FOURIER ANALYSIS PART 2: Technicalities, FFT

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FOURIER ANALYSISPART 2 Technicalities, FFT
system analysis

• Maria Elena Angoletta
• AB/BDI
• DISP 2003, 27 February 2003

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TOPICS

• 1. DFT windows
• 2. DFT resolution - improvement
• 3. Efficient DFT calculation FFT
• 4. Hints on system spectral analysis

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DFT Window characteristics

• Finite discrete sequence ? spectrum convoluted
  with rectangular window spectrum.
• Leakage amount depends on chosen window on how
  signal fits into the window.

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DFT of main windows
Windowing reduces leakage by minimising sidelobes
magnitude.
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DFT - Window choice
Common windows characteristics
NB Strong DC component can shadow nearby small
signals. Remove it!
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DFT - Window loss remedial
Smooth data-tapering windows cause information
loss near edges.

• Attenuated inputs get next windows full gain
  leakage reduced.
• Usually 50 or 75 overlap (depends on main lobe
  width).

Drawback increased total processing time.
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Zero padding
Improves DFT frequency inter-sampling spacing
(resolution).
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Zero padding -2
DFT spectral resolution
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DFT - scalloping loss (SL)
Input frequency f0 btwn. bin centres causes
magnitude loss
Worst case when f0 falls exactly midway between 2
successive bins (r½)
r ? ½
f0 (kmax r) fS/N
Frequency error ?f r fS/N, relative error
?R?f / f0 r/(kmaxr) ?R ? 1/(12 kmax)
kmax
f0
Note Non-rectangular windows broaden DFT main
lobe ? SL less severe Correction depends on
window used.
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DFT - SL Example
DC bias correction, Rectang. window, zero
padding, FFT
DC bias correction, Hanning window, zero
padding, FFT
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DFT - parabolic interpolation
Rectangular window
Hanning window

• Parabolic interpolation often enough to find
  position of peak (i.e. frequency).
• Other algorithms available depending on data.

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DFT averaging
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Efficient DFT calculation FFT
VERY BAD !
Algorithms ( Fast Fourier Transform) developed
to compute N-points DFT with Nlog2N
multiplications (complexity O(Nlog2N) ).
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FFT advantages
NB Usually you dont want to write an FFT
algorithm, just to borrow it !!! Go
shopping onto the web!
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FFT philosophy
General philosophy (to be applied recursively)
divide conquer.
Step 2 1-point input spectra calculation.
(Nothing to do!)
Step 3 Frequency-domain synthesis.
N spectra synthesised into one.
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FFT family tree
Divide conquer
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(Some) FFT concepts notes
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Systems spectral analysis (hints)
System analysis measure input-output
relationship.
yn predicted from xn, ht
H(f) LTI transfer function
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Estimating H(f) (hints)
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References - 1
Papers

• Tom, Dick and Mary discover the DFT, J. R. Deller
  Jr, IEEE Signal Processing Magazine, pg 36 - 50,
  April 1994.
• On the use of windows for harmonic analysis with
  the Discrete Fourier Transform, F. J. Harris,
  IEEE Proceedings, Vol. 66, No 1, January 1978.
• Some windows with a very good sidelobe behaviour,
  A. H. Nuttall, IEEE Trans. on acoustics, speech
  and signal processing, Vol ASSP-29, no. 1,
  February 1981.
• Some novel windows and a concise tutorial
  comparison of windows families, N. C. Geckinli,
  D. Yavuz, IEEE Trans. on acoustics, speech and
  signal processing, Vol ASSP-26, no. 6, December
  1978.
• Study of the accuracy and computation time
  requirements of a FFT-based measurement of the
  frequency, amplitude and phase of betatron
  oscillations in LEP, H.J. Schmickler, LEP/BI/Note
  87-10.
• Causes et corrections des erreurs dans la mesure
  des caracteristiques des oscillations
  betatroniques obtenues a partir dune
  transformation de Fourier, E. Asseo, CERN PS 85-9
  (LEA).

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References - 2

• Precise measurements of the betatron tune, R.
  Bartolini et al., Particle Accel., 1996, vol. 55,
  pp 247-256.
• How the FFT gained acceptance, J. W. Cooley, IEEE
  Signal Processing Magazine, January 1992.
• A comparative analysis of FFT algorithms, A.
  Ganapathiraju et al., IEEE Trans.on Signal
  Processing, December 1997.

Books

• The Fourier Transform and its applications, R. N.
  Bracewell, McGraw-Hill, 1986.
• A History of scientific computing, edited by S.
  G. Nash, ACM Press, 1990.
• Introduction to Fourier analysis, N. Morrison,
  John Wiley Sons, 1994.
• The DFT An owners manual for the Discrete
  Fourier Transform, W. L. Briggs, SIAM, 1995.
• The FFT Fundamentals and concepts, R. W.
  Ramirez, Prentice Hall, 1985.