So say I do a fourier transform of a rectangular function with magnitude 1 from (0, NT). The fourier transform of this would be:(adsbygoogle = window.adsbygoogle || []).push({});

[itex]f(jΩ) = \frac{1-e^{-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{1-e^{-jwN}}{1-e^{-jw}} = e^{-jw(N-1)/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(N-1)/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 1-e^{-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.

**Physics Forums | Science Articles, Homework Help, Discussion**

Dismiss Notice

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Fourier of boxcar vs rectangular

Can you offer guidance or do you also need help?

Draft saved
Draft deleted

**Physics Forums | Science Articles, Homework Help, Discussion**