Time-domain and frequency-domain transformation --- mathematical derivation of Fourier Series

Not to mention nonsense:

0. Overview-Demand Analysis-function description-restrictions and defects improvement + knowledge preparation

1. essential differences between Taylor series and Fourier series, Taylor's expansion

2. Function projection and vector orthogonal

3. the derivation of the two invariant functions is e ^ X, SiNx, and cosx, Which is why Fourier conversion is required!

4. Fourier technology push process

5. Appendix references

0. In some cases, especially in image processing, the amount of matrix computing data is too large and feature extraction is large. In this case, the time domain to frequency domain can be used to reduce the amount of computing, and this conversion will not lose data integrity.

The time-domain to-frequency method involves the use of Fourier Technology for periodic functions, and non-periodic functions (functions without consecutive points) Use Fourier transformation, which is similar to histogram analysis.

Two corners and Formula
Tan (α + β) = (TAN α + Tan β)/(1-tan α tan β)
Tan (α-β) = (TAN α-tan β)/(1 + Tan α tan β)
Cos (α + β) = cos α cos β-sin α sin β
Cos (α-β) = cos α cos β + sin α sin β
Sin (α + β) = sin α cos β + cos α sin β
Sin (α-β) = sin α cos β-Cos α sin β

Product and Difference
Sin α sin β =-[cos (α + β)-cos (α-β)]/2
Cos α cos β = [cos (α + β) + cos (α-β)]/2
Sin α cos β = [sin (α + β) + sin (α-β)]/2
Cos α sin β = [sin (α + β)-sin (α-β)]/2

Binarization Formula
Sine
Sin2a = 2sina · cosa
Cosine
1. cos2a = cos ^ 2 (a)-Sin ^ 2 ()
2. cos2a = 1-2sin ^ 2 ()
3. cos2a = 2cos ^ 2 (a)-1
That is, cos2a = cos ^ 2 (a)-Sin ^ 2 (A) = 2cos ^ 2 (a)-1 = 1-2sin ^ 2 ()
Tangent
Tan2a = (2 Tana)/(1-tan ^ 2 ())

Square relationship:
Sin ^ 2 (α) + cos ^ 2 (α) = 1
1 + Tan ^ 2 (α) = sec ^ 2 (α)
1 + cot ^ 2 (α) =  ^ 2 (α)
Two common formulas for different conditions
Sin ^ 2 (α) + cos ^ 2 (α) = 1
Tan α * cot α = 1

(1 ).

The Taylor series is the changing feature function and of the derivative function. The intensity of the reaction changes

Fourier series is the trigonometric function and of spectral superposition. The essential attribute of the frequency of reaction change

(2 ).

From a geometric point of view. Fourier tells us that f (x) can be expanded using the following "orthogonal basis" composed of an infinite number of trigonometric functions (including constants,

 

Fourier series expansion is actually just an action, that is, to "project" the function to a series of "coordinate axes" composed of trigonometric functions.

(3 ).

Sine and Cosine are second-order partial differential equations (Equations containing capacitor and other components), while capacitors can communicate through DC.

(4 ).

First, we will introduce the Fourier series formula. However, this item belongs to the "Cultural Relics" level and was born in the early 19th century because the old man of Fourier was born in 1768 and died in 1830.

However, Fourier series is widely used in number theory, composite mathematics, signal processing, probability theory, statistics, cryptography, acoustics, optics, and other fields. As soon as I opened books such as signal and system and the principle of the Phase-Locked Loop, I jumped out of a "Fourier series" or "Fourier transformation" and made a long string of formulas to make people confused.

The formula for Fourier series is as follows:

It can be said that this formula is like a "Smelly mother-in-law-smelly and long", and its origins are quite strange. I don't know when the Fourier will shine, write a periodic function f (t) into a bunch of things. Simply look at the ① formula, that is, the periodic function f (t) describes a series of sub-types, such as sin and cos functions with a constant coefficient A0, and 1 x ω, sin and cos functions with 2 x ω, and sin and cos functions with N x ω. and, each item has different coefficients, namely, an and BN. As for these coefficients, you need to use integral points for solving them, that is, ② ③ ④ formula. However, for the convenience of integral points, the integral interval is generally set[-π, π]Is also equivalent to the width of the cycle T.

Can this formula be derived from a mathematical point of view, so that Fourier series can be understood, so that I can understand its past and present? The following describes in detail how this formula is obtained:

 

1. represent a periodic function as a triangular series:

First, the cyclic function is a mathematical expression of Periodic Motion in the objective world. For example, an object is attached to a spring for simple harmonic vibration, single pendulum vibration, and electronic oscillation of a radio electronic oscillator:

  F (x) = A sin (ω T + psi)

Here t indicates the time, a indicates the amplitude, ω indicates the angle frequency, and PSI indicates the initial phase (it is related to the origin position set during the test ).

