What is the Fourier transform? You can understand at a glance, and the writing is superb!

Author: Han Hao

Know: Heinrich

Weibo: @Peanut Oil Workers

Know the column: Stories that have nothing to do with time

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

Students who reprinted please keep the above sentence, thank you. I would be even more grateful if I could keep the source of the article.

——Updated in 2014.6.6, students who want to see the update directly can skip to Chapter 4————

I promise that this article is different from all the articles you have read before. It was written when I was still in the nutshell in 12 years, but I went abroad before I had time to finish it... So it was delayed for two years, um, I am Procrastination patients...

The core idea of ​​this article is:

Let readers understand Fourier analysis without looking at any mathematical formulas

Fourier analysis is not only a mathematical tool, but also a mode of thinking that can completely subvert a person's previous worldview. Unfortunately, the formula of Fourier analysis seems too complicated, so many freshmen came up in confusion and hated it ever since. To be honest, such an interesting thing has become a killer course in the university, and I have to blame the person who compiled the textbook is too serious. (Will you die if you write the textbook a little more fun? Will you die?) So I always wanted to write an interesting article to explain Fourier analysis, one that can be understood by high school students if possible . So, no matter what kind of work you are doing after reading this, I guarantee that you can understand it, and you will definitely experience the thrill of seeing the world in another way through Fourier analysis. As for friends who already have a certain foundation, I also hope that they will not be able to read the meeting place quickly and read carefully, and there will be new discoveries.

————The above is the poem of the final stage————

Enter the topic below:

I'm sorry, but I still have to make a long-winded sentence: In fact, learning is not easy. My original intention of writing this article is to hope that everyone will learn more easily and full of fun. But be sure! Don't save this article or save the address, thinking in your heart: I will read it later when I have time. There are too many examples of this, and maybe you will not open this page again in a few years. In any case, be patient and read on. This article is much easier and more fun than reading a textbook...

ps In this article, whether it is cos or sin, the term "sine wave" (Sine Wave) is used to represent simple harmonics.

1. What is the frequency domain

From the time we were born, the world we see runs through time. The trend of stocks, the height of people, and the trajectory of cars will change over time. This method of observing the dynamic world with time as a reference is called time domain analysis. And we also take it for granted that everything in the world is constantly changing over time and will never stand still. But if I tell you to observe the world in another way, you will find that the world is eternal, will you think I am crazy? I'm not crazy, this static world is called the frequency domain.

Let me give you an example that is not very appropriate in terms of formula , but not in sense:

In your understanding, what is a piece of music?

This is our most common understanding of music, a vibration that changes over time. But I believe that the more intuitive understanding of music for young musical instrument players is this:

Okay! Goodbye, classmates after class.
Yes, in fact, this paragraph can be over here. The picture above shows the appearance of music in the time domain, while the picture below shows the appearance of music in the frequency domain. Therefore, the concept of frequency domain is never unfamiliar to everyone, but never realized it.

Now we can go back and revisit the idiotic dream-like sentence at the beginning: The world is eternal.

Simplify the above two figures:

Time domain:

Frequency domain:

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

and so

In your eyes, the seemingly changing world with fallen leaves is actually just lying in the arms of God a piece of music that has already been composed.
Sorry, this is not a sentence of chicken soup, but a conclusive formula on the blackboard: Student Fourier told us that any periodic function can be regarded as a superposition of sine waves of different amplitudes and phases. In the first example, we can understand that by using different keys with different strengths and different time points, any piece of music can be combined.

One of the methods that penetrates the time domain and frequency domain is the Fourier analysis mentioned in the biography. Fourier analysis can be divided into Fourier series (Fourier Serie) and Fourier transform (Fourier Transformation). Let's start with the simple one.

Second, the frequency spectrum of Fourier Series (Fourier Series)

It is better to give a chestnut and have a picture and the truth to understand.

If I said that I could superimpose a rectangular wave with a 90 degree angle with the sine wave mentioned above, would you believe it? You won't, just like me back then. But look at the picture below:

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

The second picture is the superposition of 2 cute sine waves cos(x)+a.cos(3x)

The third picture is the superposition of 4 spring sine waves

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

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, the bend can be straightened!)

