Fourier analysis (Popular interpretation)

The core idea of this article is:

Let the reader understand the Fourier analysis without looking at any mathematical formula.

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 Fourier analysis formula looks too complex, so many freshmen come up to the circle and abhor it. To be honest, such an interesting thing actually became a college killer course, and the people who had to be blamed for compiling the textbook were too serious. (Will you die if you write the textbook funny?) Are you going to die? So I've always wanted to write an interesting article explaining the Fourier analysis, and possibly the kind that high school students can read. So, no matter what kind of work you're doing here, I'm sure you'll be able to read it and be sure to appreciate the thrill of seeing the world another way through Fourier analysis. As for the friends already have a certain basis, also hope not to see the place on the hurried back, careful reading will certainly have a new discovery.

———— above are recite ————

Below to get to the chase:

Sorry, or to be wordy: in fact, learning is not easy, I write this article is the original intention is also hope that we learn more relaxed, full of fun. But be sure! Do not put this article in the collection, or save the address, in mind: there is time to see later. There are so many examples that you may not have opened this page in a few years. In any case, withstand the heart, read on. This article is much easier and happier than reading a textbook ...

P.S. Both cos and sin, the term "sine wave" (Sine wave) is used to represent simple harmonics.

First, what is the frequency domain

From the moment we were born, the world we saw was all about time, the movement of stocks, the height of people, and the trajectory of cars would change over time. This method of observing the dynamic world with time as a reference is called its last domain analysis. And we take it for granted that everything in the world is changing over time and will never rest. But if I told you to look at the world in another way, you would find that the world was immutable, wouldn't you think I was crazy? I am not crazy, this static world is called frequency domain.

Let's start with an example of a formula that is not very appropriate, but that is more appropriate in its 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 for the small players, the music is more intuitive to understand that:

Good! class, see you later, boys and girls.

Yes, in fact, this paragraph has been written so that it can be finished. It's the way music looks in the time domain, but it's the way music looks in the frequency domain. So the concept of frequency domain is not unfamiliar to everyone, just never realize it.

Now we can go back and look at the beginning of the silly words: the world is eternal.

Simplify the above two graphs:

Despoiling

Frequency domain:

In the time domain, we observe that the strings of the piano will swing in a moment, just like the movement of a stock; In the frequency domain, there is only one eternal note.

So

The seemingly falling leaves of your eyes in the changeable world, is actually just lying in the bosom of God a well-known music.

Sorry, this is not a chicken soup, but a solid formula on the blackboard: Fourier told us that any periodic function, can be seen as a different amplitude, different phase sine wave superposition . In the first example, we can understand that by using different strokes of different keys and different points of time, you can combine any piece of music.

One of the methods through time domain and frequency domain is Fourier analysis of crossing. Fourier analysis can be divided into Fourier series (Fourier Serie) and Fourier transform (Fourier transformation), we start from a simple beginning.

Spectrum of the second, Fourier series (Fourier series)

Or a chestnut and a picture of the truth to understand.

Would you believe me if I said I could stack a rectangular wave with a 90-degree angle with the sinusoidal wave I said earlier? You won't, just like I was when I was. But look at:

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

The second picture is the superposition of the 2-selling sine wave cos (x) +a.cos (3x)

The third picture is a superposition of 4 spring sine waves

The four images are the superposition of 10 constipation sine waves

As the number of sine waves increases gradually, they eventually stack up into a standard rectangle, and what do you get from it?

(as long as the effort, bent can be broken straight!) ）

As the stack increases, all the rising parts of the sine wave gradually taper the originally slowly increasing curve, and all the falling parts of the sine wave offset the part that continues to rise at the top to make it a horizontal line. A rectangle is stacked up like this. But how many sine waves can be stacked up to form a standard 90-degree rectangular wave? Unfortunately, the answer is infinitely multiple. (God: Can I make you guess me?) ）

No more than rectangles, any waveform you can think of can be superimposed in such a way with sine waves. This is not
The first difficulty with the intuition of people who have been exposed to Fourier analysis, but once this is accepted, the game begins to get interesting.

Still, the sine wave accumulates into a rectangular wave, and we look at it from a different angle:

In these pictures, the first black line is the sum of all the sine waves superimposed, that is, the graph that is getting closer to the rectangular wave. The sine wave, which is arranged in different colors, is a combination of the various components of a rectangular wave. These sine waves are arranged from low to high and the amplitude of each wave is different. There must be a careful reader to find that every two sine waves have a straight line, that is not the split line, but the amplitude of 0 sine wave! In other words, in order to form a special curve, some sine wave components are not required.

Here, sine waves of different frequencies we become frequency components.

Okay, here's the key!!

If we consider the first frequency component with the lowest frequency as "1", we have the most basic unit to build the frequency domain.

