2018-08-12

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
  - $distance = rate \times time$
  - - $v$ -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 $C_k = \int_{0}^1 e^{-2\pi ikt} f(t)dt$

2018-08-12

2018-08-12

Fourier series

The first class

The second class

Fourier transform

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