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).

---

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.