1.

What is the value of gain at the pass band frequency, i.e., what is the value of KP?(a) -10 \(log⁡ [1-(\frac{\Omega_P}{\Omega_C})^{2N}]\)(b) -10 \(log⁡ [1+(\frac{\Omega_P}{\Omega_C})^{2N}]\)(c) 10 \(log⁡ [1-(\frac{\Omega_P}{\Omega_C})^{2N}]\)(d) 10 \(log⁡ [1+(\frac{\Omega_P}{\Omega_C})^{2N}]\)I had been asked this question in an interview.This interesting question is from Design of Low Pass Butterworth Filters in section Digital Filters Design of Digital Signal Processing

Answer»

The correct answer is (B) -10 \(log⁡ [1+(\frac{\Omega_P}{\Omega_C})^{2N}]\)

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= KP at Ω=ΩP

=> KP=-10 \(log⁡[1+(\frac{\Omega_P}{\Omega_C})^{2N}]\).


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