Digital Signal Processing

Design of Low Pass Butterworth Filters

Question 1
Marks : +2 | -2
Pass Ratio : 100%
Which of the following equation is True?
\\([\\frac{\\Omega_P}{\\Omega_C}]^{2N} = 10^{-K_P/10}+1\\)
\\([\\frac{\\Omega_P}{\\Omega_C}]^{2N} = 10^{K_P/10}+1\\)
\\([\\frac{\\Omega_P}{\\Omega_C}]^{2N} = 10^{-K_P/10}-1\\)
None of the mentioned
Explanation:
We know that,
Question 2
Marks : +2 | -2
Pass Ratio : 100%
Which of the following equation is True?
\\([\\frac{\\Omega_S}{\\Omega_C} ]^{2N} = 10^{-K_S/10}+1\\)
\\([\\frac{\\Omega_S}{\\Omega_C} ]^{2N} = 10^{K_S/10}+1\\)
\\([\\frac{\\Omega_S}{\\Omega_C} ]^{2N} = 10^{-K_S/10}-1\\)
None of the mentioned
Explanation:
We know that,
Question 3
Marks : +2 | -2
Pass Ratio : 100%
What is the system function of the Butterworth filter with specifications as pass band gain KP=-1 dB at ΩP=4 rad/sec and stop band attenuation greater than or equal to 20dB at ΩS=8 rad/sec?
\\(\\frac{1}{s^5+14.82s^4+109.8s^3+502.6s^2+1422.3s+2012.4}\\)
\\(\\frac{1}{s^5+14.82s^4+109.8s^3+502.6s^2+1422.3s+1}\\)
\\(\\frac{2012.4}{s^5+14.82s^4+109.8s^3+502.6s^2+1422.3s+2012.4}\\)
None of the mentioned
Explanation:
From the given question,
Question 4
Marks : +2 | -2
Pass Ratio : 100%
If H(s)=\\(\\frac{1}{s^2+s+1}\\) represent the transfer function of a low pass filter (not Butterworth) with a pass band of 1 rad/sec, then what is the system function of a low pass filter with a pass band 10 rad/sec?
\\(\\frac{100}{s^2+10s+100}\\)
\\(\\frac{s^2}{s^2+s+1}\\)
\\(\\frac{s^2}{s^2+10s+100}\\)
None of the mentioned
Explanation:
The low pass-to-low pass transformation is
Question 5
Marks : +2 | -2
Pass Ratio : 100%
What is the value of gain at the pass band frequency, i.e., what is the value of KP?
-10 \\(log⁡ [1-(\\frac{\\Omega_P}{\\Omega_C})^{2N}]\\)
-10 \\(log⁡ [1+(\\frac{\\Omega_P}{\\Omega_C})^{2N}]\\)
10 \\(log⁡ [1-(\\frac{\\Omega_P}{\\Omega_C})^{2N}]\\)
10 \\(log⁡ [1+(\\frac{\\Omega_P}{\\Omega_C})^{2N}]\\)
Explanation:
We know that the formula for gain is K = 20 log|H(jΩ)|
Question 6
Marks : +2 | -2
Pass Ratio : 100%
The cutoff frequency of the low pass Butterworth filter is the arithmetic mean of the two cutoff frequencies as found above.
True
False
Explanation:
The arithmetic mean of the two cutoff frequencies as found above is the final cutoff frequency of the low pass Butterworth filter.
Question 7
Marks : +2 | -2
Pass Ratio : 100%
If H(s)=\\(\\frac{1}{s^2+s+1}\\) represent the transfer function of a low pass filter (not Butterworth) with a pass band of 1 rad/sec, then what is the system function of a band pass filter with a pass band of 10 rad/sec and a center frequency of 100 rad/sec?
\\(\\frac{s^2}{s^4+10s^3+20100s^2+10^5 s+1}\\)
\\(\\frac{100s^2}{s^4+10s^3+20100s^2+10^5 s+1}\\)
\\(\\frac{s^2}{s^4+10s^3+20100s^2+10^5 s+10^8}\\)
\\(\\frac{100s^2}{s^4+10s^3+20100s^2+10^5 s+10^8}\\)
Explanation:
The low pass-to-band pass transformation is
Question 8
Marks : +2 | -2
Pass Ratio : 100%
What is the expression for cutoff frequency in terms of pass band gain?
\\(\\frac{\\Omega_P}{(10^{-K_P/10}-1)^{1/2N}}\\)
\\(\\frac{\\Omega_P}{(10^{-K_P/10}+1)^{1/2N}}\\)
\\(\\frac{\\Omega_P}{(10^{K_P/10}-1)^{1/2N}}\\)
None of the mentioned
Explanation:
We know that,
Question 9
Marks : +2 | -2
Pass Ratio : 100%
What is the value of gain at the stop band frequency, i.e., what is the value of KS?
-10 \\(log⁡[1+(\\frac{\\Omega_S}{\\Omega_C})^{2N}]\\)
-10 \\(log⁡[1-(\\frac{\\Omega_S}{\\Omega_C})^{2N}]\\)
10 \\(log⁡[1-(\\frac{\\Omega_S}{\\Omega_C})^{2N}]\\)
10 \\(log⁡[1+(\\frac{\\Omega_S}{\\Omega_C})^{2N}]\\)
Explanation:
We know that the formula for gain is
Question 10
Marks : +2 | -2
Pass Ratio : 100%
If H(s)=\\(\\frac{1}{s^2+s+1}\\) represent the transfer function of a low pass filter (not Butterworth) with a pass band of 1 rad/sec, then what is the system function of a high pass filter with a cutoff frequency of 1rad/sec?
\\(\\frac{100}{s^2+10s+100}\\)
\\(\\frac{s^2}{s^2+s+1}\\)
\\(\\frac{s^2}{s^2+10s+100}\\)
None of the mentioned
Explanation:
The low pass-to-high pass transformation is