Answer:
![G \sqrt{1 +(\frac{f}{f_c})^{2n}} = 1](https://tex.z-dn.net/?f=%20G%20%5Csqrt%7B1%20%2B%28%5Cfrac%7Bf%7D%7Bf_c%7D%29%5E%7B2n%7D%7D%20%3D%201)
If we square both sides we got:
![G^2 (1+\frac{f}{f_c})^{2n}= 1](https://tex.z-dn.net/?f=%20G%5E2%20%281%2B%5Cfrac%7Bf%7D%7Bf_c%7D%29%5E%7B2n%7D%3D%201)
We divide both sides by
and we got:
![(1+\frac{f}{f_c})^{2n} = \frac{1}{G^2}](https://tex.z-dn.net/?f=%20%281%2B%5Cfrac%7Bf%7D%7Bf_c%7D%29%5E%7B2n%7D%20%3D%20%5Cfrac%7B1%7D%7BG%5E2%7D)
Now we can apply log on both sides and we got:
![2n ln(1+\frac{f}{f_c}) = ln (\frac{1}{G^2})](https://tex.z-dn.net/?f=%202n%20ln%281%2B%5Cfrac%7Bf%7D%7Bf_c%7D%29%20%3D%20ln%20%28%5Cfrac%7B1%7D%7BG%5E2%7D%29)
And solving for n we got:
![n = \frac{ ln (\frac{1}{G^2})}{2ln(1+\frac{f}{f_c})}](https://tex.z-dn.net/?f=n%20%3D%20%5Cfrac%7B%20ln%20%28%5Cfrac%7B1%7D%7BG%5E2%7D%29%7D%7B2ln%281%2B%5Cfrac%7Bf%7D%7Bf_c%7D%29%7D)
And replacing we got:
![n = \frac{ln (\frac{1}{0.1^2})}{2ln(1+\frac{60}{10})}](https://tex.z-dn.net/?f=n%20%3D%20%5Cfrac%7Bln%20%28%5Cfrac%7B1%7D%7B0.1%5E2%7D%29%7D%7B2ln%281%2B%5Cfrac%7B60%7D%7B10%7D%29%7D)
![n = \frac{4.60517}{3.8918}=1.18](https://tex.z-dn.net/?f=%20n%20%3D%20%5Cfrac%7B4.60517%7D%7B3.8918%7D%3D1.18)
And since n needs to be an integer the correct answer would be n=2 for the filter order.
Explanation:
For this case we can use the formula for the Butterworth filter gain given by:
[tec] G = \frac{1}{\sqrt{1 +(\frac{f}{f_c})^{2n}}}[/tex]
Where:
G represent the transfer function and we want that G =0.1 since the desired signal is less than 10% of it's value
represent the corner frequency
represent the original frequency
n represent the filter order and that's the variable that we need to find
![G \sqrt{1 +(\frac{f}{f_c})^{2n}} = 1](https://tex.z-dn.net/?f=%20G%20%5Csqrt%7B1%20%2B%28%5Cfrac%7Bf%7D%7Bf_c%7D%29%5E%7B2n%7D%7D%20%3D%201)
If we square both sides we got:
![G^2 (1+\frac{f}{f_c})^{2n}= 1](https://tex.z-dn.net/?f=%20G%5E2%20%281%2B%5Cfrac%7Bf%7D%7Bf_c%7D%29%5E%7B2n%7D%3D%201)
We divide both sides by
and we got:
![(1+\frac{f}{f_c})^{2n} = \frac{1}{G^2}](https://tex.z-dn.net/?f=%20%281%2B%5Cfrac%7Bf%7D%7Bf_c%7D%29%5E%7B2n%7D%20%3D%20%5Cfrac%7B1%7D%7BG%5E2%7D)
Now we can apply log on both sides and we got:
![2n ln(1+\frac{f}{f_c}) = ln (\frac{1}{G^2})](https://tex.z-dn.net/?f=%202n%20ln%281%2B%5Cfrac%7Bf%7D%7Bf_c%7D%29%20%3D%20ln%20%28%5Cfrac%7B1%7D%7BG%5E2%7D%29)
And solving for n we got:
![n = \frac{ ln (\frac{1}{G^2})}{2ln(1+\frac{f}{f_c})}](https://tex.z-dn.net/?f=n%20%3D%20%5Cfrac%7B%20ln%20%28%5Cfrac%7B1%7D%7BG%5E2%7D%29%7D%7B2ln%281%2B%5Cfrac%7Bf%7D%7Bf_c%7D%29%7D)
And replacing we got:
![n = \frac{ln (\frac{1}{0.1^2})}{2ln(1+\frac{60}{10})}](https://tex.z-dn.net/?f=n%20%3D%20%5Cfrac%7Bln%20%28%5Cfrac%7B1%7D%7B0.1%5E2%7D%29%7D%7B2ln%281%2B%5Cfrac%7B60%7D%7B10%7D%29%7D)
![n = \frac{4.60517}{3.8918}=1.18](https://tex.z-dn.net/?f=%20n%20%3D%20%5Cfrac%7B4.60517%7D%7B3.8918%7D%3D1.18)
And since n needs to be an integer the correct answer would be n=2 for the filter order.