### Fourier Series Classically

**Chapter 1 Introduction**It is also an introduction to the next courses “

**Fourier Transform”**and

**“Laplace Transform”**. The latter will appear in the future.

**Chapter 1.1 Any time function f(t) can be decomposed into sine waves!**

**Maybe**not

**every***, but let’s not be petty. It is easiest when we are dealing with

**periodic**functions

**f(t)**lasting from

**minus to plus infinity**as in

**Fig. 1-2.c**. This is where the

**Fourier Series**comes in.

**And when**the function runs in

**finite**time, e.g. a

**single rectangular pulse**? Or in

**infinite time**but the energy of the pulse is

**finite**, e.g. discharge voltage of the

**RC**system. This is what the

**Fourier Transform**course, which can be found on the main page.

**And when**the function runs in

**infinite**time and has

**infinite**energy, e.g. a

**unit step**or a

**quadratic**function? Here is a job for the

**Laplace Transform**, which will appear on this website someday..

**Note**

**Fourier Series, Fourier Transform**or

**Laplace Transform**decomposes almost any

*****function

**f(t)**into a sum of sine waves with different frequencies, amplitudes and phase shifts!

*****except for this type of freaks.

**Fig. 1-1**

A function that is not subject to

**Series**or

**Fourier**or

**Laplace Transform**.

Fortunately, such functions are rarely associated with real physical phenomena.

**Chapter 1.2 Trigonometric Fourier Series**

The periodic function **f(t)** can be approximated by a **FourierSeries**.

Here **f(t)** is just a function of time **t**, but it could just as well be a function of position **x**, which is **f(x)**.**Fig.1-2****Trigonometric** decomposition of the periodic function **f(t)** into a **Fourier Series**.**Note:**

For descriptions of alternating currents, the frequency **f** is used as the **reciprocal** of the period **T**, or the so-called pulsations**–>**angular velocity** ω**. The frequency **f** is used by ordinary people, **e.g. f=50 Hz**, while the pulsation **ω** is intended for **electricians**. Thanks to this, the formulas are a little shorter, because instead of **2πf** we have only** ω**.**Fig. 1-2a**

Formula using sum-**sigma** sign**Fig. 1-2b**

The formula without this **sigma**, which is perhaps clearer. The **ω** pulsation of the **first** harmonic is better seen here. The **pulsations** of successive **harmonics** are their** multiples**. The coefficients **a1,b1,a2,b2**… are the **amplitudes** of the successive **sine/cosines**. These are constants e.g. **a1=1 ,b1=-0.5,a2=0 ,b2=1.25,a3=0.1**, … which we will discuss in **Chapter 2**.**Fig. 1-2c**

An example of a periodic function **f(t)** with period **T=2π sec** corresponding to the pulsation **ω=1/sec**.

The above formulas are most often accurate for **n=∞**. The value of **n** can also be finite, e.g. **n=3** in **Fig. 2-3** in **Chapter** **2**.

Chapter 1.3 Complex Fourier Series

It’s just aotherwise written **Trigonometric Fourier Series**.

At first, they were less understandable, because some **complex numbers** instead of good **real** ones. But many things calculate easier. Not only that, it’s more **intuitive**! Especially after watching the animation. You will notice, for example, an analogy to the **Solar** System, in which the **Earth** revolves around the **Sun**, the **Moon** revolves around the **Earth**, etc… More on this in **Chapter 3**.**Fig. 1-3****Complex** decomposition of the periodic function **f(t)** into a **Fourier Series**.**Fig. 1-3a** The sum**-sigma** formula**Fig. 1-3b** Formula **without** this sign. Probably more transparent. The formulas are clearly simpler than in **Fig. 1-2**. Unfortunately, the coefficients **cn** used are **complex numbers**. To remind them, see the **Complex Numbers** course on the home page. Notice that on the right of the equations there is a rather complicated** sum** of **complex numbers**, the result of which is the real function **f(t)**! The **imaginary** components compensated each other!**Fig. 1-4****Fourier Series** coefficients as **complex numbers**.

**Chapter 1.4 What is all this for?****Fourier Series** appeared over **200** years ago, but they are also used in our times. And the so-called The **Fast Fourier Transform**, which is just a digital algorithm for calculating the **Fourier Transform**, was only published in **1965** by **Cooley** and** Tuckey**. What is all this for? It turns out that it is difficult to find a field in which there would be no **Series** or **Fourier Transform.****An example** from **acoustics**. The word “hello” was registered with a microphone as a function of time** f(t)**.

