https://file.aiquickdraw.com/mathbot/file/1725820168329hnwenupy.jpg

The problem you provided asks to determine whether the given signals are energy or power signals and compute the corresponding energy or average power.

The two signals given in the image are:
1. \( x(n) = 8 \cdot \sin(\pi n) \)
2. \( x(n) = 4e^{2j n} \cdot u(n) \)

### Definitions:
- **Energy Signal**: A signal is classified as an energy signal if its total energy is finite and its average power is zero.
- **Power Signal**: A signal is classified as a power signal if it has finite and non-zero average power, but infinite energy.

To compute the energy and power, we use:
- **Energy**: \( E_x = \sum_{n=-\infty}^{\infty} |x(n)|^2 \)
- **Power**: \( P_x = \lim_{N \to \infty} \frac{1}{2N+1} \sum_{n=-N}^{N} |x(n)|^2 \)

Let's analyze each signal individually:

### 1. \( x(n) = 8 \sin(\pi n) \)
- \( \sin(\pi n) = 0 \) for all integer values of \( n \), which means the signal \( x(n) \) is zero at all points.
- Since \( x(n) = 0 \), both energy and power will be zero.

Thus, this is **neither an energy nor a power signal** because it is a trivial zero signal.

### 2. \( x(n) = 4e^{2j n} u(n) \)
- This is a complex exponential signal, modulated by the unit step function \( u(n) \), which means it exists only for \( n \geq 0 \).
- The magnitude of this signal is constant: \( |4e^{2jn}| = 4 \) for all \( n \geq 0 \).

Now, we calculate its power:
\[
P_x = \lim_{N \to \infty} \frac{1}{2N+1} \sum_{n=0}^{N} |4|^2 = \lim_{N \to \infty} \frac{1}{2N+1} \sum_{n=0}^{N} 16 = \lim_{N \to \infty} \frac{16(N+1)}{2N+1} = 8
\]
Hence, this signal is a **power signal** with average power \( P_x = 8 \).

### Conclusion:
- \( x(n) = 8 \sin(\pi n) \) is **neither an energy nor a power signal**.
- \( x(n) = 4e^{2jn}u(n) \) is a **power signal** with average power \( P_x = 8 \).

---

Would you like further clarification on any part of the solution?

Here are 5 related questions to expand your understanding:
1. What are the differences between energy and power signals in continuous-time systems?
2. How would you calculate energy and power for a continuous-time signal?
3. What happens to the classification if the sinusoidal function is non-zero?
4. Can a signal be both an energy and a power signal?
5. How does the unit step function \( u(n) \) affect the energy and power of a signal?

**Tip**: Always remember that periodic signals are typically power signals, while signals that decay over time are usually energy signals.

Check whether the following signal is energy or power and compute the corresponding energy or average power for x(n) = 8 sin(πn) and x(n) = 4e^(2jn)u(n).

Analyze whether the signals x(n) = 8sin(πn) and x(n) = 4e^(2jn)u(n) are energy or power signals. Learn how to compute the corresponding energy or average power, applying key formulas for discrete signals. Suitable for undergraduate students studying signal processing.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation