**Rotating Fourier Series**

**Chapter 12. Fourier Series Classically**

**Chaptere 12.1 Introduction**Actually, all the formulas related to the

**Fourier Series**are presented in

**Fig. 7.2 in Chapter 7.**They were based on the fact that the

**nth**harmonic was a

**doubled**vector indicating the

**nth**centroid

**scn**of the trajectory

**F(njω0t)=f(t)*exp(njω0t)**. More precisely, it was the

**complex**amplitude of the

**nth**harmonic of the

**f(t)**.

Now I will present formulas in the forms most often found in the literature, i.e.

**-complex**version with positive pulsations–>

**Ch. 12-6**

**-trigonometric**version with

**positive**pulsations–>

**Ch. 12-7**

**-complex**version with

**positive**and

**negative**pulsations–>

**Ch. 12-8**

**-trigonometric**version with

**positive**and

**negative**pulsations–>

**Ch. 12-9**

Most often,

**Fourier Series**lectures start with

**trigonometric formulas**and only then move on to the

**complex**version. For me it’s the

**opposite**and probably more

**intuitive**. I start from

**F(njω0t)**with centroids

**scn**, i.e. in the

**complex**version and finish classically-

**trigonometrically**.

**Chapter 12.2 Relationship of the centroid scn of the trajectory with the nth harmonic**

All **4** versions from the **Introduction** result, of course, from the **nth** centroids **scn** of the trajectories

in **Fig. 7-2 of Chapter 7.2**. The most important formulas that have happened in my life and are understandable.

**Fig. 12-1**

Relationship of the centroid **scn** of the trajectory** F(njω0t)** with the **complex Fourier** coefficients **c0, cn=an-jbn** and harmonics **hn(t)** of the function **f(t)**.**Fig. 12-1a**

Trajectory **F(njω0t)****Fig. 12-1b**

General formula for the centroid **scn** of the **nth** trajectory **F(njω0t)** for** f(t)** with any pulsation **n*ω0** or period **T**.**Fig. 12-1c**

A more convenient formula for **scn** of the trajectory is when **ω0=1/sec** (i.e. **T=2π sec**).

This is almost, with some reservations, the general formula–> see** Chapter 7.6.****Fig. 12-1d****nth** Fourier coefficient as a** complex** number in different versions**Fig. 12-1e****nth** Fourier coefficient as doubled centroid **scn** of **nth** trajectory **F(njω0t)**.

It is shown how the **coefficient** was decomposed into** cosine** and **snusoidal** components. Differently – **real** and **imaginary**.**Fig.12-1f**

The formula for the** constant** component is the coefficient of **c0=a0** of the Fourier Series.**Fig. 12-1g**

The formula for the **an**, or **cosine**, component of the **Fourier Series**. Red arrow shows “origin”**Fig. 12-1h**

The formula for the **bn**, or **sine**, component of the **Fourier Series**. Red arrow shows “origin”**Fig. 12-1i****nth** harmonic** hn(t)** as the sum of the **cosine** and **sine** components.**Fig. 12-1j****nth** harmonic **hn(t)** as **cosine** with **phase** shift **ϕ**. Module **|cn|** is “Pythagoras” with **an** and **bn**.

**Chapter 12.3 Test function f(t) for testing Fourier Series**

We will find the Fourier Series for:**f(t)=0.25+1cos(1t)+0.3sin(1t)+0.6cos(2t)-0.4sin(2t)+0.4cos(3t)+0.2sin(3t).** **Cosine** and **sine** harmonics are visible. The **Fourier Series** formulas should confirm them. Then we will generalize the formula to any periodic function **f(t)**.

**Fig. 12-2****f(t)=0.25+1cos(1t)+0.3sin(1t)+0.6cos(2t)-0.4sin(2t)+0.4cos(3t)+0.2sin(3t)** with pulsation period** T=2π sec****c0=a0=+0.25** constant component**cn=an-jbn** i.e when **n=1,2,3****c1=1-j0.3** i.e. **a1=+1** and** b1=+0.3****c2=0.6+j0.4** i.e.** a2=+0.6 **and** b2=-0.4****c3=0.4-j0.2** i.e. **a3=+0.4** and **b3=+0.2**

**Chapter 12.4 Sum of cosines and sines as rotating vectors**

From **chapter 2 Fig. 2-3** shows that the rotating vector **exp(1j1t)**:

-is a **two-dimensional** model of the function **f(t)=cos(1t)**

-the projection of the rotating vector **exp(1j1t)** onto the real **axis Re z** is a function **f(t)=cos(1t)**

Similarly to **chapter 2 Fig. 2-4** shows that the rotating vector **-j*exp(1j1t)**:**-is** a **two-dimensional** model of the function** f(t)=sin(1t)****-the** projection of the rotating vector **exp(-1j1t)** onto the **real axis Re z** is a function **f(t)=sin(1t)**

The above can be generalized, for example, into the **three** vectors from **Fig. 12-3c** rotating at speeds **1/sec, 2/sec **and** 3/sec** and the appropriate linear combination of **cosines** and** sines**

**Fig. 12-3****a. **Function **f(t)** as a linear combination of **cos(1t),cos(2t),cos(3t)** and **sin(1t),sin(2t),sin(3t)** and the constant **a0**.**b.** Function **f(t)** as a projection of the rotating vectors **c1*exp(1j1t), c2*exp(2j1t) **and** c3*exp(3j1t)** and the constant** a0** on the **real axis**, i.e. **f(t)=Re{a0+…}****c. Rotating** vectors with initial states** c1=a1-jb1, c2=a2-jb2 **and** c3=a3-jb3** and the constant **a0** are a model of the function** f(t)** from **Fig a**. Associate the parameters** an** and **bn** with the coefficients** an** at **cosines** and **bn** at the **sines** in **Fig. a.****d.** The specific case of the function when** f(t)** is from **Fig. 12-2****e.** Rotating vectors for the specific function in **Fig. 12-2** as its model.