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

The basic unit of the time domain is "1 seconds", if we consider a sine wave cos (t) with angular frequency as the basis, then the basic unit of the frequency domain is.

With "1", but also to have "0" to constitute the world, then the frequency domain of "0" what is it? Cos (0t) is an infinitely long period sine wave, which is a straight line! So in the frequency domain, the 0 frequency is also called the DC component, in the superposition of the Fourier series, it only affects the whole waveform relative to the axis of the overall upward or downward without changing the shape of the wave.

Next, let's go back to junior high school, recall the eight commandments that have died, ah no, the dead teacher is how to define the sine wave bar.

A sine wave is a projection of a circular motion in a straight line. So the basic unit of the frequency domain can also be understood as a circle that is always rotated.

After introducing the basic elements of the frequency domain, we can look at a rectangular wave, another pattern in the frequency domain:

What kind of strange thing is this?

This is the appearance of the rectangular wave in the frequency domain, is not completely recognized? Textbooks are usually given here and then left to the reader endless reverie, and endless spit groove, in fact, as long as the textbook to fill a picture is enough: frequency domain image, known as the spectrum, is--

A little clearer:

It can be found that in the spectrum, the amplitude of even-numbered items is 0, which corresponds to the color line in the graph. Sine wave with an amplitude of 0.

To be honest, when I was studying the Fourier transform, this diagram of the wiki had not yet appeared, and then I thought of the expression, and then added another spectrum-the phase spectrum-that the wiki did not express.

But before we talk about the phase spectrum, let's look at what this example really means. Remember the phrase "The world is still" before? It is estimated that a lot of people have been spitting out the sentence for half a day. Imagine that every seemingly chaotic appearance in the world is actually an irregular curve on a timeline, but the actual curves are made up of these endless sine waves. What we seem to be irregular is the regular sine wave in the time domain projection, and the sine wave is a rotating circle in the line projection. So what's the picture in your head?

The world in our eyes is like the big curtain of shadow puppet, there are countless gears behind the curtain, big gears drive small gears, and small gears drive smaller. There's a villain on the outermost pinion--that's who we are. We only saw the villain performing in front of the curtain without any regularity, but could not predict where he would go next. And the gears behind the curtain are always spinning like that, never stopping. There is a sense of fatalism in this way. To tell the truth, this depiction of life is a friend of mine when we are all high school, when the sigh, think of indefinitely, until one day I learned the Fourier series ...

Phase spectra of three, Fourier series (Fourier series)

The key words in the previous chapter are: Look from the side. The key words in this chapter are: look from below.

At the very beginning of this chapter, I would like to answer a question from a lot of people: what is Fourier analysis for? This section is relatively boring, already know the classmate can jump directly to the next split line.

First of all, one of the most direct uses. Whether listening to the radio or watching TV, we must be familiar with a word-channel. Channel channel, is the frequency channel, different channels are different frequencies as a channel to carry out information transmission. Let's try one thing below:

Draw a sin (x) on the paper first, not necessarily the standard, the meaning is almost OK. It's not that hard.

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

Don't say the standard is not standard, when the curve when the rise when the fall you do not necessarily draw, right?

Well, it doesn't matter, I give you the curve of sin (3x) +sin (5x), but only if you don't know the equation of the curve, now you need to get sin (5x) out of the picture and see what's left. This is basically impossible to do.

But in the frequency domain? It's simple, it's just a few vertical lines.

So many of the mathematical operations that seem impossible in the time domain are quite easy in the frequency domain. This is where the Fourier transform is needed. In particular, the removal of 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 done easily in the frequency domain.

Another more important, but slightly more complex, use-solving differential equations. (This paragraph is a bit difficult, can not understand directly skip this paragraph) the importance of differential equations I don't have to introduce too much. All walks of life are used to. But solving the differential equation is a rather cumbersome task. Because in addition to calculating subtraction, we also calculate the differential integrals. The Fourier transform can make the differential and integral in the frequency domain into multiplication and division, the University Mathematics instantaneous Change Elementary School arithmetic has no.

Fourier analysis of course there are other more important uses, as we talk along with the mention.

————————————————————————————————————

Let's continue to say the phase spectrum:

By the time domain to the frequency domain transformation, we get a side view of the spectrum, but this spectrum does not contain all the information in the time domain. Because the spectrum represents only the amplitude of each corresponding sine wave, without mentioning the phase. The fundamental sine wave a.sin (wt+θ), amplitude, frequency, phase integral, different phase determines the position of the wave, so for the frequency domain analysis, only the spectrum (amplitude spectrum) is not enough, we also need a phase spectrum. So where is this phase spectrum? We see, this time in order to avoid the picture is too mixed, we use 7 waves superimposed figure.

