Question

What is the Fourier transform of rectangular window of length L?

(a) \(\frac{sin⁡(\frac{ωL}{2})}{sin⁡(\frac{ω}{2})} e^{jω(L+1)/2}\)

(b) \(\frac{sin⁡(\frac{ωL}{2})}{sin⁡(\frac{ω}{2})} e^{jω(L-1)/2}\)

(c) \(\frac{sin⁡(\frac{ωL}{2})}{sin⁡(\frac{ω}{2})} e^{-jω(L-1)/2}\)

(d) None of the mentioned

I had been asked this question in examination.

My question is taken from Frequency Analysis of Signals Using DFT in section Discrete Fourier Transform – Properties and Applications of Digital Signal Processing

Answer 1

Correct choice is (c) \(\frac{sin⁡(\frac{ωL}{2})}{sin⁡(\frac{ω}{2})} e^{-jω(L-1)/2}\)

The best explanation: We know that the equation for the rectangular window w(n) is given as

w(n)=1, 0≤ n≤ L-1

=0, otherwise

We know that the Fourier transform of a signal x(n) is given as

X(ω)=\(\sum_{n=-∞}^∞ x(n)e^{-jωn}\)

=>W(ω)=\(\sum_{n=0}^{L-1} e^{-jωn}=\frac{sin⁡(\frac{ωL}{2})}{sin⁡(\frac{ω}{2})} e^{-jω(L-1)/2}\)