Since we were born, the world we saw runs through time, and the trend of stocks, the height of people and the trajectory of cars will change with time. This method of observing the dynamic world with time as a reference is called time domain analysis. And we take it for granted that everything in the world is constantly changing with time and will never stop. But if I tell you to look at the world from another angle, you will find that the world is eternal. Do you think I'm crazy? I'm not crazy. This still world is called frequency domain.

Second, the spectrum of Fourier series.

It is better to give a chestnut, and you can understand it only if you have a picture and the truth.

Would you believe me if I said that I could superimpose a rectangular wave and the sine wave mentioned above into a 90-degree angle? You won't, just like me. But look at the picture below:

The first picture is a descending sine wave cos(x)

The second picture is the superposition of two lovely sine waves cos (x) +a.cos (3x).

The third picture is the superposition of sine waves of four springs.

The fourth picture is the superposition of 10 constipation sine waves.

As the number of sine waves increases, they will eventually be superimposed into a standard rectangle. What did you learn from it? As long as you work hard, you can bend straight! )

With the increase of superposition, the rising part of all sine waves gradually steepens the curve that was originally slowly rising, while the falling part of all sine waves offsets the part that continues to rise when it rises to the highest point, making it a horizontal line. This adds a rectangle. But how many sine waves need to be superimposed to form a standard 90-degree rectangular wave? Unfortunately, the answer is infinite. God: Can I let you guess me? )

Not just rectangles, but any waveform you can think of can be superimposed with sine waves in this way. This is the first intuitive difficulty for people who have not been exposed to Fourier analysis, but once you accept this setting, the game becomes interesting.

Or the sine wave in the above picture accumulates into a rectangular wave. Let's look at it from another angle:

In these figures, the black line in front is the sum of all sine waves, that is, the figure that is getting closer and closer to rectangular waves. Sinusoidal waves arranged in different colors are components of rectangular waves. These sine waves are arranged from front to back according to the frequency from low to high, and the amplitude of each wave is different. Careful readers must have found that there is a straight line between every two sine waves, which is not a dividing line, but a sine wave with an amplitude of 0! In other words, some sine wave components are unnecessary in order to form a special curve.

Here, sine waves with different frequencies become frequency components.

Well, here's the key! !

If we regard the first frequency component with the lowest frequency as "1", we have the most basic unit to construct frequency domain. For our most common rational axis, the number "1" is the basic unit of rational axis.

(Well, it's mathematically called -base. At that time, there was no other strange explanation for the word, and there were words like orthogonal base behind it. Can I say something? )

The basic unit of time domain is "1" seconds. If we take a sine wave COS (ω0t) with an angular frequency ω0 as the basis, then the basic unit of frequency domain is ω 0.

The world is composed of "1" and "0". What is the "0" in frequency domain? Cos(0t) is an infinite period sine wave, a straight line! Therefore, in the frequency domain, zero frequency is also called DC component. In the superposition of Fourier series, it only affects the whole waveform up or down relative to the number axis, without changing the shape of the waveform.

Next, let's go back to junior high school and recall the dead Bajie. Oh, no, how did the dead teacher define sine wave?

Sine wave is the projection of circular motion on a straight line. So the basic unit of frequency domain can also be understood as a circle that has been rotating.

Fourier series square wave circle animation. gif

[Fourier series sawtooth wave circle animation. gif]

After introducing the basic components of frequency domain, we can look at another expression of rectangular wave in frequency domain:

What is this strange thing?

This is what rectangular waves look like in frequency domain. Is it completely unrecognizable? Textbooks are generally given here, and then left to readers endless reverie and endless spit. In fact, it is enough to add a picture to the textbook: frequency domain image, also called frequency spectrum, is-

To be clear:

It can be found that in the spectrum, the amplitudes of even terms are all zero, which corresponds to the colored straight lines in the graph. A sine wave with an amplitude of 0.

Fourier series sum transformation

To tell the truth, when I was studying Fourier transform, this diagram of Wiki had not appeared yet, and I thought of this expression at that time. Another spectrum that Wiki did not express-phase spectrum, will be supplemented later.

