Fourier Analysis

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Fourier Analysis

• By Jesse Lawrence

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The Basics

• Represent a series as frequency or wavenumber.

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Basics 2

• Any series can be represented as a series of sin
  and cosine waves with different offsets (time or
  distance) for each frequency or wavenumber.

Time
Distance
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Basics 3

• The imaginary exponential is useful because it
  composes two parts
• Real (cos)
• Imaginary (sin)
• So for each frequency and time, there is a
  different pair of imaginary and a real components.

Fn is complex
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Inverse Fourier Transform
1D Continuous

• Any signal is really composed of continuous
  spectra, not discrete spectra.
• The Fourier transform is not limited to 1D.
• Dot product aligns wavelengths in proper
  directions.

2D Continuous
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Forward Fourier Transform
1D Continuous

• A spectrum can be determined for each real series.

2D Continuous
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Amplitude Phase

• Part of the frequency content indicates amplitude
• Part of the frequency content tells about phase
  or time shift.

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Example
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Delta Function
y(x) ?(x0)

• A delta function is represented by all
  frequencies with the same amplitude having a
  linear change in phase shift w.r.t.
  wavenumber/frequency.

Amplitude
x0
Distance/Time
x ? x0
x x0
Wavenumber/Frequency
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A sin Wave
y(x) cos(2?k0x)

• A sin wave is composed of only one frequency, so
  it is a delta function in the frequency domain

Amplitude
Distance/Time
k k0
Amplitude
k0
k ? k0
Wavenumber/Frequency
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Useful Shapes

• Every shape in the space domain has an equivalent
  (but different) shape in the frequency domain.
• Some are useful for calculations.
• Some are useful for symmetry.
• Some are dangerous and cause havoc in many
  studies.

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2D Shapes

• Every shape in the space domain has an equivalent
  (but different) shape in the frequency domain.
• Some are useful for calculations.
• Some are useful for symmetry.
• Some are dangerous and cause havoc in many
  studies.

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Odd Even Components
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Convolution
indicates convolution

• Convolution, which is a slow process in the time
  domain, but quick in the frequency domain.

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Deconvolution
indicates convolution

• No direct time domain calculation of inverse
  operation of convolution that leads to a unique
  solution. In the frequency domain there is.

-1
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UncertaintyPrinciple

• "The more precisely the position is
  determined,the less precisely the time/momentum
  is known
• -Heisenberg

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Discrete Series
1D

• We live in a world where computers rule.
• Computers work with discrete packets
  (bits/bytes).
• Sample gravity at discrete intervals.
• Sample ground motion at discrete times.
• Easiest to work with evenly sampled data.

Or