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