But before we discuss the phase spectrum, let's review the meaning of this example. Remember the phrase "the world is still" mentioned earlier? This sentence is estimated that many people have been complaining for a long time. Imagine that every seemingly chaotic appearance in the world is actually an irregular curve on the time axis, and these curves are actually composed of these endless sine waves. What we look irregular is the projection of regular sine wave in time domain, and sine wave is the projection of a rotating circle on a straight line. So what kind of picture will come to your mind?

The world in our eyes is like the big curtain of shadow play. There are countless gears behind the scenes. The big gear drives the pinion, and the pinion drives the pinion. There is a little man on the outermost pinion-that is ourselves. We only see the little man performing irregularly in front of the curtain, but we can't predict where he will go next. But the gear behind the curtain always keeps spinning like that and never stops. That's a bit fatalistic. To tell the truth, this description of life was lamented by a friend of mine when we were both high school students. At that time, I thought it was incomprehensible until one day I learned Fourier series. ...

Third, the phase spectrum of Fourier series.

The key words in the previous chapter are: from the side. The key words in this chapter are as follows.

At the beginning of this chapter, I want to answer a question asked by many people: what is Fourier analysis for? This paragraph is boring, and students who already know it can jump directly to the next dividing line.

Let me say one of the most direct uses. Whether listening to the radio or watching TV, we must be familiar with one word-channel. Channel channels are frequency channels, and different channels use different frequencies as a channel to transmit information. Let's try one thing:

Draw a sin (X) on the paper first, which is not necessarily standard, but the meaning is similar. It's not that hard. Ok, let's draw a graph of sin(3x)+sin(5x). Don't say that the standard is not standard. You don't have to draw a curve when you go up or down, do you?

Well, it doesn't matter if you can't draw it. I'll give you the curve of sin(3x)+sin(5x), but only if you don't know the equation of this curve. Now I need you to remove sin(5x) from the picture and see what's left. This is basically impossible. But in the frequency domain? It's simple. It's just a few vertical lines.

Therefore, many seemingly impossible mathematical operations in the time domain can be easily reversed in the frequency domain. This is where the Fourier transform is needed. Especially, removing some specific frequency components from a curve, which is called filtering in engineering, is one of the most important concepts in signal processing, and can only be easily done in frequency domain.

Let's talk about a more important but slightly complicated usage-solving differential equations. I don't need to introduce the importance of differential equations too much. It is used in all walks of life. But solving differential equations is a troublesome thing. Because in addition to calculating addition, subtraction, multiplication and division, differential integral is also calculated. Fourier transform can make differential and integral become multiplication and division in frequency domain, and college mathematics instantly becomes elementary school arithmetic.

Of course, Fourier analysis has other more important uses, which we will mention in our speech.

Let's continue to discuss the phase spectrum:

Through the transformation from time domain to frequency domain, we get a spectrum from the side, but this spectrum does not contain all the information in time domain. Because the spectrum only represents the amplitude of each corresponding sine wave, there is no mention of phase. In the basic sine wave A.sin(wt+θ), amplitude, frequency and phase are indispensable, and different phases determine the position of the wave. So for frequency domain analysis, only frequency spectrum (amplitude spectrum) is not enough, we need phase spectrum. So where is this phase spectrum? Let's look at the picture below. This time, in order to avoid the confusion of the picture, we use a picture with seven waves superimposed.

Since sine wave is periodic, we need to set something to mark the position of sine wave. Those in the picture are little red dot. Little red dot is the peak closest to the frequency axis. How far is this peak from the frequency axis? In order to see more clearly, we project the red dot to the lower plane, and the projection point is represented by a pink dot. Of course, these pink dots only represent the distance from the peak to the frequency axis, not the phase.

A concept needs to be corrected here: time difference is not phase difference. If all periods are regarded as 2Pi or 360 degrees, the phase difference is the proportion of the time difference within a period. We divide the time difference by the period and multiply it by 2Pi to get the phase difference.

In the complete stereogram, we divide the time difference obtained by projection by the period of frequency in turn to get the lowest phase spectrum. Therefore, look at the spectrum from the side and the phase spectrum from the bottom. The next time you are caught peeking at a girl's skirt, you can tell her, "I'm sorry, I just want to see your photo album."

Note that all phases in the phase spectrum are π except 0. Because cos(t+Pi)=-cos(t), in fact, the wave with phase Pi just flips up and down. For the Fourier series of periodic square waves, such a phase spectrum is very simple. In addition, it is worth noting that because cos(t+2Pi)=cos(t), the phase difference is periodic, and Pi is the same as 3pi, 5pi and 7pi. The range of the artificially defined phase spectrum is (-π, π), so the phase difference in the figure is π.

Finally, a big collection:

Fourth, Fourier transformation (Fourier transformation)

Fourier transform is actually the Fourier transform of an infinite periodic function.

Therefore, the piano score is actually not a continuous spectrum, but many discrete frequencies in time, but it is really difficult to find a second one with such an appropriate metaphor.

Therefore, Fourier transform changes from discrete spectrum to continuous spectrum in frequency domain. So what does the continuous spectrum look like?

Have you ever seen the sea?

For the convenience of comparison, let's look at the spectrum from another angle this time, or the picture that Fourier series is used the most. Let's look at the direction with higher frequency.

The above is a discrete spectrum. What does a continuous spectrum look like?

Give full play to your imagination and imagine that these discrete sine waves are getting closer and closer and gradually become continuous. ...

Until it becomes like a raging sea:

Sorry, in order to see these waves more clearly, I didn't choose the correct calculation parameters, but chose some parameters to make the picture more beautiful, otherwise the picture would look like shit.

But comparing these two pictures, you should understand how to change from discrete spectrum to continuous spectrum, right? The superposition of discrete spectra becomes the accumulation of continuous spectra. Therefore, in calculation, it has changed from a summation symbol to an integral symbol.

However, the story is not over yet. Next, I promise to show you a more beautiful and spectacular picture than the above, but here you need to introduce a mathematical tool to continue the story. This tool is-

Fifth, the first formula of the universe: Euler formula

The concept of imaginary number was touched by everyone in high school, but at that time we only knew that it was the square root of-1, but what was its real meaning?

Here is a number axis with a red line segment on it. Its length is 1. When multiplied by 3, its length changes and becomes a blue line segment, and when 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 multiplying-1 is actually multiplying I twice to make the line segment rotate 180 degrees, and multiplying I once-the answer is simple-to rotate 90 degrees.

At the same time, we get a vertical imaginary axis. The real axis and the imaginary axis are isomorphic to form a complex plane, also known as a complex plane. In this way, we know that a function is multiplied by the imaginary I- rotation.

Now, please welcome the Euler formula, the first formula of the universe-

This formula is far more significant in the field of mathematics than Fourier analysis, but because of its special form-when x equals π, it is the first formula in the universe.

In order to show their academic skills, students of science and engineering often use this formula to explain the beauty of mathematics to their sisters: "Look, Sister Pomegranate, there are natural number base E, natural number 1 and 0, imaginary number I and pi in this formula. It's so simple and beautiful! " But girls often have only one sentence in their hearts: "Smelly diaosi ..."

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

Euler's formula describes that a point moves in a circle with time on the complex plane and turns into a spiral with time on the time axis. If we only look at its real part, that is, the projection of the left helix, it is the most basic cosine function. The projection on the right is a sine function.

For a deeper understanding of complex numbers, please refer to:

What is the physical meaning of complex numbers?

Needless to say, it is too complicated here, and the following content is enough for everyone to understand.

Exponential Fourier transform of intransitive verbs

With the help of Euler formula, we know that the superposition of sine waves can also be understood as the projection of spiral superposition in real space. And if you use the image of a chestnut to understand, what is the superposition of spirals?

light wave

We learned in high school that natural light is composed of different colors of light. The most famous experiment is Master Newton's prism experiment:

So in fact, we have been exposed to the spectrum of light for a long time, but we just don't understand the more important significance of spectrum.

But the difference is that the spectrum after Fourier transform is not only the superposition of visible light in a limited frequency range, but the combination of all frequencies from 0 to infinity.

Here, we can understand the sine wave from two aspects:

The first one, as mentioned before, is the projection of the helix on the real axis.

The other needs to be understood by another form of Euler's formula:

Add the above two formulas and divide by 2 to get:

How to understand this formula?

