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

Welcome back, periodic readers.

Link to Part A

One more groovy spectrum

Chapter 3

I know that you’ve been wondering for so long, “What exactly does Fourier analysis do?”

Here, I will give you three easy mathematical challenges; please prepare for them.

Firstly, on any surface of your choice, try drawing a part of y=sin(x), not necessarily precisely.

It’s not difficult, isn’t it?

All right! Second challenge! Now try drawing y=sin(7x)+(sin3x), by hand. (The usage of rulers is permitted, at most. I am generous.)

Let alone precision – can you even state immediately when the curve rises and when it goes down?

Third and last. Consider if I give you this curve below without its equation and ask you to find the resultant curve if you remove a y=sin(3x) from it.

pasted-graphic

Manually, it is fair to say that doing this is hard and senseless.

Hang on a minute! What if we viewed the questions in the frequency domain?

Easy as – the curve is just a few vertical lines.

Some tasks seemingly insurmountable in the time domain are easy to deal with in the frequency domain – this is exactly where we need the techniques named after Baron Jean Baptiste Joseph Fourier.

Getting rid of some specific frequencies in a function (as seen in Challenge 3) is called wave filtering. It is not hard to see that it is ideal if the original signal is first Fourier transformed and broken down into its frequency components.

We will soon see this in action. Before that, let’s continue clearing some basics.

I am not sure about you, but when I was plotting those graphs in Part A, I felt that the gist of the entirety of the frequency domain was to look “sideways”, to enumerate all the additive components of a curve you’d say. And now, we are looking from above – or below.

Let’s review what the term “phase” refers to.pasted-graphic-5The frequency spectrum we discussed before does not contain all the necessary information to reconstruct a function, because for each component, amplitude, frequency and phase are equally important. We’ll attempt to construct a spectrum diagram representing all of those quantities.

Pasted Graphic 1.png

More specifically, the black curve is formed by the addition of the three sine waves behind it, arranged in the order of frequency. Their phases are projected onto a plane.

Time for a summary,

未标题-1.png

Fourier Transformation

Chapter 4

If you have been following, I believe that the first three chapters might have reshaped your ideas about the frequency domain and Fourier analysis. However, in the early example of chapter 1 about music, I said that it was mathematically incorrect. Why?

The essence of a Fourier series lies in the fact that it breaks down a periodic signal into an infinite sum of (discrete) harmonic waves. However, the evolving and chaotic universe does not seem to be periodical; time never stops its pace; some things are destined to be once-in-a-lifetime. They happen and don’t return again, no matter how eagerly (or not) we want them to, and no matter how clearly we memorise them during their occurrence. On the other hand, as things take place and go, much of our mundane life is going to be forgotten, while only the most significant events fracture into pieces of stories that occasionally haunt our minds. In all, memories work such that they are a real-life transformation between two kinds of signals:

Abstractly, experiences in the real world form a continuous non-periodic signal, and the memory is a periodic discrete signal.

idle-time-4eee724792327_hires

Relatedly, Vsauce has good episodes about memory (and that’s why I devised the entire example)

It is a bummer, but there are NO mathematical tools capable of converting between them – between continuous non-periodic signals and periodic discrete signals.

Take the Fourier series as an example; it carries out the conversion between continuous periodic signals and non-periodic discrete signals. I seem to intentionally bend your mind here… Please, just revisit the diagrams in Chapter 2 and you will get what I am saying.

Contrarily, Fourier transformation is a conversion between a time-wise non-periodic continuous function and a frequency-wise non-periodic continuous function.

To better aid your understanding, it’s diagram time. The breakdown of a square wave – our old friend – looks like this:

Pasted Graphic 4.png

Now imagine, as the components get closer and closer, neighbouring waves start to merge, and the sum symbol transitions into an integration symbol.

A patch of tremulous sea, this is what you will get.

pasted-graphic-3

All this, gradually dotted with fancy maths, is not the entire story. To proceed further we would have to pay Euler a visit. And I will try to finish that section between my MATHS 250 and MATHS  260 exams. It’s more of a personal revision.

A hint of where we are going:

Pasted Graphic 6.png

I could have done the image with only one wavelength…

2 thoughts on “Fourier Analysis – A Student’s Introduction (Part B)

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