1}},\) ROC: \(|z| > |a|\) But \(x_2[n]\) is shifted by 2. \[ x_2[n] = (.7)^{n-2}u[n-2} \cdot (.7)^{-1} \] So, \[ \mathcal{Z}\{ a^{n-k}u[n-k] \} = z^{-k} \frac{1}{1 - az^{-1}},\quad |z| > |a| \] Thus, \[ X_2(z) = (.7)^{-1} z^{-2} \frac{1}{1 - .7z^{-1}} \] Or, \[ X_2(z) = \frac{z^{-2}}{.7 (1 - .7z^{-1})} \] ### **Step 3: ROC** - **ROC:** \(|z| > .7\) --- ## (iii) \( x_3[n] = (-1)^n (n+1) (.6)^n u[n] \) ### **Step 1: Combine terms** \[ x_3[n] = (n+1)[(-.6)^n]u[n] \] ### **Step 2: Z-Transform** Recall: \(\mathcal{Z}\{ (n+1)a^n u[n] \} = \frac{1 - az^{-1}}{(1 - az^{-1})^2} \), ROC: \(|z| > |a|\) Here, \(a = -.6\): \[ X_3(z) = \frac{1 - (-.6)z^{-1}}{(1 + .6z^{-1})^2} \] ### **Step 3: ROC** - **ROC:** \(|z| > .6\) --- ## (iv) \( x_4[n] = (1.25)^n u[-n-2] + (.5)^n u[n-1] \) ### **Step 1: First term: \( (1.25)^n u[-n-2] \)** - \(u[-n-2] = 1\) for \(n \leq -2\) - This is a **left-sided** sequence. Recall: \[ \mathcal{Z}\{ a^n u[-n-k] \} = \frac{a^{-k} z^{k}}{1 - a^{-1}z},\quad |z| < |a| \] For \(k=2\), \(a=1.25\): \[ \mathcal{Z}\{ (1.25)^n u[-n-2] \} = (1.25)^{-2} z^{2} \cdot \frac{1}{1 - (1/1.25)z} = .64 \frac{z^2}{1 - .8z} \] --- ### **Step 2: Second term: \( (.5)^n u[n-1] \)** Shifted right by 1: \[ \mathcal{Z}\{ a^n u[n-k] \} = z^{-k} \frac{1}{1 - a z^{-1}},\quad |z| > |a| \] So: \[ \mathcal{Z}\{ (.5)^n u[n-1] \} = z^{-1} \frac{1}{1 - .5z^{-1}} \] --- ### **Step 3: Combine** \[ X_4(z) = .64 \frac{z^2}{1 - .8z} + \frac{z^{-1}}{1 - .5z^{-1}} \] Or, if you want all in terms of \(z^{-1}\): \[ .64 \frac{z^2}{1 - .8z} = .64 \frac{1}{z^{-2} - .8z^{-1}} \] --- ### **Step 4: ROC** - First term: left-sided, **ROC:** \(|z| < 1.25\) - Second term: right-sided, **ROC:** \(|z| > .5\) - **Overall ROC:** The intersection is **NULL** (no \(z\) satisfies both), so ROC is **empty** unless \(x_4[n]\) is zero everywhere. --- # **Final Answers Summary** | Sequence | Z-Transform | ROC | |:-----------:|:--------------:|:----------------:| | (i) | \(z^2 + 2 - 3z^{-1}\) | All \(z\) except \(z=,\,\infty\) | | (ii) | \(\frac{z^{-2}}{.7(1-.7z^{-1})}\) | \(|z| > .7\) | | (iii) | \(\frac{1 + .6 z^{-1}}{(1 + .6 z^{-1})^2}\) | \(|z| > .6\) | | (iv) | \(.64\frac{z^2}{1 - .8z} + \frac{z^{-1}}{1 - .5z^{-1}}\) | **Empty (no overlap)** | --- Let me know if you need further clarifications on any part!

