Fourier series Fourier Transformation

Fourier analysis's cutting-edge tutorial (full version) was updated 4 months ago at 2014.06.06heinrich

Author: Han Hao

Zhihu: Heinrich

Weibo: @ peanut oil workers

Zhihu column: Stories irrelevant to time

I would like to give this document to Wu Nan, Liu Xiaoming, Wang Xinnian, and Zhang jingbo of Dalian Maritime University.

Reprinted personnel should keep the above sentence. Thank you. If you can retain the source of the article, you will be more grateful.

-- Updated on 2014.6.6. If you want to directly view the update, you can directly jump to Chapter 4 ----

I promise this article is different from all the articles you have read before. It was written during the last 12 years, but I went abroad before I could write it ...... So it took two years. Well, I'm a procrastinator ......

The core idea of this article is:

Let readers understand Fourier analysis without looking at any mathematical formula.

Fourier analysis is not only a mathematical tool, but also a way of thinking that can thoroughly subvert a person's previous world view. However, unfortunately, the Fourier analysis formula seems too complex, so many new students have come up with a circle and have been suffering from it. To be honest, such an interesting thing has actually become a killer course in the university, and it is too serious to blame the people who compile teaching materials. (Are you sure you want to write your teaching materials? Will it die ?) So I have always wanted to write an interesting article to explain Fourier analysis. If possible, high school students can understand it. Therefore, no matter what kind of work you are engaged in here, I promise you can understand it, and you will surely experience the pleasure of seeing the world look like another through Fourier analysis. As for friends who already have a certain degree of foundation, I also hope that they will not jump back and read it carefully, so there will be new discoveries.

---- The above is dingchang poetry ----

Enter the subject below:

Sorry, it's not easy to learn. The original intention of writing this article is to hope that everyone can learn more easily and enjoy it. But 10 million! Do not add this article to favorites or save it as an address. You may want to read it later. There are too many such examples. Maybe you haven't opened this page a few years later. In any case, be patient and read it. This article is much easier and happier than reading textbooks ......

P.s. Both COs and sin use the word "Sine Wave" to represent the harmonic.

I. What is frequency domain?

Since we were born, we have seen the world run through time, and the trend of stocks, human height, and vehicle trajectory will change over time. This method of observing the dynamic world with time as a reference is called Time Domain Analysis. We also take it for granted that everything in the world is constantly changing over time and will never remain static. But if I tell you another way to observe the world, you will find that the world remains the same forever. Do you think I am crazy? I'm not crazy. This static world is called the frequency domain.

Let's start withThe formula is not very appropriate.But in the sense, the following is an example:

In your understanding, what is a piece of music?

This is our most common understanding of music, a vibration that changes over time. However, I believe that the following is a more intuitive understanding of Music:

Okay! Goodbye to the class.

Yes. In fact, this section can be written here. It is the appearance of music in the time domain, but the appearance of music in the frequency domain. Therefore, the concept of frequency domain is no stranger to everyone, but never realized it.

Now, let's look back and look at the first crazy dream: The world is everlasting.

Simplify the above two figures:

Time Domain:

Frequency Domain:

In the time domain, we observe the swing of the piano strings in a moment, just like the trend of a stock. in the frequency domain, there is only one permanent note.

So

In your eyes, the fallen leaves seem to change the impermanence of The World. Actually, it is just a piece of music that has already been written in the hearts of God.

Sorry, this is not a chicken Article, but a definite formula on the blackboard: Fourier tells us that any periodic function can be considered as a superposition of sine waves of different amplitude and phases. In the first example, we can understand that you can combine any piece of music by tapping different keys at different times.

One of the methods that run through the time domain and frequency domain is the Fourier analysis in the text of the biography. Fourier analysis can be divided into Fourier series (Fourier Serie) and Fourier Transformation (Fourier transformation). Let's start with a simple introduction.

2. Spectrum of Fourier Series

Let's just give a chestnut and understand it with a picture of the truth.

If I say that I can use the sine wave mentioned above to create a rectangular wave with a 90 degree angle, will you believe it? You won't, just like me. But let's see:

The first figure is a depressing sine wave cos (X)

The second figure is the superposition of two cute sine waves cos (x) + A. Cos (3x)

The third figure is the superposition of four spring sine waves.

The fourth figure is the superposition of 10 constipation sine waves

As the number of sine waves gradually increases, they will eventually be superimposed into a standard rectangle. What do you learn from this?

(As long as you work hard, you can bend straight !)

As the superposition increases, all the rising parts of the sine wave gradually make the originally slowly increasing curve steep, the falling part of all sine waves offset the part that continues to rise when it rises to the highest position and changes it to a horizontal line. A rectangle is superimposed. But how many sine waves need to be superimposed to form a rectangular wave with a standard 90 degree angle? Unfortunately, the answer is infinite. (God: Can you guess me ?)

