https://file.aiquickdraw.com/mathbot/file/1729063244958xo5ikc86.jpg

Let's break down the problem in steps to calculate the output \( y(n) \) based on the provided information.

### Problem Breakdown:

#### 1. **Input Signal \( x(n) \)**:
The input signal \( x(n) \) is given as:
\[
x(n) = 12 \cos\left(\frac{\pi n}{2}\right) + \frac{3}{2} \cos\left(\frac{2\pi n}{3}\right)
\]
This is a sum of two cosine signals with different frequencies.

- The first term \( 12 \cos\left(\frac{\pi n}{2}\right) \) has a frequency of \( \frac{1}{4} \) (since \( \cos(2\pi f n) \), so \( f_1 = \frac{1}{4} \)).
- The second term \( \frac{3}{2} \cos\left(\frac{2\pi n}{3}\right) \) has a frequency of \( \frac{1}{3} \) (so \( f_2 = \frac{1}{3} \)).

#### 2. **Channel's Amplitude Response \( S(f) \)**:
The channel is modeled as a linear time-invariant (LTI) system with an amplitude response given by:
\[
S(f) = \frac{2f}{f + 2}
\]

This response will affect each frequency component of the input signal \( x(n) \).

#### 3. **Effect of Channel on Each Frequency Component**:
For each frequency component of \( x(n) \), the channel will scale it based on its amplitude response \( S(f) \).

- For the first term \( 12 \cos\left(\frac{\pi n}{2}\right) \), the frequency is \( f_1 = \frac{1}{4} \). So, the amplitude response will be:
  \[
  S\left(\frac{1}{4}\right) = \frac{2 \times \frac{1}{4}}{\frac{1}{4} + 2} = \frac{\frac{1}{2}}{\frac{9}{4}} = \frac{1}{2} \times \frac{4}{9} = \frac{2}{9}
  \]
  The new amplitude for this term will be:
  \[
  12 \times \frac{2}{9} = \frac{24}{9} = \frac{8}{3}
  \]
  So the output for this term will be \( \frac{8}{3} \cos\left(\frac{\pi n}{2}\right) \).

- For the second term \( \frac{3}{2} \cos\left(\frac{2\pi n}{3}\right) \), the frequency is \( f_2 = \frac{1}{3} \). So, the amplitude response will be:
  \[
  S\left(\frac{1}{3}\right) = \frac{2 \times \frac{1}{3}}{\frac{1}{3} + 2} = \frac{\frac{2}{3}}{\frac{7}{3}} = \frac{2}{7}
  \]
  The new amplitude for this term will be:
  \[
  \frac{3}{2} \times \frac{2}{7} = \frac{6}{14} = \frac{3}{7}
  \]
  So the output for this term will be \( \frac{3}{7} \cos\left(\frac{2\pi n}{3}\right) \).

#### 4. **Final Output \( y(n) \)**:
The final output \( y(n) \) will be the sum of the two terms after being affected by the channel:
\[
y(n) = \frac{8}{3} \cos\left(\frac{\pi n}{2}\right) + \frac{3}{7} \cos\left(\frac{2\pi n}{3}\right)
\]

This is the output signal after passing through the LTI system.

---

Do you need more details on any step or further clarifications?

### Related Questions:
1. How do you find the frequency of a cosine signal from its equation?
2. What is the role of an amplitude response in an LTI system?
3. How do you apply a system's amplitude response to a given signal?
4. Can you explain why different frequencies are affected differently by the channel?
5. How would the output change if the channel response \( S(f) \) was different?

**Tip**: Always identify the frequencies of the input signal's components before applying the system's response to each.

The input to the channel is given as: x(n) = 12 cos(πn/2) + (3/2) cos(2πn/3), and the channel is modeled as an LTI system with amplitude response S(f) = 2f/(f+2). Calculate the output y(n).

Solve the problem of calculating the output y(n) when the input signal x(n) = 12 cos(πn/2) + (3/2) cos(2πn/3) passes through a Linear Time-Invariant (LTI) system with an amplitude response S(f) = 2f/(f+2). This problem covers key signal processing concepts such as frequency response and system behavior for cosine signals.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation