Fourier Analysis – A Student’s Introduction (Part A)

Illustration-wise, this scintillating short post is going to be a medley of Wikipedia graphs and my own plots. Please have some patience for the all the gifs to fully load.

I could not progress very far tonight (that is, 2:10 am, no caffeine), so please expect part B to follow up in another post later.

CC-BY-SA

Don’t Lie

Chapter 0

“Feel free to stay silent. But whenever you speak, don’t lie.”

That previous statement was irrelevant to today’s topic in many ways but two.

Firstly, that sentence and this article, I wrote both on the same day. One was account for some fretful matters that involved text messages; the other was the fruit of three months of procrastination.

Secondly, I spoke, consequently not lying. I want to make a promise. Whoever you are, my goal in this series is:

To give you an overall understanding of Fourier analysis without mentioning any mathematical formulae.

Nobody said studying was easy, and all I wanted to do is to bring up some directions to find some unusual delight, to provide you with some new reasons for curiosity maybe. I feel unqualified to deliver the most elegant and precise ideas, but I will try not to be incompetent. This informal essay is going to be more pleasant and comfortable to read than most academic textbooks (the few Chinese ones I have seen, at least).

For convenience, we call all of the functions in the form of y=Asin(⍵t+ϕ) sine waves. This will include all simple harmonic (that is, sine and cosine) waves.

The Frequency Domain

Chapter I

The world in our eyes is full of stories connected by the flow of time. Ever since our birth, our heights, stock markets, paths of cars passing on the road… Things change as time passes by; we live in a dynamic world.

Naturally, we perform most of our daily observations and reasoning based on the passage of time, believing that everything changes steadily and will never stop. We shall call this analysis on the Time Domain.

What if, please consider, I tell you that there is another substantial way to observe the world as an eternally stationary object? Would you think that I am insane?

I am not, and that static world is called the Frequency Domain.

Here is a mathematically incorrect (but good) example that precisely represents my meaning.

In your mind, what is music?

Picture1.png

This is the most general understanding of any sound, a vibration that changes along time. However, as far as I am concerned, to all the marvellous musicians as you are, music will seem more familiar if represented in this way:

2.png

“Class’s over.”

Yes, we can end this chapter now. The two pictures above show the same piece of music, first in the Time Domain and later in the Frequency Domain.

The frequency domain is not a fantastic concept whatsoever. We are familiar with it all along, and just seldom realised that.

Now we can look at the sentence in the beginning of the chapter:

“Reality is eternally stationary.”

We simplify the graphs above:

Time:

3.png

 

Frequency:

4.png

In the time domain, we will see the strings of the piano or violin oscillate up and down, just like the way some particular stock prices vibrate. However, in the frequency domain, that single key is all we will ever need to represent everything, the vibration you see and the note you hear, and the cash you gain.

Therefore.

No matter how fickle the world appears before your eyes, surprisingly enjoyable events, the cosmic delicacy, dancing leaves and the raindrops on your way home, every serendipitous change around that we had ever or had ever not noticed … They all could be something prepared beforehand for us, in a score finished outside the realm of time.

Probably, it sounds poetic – because it does not; it is undeniable, mathematics.

Fourier tells us, no matter how complicated, all continuous periodic functions can be written as the sum of sine waves of different phases and frequencies.

In the context of our original mathematically false example, by striking distinct keys at various points of time with different forces, you can produce any melody.

Luckily for us, we have tools to transcend the domains of time and frequency, one of which being the famous Fourier Analysis, our topic today.

Carry on, shall we?

The Frequency Spectrum Of Fourier Series

Chapter 2

Maths is usually better served with examples and illustrations. To me at least, here at f-STEM.

I now claim, that with the curvy sine waves, we can construct a flat, blocky “square wave”.

6.pngHow is that possible? You would not believe in that, just like myself a few months ago (duh!).

Now let’s look at the highly original anthropomorphism below:5.png

As the number of waves increase, they will finally form a standard series of rectangles. The more waves there are, the region where the original graph increases or decreases becomes steeper, while the effects around tops and bottoms cancel, making it “flatter” there.

Fourier_series_for_square_wave.gif

Repeat this process until some point, we will eventually get a rectangle. How many, you would probably ask, waves do we need to make a perfect 90º rise?