Not just a rectangle, but any waveform you can think of can be superimposed with a sine wave in this way. This is the first intuitive difficulty for people who have never been familiar with Fourier analysis, but once they accept this setting, the game begins to become interesting.

Still the sine wave is added to the Rectangular Wave. Let's take a look at it from another angle:

In these images, the front black line is the sum of all sine waves, that is, the image that is closer and closer to the Rectangular Wave. The sine wave arranged by different colors is the component of the Rectangular Wave. These sine waves are arranged from low to high from front to back, and the amplitude of each wave is different. A careful reader must have discovered that there is a straight line between every two sine waves. It is not a split line, but a sine wave whose amplitude is 0! That is to say, in order to form a special curve, some sine wave components are not required.

Here, sine waves of different frequencies become frequency components.

Okay, the key point is coming !!

If we regard the first lowest frequency component as "1", we have the most basic unit for building the frequency domain.

For the most common rational number axis, the number "1" is the basic unit of the rational number axis.

The basic unit in the time domain is "1 second". If we regard a sine wave cos (T) with an angle frequency as the basis, the basic unit in the frequency domain is.

With "1" and "0", what is the "0" in the frequency field? Cos (0 T) is a sine wave with an infinite cycle, that is, a straight line! Therefore, in the frequency field, the 0 frequency is also called the DC component. In the superposition of Fourier series, it only affects the overall upward or downward direction of all waveforms relative to the number axis without changing the wave shape.

Next, let's go back to junior high school and recall the dead Eight Precepts. No, how do dead teachers define a sine wave.

A sine wave is the projection of a circular motion on a straight line. Therefore, the basic unit in the frequency domain can be understood as a circle that is always rotating.

It's a pity that you cannot upload a dynamic graph ......

For more information about motion graphs, click here:

File: Fourier series square wave circles animation.gif

And here:

File: Fourier series sawtooth wave circles animation.gif

If you click out, don't be turned off and run on Wiki. Where are the articles written on Wiki.

After introducing the basic component units in the frequency field, we can take a look at a Rectangular Wave in the frequency field:

What is this strange thing?

This is what the Rectangular Wave looks like in the frequency field. Isn't it completely recognizable? Textbooks are generally given here and left with endless fantasies and endless sputation. In fact, it is enough for the textbook to make up a picture: the frequency-domain image, also known as the spectrum, is --

A little clearer: we can find that in the spectrum, the amplitude of the even number is 0, which corresponds to the color line in the figure. A sine wave whose amplitude is 0.

Animation stamp:

File: Fourier series and transform.gif

Honestly, when I was learning Fourier transform, this image of Wikipedia had not yet appeared. At that time, I came up with this expression method, the phase spectrum, which is not represented by the Wikipedia, will be added later.

But before talking about the phase spectrum, let's first review what this example actually means. Do you remember the saying "the world is static? It is estimated that many people have been talking about this sentence for half a day. Imagine that every seemingly chaotic appearance in the world is actually an irregular curve on the timeline, but actually these curves are composed of these endless sine waves. What seems irregular is the projection of a regular sine wave in the time domain, while a sine wave is a projection of a rotating circle on a straight line. So what kind of picture will be generated in your mind?

The world in our eyes is like the backdrop of a movie. There are countless gears behind the backdrop. The gears drive the gears and the gears drive the smaller ones. There is a villain on the outmost gear-we are ourselves. We can only see this little guy performing in front of the backdrop without being able to predict where he will go next. The gears behind the backdrop are always rotating and never stopping. In this way, I feel a little fatalistic. To be honest, this kind of depiction of life is what a friend of mine lamented when we were both high school students. At that time, it seemed like I understood it, until one day I learned the Fourier series ......

3. phase spectrum of Fourier Series

The keywords in the previous chapter are as follows. The key words in this chapter are as follows.

At the beginning of this chapter, I want to answer a question from many people: What is Fourier analysis? This section is relatively boring. If you already know it, You can directly jump to the next split line.

Let's talk about the most direct purpose. Whether we listen to broadcast or watch TV, we must be familiar with a channel. A channel is a channel of frequency. Different channels use different frequencies as a channel for information transmission. Let's try one thing:

First draw a sin (X) on the paper, which is not necessarily a standard, and the meaning is almost the same. Not very difficult.

Next, draw a sin (3x) + sin (5x) image.

Don't say the standard is not standard. When will the curve rise or fall?

Well, it doesn't matter if you can't draw it. I will give you the sin (3x) + sin (5x) curve, but only if you don't know the equation of this curve, now you need to take sin (5x) out of the queue to see what is left. This is basically impossible.

