Fourier Transforms

Jose Jesus Fernandez jose at indy.cnb.uam.es
Fri Jun 30 04:46:05 EST 1995


The problem is that the DFT of a sampled function is not necessarily in close 
agreement with the Fourier transform of the function itself (due to effects of
finite length, windowing, etc). Then, if we want the Fourier transform of
the continuous function, we have to estimate it from the DFT. How??
It has been shown that the Power Spectrum of a sampled function is a correct
estimate of the Power Spectrum of the function itself. And it's known that
the Power Spectrum is the square Amplitude of the Fourier transform. So, from
the DFT we can obtain the Power Spectrum of the sampled function, which is a
estimate of Power Spectrum of the function itself. Then, we calculate the
square
root of this Power Spectrum, and in this way we get a correct estimate of the
real continuous Fourier transform of the function itself (Only amplitudes,
phases are lost).

References:
- The Fourier Transform and its applications. Ronald N. Bracewell. 1986.
  McGraw-Hill. ISBN 0-07-Y66454-4
- Introduction to Discrete-Time Signal Processing. Steven A. Tretter. 1976.
  John Wiley & Sons. ISBN 0-471-88760-9
- Discrete-Time Signal Processing. Oppenheim & Schafer. 1989. Prentice-Hall.
  ISBN 0-1-216771-9


Hoping it helps you,

Jose.





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