Continious FT of a rectangle waveform is real valued, but the DFT of it is not?

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SUMMARY

The Continuous Fourier Transform (CFT) of a rectangular waveform with amplitude 1 between [-u, u] results in a real-valued sinc function. In contrast, the Discrete Fourier Transform (DFT), computed using the Fast Fourier Transform (FFT) algorithm, yields complex numbers, which are complex conjugates. This discrepancy arises because the DFT captures both amplitude and phase information, influenced by the time axis definition. Specifically, altering the rectangular function's definition from [-u, u] to [0, 2u] also results in complex outputs.

PREREQUISITES
  • Understanding of Continuous Fourier Transform (CFT)
  • Familiarity with Discrete Fourier Transform (DFT) and Fast Fourier Transform (FFT)
  • Knowledge of sinc functions and their properties
  • Basic concepts of complex numbers and their conjugates
NEXT STEPS
  • Study the properties of the sinc function in signal processing
  • Learn about the implications of phase information in DFT outputs
  • Explore the differences between CFT and DFT in detail
  • Investigate the effects of different time axis definitions on Fourier Transforms
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Signal processing engineers, electrical engineers, and students studying Fourier analysis who seek to understand the differences between continuous and discrete Fourier transformations.

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Continious FT of a rectangle is real valued but DFT of it is not!?

Continious Fourier Transform of a rectangle with amplitude of 1 between [-u,u] is a real valued function (u is a positive number). Actually it is a sinc function.

However when I use discrete Fourier Transform (fft) I obtain complex numbers. I know they are complex conjugates but according to the formulas of continious and discrete versions of the transformation, the only difference should have been the dt factor. Am I missing something?
 
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The DFT returns both amplitude and phase information.
 
It's a matter of picking your time axis.
If your rectangular function is defned from 0 to 2u instead of -u to u you will get a complex number too.
 

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