But in the frequency domain? It is very simple, just a few vertical lines.

Therefore, many mathematical operations that seem impossible in the time domain are very easy in the frequency domain. This is where Fourier transformation is needed. In particular, some specific frequency components are removed from a curve. This is called filtering in engineering and is one of the most important concepts in signal processing. It can be easily implemented only in the frequency domain.

Another more important thing, but a little more complex, is to solve the differential equation. (This section is a little difficult. If you cannot understand it, you can skip this section.) The importance of the differential equation does not need to be described too much. All walks of life. However, solving a differential equation is quite troublesome. In addition to addition, subtraction, multiplication, division, and differential integral calculation. However, Fourier transformation can convert the differentiation and integral into multiplication and division in the frequency domain. In college mathematics, it instantly changes to elementary arithmetic.

Fourier analysis also has other more important functions, which we will discuss later.

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Next we will continue to talk about the phase spectrum:

Through the time domain to the frequency domain transformation, we get a spectrum from the side, but this spectrum does not contain all the information in the time domain. Because the spectrum only represents the amplitude of each corresponding sine wave, not the phase. Base sine wave. in sin (wt + θ), the amplitude, frequency, and phase are indispensable. Different Phases determine the position of the wave. Therefore, for frequency domain analysis, only the spectrum (amplitude spectrum) is not enough, we also need a phase spectrum. So where is the phase spectrum? We can see that in order to avoid image mixing, we use seven waves of overlapping images.

Given that the sine wave is periodic, we need to set something to mark the position of the sine wave. The red dots in the figure. A small red dot is the peak closest to the frequency axis. How far is the peak located from the frequency axis? To make it clearer, we projected the red points to the lower plane. The projection points are represented by pink points. Of course, these pink points only indicate the distance from the peak to the frequency axis, not the phase.

Here, we need to correct the concept that time difference is not a phase difference. If the entire cycle is regarded as 2 PI or 360 degrees, the phase difference is the proportion of the time difference in a cycle. After dividing the time difference by the period and then multiplying the 2PI, the phase difference is obtained.

In the complete three-dimensional graph, we divide the time difference obtained by projection by the cycle of the frequency to obtain the lowest phase spectrum. Therefore, the spectrum is from the side, and the phase spectrum is from the following. The next time you peek at a girl's dress, you can tell her: "Sorry, I just want to see your phase spectrum ."

Note that the phase in the phase spectrum is pi except 0. Because cos (t + PI) =-cos (t), in fact, the wave whose phase is pi is only flipped up and down. For the Fourier series of the periodic square wave, such a phase spectrum is very simple. In addition, because cos (t + 2PI) = cos (t), the phase difference is periodic. Pi and 3pi, 5pi, and 7pi are both the same phase. The value range of the manually defined phase spectrum is (-Pi, Pi], so the phase difference in the figure is pi.

Finally, let's look at a large collection:

Iv. Fourier Transformation)

I believe that through the previous three chapters, we have a new understanding of the frequency and Fourier series. However, I have mentioned at the beginning that the example of piano music is a formula error, but a typical concept example. Where is the so-called formula error?

The essence of Fourier series is to divide a periodic signal into an infinitely multiple separated (discrete) sine wave, but the universe does not seem to be cyclical. I used to write a write-down poem when I was learning digital signal processing:

In the past, consecutive non-cyclic events and non-consecutive recall cycles cannot be restored if you use ZT or DFT.

(Please ignore the literary level like my scum ......)

In this world, some things will never come again after a while, and time will never stop marking those unforgettable past records on time points. However, these things often become our extremely valuable memories. In our brains, they will pop up cyclically over a period of time. Unfortunately, these memories are scattered fragments, there are often only the happiest memories, and the plain memories are gradually forgotten. Because the past is a continuous non-periodic signal, and recall is a periodic discrete signal.

Is there a mathematical tool for converting continuous non-periodic signals into periodic discrete signals? Sorry, no.

For example, Fourier series is a periodic and continuous function in the time domain, and a non-periodic discrete function in the frequency domain. This sentence is a bit of a detour. You can simply recall the first chapter of the picture.

The next step we will talk about is to convert a non-periodic continuous signal in the time domain into a non-periodic continuous signal in the frequency domain.

Forget it. Let's take a look at the previous figure for your convenience:

Or we can also understand from another angle: Fourier transformation is actually a Fourier transformation of a function with an infinite periodic size.

Therefore, the piano spectrum is not a continuous spectrum, but a lot of discrete frequencies in time. However, such an apt metaphor is really hard to find the second one.