Given that 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 red dot is the crest closest to the frequency axis, and how far is the crest located from the frequency axis? To see more clearly, we projected the red dots to the lower plane, and the projection points are indicated by the pink dots. Of course, these pink dots only mark the distance from the frequency axis of the crest, not the phase.

A concept needs to be corrected here: The time difference is not phase difference. If all cycles are considered to be 2Pi or 360 degrees, the phase difference is the ratio of the time difference to one cycle. We will get the phase difference in addition to the period of time and then multiply by 2Pi.

In the complete stereoscopic chart, we divide the projected time difference by the period of the frequency, and then we get the bottom phase spectrum. So, the spectrum is viewed from the side, and the phase spectrum is viewed from below. The next time you peek at a girl's skirt, you can tell her: "Sorry, I just want to see your phase spectrum." ”

It is noted that the phase spectrum is in addition to 0, which is pi. Because cos (T+PI) =-cos (t), the actually phase Pi wave is just upside down. For the Fourier series of periodic square waves, such phase spectra are already very simple. It is also noteworthy that due to the cos (T+2PI) =cos (t), the phase difference is periodic and the Pi and 3pi,5pi,7pi are the same phase. The domain value of the man-defined phase spectrum is (-PI,PI], so the phase difference in the graph is pi.

Finally come a large collection:

Four, Fourier transform (Fourier transformation)

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

The essence of the Fourier series is the decomposition of a periodic signal into an infinite number of separate (discrete) sine waves, but the universe does not seem to be cyclical. When I was learning digital signal processing, I wrote a limerick:

The past is a continuous non-cyclical,

The memory cycle is discontinuous,

Let your ZT, DFT,

Restore does not go back.

(Please disregard my standard of literature.) ）

In this world, some things in a moment, never again, and time has never ceased to those unforgettable past continuous mark on the point of time. But these things often become our extra precious memories, in our brains in a period of time will be periodically jumping out, but these memories are scattered fragments, often only the happiest memories, and dull memories gradually forgotten by us. Because, the past is a continuous non-periodic signal, and memory is a periodic discrete signal.

Is there a mathematical tool that transforms a continuous non-periodic signal into a periodic discrete signal? I'm sorry, I don't.

For example, the Fourier series, in the time domain is a periodic and continuous function, and in the frequency domain is a non-periodic discrete function. This sentence is more raozui, really look at the trouble can simply recall the first chapter of the picture.

And in the Fourier transform we're going to talk about, we're going to convert a time-domain non-cyclical continuous signal into a non-periodic signal in the frequency domain.

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

Or we can take a different view: The Fourier transform is actually a Fourier transform of a function with an infinitely large period.

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

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

Have you ever seen the sea?

For the sake of comparison, we look at the spectrum from another angle, or the most used one in the Fourier series, from a higher frequency.

The above is a discrete spectrum, so what does continuous spectrum look like?

Make the most of your imagination and imagine that these discrete sine waves are getting closer to each other and gradually becoming continuous ...

Until it becomes like a heaving sea:

I'm sorry, in order to make these waves clearer to see, I did not choose the correct calculation parameters, but chose some to make the picture more beautiful parameters, otherwise this figure looks like a excrement.

But by comparing these two graphs, you should be able to understand how to turn from discrete spectrum to continuous spectrum. The superposition of the original discrete spectrum becomes the accumulation of continuous spectrum. So in the calculation also from the summation symbol into the integral symbol.

However, this story is not finished, and then, I promise you to see a more beautiful than a spectacular picture, but here need to introduce a mathematical tool to continue the story, this tool is

Five, the Universe play handsome first formula: Euler formula

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

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

We know that multiply-1 is actually two times I make the segment rotate 180 degrees, then multiply I--the answer is simple--rotate 90 degrees.

At the same time, we get a vertical imaginary axis. The real and imaginary axes together form a complex plane, also called a complex plane. So we know that a function of headed number I--rotation.

Now, please let the universe first play handsome formula Euler formula grand debut--

This formula is much more important than Fourier analysis in the field of mathematics, but it is the first handsome formula for the universe because of its special form-when x equals Pi.

Often have science and engineering students in order to show their academic foundation with sister, with this formula to explain the beauty of mathematics: "Pomegranate sister you see, this formula has a natural base e, the natural number 1 and 0, the imaginary I also have pi pi, it is so concise, so beautiful ah!" "But the girls often have only one sentence:" The smelly cock 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:

The Euler formula depicts a point that, over time, makes a circular motion on the complex plane, which, over time, becomes a spiral on the timeline. If you look only at its real number, which is the projection of the helix 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?

There is no need to talk too complex, enough to let you understand the content behind it.

Fourier transform in exponential form

With the help of Euler's formula, we know that the superposition of the sine wave can also be understood as the projection of the helix superimposed on the real space. And what is the superposition of the helix if it is understood by an image of chestnuts?