# Z-Transform & ROC for Given Sequences This document provides the step-by-step solutions for finding the Z-transform and indicating the Region of Convergence (ROC) for each specified sequence. --- ## (i) \( x_1[n] = \delta[n+2] + 2\delta[n] - 3\delta[n-1] \) ### Step 1: Z-Transform Recall that the Z-transform of \(\delta[n-k]\) is \(z^{-k}\): \[ \begin{align*} X_1(z) &= \mathcal{Z} \{ \delta[n+2] \} + 2\mathcal{Z} \{ \delta[n] \} - 3\mathcal{Z} \{ \delta[n-1] \} \\ &= z^{2} + 2 \cdot 1 - 3z^{-1} \\ &= z^2 + 2 - 3z^{-1} \end{align*} \] ### Step 2: Region of Convergence (ROC) For finite-length sequences, the ROC is **all \(z\) except possibly \(z=0\) and \(z=\infty\)**. - **ROC:** All \(z\) except \(z=0, \infty\) --- ## (ii) \( x_2[n] = (0.7)^{n-3}u[n-2] \) ### Step 1: Simplify the Sequence Let \( k = n-2 \), thus \( n = k + 2 \): \[ x_2[n] = (0.7)^{n-3}u[n-2] = (0.7)^{(n-2)-1} u[n-2] = (0.7)^{-1} (0.7)^{n-2} u[n-2] \] ### Step 2: Z-Transform Using the Z-transform property for shifted sequences: \[ X_2(z) = (0.7)^{-1} z^{-2} \frac{1}{1 - 0.7z^{-1}} \] ### Step 3: ROC - **ROC:** \(|z| > 0.7\) --- ## (iii) \( x_3[n] = (-1)^n (n+1) (0.6)^n u[n] \) ### Step 1: Combine Terms \[ x_3[n] = (n+1)(-0.6)^n u[n] \] ### Step 2: Z-Transform Using the property for sequences of the form \((n+1)a^n u[n]\): \[ X_3(z) = \frac{1 - (-0.6)z^{-1}}{(1 + 0.6z^{-1})^2} \] ### Step 3: ROC - **ROC:** \(|z| > 0.6\) --- ## (iv) \( x_4[n] = (1.25)^n u[-n-2] + (0.5)^n u[n-1] \) ### Step 1: Z-Transform for Each Term #### First Term: \( (1.25)^n u[-n-2] \) This term is left-sided: \[ \mathcal{Z}\{ (1.25)^n u[-n-2] \} = \frac{(1.25)^{-2} z^2}{1 - (1/1.25)z} \] #### Second Term: \( (0.5)^n u[n-1] \) This term is right-sided: \[ X_4(z) = z^{-1} \frac{1}{1 - 0.5z^{-1}} \] ### Step 2: Combine The overall Z-transform is: \[ X_4(z) = \frac{.64 z^2}{1 - .8z} + \frac{z^{-1}}{1 - 0.5z^{-1}} \] ### Step 3: ROC - First term: left-sided, **ROC:** \(|z| < 1.25\) - Second term: right-sided, **ROC:** \(|z| > 0.5\) - **Overall ROC:** No intersection (ROC is empty). --- # Final Answers Summary | Sequence | Z-Transform | ROC | |:--------:|:-----------:|:---:| | (i) | \(z^2 + 2 - 3z^{-1}\) | All \(z\) except \(z=0, \infty\) | | (ii) | \(\frac{z^{-2}}{0.7(1-0.7z^{-1})}\) | \(|z| > 0.7\) | | (iii) | \(\frac{1 + 0.6 z^{-1}}{(1 + 0.6 z^{-1})^2}\) | \(|z| > 0.6\) | | (iv) | \(\frac{0.64 z^2}{1 - 0.8z} + \frac{z^{-1}}{1 - 0.5z^{-1}}\) | Empty | --- Let me know if you need further clarifications on any part!