SquareWaveFourierArrows.gif

Bad news. The answer is infinitely many. “I am not letting you just get my answers.” — said Mathematics.

In fact, not only rectangles waves, all waves you can think of are all possible to be made in this way. This is the most difficult part to get your head around when starting to learn Fourier analysis. However, if you can accept this picture, you will be able to see things from a whole new perspective.

Here comes that perspective, haha. Let’s look at the wave addition comic from another angle. Imagine we reach around the graphs, to try to see behind.

 

333.png
The sum of 2 sine waves.

8.pngThe sum of 3 sine waves.

7.pngThe sum of 10 sine waves, with its non-zero components listed behind it.

In the graphs above, the white curve is the sum of all components behind it. The components are listed in terms of increasing frequencies. They also each have different amplitudes.

You may also find some yellow lines between the curves in the first two graphs (I was too lazy to draw them for the third one, picture that there is one straight line between every two green curves), they are not separators, though. They are sine waves with zero amplitudes! Obviously, in the summing process to form a particular curve, some (many) frequencies are not required. Actually, only odd (1,3,5,7… times) multiples of the original frequency is needed in the case of square waves.

We call each of the non-zero sine wave a frequency component.

Here comes the crucial part! If we pay special attention the frequency component with the lowest frequency, we would have made ourselves a fundamental unit to construct the frequency domain.

Before we go to explore how to use that unit, it is helpful to notice that on the real number axis that we are all familiar with, “1” is the unit of the axis. With 1, we can construct any real number.

Now that we have “1”, we will also need an “0” to complete inventing the universe. What is “0” then?

Notice y=Acos(0t) is a sine wave with an infinite period, i.e. it is the straight line y=A! In our summation, it shifts the entire curve up and down without altering its shape, while itself being the equilibrium position of vibration, a perfect candidate for “0”.

Now let’s briefly recall how the trigonometrical functions were defined in our school days:

Circle_cos_sin.gif

Sine waves is simply the projection of a circular motion onto a certain line. Similarly, the units of the frequency domain can also be seen as rotating circles, a series of them!

Fourier_series_square_wave_circles_animation.gif

We now know what the basic components of the frequency domain are, and eventually can understand this picture (that I happen to find on the Internet): a.png

It is just the amplitudes of each frequency component used! Some textbook just leave you with imagination on this point. But all we need is really the diagram below to summarise all the relationships we have seen so far. b.png

This animation sums all the theoretical ideas of this chapter up:

Fourier_series_and_transform.gif

Before we go even further, let’s briefly reflect on what that example means at all.

Do you remember, that “the world is eternally static”? I know that you probably have been complaining about that argument. Imagine, every chaotic phenomena on the time axis is just another irregular curve, one that can also be written as an endless sum of sine waves. What we perceive as irregularities are results from one sine wave and another, and they are of the most regular things there could be. The sine wave themselves are projections of a rotating circle…

Our explo experience is really the silver screen, on which was a projection from an enormous machine made up with countless cogs. Big gears drive smaller ones, smaller ones drive even smaller ones, and on the smallest gear, there is an attached figurine – maybe that’s ourselves. We see the figurine’s shadow projected onto the screen. It dances around, performing, bewildering us; we cannot predict where it is going next; the gears behind rotates on, without hesitation.

This sounded too depressingly deterministic to me … I think that this picture came to me during my readings in high school, and I wasn’t quite clear of how this picture formed, let alone knew why it formed … This lasted, until I came across the Fourier series.

This is the 1/3 mark. We will continue soon.

Midnight Discussions

I wanted to plot some irregularity with only ten components, and then my computer for a while froze:c.png

d.png

Not quite as beautiful as the maths. Just for fun. Do exercises. Have a good weekend.

Why is the introductory example mathematically improper? In Chapter 3 (Work in progress), I will elaborate on that of course.

The musics is actually Coldplay’s Paradise, digital piano cover by me

I thought that I would try to make this topic feel as friendly as possible and find a balance between intuition and simplification. However, studying requires far more rigorous attitudes and constant attempts to understand how to use the knowledge in various situations.

On mobile phones, there’s a nice app called MultiWave Generator, with which you can manipulate and combine digital waves. Googling it is a good idea if you are interested.

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