1. **Find the z-transform of \(\delta(n)\).** The sequence \(\delta(n)\) is the unit impulse, which is 1 at \(n=0\) and 0 elsewhere. The z-transform is defined as: $$X(z) = \sum_{n=-\infty}^{\infty} x(n) z^{-n}$$ For \(\delta(n)\), only \(n=0\) term contributes: $$X(z) = \delta(0) z^{0} = 1$$ 2. **Determine z-transform and ROC of sequence \(x(n) = \{1, 2, 3\}\).** Assuming \(x(0)=1, x(1)=2, x(2)=3\), zero elsewhere. $$X(z) = 1 \cdot z^{0} + 2 \cdot z^{-1} + 3 \cdot z^{-2} = 1 + 2z^{-1} + 3z^{-2}$$ Since finite length, ROC is entire \(z\)-plane except possibly \(z=0\). 3. **Find z-transform of (a) \(\delta[n-k]\), (b) \{1, 2, 3, 0, 5\}.** (a) \(\delta[n-k]\) is impulse shifted by \(k\): $$X(z) = z^{-k}$$ (b) For \(x(n) = \{1, 2, 3, 0, 5\}\) with \(n=0\) to 4: $$X(z) = 1 + 2z^{-1} + 3z^{-2} + 0 \cdot z^{-3} + 5z^{-4} = 1 + 2z^{-1} + 3z^{-2} + 5z^{-4}$$ 4. **Find z-transform of \(3^{u(n)}\).** Here \(u(n)\) is unit step, so \(x(n) = 3^n\) for \(n \geq 0\). $$X(z) = \sum_{n=0}^\infty 3^n z^{-n} = \sum_{n=0}^\infty (3 z^{-1})^n = \frac{1}{1 - 3 z^{-1}} = \frac{z}{z - 3}$$ ROC: \(|z| > 3\). 5. **Compute Z-transform of \(x(n) = \{-4, 1, 6, 0, 0, 0, 0, 3\}\).** Assuming \(n=0\) to 7: $$X(z) = -4 + 1 z^{-1} + 6 z^{-2} + 0 + 0 + 0 + 0 + 3 z^{-7} = -4 + z^{-1} + 6 z^{-2} + 3 z^{-7}$$ 6. **Find Z-transform and plot pole-zero of \(x[n] = \{1,2,3,0,-3,-2,-1\}\).** Assuming \(n=0\) to 6: $$X(z) = 1 + 2 z^{-1} + 3 z^{-2} + 0 + (-3) z^{-4} + (-2) z^{-5} + (-1) z^{-6}$$ ROC is entire \(z\)-plane except possibly \(z=0\). Pole-zero plot would show zeros at roots of numerator polynomial. 7. **Determine Z-transform and ROC of \(x(n) = [2(3)^n - 5(4)^n] u(n)\).** Since \(u(n)\) is unit step: $$X(z) = 2 \sum_{n=0}^\infty (3^n) z^{-n} - 5 \sum_{n=0}^\infty (4^n) z^{-n} = 2 \frac{1}{1 - 3 z^{-1}} - 5 \frac{1}{1 - 4 z^{-1}} = \frac{2z}{z - 3} - \frac{5z}{z - 4}$$ ROC: \(|z| > 4\) (intersection of both). 8. **Determine Z-transform and ROC of finite duration signal \(x(n) = \{1, 2, 5, 4, 0, 7\}\).** Assuming \(n=0\) to 5: $$X(z) = 1 + 2 z^{-1} + 5 z^{-2} + 4 z^{-3} + 0 + 7 z^{-5}$$ ROC is entire \(z\)-plane except possibly \(z=0\). 9. **Determine Z-transform and ROC of \(x(n) = \alpha^n u(n) + \beta^n u(-n-1)\).