To solve for the $Z$-transform of each given function, we can apply the $Z$-transform definitions and common properties. Here's a breakdown of each one:

1. $n \cos(n\theta)$

The $Z$-transform of $n \cos(n\theta)$ can be derived using the property of the $Z$-transform of a sequence with a cosine term: $Z\{n \cos(n\theta)\} = -\frac{z \cos(\theta) + z^2}{(z - e^{j\theta})(z - e^{-j\theta})^2}$

2. $\sin\left(\frac{n\pi}{2} + \frac{\pi}{4}\right)$

For this trigonometric sequence, the term inside the sine function can be simplified based on the angle addition formula. Generally, we can write: $Z\{\sin\left(\frac{n\pi}{2} + \frac{\pi}{4}\right)\} = \text{standard trigonometric series}$ However, the exact $Z$-transform depends on applying the trigonometric identity in terms of a sinusoidal series expansion, which would generally yield complex expressions.

3. $e^{-2n}$

The $Z$-transform of an exponential sequence $e^{-an}$ is given by: $Z\{e^{-2n}\} = \frac{1}{z - e^{-2}}$

4. $a^{|n|}$

For $a^{|n|}$, the $Z$-transform needs to consider values of $n$ for both positive and negative cases:

• For $n \geq 0$: $Z\{a^n\} = \frac{z}{z - a}$
• For $n < 0$: $Z\{a^{-n}\} = \frac{z}{z - a^{-1}}$

Combining both cases would yield a more complex expression depending on whether $n$ is positive or negative.

Would you like to see a deeper explanation for any of these specific transforms or examples of their applications?

Follow-Up Questions

1. How do initial conditions affect the $Z$-transform for causal systems?
2. What is the role of the region of convergence (ROC) in determining the $Z$-transform's stability?
3. How does the $Z$-transform differ when applied to left-sided sequences versus right-sided sequences?
4. Can the $Z$-transform be used to solve differential equations, and if so, how?
5. What is the inverse $Z$-transform, and how can it be used to reconstruct signals?

Tip:

When working with trigonometric functions in the $Z$-domain, using Euler's formulas for sine and cosine can simplify the process by converting the terms into exponential forms.