Fourier Transforms: Definition, Applications, and Properties, Slides of Electrical Engineering

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Fourier Transform

and its applications

Fourier Transforms are used in

• X-ray diffraction
• Electron microscopy (and diffraction)
• NMR spectroscopy
• IR spectroscopy
• Fluorescence spectroscopy
• Image processing
• etc. etc. etc. etc.

Fourier Transforms

• Different representation of a function
  • time vs. frequency
  • position (meters) vs. inverse wavelength
• In our case:
  • electron density vs. diffraction pattern

Discrete Fourier Transforms

• Function sampled at N discrete points
  • sampling at evenly spaced intervals
  • Fourier transform estimated at discrete values:
  • e.g. Images
• Almost the same symmetry properties as the continuous Fourier transform

..., 3 , 2 , 1 , 0 , 1 , 2 , 3 ,...

( ) = − − −

= ∆ n

hn h n = (^) N ∆ f (^) n n 2 ,..., 2 n =− N^ N

DFT formulas

[ ] [ ]

h [ ikn N ]

H f h t if t dt h if t N k k

n n

N n n k k

exp 2 /

( ) ( )exp 2 exp 2 1 0

1 0 π

π π

−

−

∞ −∞ = ∆

= ≈ ∆

∑^ [^ ]

−

≡

1 0

exp 2 /

N k

Hn hk π ikn N H ( fn ) ≈ ∆ Hn

∑^ [^ ]

−

= −

1 0

(^1) exp 2 / N k (^) n n H ikn N N h π