As the superposition increases, the rising part of all sine waves gradually steepens the originally slowly increasing curve, and the falling part of all sine waves offsets the part that continues to rise when it rises to the highest point and turns it into a horizontal line. A rectangle is superimposed in this way. But how many sine waves can be superimposed to form a rectangular wave with a standard 90 degree angle? Unfortunately, the answer is infinite. (God: Can I let you guess me?)

Not just rectangles, 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 they accept this setting, the game becomes interesting.

Or the sine waves in the above figure are accumulated into a rectangular wave. Let’s look

at it from another angle: In these pictures, the black line at the top is the sum of all sine waves, which is getting closer and closer to the rectangular wave. That graphic. The following sine waves arranged in different colors are the components of the rectangular wave combined. These sine waves are arranged in frequency from low to high and 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 dividing line, but a sine wave with an amplitude of 0! In other words, in order to compose a special curve, some sine wave components are not needed.

Here, sine waves of different frequencies become frequency components.

Okay, here comes the key! !

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

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

The basic unit of the time domain is "1 second". If we take a sine wave cos ([formula]t) with an angular frequency of [formula] as the basis, then the basic unit of the frequency domain is [formula].

With "1", there must be "0" to form the world, so what is the "0" in the frequency domain? cos(0t) is a sine wave with an infinite period, that is, a straight line! Therefore, in the frequency domain, the 0 frequency is also called the DC component. In the superposition of the Fourier series, it only affects the overall upward or downward direction of the entire waveform with respect to the number axis without changing the shape of the wave.

Next, let us go back to junior high school and recall the dead Bajie, ah no, how did the dead teacher define the sine wave.

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

. It is a pity that I can't transmit a dynamic picture...

Those who want to see the animation, please click here:
File:Fourier series square wave circles animation.gif

And here:

File:Fourier series sawtooth wave circles animation.gif

Friends who clicked out should not be abducted by the wiki. There are no articles written on the wiki so unscrupulous, right?

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

what is this strange thing?
This is what the rectangular wave looks like in the frequency domain. Is it completely unrecognizable? Textbooks are generally given here and left to readers endless daydreams and endless complaints. In fact, textbooks only need to add a picture: frequency domain image, which is commonly known as frequency spectrum, is -

To be more clear: it

can be found that in the frequency spectrum, the amplitude of the even-numbered items is all 0, which corresponds to the colored straight line in the figure. A sine wave with an amplitude of 0.
Please poke the animated picture:

File:Fourier series and transform.gif

To be honest, when I was learning the Fourier Transform, the Wiki chart hadn’t appeared yet. At that time, I thought of this way of expression, and I will add another spectrum that Wiki does not show—the phase spectrum.

But before we talk about the phase spectrum, let’s review what this example just means. Remember the phrase "the world is static" that I said earlier? It is estimated that many people have been complaining about this sentence for a long time. Imagine that every seemingly chaotic appearance in the world is actually an irregular curve on the time axis, but in reality these curves are composed of these endless sine waves. What we seem to be irregular is the projection of a regular sine wave on the time domain, and the sine wave is the projection of a rotating circle on a straight line. So what picture will appear in your mind?

The world in our eyes is like the big curtain of a shadow play. Behind the curtain are countless gears. The big gear drives the small gears, and the small gears drive the smaller ones. There is a small person on the outermost small gear-that is ourselves. We only saw this little man performing in front of the curtain irregularly, but we couldn't predict where he would go next. But the gears behind the curtain always keep spinning like that, never stopping. In this way, it feels a bit fatalistic. To be honest, this kind of depiction of life was sighed by a friend of mine when we were both high school students. At that time, I thought about it, but I didn’t understand it until I learned the Fourier series one day...

3. Phase spectrum of Fourier Series (Fourier Series)

The key word in the previous chapter is: looking at it from the side. The key word in this chapter is: look from below.

At the beginning of this chapter, I want to answer a question from many people: What is Fourier analysis used for? This period is relatively boring, and students who already know it can skip directly to the next dividing line.