As we said just now, e^(it) can be understood as a spiral rotating counterclockwise, then E (-it) can be understood as a spiral rotating clockwise. Cos (t) is half of the superposition of these two spirals with different rotations, because the imaginary parts of these two spirals cancel each other out!

For example, two light waves with different polarization directions, the magnetic field cancels and the electric field doubles.

Here we call counterclockwise rotation a positive frequency and clockwise rotation a negative frequency (note that it is not a complex frequency).

Well, just now we have seen the ocean-continuous Fourier transform spectrum. Now think about what a continuous spiral will look like:

Imagine refusing again:

Isn't it beautiful?

Can you guess what this graph looks like in time domain?

Haha, did you feel slapped? Mathematics is such a thing that makes simple problems very complicated.

By the way, like the picture of the big conch, for the convenience of viewing, I only showed the positive frequency part, but not the negative frequency part.

If you look carefully, you can see every spiral on the conch map clearly. Each spiral has a different amplitude (rotation radius), frequency (rotation period) and phase. Connecting all the spirals into a plane is this conch picture.

Well, at this point, I believe everyone has an image understanding of Fourier transform and Fourier series. Finally, summarize it with a picture:

Well, Fourier's story is finally over. Let's tell my story:

You'll never guess where this article was first uninstalled. It was on an advanced mathematics test paper. At that time, in order to score points, I retaken advanced mathematics (Volume I), but because of the tight time, I didn't review at all, so I went to the examination room with the mentality of naked examination. But when I arrived at the examination room, I suddenly realized that I would never do better than last time, so I just wrote some thoughts about mathematics. So it took an hour or so to write the first draft of this article on the test paper.

Guess what my score is?

6 points

Yes, this is the number. The result of this 6 points is because I was really bored at last, and I filled in all the multiple-choice questions with C. I should have won two and got this precious 6 points. To tell you the truth, I really hope that paper is still there, but it should be impossible.

Then guess how many points I got in the first signal and system exam?

45 points

Yes, just enough to make up the exam. But I didn't take the exam and decided to retake it. Because I was busy with other things that semester, I really forgot to study. But I know this is a very important course, and I will understand it thoroughly anyway. Seriously, the course Signal and System is the foundation of almost all engineering courses, especially the communication major.

In the process of reconstruction, I carefully analyzed each formula and tried to give it an intuitive understanding. Although I know that this learning method has no future for people who study mathematics, because with the concept becoming more and more abstract and the dimension getting higher and higher, this image or model understanding method will completely lose its function. But for an engineering student, it is enough.

Later, I came to Germany, and when the school here asked me to rebuild the signal and system, I was completely speechless. But I can't. Germans sometimes despise China people and think your education is unreliable. So there is no way. Let's do it again.

I got full marks this time, and the passing rate was only half.

To tell the truth, mathematical tools have completely different meanings for engineering students and science students. It is enough for engineering students to understand, use and check. However, many colleges and universities teach these important mathematics courses to teachers of mathematics departments. So there's a problem. The math teacher talks nonsense, makes sense and proves it. There is only one sentence in the students' minds: What's the use of learning this product?

Education without goals is a complete failure.

At the beginning of learning a mathematical tool, students have no idea about the function and practical significance of this tool. There are only obscure concepts, attributes of more than 20 words, and formulas that make people dizzy. Learning interest is strange!

Luckily, I was lucky enough to meet Wu Nan, a teacher from Dalian Maritime University. There are two clues in his class, one is top-down and the other is bottom-up. Firstly, explain the significance of this course, and then point out what problems this course will encounter, so that students can know the role of some knowledge they have learned in reality. Then start from the basics and comb the knowledge tree until the questions raised in another clue are perfectly connected!

This teaching mode, I think, should appear in universities.

Finally, write and leave messages to all my classmates who like me. I really appreciate your support, and I'm sorry I can't reply one by one. Because the news of Zhihu column has to be loaded one by one, and it has to be loaded many times to see the last point. Of course, I insisted on reading it, but I just couldn't reply one by one.

This paper only introduces a novel understanding method of Fourier analysis. For learning, it is still necessary to find out the formulas and concepts in a down-to-earth manner. There is really no shortcut to learning. But at least through this article, I hope to make this long road more interesting.

Finally, I wish everyone can find fun in their study …