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2018-08-12

2018-08-12

作者: 快乐的大脚aaa | 来源:发表于2018-08-12 18:37 被阅读0次

    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
        • distance = rate \times time
        • \lambda = v \times \frac{1}{\nu}
          • v-the velocity of wave(rate)
        • v = \lambda \times \nu
        • reciprocal relationship between the frequency and wavelength

    The second class

    • How can use such simple functions \sin t \cos t to model complex phenomenon
    • can modify \sin 2\pi t and \cos 2\pi t to model very general phenomena of period one
    • one period ,many frequencies
      • \sin 2\pi t period 1 frequency 1
      • \sin 4\pi t period \frac{1}{2} frequency 2
      • \sin 6\pi t period \frac{1}{3} frequency 3
      • combination
        • \sin 2\pi t + \sin6 \pi t + \sin4\pi t
        • 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
        • \sum_{k=1}^N A_ksin(2\pi k t + \phi _k)
        • \sin(2\pi kt +\phi_k) = \sin 2\pi kt\cos\phi_k +cos2\pi kt \sin\phi_k
        • \sum_{k = 1} ^ n(a_k\cos2\pi kt +b_k\sin2\pi kt)
        • \frac{a_0}{2} +\sum_{k = 1} ^ n(a_k\cos2\pi kt +b_k\sin2\pi kt)
      • By far before
        • e^{2\pi ikt} = \cos 2\pi kt +\sin2\pi kt
        • i = \sqrt{-1}
        • Euler's formula
          • \cos2 \pi kt = \frac{e^{2\pi ikt} + e^{-2\pi ikt}}{2}
          • \sin2 \pi kt = \frac{e^{2\pi ikt} - e^{-2\pi ikt}}{2i}
      • you can convert a trigonometric sum as before to the form sum
        • \sum_{k = -n}^nC_ke^{2\pi ikt}
        • C_k's are complex
      • the symmetry property
      • C_{-k} = \overline{C_k}
      • f(t)periodic of perid 1 can be write f(t) = \sum_{k = -n}^nC_ke^{2\pi ikt}
      • C_{-0} = C_{0} = \overline{C_0}
      • Suppose you can write f(t)= \sum_{k = -n}^nC_ke^{2\pi ikt}
      • f(t) = ...+ C_me^{2\pi imt}+...
      • C_me^{2\pi imt} = f(t)-\sum_{k \neq m}^nC_ke^{2\pi ikt}
      • C_m = e^{-2\pi imt} (f(t)-\sum_{k \neq m}^nC_ke^{2\pi ikt})
      • C_m = e^{-2\pi imt} f(t)-\sum_{k \neq m}^nC_ke^{2\pi i(k-m)t}
      • Everything's periodic at period one
      • \int_{0}^1C_m dt = C_m
      • C_m =\int_{0}^1 e^{-2\pi imt} f(t)dt -\sum_{k \neq m}^nC_k\int_{0}^1e^{2\pi i(k-m)t}
      • \int_{0}^1e^{2\pi i(k-m)t}
      • \frac{1}{2\pi i(k-m)}e^{2\pi i(k-m)t}|_{0}^1
      • \frac{1}{2\pi i(k-m)} (e^{2\pi i(k-m)} - e^0)
      • C_m = \int_{0}^1 e^{-2\pi imt} f(t)dt
    • Given f(t) = \sum_{k = -n}^nC_ke^{2\pi ikt} periodic of period one then C_k = \int_{0}^1 e^{-2\pi ikt} f(t)dt

    Fourier transform

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