Let me talk about the most direct use. 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 one channel for information transmission. Let's try one thing:
first draw a sin(x) on the paper, not necessarily standard, the meaning is almost the same. It's not difficult.

OK, then draw a graph of sin(3x)+sin(5x).

Don't say that the standard is not standard. You don't necessarily draw when the curve rises or falls, right?

Okay, it doesn’t matter if you can’t draw it. I’ll give you the sin(3x)+sin(5x) curve, but the premise is that you don’t know the equation of this curve. Now you need to give me sin(5x) and take it out of the picture. And see what's left. This is basically impossible.

But in the frequency domain? It's very simple, nothing more than a few vertical lines.

So many mathematical operations that seem impossible in the time domain are easy to reverse in the frequency domain. This is where the Fourier transform is needed. Especially removing some specific frequency components from a certain curve. This is called filtering in engineering. It is one of the most important concepts in signal processing, and it can be done easily only in the frequency domain.

There is another more important, but slightly more complicated purpose-solving differential equations. (This paragraph is a bit difficult, you can skip this paragraph if you don’t understand it) I don’t need to introduce the importance of differential equations too much. Used in all walks of life. But solving differential equations is quite troublesome. Because in addition to calculating addition, subtraction, multiplication and division, it is also necessary to calculate differential integral. The Fourier transform can make differentiation and integration into multiplication and division in the frequency domain, and college mathematics instantly becomes elementary school arithmetic.

Of course, Fourier analysis has other more important uses, which we will mention as we talk.

————————————————————————————————————

Let's continue to talk about the phase spectrum:

Through the transformation from the time domain to the frequency domain, we get a spectrum viewed from the side, but this spectrum does not contain all the information in the time domain. Because the frequency spectrum only represents the amplitude of each corresponding sine wave, without mentioning the phase. In the basic sine wave A.sin(wt+θ), amplitude, frequency, and phase are indispensable. Different phases determine the position of the wave. Therefore, for frequency domain analysis, only the frequency spectrum (amplitude spectrum) is not enough. A phase spectrum is required. So where is this phase spectrum? Let's look at the picture below, this time in order to avoid the picture being too confusing, we use a picture with 7 waves superimposed.

Since the sine wave is periodic, we need to set something to mark the position of the sine wave. In the picture are the little red dots. The small 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 point to the lower plane, and the projection point is represented by the pink point. Of course, these pink dots only mark the distance between the peak and the frequency axis, not the phase.

A concept needs to be corrected here: the time difference is not the phase difference. If the whole period is regarded as 2Pi or 360 degrees, the phase difference is the ratio of the time difference in a period. We divide the time difference by the period and multiply it by 2Pi to get the phase difference.
In the complete three-dimensional image, we divide the time difference obtained by the projection by the period of the frequency in turn to get the lowest phase spectrum. Therefore, the spectrum is viewed from the side, and the phase spectrum is viewed from below. Next time you peek at the bottom of a girl’s skirt and be found, you can tell her: "Sorry, I just want to see your phase spectrum."

Note that, except for 0, the phase in the phase spectrum is Pi. Because cos(t+Pi)=-cos(t), the wave with the phase of Pi is actually only flipped up and down. For the Fourier series of periodic square waves, such a phase spectrum is very simple. It is also worth noting that since cos(t+2Pi)=cos(t), the phase difference is periodic, and pi is the same phase as 3pi, 5pi, and 7pi. The value range of the phase spectrum is artificially defined as (-pi, pi], so the phase difference in the figure is all Pi.

Finally, a big collection:

Four, Fourier Transformation (Fourier Transformation)

I believe that through the previous three chapters, everyone has a new understanding of the frequency domain and Fourier series. But at the beginning of the article about the piano score example, I said that this chestnut is a formula error, but a typical example of the concept. Where is the so-called formula error?

The essence of Fourier series is to decompose a periodic signal into infinitely separated (discrete) sine waves, but the universe does not seem to be periodic. I once wrote a limericks poem when I was learning digital signal processing:

In the past, continuous non-periodical,

The recall cycle is not continuous,

No matter what you ZT, DFT,

Restore will not go back.

