Question

What is the width of the main lobe of the frequency response of a rectangular window of length \( M-1 \)?

• A. \( \frac{\pi}{M} \)
• B. \( \frac{2\pi}{M} \)
• C. \( \frac{4\pi}{M} \)
• D. \( \frac{8\pi}{M} \)

Hint:

For a rectangular window, the main lobe width of the frequency response is \( \frac{4\pi}{M} \). The wider the window, the narrower the main lobe.

Answer 1

The Correct Option is C

Step 1: The frequency response of a rectangular window is given by the sinc function: \[ W(f) = \frac{\sin(\pi M f)}{\sin(\pi f)} \] where \( M \) is the window length. 
Step 2: The main lobe width of the sinc function is determined by the first zero crossings, which occur at: \[ f = \pm \frac{1}{M} \] 
Step 3: The total width of the main lobe in the frequency domain is: \[ \Delta \omega = \frac{4\pi}{M} \] 
Step 4: Evaluating options: 
- (A) Incorrect: \( \frac{\pi}{M} \) is too narrow. 
- (B) Incorrect: \( \frac{2\pi}{M} \) does not match the main lobe width. 
- (C) Correct: \( \frac{4\pi}{M} \) matches the correct main lobe width. 
- (D) Incorrect: \( \frac{8\pi}{M} \) is too wide.