Topic: Engineering Mathematics
Find the z- transform of $\cosh(n\theta)$
- A. $\frac{z(z-\cosh\theta)}{(z^{2}-2z\cosh\theta+1)}$
- B. $\frac{z(z-\cosh\theta)}{(z^{2}-2z\cosh\theta)}$
- C. $\frac{(z-\cosh\theta)}{(z^{2}-2z\cosh\theta+1)}$
- D. $\frac{z(z-\coth\theta)}{(z^{2}-2z\coth\theta+1)}$
Answer: A. $\frac{z(z-\cosh\theta)}{(z^{2}-2z\cosh\theta+1)}$
Explanation: Using the exponential definition, $\cosh(n\theta) = \frac{e^{n\theta} + e^{-n\theta}}{2}$.
$Z[\cosh(n\theta)] = \frac{1}{2} \left( Z[e^{n\theta}] + Z[e^{-n\theta}] \right) = \frac{1}{2} \left( \frac{z}{z-e^{\theta}} + \frac{z}{z-e^{-\theta}} \right)$.
$= \frac{1}{2} \left[ \frac{z(z-e^{-\theta}) + z(z-e^{\theta})}{z² - z(e^{\theta}+e^{-\theta}) + 1} \right] = \frac{1}{2} \left[ \frac{2z^2 - z(e^{\theta}+e^{-\theta})}{z² - 2z\cosh\theta + 1} \right] = \frac{z² - z\cosh\theta}{z² - 2z\cosh\theta + 1} = \frac{z(z-\cosh\theta)}{z² - 2z\cosh\theta + 1}$.