Question

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 1

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