What is the value of gain at the stop band frequency, i.e., what is the value of KS?(a) -10 \(log⁡[1+(\frac{\Omega_S}{\Omega_C})^{2N}]\)(b) -10 \(log⁡[1-(\frac{\Omega_S}{\Omega_C})^{2N}]\)(c) 10 \(log⁡[1-(\frac{\Omega_S}{\Omega_C})^{2N}]\)(d) 10 \(log⁡[1+(\frac{\Omega_S}{\Omega_C})^{2N}]\)This question was posed to me in exam.My question is based upon Design of Low Pass Butterworth Filters in portion Digital Filters Design of Digital Signal Processing

What is the value of gain at the stop band frequency, i.e., what is th

1.	What is the value of gain at the stop band frequency, i.e., what is the value of KS?(a) -10 \(log⁡[1+(\frac{\Omega_S}{\Omega_C})^{2N}]\)(b) -10 \(log⁡[1-(\frac{\Omega_S}{\Omega_C})^{2N}]\)(c) 10 \(log⁡[1-(\frac{\Omega_S}{\Omega_C})^{2N}]\)(d) 10 \(log⁡[1+(\frac{\Omega_S}{\Omega_C})^{2N}]\)This question was posed to me in exam.My question is based upon Design of Low Pass Butterworth Filters in portion Digital Filters Design of Digital Signal Processing
Answer» The correct option is (a) -10 \(log⁡[1+(\frac{\Omega_S}{\Omega_C})^{2N}]\) To explain: We know that the formula for GAIN is K = 20 log\|H(jΩ)\| We know that \(\|H(j \Omega)\|=\frac{1}{\SQRT{(1+(\frac{\Omega}{\Omega_C})^{2N}}}\) By applying 20log on both sides of above equation, we get K = 20 \(log\|H(j\Omega)\|=-20 [log⁡[1+(\frac{\Omega}{\Omega_C})^{2N}]]^{1/2}\) = -10 \(log⁡[1+(\frac{\Omega}{\Omega_C})^{2N}]\) We know that K= KS at Ω=ΩS => KS=-10 \(log⁡[1+(\frac{\Omega_S}{\Omega_C})^{2N}]\).

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