Classical Algorithm Research Series: 10. A thorough understanding of the Fourier Transform Algorithm from start to end

Author: July, dznlong February 22, 2011

Recommended reading:The Scientist and Engineers Guide to Digital Signal Processing<, By Steven W. Smith, Ph. D.Book address:Http://www.dspguide.com/pdfbook.htm.
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A thorough understanding of the Fourier transform algorithm
Preface
Part 1, DFT
Chapter 1: Evolution of Fourier Transformation
Chapter 2 Real number form discrete Fourier transform (Real DFT)

A thorough understanding of the Fourier transform algorithm
Chapter 3, plural
Chapter 4. complex form discrete Fourier Transformation

Preliminary review: in the previous article: html "target = _ blank> 10. A thorough understanding of the Fourier Transform Algorithm from start to end, we have discussed the origins of Fourier transformation and the Real number form discrete Fourier Transformation (Real DFT,
In this article, we will focus on two problems, namely, the complex number and the discrete Fourier transformation of the complex number form.

 

Chapter 3, plural

The plural number extends the concept of a number that we can generally understand. The plural number contains the real number and the imaginary number, the expression represented by two variables can be expressed by a variable (complex variable) in the form of a complex number, making processing more natural and convenient.
We know that the result of Fourier transformation is composed of two parts. The complex form can shorten the transformation expression, this allows us to process a variable separately (which we can know more precisely in the subsequent description), and the fast Fourier transformation is based on the complex form, therefore, almost all the descriptions of Fourier transformation form are in the form of plural.
However, the concept of the plural is more than what we can understand in our daily life. It is more difficult to understand the plural. Therefore, before understanding the complex Fourier transformation, we should first review the knowledge about the plural, this is very important for later understanding.
 
I. Proposal of plural numbers
 
Here, let's take a look at a physical experiment: throw a ball from a certain point, and then calculate the height of the Ball Based on the initial speed and time. This method can be calculated based on the following formula:

H indicates the height, g indicates the acceleration of gravity (9.8 m/s2), v indicates the initial speed, and t indicates the time. Now, if we know the height and need to calculate the time required for this height, we can calculate it using the following formula:

After calculation, we can know that when the height is 3 meters, there are two time points to reach this height: when the ball is up, the time is 0.38 seconds, the time when the ball moves downward is 1.62 seconds. But if the height is equal to 10, what is the result? Based on the formula above, we can find that there is an open square operation on the negative number. We know this is definitely unrealistic.
The first time I used this unusual formula was the Italian mathematician Girolamo Cardano (1501-1576). Two centuries later, the German great mathematician Carl Friedrich Gause (1777-1855) the concept of plural is proposed, paving the way for later application. He expressed the plural as follows: the plural is composed of real and imaginary, negative 1 of the root number in the imaginary number is represented by I (Here we use j, Because I represents the meaning of current in Electric Power ).
 
We can represent the X coordinate as the real number, and the Y coordinate as the virtual number, then the vector of each point in the coordinate can be expressed by the plural number, for example:

The three vectors ABC in can be expressed as the following formula:
 
A = 2 + 4.7
B =-4-1.5j
C = 3-7j
 
In this way, it is convenient to use a symbol to combine two arrays that were originally hard to associate. What is inconvenient is that we need to identify which is a real number and which is a virtual number, we generally use Re () and Im () to represent the real number and virtual number, such:
 
Re A = 2 Im A = 6
Re B =-4 Im B =-1.5
Re C = 3 Im C =-7
 
You can also perform addition, subtraction, multiplication, and Division operations between multiple numbers:


Here is a special note that j2 is equal to-1. In the fourth formula above, the numerator and denominator are multiplied by c-dj at the same time, so that j in the denominator can be eliminated.
 
The plural also conforms to the exchange law, combination law, and allocation Law in the algebraic operation:
 
A B = B
(A + B) + C = A + (B + C)
A (B + C) = AB + AC
 
 
2. Polar representation of the plural number
 
As mentioned above, Cartesian coordinates are used to represent the plural. In fact, polar coordinates are more commonly used, for example:

M in is the magnproduct, which indicates the distance from the origin point to the coordinate point. θ is the phase angle, which indicates the angle from the square of the X axis to a certain vector, the following four formulas are the calculation methods:

We can also convert polar coordinates to Cartesian coordinates using the following formula:
 
A + jb = M (cos θ + j sin θ)

In the above equation, the left side is the Cartesian expression, and the right side is the polar expression.
 
There is also a more important equation-Euler's equation (Euler, Switzerland's famous mathematician, Leonhard Euler, 1707-1783 ):
Ejx = cos x + j sin x
 
This equation can be proved from the following series Transformation:

The two formulas on the right of the preceding table are the Taylor series of cos (x) and sin (x) respectively.
 

In this way, we can express the expression of the plural number as an index:
 
A + jb = M ejθ (this is the two expressions of the plural number)
 
The exponential form is the pillar of mathematical methods in digital signal processing. Perhaps it is because the multiplication and division operations of the plural form are extremely simple:

Iii. Plural is a tool in Mathematical Analysis
 
Why use the plural number? In fact, it is just a tool, just like the relationship between a nail and a hammer. The plural is like the hammer, which is used as a tool. We implement the complex form of the problem table to be solved (because some problems are more convenient to perform operations in the form of the plural), and then perform operations on the plural, finally, convert it back to get the results we need.
 
There are two ways to use the plural, one is to use the plural for simple replacement, such as the vector expression method mentioned above and the Real-domain DFT discussed in the previous section, the other