Fourier series

The first class

• math and engineering of regularly repeating patterns
• periodic phenomenon
• periodic phenomena often are either periodicity in time a pattern repeats in time over and over again
  • e.g. harmonic motion, or periodicity in space
• physical quantity distributed over a region with symmetry
• so periodicity arises from the symmetry
  • e.g. the distributed of heat on a circular ring
  • the temperature is periodic as a function periodicity in space
• fourier analysis is often associated with problems that have some sort of underlying symmetry
• math descriptors of periodicity
  • periodic in time or a phenomenon
    • frequency number of repetitions of patterns in one second or over time
  • periodic in space
    • measurement how big the pattern is that repeats
    • use the period
  • come together in e.g. wave motion
    • a regularly moving
    • frequency in time cycles per second
    • wavelength one complete pattern
    • in space fix the time see the pattern distributed over space
    • fraquency and wavelegth
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      • -the velocity of wave(rate)
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    • reciprocal relationship between the frequency and wavelength

The second class

• How can use such simple functions to model complex phenomenon
• can modify and to model very general phenomena of period one
• one period ,many frequencies
  • period 1 frequency 1
  • period frequency 2
  • period frequency 3
  • combination
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    • the period of the sum is 1
    • the frequencies are 1 ,2 and 3
  • so to model complicates,perhaps,how complicated?
  • A complicated signal of period one we can sum,modify the amplitude,the ferquency,the phases of either sines or cosines
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  • By far before
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    • Euler's formula
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  • you can convert a trigonometric sum as before to the form sum
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    • 's are complex
  • the symmetry property
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  • periodic of perid 1 can be write
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  • Suppose you can write
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  • Everything's periodic at period one
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• Given periodic of period one then

Fourier transform