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

In summary, the continuous Fourier Transform of a rectangle with amplitude of 1 between [-u,u] is a real valued function, specifically a sinc function. However, when using the discrete Fourier Transform (DFT), complex numbers are obtained instead of real values. This difference may be due to the inclusion of the dt factor in the discrete version. Also, the DFT returns both amplitude and phase information, and the choice of time axis can affect the results, as seen in the example of defining the rectangle from 0 to 2u instead of -u to u.
  • #1
truva
18
1
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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  • #2
The DFT returns both amplitude and phase information.
 
  • #3
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.
 

What is the difference between continuous FT and DFT?

The continuous Fourier transform (FT) is a mathematical operation that decomposes a continuous time function into its frequency components. It is a continuous function of frequency and is defined for continuous time signals. The discrete Fourier transform (DFT) is a discrete time version of the FT, which is used to analyze discrete time signals. While the continuous FT is a continuous function, the DFT is a sequence of complex numbers.

Why is the continuous FT of a rectangle waveform always real valued?

A rectangle waveform is a periodic signal with a rectangular shape. The continuous FT of this signal is always real valued because the rectangular shape has no imaginary components and the FT of a real valued signal is also real valued.

Why is the DFT of a rectangle waveform not always real valued?

The DFT of a rectangle waveform is not always real valued because it is calculated using a finite number of samples. This means that the signal is not truly periodic and therefore, the DFT will have imaginary components in order to represent the signal accurately. As the number of samples used in the DFT increases, the imaginary components decrease and the DFT approaches the continuous FT.

Can the DFT of a rectangle waveform be made real valued?

Yes, the DFT of a rectangle waveform can be made real valued by using an even number of samples. This is because an even number of samples allows for the waveform to be symmetric, which eliminates the imaginary components in the DFT. However, this is not always practical as the number of samples needed to completely eliminate the imaginary components may be too large.

How does the sampling rate affect the DFT of a rectangle waveform?

The sampling rate affects the DFT of a rectangle waveform by changing the frequency resolution of the DFT. A higher sampling rate results in a higher frequency resolution, meaning that the DFT will have more data points and can better represent the frequency components of the signal. On the other hand, a lower sampling rate will result in a lower frequency resolution, making it more difficult to accurately represent the frequency components of the signal.

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