(Please ignore my scumbag literary level...)

In this world, some things happen for a while and never come again, and time never stops marking those unforgettable past and continuous points in time. But these things often become our extremely precious memories, which will pop out periodically in our brains after a period of time. Unfortunately, these memories are scattered fragments, often only the happiest memories, and the plain memories. Gradually forgotten by us. Because the past is a continuous non-periodic signal, and the memory is a periodic discrete signal.

Is there a mathematical tool to transform a continuous non-periodic signal into a periodic discrete signal? Sorry, no.

For example, the 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 rather circumspect, and it is really troublesome to recall the pictures in Chapter 1.

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

Forget it, the last picture is for everyone to understand:

Or we can also understand from another angle: Fourier transform is actually a Fourier transform of a function with infinite period.
So, the piano score is actually not a continuous spectrum, but a lot of discrete frequencies in time, but such an appropriate analogy is really difficult to find the second one.

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

Have you seen the sea?

In order to facilitate your comparison, we will look at the frequency spectrum from another angle this time, which is the most used picture in the Fourier series. Let's look at it from the direction of higher frequency.
The above is a discrete spectrum, so what does a continuous spectrum look like?

Give full play to your imagination, imagine these discrete sine waves getting closer and closer, gradually becoming continuous...

Until it becomes like the undulating sea:

Sorry, in order to make these waves more clearly visible, I did not choose the correct calculation parameters, but selected some parameters to make the picture more beautiful, otherwise the picture looks like shit.

But by comparing these two pictures, everyone should be able to understand how the discrete spectrum becomes the continuous spectrum, right? The superposition of the original discrete spectrum has become the accumulation of continuous spectrum. So in calculation, it has also changed from a summation symbol to an integral symbol.

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

5. The first formula for being handsome in the universe: Euler's formula

The concept of the imaginary number i has been used in high school, but at that time we only knew that it was the square root of -1, but what is its real meaning?

There is a number line, and there is a red line segment on the number line, its length it's 1. When it is multiplied by 3, its length changes and becomes a blue line segment, and when it is multiplied by -1, it becomes a green line segment, or the line segment rotates around the origin on the number axis 180 degrees.
We know that multiplying by -1 is actually multiplying i twice to rotate the line segment by 180 degrees, then multiplying i once-the answer is very simple-rotated by 90 degrees.

At the same time, we obtain a vertical imaginary axis. The real number axis and the imaginary number axis together form a complex plane, also known as the complex plane. In this way, we understand that a function of multiplying the imaginary number i-rotation.

Now, there is a grand debut of Euler’s formula, the number

one formula in the universe- this formula is far more meaningful in the field of mathematics than Fourier analysis, but multiplying it as the number one formula in the universe is because of its special form- -When x is equal to Pi.

In order to show their academic skills to girls, students of science and engineering often use this formula to explain the beauty of mathematics to girls: "Look, sister Pomegranate, this formula has natural base e, natural numbers 1 and 0, imaginary number i, and pi. Pi, it is so simple and beautiful! "But the girls often only have one sentence in their hearts: "Smelly dick silk..."

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:
Euler's formula depicts a point that changes in a circular motion on the complex plane over time. As time changes, it becomes a spiral on the time axis. If you only look at its real part, that is, the projection of the spiral on the left, it is the most basic cosine function. The projection on the right is a sine function.
For a deeper understanding of complex numbers, you can refer to:

What is the physical meaning of complex numbers?

It doesn’t need to be too complicated here, just enough for everyone to understand the rest of the content.

Six, the Fourier transform in exponential form

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

Light wave

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

So in fact, we have been exposed to the spectrum of light very early, but we didn’t understand that the spectrum is more important. Meaning.

But the difference is that the Fourier transformed spectrum is not just a superposition of limited frequency range such as visible light, but a combination of all frequencies from 0 to infinity.

Here, we can use two methods to understand sine waves:

The first one has been mentioned before, which is the projection of the helix on the real axis.

The other one needs to be understood with the help of another form of Euler's formula:
add the above two formulas and divide by 2 to get:
How can this formula be understood?

