If you read this article and you don't understand the Fourier transform, come and strangle me. (c)

I would like to dedicate this article to the Dalian Maritime University Wu Nan teacher, Liu teacher, Wang Ning teacher and Zhang Jing Park teacher.

For the full version, check out my know-how column: The Strangled tutorial of Fourier analysis

Four, Fourier transform (Fourier tranformation)

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 an aperiodic continuous signal.

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 has been unloaded you will never guess where it is, on a high-number test paper. 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.

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, the significance of this course, and then pointed out that the course will encounter some of the problems, let students know that they learn a knowledge of the role played in the real. 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.

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Author: Han Hao

Know: Heinrich

Weibo: @ Peanut oil workers

A story that has nothing to do with time

from:http://daily.zhihu.com/story/3955477

If you read this article and you don't understand the Fourier transform, come and strangle me. (c)