Pitfall-talk about the Fourier transformation of discrete time signals"

Although I have learned a lot about computer vision, I have never understood the mathematical meanings behind Fourier transform and image filtering. I am deeply disturbed by the existing formulas. Fortunately, through this internship at Huawei during this period, the basic knowledge related to digital signal processing has finally been filled in. The following are my learning achievements over the past few days, that is, my own understanding of Fourier transform.

 

 

 

I. Discrete Time Signal

To understand the Fourier transformation of discrete time signals, we must first find out what signals are and what are discrete time signals. (Although it feels like nonsense, I have no idea about these things at the beginning as a child shoes for Software Engineering. t_t)

The so-called signal is actually a function that contains one or more variables. For example, a speech signal can be expressed as a function of sound pressure changing over time, black and white photos can be represented as functions in which the brightness changes with the coordinate of two-dimensional space. (This part is from the signal and system in obenheim.)

The so-called Discrete Time Signal, also known as a digital signal, is a discrete function represented by a digital SEQUENCE {X[N]},-∞ <n <+ ∞, where n is an integer. In general, for convenience, we usually set {X[N]} is abbreviatedX[N]. Common discrete time signals include:

① Unit sampling sequence:

　　　　　　　　　　　　

② Unit order sequence:

③ Rectangular sequence:

　　　　　　　　 　　 　

(The above content comes from the digital signal processing textbook of Huazhong University of Science and Technology.)

 

Ii. Discrete Time System

Before understanding the Fourier transformation of discrete time signals, we need to clarify the concept of a discrete time system.

The discrete time system refers to the input discrete sequence.X[N] ing to output sequenceYThe unique transformation of [N], usingT.

A system that meets the linear superposition principle is called a linear system, namely:

①T[X1(N) + BX2(N)] =Y1(N) + BY2(N)

If the system response has nothing to do with the time n when the input signal is applied to the system, it is called a non-shift system.

②T[X(N)] =Y(N), then there isT[X(N-k)] =Y(N-k)

A system that meets both of the preceding conditions is called a Linear Non-shift system.

 

Iii. convolution

I believe that those who have learned computer vision will not be unfamiliar with this word. convolution is a feature of a Linear Non-shift system. Its definition is as follows:

SetT[] Is a linear non-shift system. When the input is a unit sampling sequenceDelta[N], define the output (where the symbol with an equal sign under the triangle represents "defined "),H[N] is called the unit sampling response (or unit impulse response ).X[N], the following equation is true: (no time to write, tomorrow)

 

Iv. Fourier Transformation

Fourier Transformation: any periodic function can be expressed as a sine function of different frequencies.Sin(ω x + PHI) superposition form, where each sine function has a different coefficient A, formula ω x is the frequency

According to this definition, we can extend the Fourier transformation to non-cyclic functions: Non-cyclic functions can be expressed as an infinitely multiple sine functions with different frequencies, the coefficients corresponding to each sine function are close to an infinitely small number (Note: These coefficients close to an infinitely small number are not equal ). (I have not finished writing it. I will make it up tomorrow)