** First term: $$X_1(z) = \sum_{n=0}^\infty \alpha^n z^{-n} = \frac{1}{1 - \alpha z^{-1}}, \quad |z| > |\alpha|$$ Second term: $$X_2(z) = \sum_{n=-\infty}^{-1} \beta^n z^{-n} = \sum_{m=1}^\infty \beta^{-m} z^{m} = \sum_{m=1}^\infty (z / \beta)^m = \frac{z/\beta}{1 - z/\beta} = \frac{z}{\beta - z}, \quad |z| < |\beta|$$ Overall: $$X(z) = \frac{1}{1 - \alpha z^{-1}} + \frac{z}{\beta - z}$$ ROC: \(|\alpha| < |z| < |\beta|\). 10. **Find inverse Z-transform of \(X(Z) = \frac{z+1}{(z+2)(z+3)}\).** Partial fraction decomposition: $$\frac{z+1}{(z+2)(z+3)} = \frac{A}{z+2} + \frac{B}{z+3}$$ Solve for A, B: Multiply both sides by \((z+2)(z+3)\): $$z+1 = A(z+3) + B(z+2)$$ Set \(z = -2\): $$-2 + 1 = A(1) + B(0) \Rightarrow -1 = A$$ Set \(z = -3\): $$-3 + 1 = A(0) + B(-1) \Rightarrow -2 = -B \Rightarrow B = 2$$ So: $$X(Z) = \frac{-1}{z+2} + \frac{2}{z+3}$$ Inverse Z-transform corresponds to sequences: $$x(n) = -(-2)^{-n-1} u(n) + 2(-3)^{-n-1} u(n)$$ 11. **Inverse Z-transform of \(X(Z) = \frac{z}{z - a}\) for \(|z| < |a|\).** Rewrite: $$X(Z) = \frac{z}{z - a} = \frac{1}{1 - a z^{-1}}$$ For \(|z| < |a|\), ROC inside circle, inverse transform is right-sided: $$x(n) = -a^{n-1} u(-n-1)$$ 12. **Inverse Z-transform of \(X[z] = \frac{z^2}{(z - 0.5)(z - 1)^2}\).** Partial fraction decomposition and inverse transform involve residues at poles 0.5 and 1. 13. **Inverse Z-transform of \(X(z) = \frac{1 + 3 z^{-1}}{1 + 3 z^{-1} + 2 z^{-2}}\), \(|z| > 2\).** Rewrite denominator: $$1 + 3 z^{-1} + 2 z^{-2} = (1 + z^{-1})(1 + 2 z^{-1})$$ Use long division or partial fractions to find inverse. 14. **Inverse Z-transform of \(X(Z) = 4Z^{-3} - 6Z^{-2} + 9 + 5Z + 3Z^2 - 2Z^3\).** Coefficients correspond to sequence values at respective \(n\). 15. **Unit step response of \(h(n) = 5^n u(n)\).** Unit step response is \(s(n) = \sum_{k=0}^n h(k) = \sum_{k=0}^n 5^k = \frac{5^{n+1} - 1}{5 - 1} = \frac{5^{n+1} - 1}{4}\). 16. **Determine \(x(n)\) using long division for \(X(Z) = \frac{1 + 2Z^{-1}}{1 + 2Z^{-1} + Z^{-2}}\), causal.** Perform polynomial division in \(Z^{-1}\) to find sequence coefficients. 17. **Convolution using Z-transform for \(x(n) = \{1, 2, 1\}\) and \(h(n) = \{1, 2, 3, 4, -5\}\).** Z-transform multiplication: $$X(z) = 1 + 2 z^{-1} + z^{-2}$$ $$H(z) = 1 + 2 z^{-1} + 3 z^{-2} + 4 z^{-3} - 5 z^{-4}$$ Convolution is inverse Z-transform of \(X(z) H(z)\). 18. **Inverse Z-transform of \(X(z) = \frac{1 - \frac{1}{2} z^{-1}}{1 + \frac{3}{2} z^{-1} + \frac{1}{8} z^{-2}}\).** Use partial fraction or long division to find sequence.