Question

What is the magnitude response |W(ω)| of a rectangular window function?

(a) \(\frac{|sin(ωM/2)|}{|sin(ω/2)|}\)

(b) \(\frac{|sin(ω/2)|}{|sin(ωM/2)|}\)

(c) \(\frac{|cos(ωM/2)|}{|sin(ω/2)|}\)

(d) None of the mentioned

This question was addressed to me in a national level competition.

This question is from Design of Linear Phase FIR Filters Using Windows topic in portion Digital Filters Design of Digital Signal Processing

Answer 1

The correct option is (a) \(\frac{|sin(ωM/2)|}{|sin(ω/2)|}\)

The explanation: We know that for a rectangular window

W(ω)=\(\sum_{n=0}^{M-1} w(n) e^{-jωn}=e^{-jω(M-1)/2} \frac{sin⁡(\frac{ωM}{2})}{sin⁡(\frac{ω}{2})}\)

Thus the window function has a magnitude response

|W(ω)|=\(\frac{|sin(ωM/2)|}{|sin(ω/2)|}\)