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Fourier of boxcar vs rectangular 
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#1
Jun1214, 02:03 AM

P: 25

So say I do a fourier transform of a rectangular function with magnitude 1 from (0, NT). The fourier transform of this would be:
[itex]f(jΩ) = \frac{1e^{jΩNT}}{jΩ} = NT\cdot{e^{jΩNT/2}}\cdot{sinc(ΩNT/2)}[/itex] Now say if I sample this rectangle at time T producing N samples, the DTFT of this is: [itex]f(e^{jw}) = \frac{1e^{jwN}}{1e^{jw}} = e^{jw(N1)/2}\cdot\frac{sin(wN/2)}{sin(w/2)}[/itex] Since DTFT and Fourier Transform is related by Ω = wT where [itex]f(e^{jw}) = \frac{1}{T}\sum{f(jΩ + j2\pin)}[/itex] Now if I try this method I get to this point: [itex]f(e^{jw}) = e^{jw(N1)/2}\cdot{sin(wN/2)}\cdot{\sum\frac{1}{w+2{\pi}n}}[/itex] This is where I get stuck, because that last summation needs to somehow equal sin(w/2) or 1e^{jw}. The summation is from negative infinity to infinity. Was wondering if there is some math trick that gives the result of that summation. Thanks. 


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