Optical

We learned in high school that natural light was superimposed by different colors of light, and the most famous experiment was the three prism experiment of Master Newton:

So in fact, we have a very early exposure to the spectrum of light, but do not understand the spectrum of more important significance.

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

Here, we can understand sine waves in two ways:

The first one has already been said, is the spiral in the real axis projection.

Another way to understand this is by using a different form of Euler's formula:

Add the above two formula and divide it by 2 to obtain:

How can this formula be understood?

As we have just said, e^ (it) can be understood as a spiral counterclockwise rotation, then e^ (-it) can be understood as a clockwise rotation of the helix. The cos (t) is half the number of spiral stacks with different rotation directions, because the imaginary parts of the two spirals are offset from each other!

For example, two light waves with different polarization directions, a magnetic field offset, and an electric field doubling.

Here, the counterclockwise rotation we call the positive frequency, and the clockwise rotation we call the negative frequency (note not the complex frequency).

Well, just now we've seen the sea--the continuous Fourier transform spectrum, think about what a continuous helix would look like:

Think about it and turn it down:

Isn't it beautiful?

Guess what this graphic looks like in the time domain?

Haha, is not felt to be slapped a slap. Mathematics is such a complex thing as a simple question.

By the way, the picture is like a sea snail, for the sake of viewing, I just show the part of the positive frequency, and the negative frequency part is not shown.

If you look carefully, each spiral on the conch chart can be clearly seen, each helix has a different amplitude (rotation radius), frequency (rotation period) and phase. And all the helix is connected to a plane, this is the conch chart.

Well, in this case, I believe we have an image of the Fourier transform and Fourier series, and we end up with a picture to summarize:

Well, the Fourier story is finally finished, and here's my story:

The first time this article was written down, you'll never guess where it is, it's on a test paper with a high number. At that time in order to brush points, I rebuilt the high number (on), but later time is not review, so I took the nude test mentality to the examination room. But in the examination room I suddenly realized that I would not be better than the last test, so simply write some of my own thoughts about maths. So it took about one hours to write the first draft of this paper on the paper voluminous.

How many points do you think I have?

6 min

Yes, that's the number. And this 6 score is because finally I was bored, the choice of all filled in C, should be in two, got this valuable 6 points. To tell you the truth, I really hope that the paper is still there, but it should not be possible.

So, guess what my first signal and system score was?

45 min

Yes, just enough to take the retake. But my heart did not go to test, decided to rebuild. Because the semester is busy with other things, learning is really left behind. But I know this is a very important lesson, I want to thoroughly understand it anyway. To be honest, the course of signaling and systems is almost the foundation of most engineering courses, especially in the field of communications.

In the process of rebuilding, I carefully analyzed each formula and tried to give this formula an intuitive understanding. Although I know that for those who study mathematics, such a learning method has no future at all, because as the concept becomes more abstract and the dimensions become more and more high, this image or model understanding method will completely lose its effect. But for an engineering student, that's enough.

After coming to Germany, this school asked me to rebuild the signal and system, I completely no language. But there is no way, the Germans sometimes to the Chinese is a kind of contempt, think your education is not reliable. So there is no way to do it again.

This time, I scored a full mark, and the pass rate was only half.

To be honest, the meaning of mathematical tools for engineering students and for science students is completely different. As long as the engineering students understand, will use, will check, is enough. But many colleges and universities have taught these important math courses to teachers in the math department. So there is a problem, the math teacher said the hype, but also reasoning and proof, but the students in the heart only one sentence: Learn this goods in the end why use?

The lack of a goal of education is a complete failure. (I think the author of this sentence is very good, praise!!!) )

At the beginning of learning a mathematical tool, students do not know the role of the tool, the actual meaning. And the textbook has only obscure difficult to understand, attributive on the concept of more than 20 words and see the formula of dizzying. It's strange to be able to learn an interest!

Fortunately, I was fortunate to meet the Dalian Maritime University Wu Nan teacher. The whole course of his class was two clues, one from top to bottom and one bottom. First of all, the significance of this course, and then pointed out that the course will encounter some of the problems, so that students know that they learn a certain knowledge in the real role. And then from the foundation, comb the knowledge tree, until the extension to another clue raised in the question, perfect cohesion together!

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

Finally, write to all the students who gave me the praise and message. Really thank you for your support, but also very sorry not to reply. Because the message of the column to be loaded one at a time, in order to see the last point loaded many times. Of course I have to insist on reading, just can't reply.

This article just introduces a novel method of understanding Fourier analysis, for studying, or to get a clear formula and concept, learning, there is no shortcut. But at least through this article, I hope to make this long road a bit more interesting.

Finally, I wish all of you can find fun in your study. ...

Heinrich
Links: https://zhuanlan.zhihu.com/p/19763358
Source: Know
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Fourier analysis (Popular interpretation)