The author is Erik Cheever. By the way, I highly recommend his article **https://lpsa.swarthmore.edu/Fourier/Series/WhyFS.html.**

Click the sound recorder button. You will hear a not very loud “Hello”.**Fig. 1-5**

The word “Hello” written as:**Fig. 1-5a** function of time **f(t)**

For the average bread eater, the physical phenomenon (here the word “hello”) is associated with the **f(t)** function of time, as in** Fig. 1-5a**. The function **f(t)** begins at **t=0.1 se**c and ends at** t=0.6 sec**. It is very confusing compared to, for example, a **square function**. How to analyze it?**Fig.1-5b** frequency spectrum **F(ω)**

The function** f(t)** of the word** “hello”** as a** sum** of specific** sinusoids,** each of which has its own amplitude, **ω** pulsation (**f** frequency in **Fig. 1-5b**) and **φ** phase. The graph shows only the **amplitudes**, so it does not contain the full information about **“hello”**, but it is clearly simpler than **Fig.1-5a**. The first benefit of approaching the phenomenon as a frequency distribution, i.e. spectrum** F(ω)** you will see in the next video. You will hear the same **“hello”** but with an annoying distortion.

**Fig. 1-6**

The word **“Hello”** written as:**Fig. 1-6a** function of time **f(t)**

Note that although the continuous tone disturbance is not very complicated, the timing diagram has changed quite a bit. Where in this tangle to look for disturbance?**Fig.1-6b** frequency spectrum** F(ω)**

The frequency spectrum **F(ω)** has not changed much compared to **Fig. 1-5a**. Only **2** additional sinusoidal **fringes** appeared around **300Hz** and **700Hz**. Just cut them out and you’ll get a clean **“Hello”** as before. This notch is precise if we use **digital** algorithms. In the past,** analog** filters with **coils** and **capacitances** were used, which also cut out the original frequencies, but not so precisely.

I hope you see the benefit of the spectral approach to the function of time.

**Chapter 1.5 Other domains with Fourier Series or Transform**

**Digital image processing**

You’ve probably seen a movies in which tsar

**Nicolas**or emperor

**Francis Joseph II**are alive. As if it was made yesterday from a good

**cell**. And it’s all thanks to the

**Fourier Transform**. What is it about? Each image can be decomposed into

**sinusoids**with known

**parameters**. You only need to

**save**them and

**recreate**them as an

**image**at any time. This will usually take less

**space**than remembering every

**pixel**of an image. This is an

**example**where the

**sine**waves refer to the

**x**positioning (strictly

**xy**pozitioning) and not the time t. And most importantly, it is easier to remove the

**interferences**– unnecessary additional sine waves from the imperfect

**original**from more than 100 years ago. Especially when we use

**Artificial Intelligence**. Similar to the sound in

**Fig.1-6**.

**Machine Diagnostics**

The turbo generator must be perfectly balanced. After all, these are tons of rapidly rotating mass. An ideal machine gives the appropriate spectrum

**F(ω)**for the

**frequency**varying from 0 to the

**maximum**. What if some sawdust got into the shaft? The human ear may not pick it up, but a

**distinct**band will appear in the

**spectrum**. Now finding that

**sawdust**is just a matter of looking closely at the shaft.

**Earthquake**

Fortunately, this does not apply to all, only some, for example, California residents. The so-called time chart thanks to Mr. Fourier, the seismograph can be broken down into individual harmonics. After a number of years there, it will turn out that certain frequencies predominate. And this is important information for architects. The buildings themselves must suppress them! At certain heights, large swinging masses are appropriate, which, together with the rest of the building, reduce its oscillation amplitudes. Why? Because they’re swinging ini counter-phase with the rest of the building.

**Automatics**

Control systems with and without feedback are described by

**differential**equations. Fortunately, with some simplifications, these are

**linear differential equations**. The equations will be easier when its solution, i.e. the

**f(t)**function, is decomposed into

**sinusoidal**components. And this is nothing more than a

**Series , Fourier Transform**or

**Laplace Transform**. Why easier. Because the derivative of the

**sine**is

**cosine**and the

**cosine**is

**sine**(with a minus). I will not go into details, but such equations can be converted to

**quadratic**equations or

**higher**degrees. Now we can solve and analyze them using “ordinary” algebra.

**Medicine**

More than once you’ve been lying on a doctor’s couch connected to monitors with cables.

For the

**Cardiologist**, the graph on the monitor is an

**electrocardiogram**

For a

**Neurologist**, it is an

**electroencephalogram**.

Based on them, an

**experienced**doctor knows more or less what ails you. Experience is gained with practice. Especially since the

**graphs**are not much

**different**. And here is the field for the

**Fourier Transform**, which will break the function

**f(t)**into individual

**sinusoids**–>

**frequency spectrum**.

**Spectra**for different diseases are

**different**. This is where the aforementioned Fast Fourier Transform-

**FFT**comes in handy.

An example of how neurologists go deep into the Fourier Transform is youtube

**Click.**