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To solve the problem, let's break down the information provided in the image:

### Given:
- **Sampling Rate (\( f_s \))**: 40,000 Hz
- **Filter Type**: Butterworth lowpass filter
- **Cutoff Frequency (\( f_c \))**: 8,000 Hz
- **Percentage of Aliasing Level**: Less than 1% (or \( A(f) < 0.01 \))

### Objective:
Determine the **order of the anti-aliasing lowpass filter**.

### Steps to Solve:

1. **Butterworth Filter Characteristics**:
   - The Butterworth filter is characterized by a maximally flat magnitude response in the passband and a roll-off that is proportional to the filter order \( n \).
   - The magnitude response of a Butterworth filter is given by:
     \[
     \left|\frac{H(j\omega)}{H(j\omega_c)}\right|^2 = \frac{1}{1 + \left(\frac{\omega}{\omega_c}\right)^{2n}}
     \]
   - The attenuation level \( A(f) \) is the reciprocal of the magnitude response squared.

2. **Determine the Attenuation**:
   - The attenuation must be such that the aliasing is below 1%.
   - \( A(f) < 0.01 \) corresponds to \( \frac{1}{\sqrt{1 + \left(\frac{f_a}{f_c}\right)^{2n}}} < 0.01 \), where \( f_a = f_s - f_c \).

3. **Calculate \( f_a \)**:
   - \( f_a = 40,000 \text{ Hz} - 8,000 \text{ Hz} = 32,000 \text{ Hz} \).

4. **Set Up the Inequality**:
   \[
   \frac{1}{\sqrt{1 + \left(\frac{32,000}{8,000}\right)^{2n}}} < 0.01
   \]

5. **Simplify the Expression**:
   \[
   \sqrt{1 + \left(4\right)^{2n}} > 100
   \]
   \[
   1 + 4^{2n} > 10,000
   \]
   \[
   4^{2n} > 9,999
   \]
   \[
   (2^2)^{2n} > 9,999
   \]
   \[
   2^{4n} > 9,999
   \]

6. **Solve for \( n \)**:
   - Take the logarithm base 2:
     \[
     4n \log_2(2) > \log_2(9,999)
     \]
     \[
     4n > \log_2(9,999)
     \]
   - Since \( \log_2(10,000) = 13.29 \),
     \[
     4n > 13.29
     \]
   - \( n > 3.32 \), therefore \( n \) must be at least 4.

### Conclusion:
The order of the anti-aliasing filter should be **4**.

The correct answer to the question is **option D (4)**.

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Do you need further details or have any questions about this solution? Here are five related questions that might help you delve deeper:

1. How does the order of a Butterworth filter affect its frequency response?
2. What are the general characteristics of Butterworth filters compared to Chebyshev filters?
3. How do you determine the cutoff frequency for an anti-aliasing filter in general?
4. What role does the sampling rate play in avoiding aliasing?
5. How would the solution change if the required aliasing level was stricter (e.g., 0.1%)?

**Tip**: Always remember that the higher the order of a Butterworth filter, the steeper the roll-off, but it also increases the complexity of the filter design.

Learn how to determine the order of an anti-aliasing Butterworth filter with a detailed mathematical analysis. This problem involves signal processing concepts such as digital filters and the Butterworth filter magnitude response. Suitable for advanced level learners and professionals.

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