傅里叶分析之一 傅里叶级数

sine（正弦）一词始于阿拉伯人雷基奥蒙坦。他 是十五世纪西欧数学界的领导人物，他于1464 年完成的著作《论各种三角形》，1533年开始 发行，这是一本纯三角学的书，使三角学脱离天 文学，独立成为一门数学分科。 cosine（余弦） 及cotangent（余切）为英国人根日尔首先使用， 最早在1620年伦敦出版的他所著的《炮兵测量 学》中出现
Example:
t t f (t ) cos cos 3 4
f (t ) f (t T )
Find its period.
t t 1 1 cos cos cos (t T ) cos (t T ) 3 4 3 4
Fact: cos cos( 2m)
This is important！
One
period, many frequencies.
Idea2: How complicate signal is?

How general a periodic phenomena can this formula express ?
Alternative formula：

F ( x ) = a /2 + a 1 cos x + b 1 sin x + a 2 cos 2 x + b 2 sin 2 x + ... + a n cos nx + b n sin nx + ...
Fourier synthesis

Fourier synthesis works by combining a sine wave signal and sine-wave or cosine-wave harmonics (signals at multiples of the lowest, or fundamental, frequency) in certain proportions. F ( x ) = a /2 + a 1 cos x + b 1 sin x + a 2 cos 2 x + b 2 sin 2 x + ... + a n cos nx + b n sin nx + ...

Linear Operation

Fourier synthesis and analysis based on Linear Operation: Integration and series. Fourier Transform is part of linear systems.

Fourier Series

Fourier Series
Analysis and Synthesis
Fourier analysis

The process of decomposing a musical instrument sound or any other periodic function into its constituent sine or cosine waves is called Fourier analysis
1T 2m 2T 2n
1 m 2 n
1 must be a 2 rational number
Example2:
f (t ) cos1Leabharlann Baidut cos(10 )t
Is this function a periodic one?
1 10 2 10
Periodic Phenomenon in space

Wave motion: Temporal and spatial periodicity
come together.
(1) Fixed x= xo, corresponding to the
oscillating curve (振动曲线) of medium element at position xo , i.e. y(t, xo).
Fourier Series
Fourier Series
Introduction


Suppose we have a complicated looking periodic signal f(t). Decompose a periodic input signal into primitive periodic components. Can we?

It’s more common to write a general trigonometric sum as：
if we include a constant term (n= 0), as
Notes:



The constant term with the fraction 1/2 is because it simplifies the computation. In electrical engineering the constant term is often referred to as the DC components in “direct current”. The other terms, being periodic, “alternate”, as in AC.
y
x1 A o
x2
波具有空间周期（ )

x 广镜头拍照片
periodicity in space is measured by the wave length λ
λ and v
The frequency and wavelength are related through the equation v = λν where v is the speed of propagation — this is nothing but the wave version of speed = distance/time. Thus the higher the frequency the shorter the wavelength, and the lower the frequency the longer the wavelength.
Fourier Series
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Content



Fourier Series and Fourier Transform Analysis and Synthesis Periodic Phenomenon and Function Trigonometric function Fourier Series Complex Form of the Fourier Series Detail of Fourier Series
Periodic Phenomenon in time

Time For example, you stand at a fixed point in the ocean wash over you with a regular, recurring pattern of crests and troughs. The height of the wave is a periodic function of time.

周期性是一种物理属性。 为什么能用数学描述呢？


因为有一种简单的函数能表示周期的性质，利用 这种简单的函数，就可以对周期性进行建模。 sine and cosine
Fourier Series
Trigonometric Function
History of sine and cosine
f(t)
t
A periodic sequence
T
2T
3T

Question is Solving for these coefficients. A direct approach:
Another idea is needed, and that idea is integrating both sides from 0 to 1.
Using Euler’s Formula
Complex Form

In this final form of the sum, the coefficients cn are complex numbers, and they satisfy
Therefore the sum is real:
not a rational number
Example3：
would you say it had frequency 1 Hz? I don’t think so. It has one period but you’d probably say that it has, or contains two frequencies, one cosine of frequency 1 Hz and one of frequency 2 Hz.
Periodic of sine and cosine
Question:
How
to use such simple function to build Complicated periodic function?
Answer：It All Adds Up


We can combine the basic function of period 1 such as sin2πt and cos2πt to form more complicated periodic functions. Idea1: One period, many frequencies.



More on spatial periodicity
It’s reasonable to say that one of the patterns is low frequency and that the others are high frequency, meaning roughly that there are fewer stripes per unit length in the one than in the others.
T 2m 3 T 2 n 4
T 6 m T 8n T 24 smallest T
Example1:
f (t ) cos1t cos2t
f (t ) f (t T )
Find its period.
cos1t cos2t cos1 (t T ) cos2 (t T )
The Mathematic Formulation

Any function that satisfies
f (t ) f (t T )
where T is a constant and is called the period of the function.
Why mathematics come?
y
o
A T
t
波具有时间周期（T ) 盯住一点拍电影
periodicity in time is measured by the frequency ν, with dimension 1/sec

(2) Fixed t =to, corresponding to the wave pattern curve (波形曲线) at time to.
Fourier Series
Fourier Series and Fourier Transform
Fourier

Fourier Series Almost periodic phenomenon Fourier Transform Non-periodic phenomenon 一些概念上是通用的，一些则不通用
Periodic Phenomenon & Functions
Periodic Phenomenon

Generally speaking we think about periodic phenomena according to whether they are periodic in time or periodic in space.