# Z-Transform & ROC for Given Sequences This document provides the step-by-step solutions for finding the Z-transform and indicating the Region of Convergence (ROC) for each specified sequence. --- ## (i) \( x_1[n] = \delta[n+2] + 2\delta[n] - 3\delta[n-1] \) ### Step 1: Z-Transform The Z-transform of \(\delta[n-k]\) is \(z^{-k}\): \[ \begin{align*} X_1(z) &= \mathcal{Z} \{ \delta[n+2] \} + 2\mathcal{Z} \{ \delta[n] \} - 3\mathcal{Z} \{ \delta[n-1] \} \\ &= z^{2} + 2 \cdot 1 - 3z^{-1} \\ &= z^2 + 2 - 3z^{-1} \end{align*} \] ### Step 2: Region of Convergence (ROC) For finite-length sequences, the ROC is **all \(z\) except possibly \(z=0\) and \(z=\infty\)**. - **ROC:** All \(z\) except \(z=0, \infty\) --- ## (ii) \( x_2[n] = (0.7)^{n-3}u[n-2] \) ### Step 1: Simplify the Sequence Let \( k = n-2 \), thus \( n = k + 2 \): \[ x_2[n] = (0.7)^{n-3}u[n-2] = (0.7)^{(n-2)-1} u[n-2] = (0.7)^{-1} (0.7)^{n-2} u[n-2] \] ### Step 2: Z-Transform Using the Z-transform property for shifted sequences: \[ X_2(z) = (0.7)^{-1} z^{-2} \frac{1}{1 - 0.7z^{-1}} = \frac{z^{-2}}{0.7(1 - 0.7z^{-1})} \] ### Step 3: ROC - **ROC:** \(|z| > 0.7\) --- ## (iii) \( x_3[n] = (-1)^n (n+1) (0.6)^n u[n] \) ### Step 1: Combine Terms \[ x_3[n] = (n+1)(-0.6)^n u[n] \] ### Step 2: Z-Transform Using the property for sequences of the form \((n+1)a^n u[n]\): \[ X_3(z) = \frac{1 - (-0.6)z^{-1}}{(1 + 0.6z^{-1})^2} \] ### Step 3: ROC - **ROC:** \(|z| > 0.6\) --- ## (iv) \( x_4[n] = (1.25)^n u[-n-2] + (0.5)^n u[n-1] \) ### Step 1: Z-Transform for Each Term #### First Term: \( (1.25)^n u[-n-2] \) This term is left-sided: \[ \mathcal{Z}\{ (1.25)^n u[-n-2] \} = \frac{(1.25)^{-2} z^2}{1 - (1/1.25)z} \] #### Second Term: \( (0.5)^n u[n-1] \) This term is right-sided: \[ X_4(z) = z^{-1} \frac{1}{1 - 0.5z^{-1}} \] ### Step 2: Combine The overall Z-transform is: \[ X_4(z) = (1.25)^{-2} z^2 \cdot \frac{1}{1 - 0.8z} + z^{-1} \cdot \frac{1}{1 - 0.5z^{-1}} \] ### Step 3: ROC - First term: left-sided, **ROC:** \(|z| < 1.25\) - Second term: right-sided, **ROC:** \(|z| > 0.5\) - **Overall ROC:** No intersection (ROC is empty). --- # Final Answers Summary | Sequence | Z-Transform | ROC | |:--------:|:-----------:|:---:| | (i) | \(z^2 + 2 - 3z^{-1}\) | All \(z\) except \(z=0, \infty\) | | (ii) | \(\frac{z^{-2}}{0.7(1-0.7z^{-1})}\) | \(|z| > 0.7\) | | (iii) | \(\frac{1 + 0.6 z^{-1}}{(1 + 0.6 z^{-1})^2}\) | \(|z| > 0.6\) | | (iv) | \(\frac{(1.25)^{-2} z^2}{1 - 0.8z} + z^{-1} \cdot \frac{1}{1 - 0.5z^{-1}}\) | Empty | --- Let me know if you need further clarifications on any part!