Question

What is the Fourier transform of the rectangular window of length M-1?

(a) \(e^{jω(M-1)/2} \frac{sin⁡(\frac{ωM}{2})}{sin⁡(\frac{ω}{2})}\)

(b) \(e^{jω(M+1)/2} \frac{sin⁡(\frac{ωM}{2})}{sin⁡(\frac{ω}{2})}\)

(c) \(e^{-jω(M+1)/2} \frac{sin⁡(\frac{ωM}{2})}{sin⁡(\frac{ω}{2})}\)

(d) \(e^{-jω(M-1)/2} \frac{sin⁡(\frac{ωM}{2})}{sin⁡(\frac{ω}{2})}\)

I have been asked this question by my college director while I was bunking the class.

Enquiry is from Design of Linear Phase FIR Filters Using Windows in portion Digital Filters Design of Digital Signal Processing

Answer 1

Correct choice is (d) \(e^{-jω(M-1)/2} \frac{sin⁡(\frac{ωM}{2})}{sin⁡(\frac{ω}{2})}\)

To elaborate: We know that the Fourier transform of a function w(n) is defined as

W(ω)=\(\sum_{n=0}^{M-1} w(n) e^{-jωn}\)

For a rectangular window, w(n)=1 for n=0,1,2….M-1

Thus we get

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