Question

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 1

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 \(log⁡[1+(\frac{\Omega_P}{\Omega_C})^{2N}]\).