Therefore, in the frequency field of Fourier transformation, the discrete spectrum is changed to the continuous spectrum. So what does the continuous spectrum look like?

Have you ever seen the sea?

In order to facilitate the comparison, we will look at the spectrum from another perspective, or the most frequently used graph in Fourier series. We will look at it from a high frequency perspective.

The above is the discrete spectrum, so what is the continuous spectrum?

Imagine that these discrete sine waves are getting closer and closer, and gradually become continuous ......

Until it becomes like a sea of waves:

Sorry, in order to make these waves clearer, I did not select the correct calculation parameters. Instead, I chose some parameters to make the image more beautiful, otherwise, this figure will look like shit.

However, by comparing the two images, you can understand how to change from discrete spectrum to continuous spectrum? The overlay of Discrete Spectrum turns into the accumulation of continuous spectrum. Therefore, in calculation, the sum symbol is also changed to the integral symbol.

However, this story is not complete yet. Next, I promise you to see a more beautiful and spectacular picture. But here we need to introduce a mathematical tool to continue the story, this tool is --

5. The first formula of the universe: Euler's Formula

The concept of virtual number I was introduced in high school, but at that time we only knew that it was the square root of-1. But what does it really mean?

There is a digital axis with a red line segment on it. Its length is 1. When it is multiplied by 3, its length changes to a blue line segment, and when it is multiplied by-1, it becomes a green line segment, or the line segment rotates 180 degrees around the origin on the number axis.

We know that multiplication-1 is actually a multiplication of two I to make the line segment rotate 180 degrees. So what about multiplication I? The answer is simple -- it is rotated 90 degrees.

At the same time, we obtain a vertical virtual number axis. The real number axis and the virtual number axis form a complex plane. In this way, we can see that a function of multiplication I-rotation.

Now, we have a grand debut of the world's first handsome expression Euler's formula --

This formula has much greater significance than Fourier analysis in mathematics, but it is the first handsome formula in the universe because of its special form-when X is equal to pi.

Science and engineering students often use this formula to explain the beauty of mathematics to their sisters in order to show their academic skills: "You see, this formula has a natural basis E, natural Numbers 1 and 0, virtual numbers I and PI, it is so concise, so beautiful! "But girls often have only one sentence in their hearts:" stinks ...... "

The key function of this formula is to unify the sine wave into a simple exponential form. Let's take a look at the meaning of the image:

The Euler's formula depicts a point that performs circular motion on the complex plane over time. As time changes, it becomes a spiral on the timeline. If you only look at its real number, that is, the projection of the spiral on the left side, it is a basic cosine function. The projection on the right side is a sine function.

For a deeper understanding of the plural, refer:

What is the physical meaning of a plural number?

I don't need to talk about it too much here. It's enough for everyone to understand the content below.

6. Fourier transformation in exponential form

With the help of the Euler's formula, we know:Sine Wave SuperpositionCan also be understoodSpiral superpositionProjection in real space. What is an image chestnut used to understand the superposition of the Spiral Line?

Light Wave

We learned in high school that natural light is made up of different colors of light, and the most famous experiment is the triprism experiment of Master Newton:

In fact, we have been exposed to the optical spectrum for a long time, but we have not understood the more important significance of the spectrum.

However, the difference is that the spectrum transformed by Fourier is not only a finite superposition of the visible light, but a combination of frequencies from 0 to infinity.

Here, we can use two methods to understand the sine wave:

The first is the projection of the spiral in the real axis.

Another form that needs to be understood using the Euler's formula:

Add the preceding two formulas and divide them by 2:

How can this formula be understood?

As we mentioned earlier, e ^ (IT) can be understood as a counter-clockwise spiral, and E ^ (-It) can be understood as a clockwise spiral. Cos (t) is the half of the two spiral s with different rotation directions, because the virtual part of the two spiral S is offset by each other!

For example, two optical waves with different polarization directions are used to offset the magnetic field and double the electric field.

Here, the clockwise rotation is called the positive frequency, while the clockwise rotation is called the negative frequency (note that it is not the complex frequency ).

Now, we have seen the sea-the continuous Fourier transform spectrum. Now let's think about what the continuous spiral will look like:

Let's look at it again:

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Pretty?

Guess what the image looks like in the time domain?

Haha, do you think it is a slap in the face. Mathematics is such a complicated question.

By the way, for the convenience of viewing, I only showed the positive frequency and the negative frequency.

If you take it seriously, every spiral in the conch diagram can be clearly seen. Each spiral has a different amplitude (rotation radius), frequency (rotation cycle), and phase. Connecting all the Helios to a plane is the conch chart.

Now, I believe that everyone has an image understanding of Fourier transformation and Fourier series. Let's summarize it with a figure:

Fourier series Fourier Transformation