As we said earlier, e ^{(it) can be understood as a counterclockwise spiral, and e} (-it) can be understood as a clockwise spiral. And cos(t) is half of the superposition of the two spirals with different rotation directions, because the imaginary parts of the two spirals cancel each other out!

For example, two light waves with different polarization directions will cancel out the magnetic field and double the electric field.

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

Okay, just now we have seen the sea-the continuous Fourier transform spectrum. Now think about what a continuous spiral would look like:

Imagine scrolling down again:

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Isn't it beautiful?

Guess, what does this graph look like in the time domain?

Haha, do you feel that you have been slapped in the face. Mathematics is such a thing that makes simple problems very complicated.

By the way, for the picture that looks like a large conch, I only show the positive frequency part, and the negative frequency part is not displayed for the convenience of viewing.

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

Okay, having said that, I believe everyone has a vivid understanding of Fourier transform and Fourier series, we finally use a picture to summarize:

Okay, Fourier’s story is finally over, let’s talk about my story below:

You can never guess where this article was written for the first time. It was on a high-math test paper. At that time, in order to score points, I retaken the high math (Part 1), but then I didn’t review it at all because of time constraints, so I went to the examination room with the mentality of a naked exam. But when I arrived at the exam room, I suddenly realized that in any case, I would not be better than the last time I took the exam, so just write some of my thoughts on mathematics. So it took an hour or so to write the first draft of this article eloquently on the test paper.

Guess how many points I got?

6 points

That's right, it's this number. And the 6 points is because I was really bored in the end. I filled in all the multiple-choice questions with C. I should have hit two and got this precious 6 points. Seriously, I really hope that the paper is still there, but it shouldn't be possible anymore.

So do you guys guess how many points I scored on the first signal and system test?

45 minutes

That's right, just enough to take the make-up exam. But I didn't take the exam and decided to take it again. Because I was busy with other things that semester, learning was really forgotten. But I know this is a very important course, and I have to thoroughly understand it anyway. To be honest, the course of Signals and Systems is almost the basis of most engineering courses, especially communications.

In the process of rebuilding, I carefully analyzed each formula, trying to give this formula an intuitive understanding. Although I know that for people who study mathematics, this kind of learning method has no future at all, because as the concept becomes more abstract and the dimension becomes higher and higher, this kind of image or model understanding method will completely lose its effect. But for an engineering student, it is enough.

Later, when I came to Germany, when the school here asked me to rebuild the signal and system, I was completely speechless. But there is no way. Germans sometimes despise the Chinese and feel that your education is not reliable. So no way, let's do it again.

This time, I got a perfect score, and the passing rate was only half.

To be honest, mathematical tools have completely different meanings for engineering students and for science students. As long as engineering students understand, know how to use it, and know how to check, it is enough. However, many colleges and universities teach these important mathematics courses to teachers in the mathematics department. Then there is a problem. The mathematics teacher speaks so much, and it is reasoning and proof, but the students have only one sentence in their minds: What is the purpose of learning this stuff?

Education without a goal is a complete failure.

At the beginning of learning a mathematical tool, students did not know the function of this tool and its practical meaning. On the textbooks, there are only obscure and difficult concepts, attributives of just over twenty words, and formulas that make you dizzy. It's weird to be able to learn interest!

Fortunately, I was lucky to meet Wu Nan from Dalian Maritime University. There are two clues throughout his class, one from top to bottom and the other from bottom to top. First talk about the meaning of this course, and then point out what kind of problems will be encountered in this course, so that students know the role of a certain knowledge they have learned in reality. Then start from the basics, comb the tree of knowledge, until it extends to the questions raised in another clue, and they are perfectly connected together!

This kind of teaching mode, I think, is what should appear in the university.

Finally, write to all the students who gave me likes and comments. Thank you all for your support, and I am sorry that I cannot reply one by one. Because I know that the messages of the column need to be loaded one by one, in order to see the last point, load many times. Of course I insisted on reading it, but I couldn't reply one by one.

This article only introduces a new way of understanding Fourier analysis. For studying, it is still necessary to steadfastly figure out formulas and concepts. 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 have fun in learning. …