 

However, many periodic signals in the world are not as simple as sine functions, such as square waves and triangular waves. Fu Ye thought, can we use the sum of a series of trigonometric functions, an sin (N ω T + psi), to represent the complex periodic function f (t? Sin is the simplest periodic function. Therefore, Fourier writes the following formula: (Fourier derivation is purely a conjecture)

   
Here, T is a variable, and others are constants. Compared with the simplest sine-period function above, n is in the 5 formula, and N ranges from 1 to infinity. Here, f (t) is a known function, that is, the original periodic function to be decomposed. In formula 5, Fourier is used to represent a periodic function as linear superposition of many sine functions. These many sine functions have different amplitude components (I .e., an in formula), has different cycles or frequency (an integer multiple of the original cyclic function, that is, n), has different initial phase angles (that is, PSI ), of course, there is also a constant term (A0 ). What's terrible is that this n is from 1 to infinity, that is, an infinite series.

It should be said that Fourier was a genius and thought so complicated. Generally, it is difficult to make a simple periodic function into such a complex representation. However, Fourier believes that a lot of functions on the right side of the formula are actually the simplest sine functions, which are conducive to subsequent analysis and computation. Of course, the key to the establishment of this formula is that each item in the series has an unknown coefficient, such as A0 and an. If we can find these coefficients, the formula 5 can be true. Of course, in formula 5, the only known function is the original periodic function f (t), so we only need to use the known function f (t) To express the coefficients. the above formula can be established, it can also be calculated.

Then, Fourier first performs the following deformation on formula 5:

In this way, formula 5 can be written in the form of formula 6 as follows:

This formula 6 is the triangle series in the general form. The next task is to express the coefficients an, bn, and A0 using the known function f (t.

 

2. Orthogonal trigonometric functions:

This is the preparation knowledge for the points used for the next step of Fourier series expansion.A trigonometric function: 1, cosx, SiNx, cos2x, sin2x ,... , Cosnx, sinnx ,... If the product of any two different functions in the heap function (including constant 1) on the interval [-π, π] is equal to zero, the trigonometric function is in the interval [-π, π] Forward, That is, the following formula:

The above types are in the range[-π, π]The integral values of are all 0, and the 1st 2nd formula can be regarded as the integral values multiplied by the trigonometric function COs and sin and 1. The 3-5 formula is the integral formula multiplied by different combinations of sin and cos. In addition to the five formulas, there cannot be any other combinations. Note: In the 4th 5th formula, K cannot be equal to N. Otherwise, it is not defined as "any two different functions in the trigonometric function system" and becomes the square of the same function. But in formula 3rd, K and N can be equal. If they are equal, they are two different functions. The correctness of the formula 4th is verified by calculating the formula 4th. In the formula, the multiplication of two functions can be written:

  
Visible in the specified[-π, π]In the range. Other methods can also be verified one by one.

 

3. Expand the function into Fourier series:

First, the Fourier series is represented as the following formula, that is, formula 6:

For type 6 slave[-π, π]Points:

The expression of the first coefficient A0 is obtained, that is, the ② formula in the top Fourier series formula. Next, evaluate the expressions of an and bn. UseCos (k ω T)Take the two sides of the formula 6:

So far, we have obtained the expression of each coefficient in the Fourier series. as long as these points exist, the Fourier series represented on the right of the equal sign of formula 6 can be used to express the original function f (t ). The above process is the derivation of the entire Fourier series. In fact, if we can write the formula 6, it is not difficult to find the expressions of various coefficients. The key is that people don't think that a periodic function can be expressed by some simple sine or Cosine functions, this expression is an infinite series. Of course, this is the genius of the mathematician Fourier. I have to try my best to understand it.

 

In summary, the generation process of Fourier series can be divided into the following three steps:

1. Imagine that a periodic function f (t) can be expressed by the simplest series of sine functions, that is, formula 5;

2. After deformation, it is represented by triangular series (including sin and COS;

3. Use the integral formula of f (t) To express the unknown coefficients through points;

4. The last four expressions are Fourier series formulas.

 

In electronics, Fourier series is a frequency-domain analysis tool. It can be understood as a complex cyclic wavelength division to form a DC term, a fundamental wave (the angle frequency is ω), and various harmonic waves (the angle frequency is N ω) the sum of, that is, the items in the series. Generally, with the increase of N, the energy of each harmonic gradually degrades, so the sum of the First n items from the series can be very close to the original cycle waveform. This is an important application of Fourier series in electronics analysis.

Of course, there is another question about the difference between the period value L and T. Some are W = l/2, and some are W =.

:

0. Fourier is right. f (x) is divided into the sum of trigonometric functions at different frequencies.

1. Use the Inner Product Method to separate the coefficients of each component

 

2. calculation coefficient of grounding gas

In the textbook of high numbers, the DC component is a/2 in the section of Fourier transformation in the book "Signals and Systems, but the formula for solving a and B is the same. What is the problem? The expected results must be different,

(5 ).

Http://www.360doc.com/content/13/0328/12/202?_27444=7.shtml;

Http://blog.sina.com.cn/s/blog_57ad1bd20100txgs.html

Http://www.360doc.com/content/13/0301/19/202378_268716061.shtml (geometric formula)

Http://blog.renren.com/share/343320656/15540620254 (projection + orthogonal)

Http://www.360doc.com/content/11/0126/14/2355320_89141366.shtml (Taylor formula)

Http://www.360doc.com/content/12/0120/16/99504_180542746.shtml (Taylor series)

Http://www.360doc.com/content/13/0205/02/1489589_264295813.shtml# (Fourier series of non-sine period)