**Fig.12-4****f(t)=Re{0.25+1cos(1t)+0.3sin(1t)+0.6cos(2t)-0.4sin(2t)+0.4cos(3t)+0.5sin(3t)**

Interpretation of **an** animation lasting **T=2πsec****+0.25** stationary vector, i.e. constant **a0=+0.25****+(1-0.3j)exp(1j1t)**** first** harmonic vector rotating at **1ω0=1/sec**. In time **T=2πsec** it will make **1** revolution**+(0.6+0.4j)exp(2j1t)** **second** harmonic vector rotating at **2ω0=2/sec**. In time **T=2πsec** it will make **2** revolutions**+(0.4-0.2j)exp(3j1t)** **third** harmonic vector rotating at **3ω0=3/sec**. In time **T=2πsec** it will make **3** revolutions

And what will be the projections of these vectors in time? Return for a moment to **Chapter 2.6** where you will find out that:**Re (a-jb)*exp(jω0t)=a*cos(ω0t)+b*sin(ω0t**).

That is**Re** **(1-0.3j)exp(1j1t)**=**1cos(1t)+0.3sin(1t)****Re** **(0.6+0.4j)exp(2j1t)****=0.6cos(2t)-0.4sin(2t)****Re** **(0.4-0.2j)exp(3j1t)****=0.4cos(3t)+0.2sin(3t)**

and**Re {+0.25}=+0.25** which is obvious

That is, the **sum** of the** projections** of rotating vectors on the **Re axis** is a function of **f(t)** from **Fig. 12-2**!**Or** whatever comes to the same thing

The **projection** of the **sum** of rotating vectors onto the **Re axis** is a function of **f(t)** in **Fig. 12-2**!**Otherwise**

The real part, i.e. **Re** of the sum of the sum of all rotating vectors on the** Re axis**, is the function **f(t)** from **Fig. 12-2**!

**Chapter 12.5 Rotating vector as a model of the function** **f(t)=an*cos(n*ω0t)+bn*sin(n*ω0t)**

When, for example, **n=1** and **ω0=1/sec** then **a1=1, b1=-0.3 ** i.e. **f(t)=1*cos(1t)+0.3*sin(1t)**

**Fig. 12-5**

Rotating vectors as a model of the function **f(t)=1cos(1t)+0.3sin(1t)****Fig. 12-5.1****Single** rotating vector as** f(t)** model**a**. Rotating vector** +(1-0.3j)*exp(1t)****b.** **f(t)=Re{(1-0.3j)exp(1j1t)}**

Otherwise, the **projection** of the rotating **(1-0.3j)*exp(1j1t)** on the **real axis Re z** is a function** f(t)****Fig. 12-5.2****A pair** of contrary rotating vectors as a model** f(t)****a.** Rotating vector**+(0.5-0.15j)*exp(1t)**. This is **one-half** of the rotating vector in **Fig. 12-5.1a****b.** Above rotating vector but in the opposite direction **+(0.5+0.15j)*exp(-1j1t)**.

As a **complex** number, it is at any moment a **conjugate number** with respect to the rotating vector** a**.**Note****Complex** numbers, e.g. **z=5+3j** and** z*=5-3j**, are so-called** conjugate numbers**. Note that **z*** is a mirror image with respect to **z** when the “mirror” is the **real** axis **Re z**.**c.** **f(t)=(0.5-0.15j)*exp(1t)+(0.5+0.15j)*exp(-1j1t)**

In other words

The** sum** (complex or vector) of oppositely rotating vectors **a** and **b** is a real function **f(t)**.

Indeed, vector **b** on **Fig. 12-5.1b **moves identically to vector **c** on **Fig. 12-5.2b**. Note that we do not need to derive the function **f(t)** as the** real part** of a complex number!**Conclusions****1.** Both models, i.e. **a single rotating vector** and a pair of **rotating vectors**, describe the same function** f(t)**, i.e.**f(t)=an*cos(n*ω0t)+bn*sin(n*ω0t)****2. Single rotating vectors** are used in **Chapter 12.6** Complex Fourier Series with **Positive** Pulsations**3. Pairs of rotating vectors** are used in** Chapter 12.8** Complex Fourier Series with **Positive** and **Negative** Pulsations

**Chapter 12.6 Complex Fourier Series with Positive Pulsations**This is a generalized formula for single rotating vectors from

**Fig. 12-3e**and the animation of

**Fig. 12-4**for

**n=∞**

**Fig. 12-6****Complex Fourier Series** with **positive** pulsations**Any** (almost, but let’s not go into details) **periodic** function **f(t)** can be represented as an **infinite** series of** rotating** vectors plus a **constant** component **co**.**a. Single** rotating vectors with complex amplitudes **c1, c2 …cn** with a constant component **c0**. The projection of these vectors onto the **real axis Re z** (i.e. **Re{…}** is the function **f(t)**. The speed **ω0**, e.g. **ω0=1/sec**, corresponds to the pulsation of the periodic function **f(t)**.**b.** As above, only complex amplitudes as **c0=a0, c1=a1-jb1, c2=a2-jb2,…cn=an-jbn**

For example, for the animation in **Fig. 12-4****a0=+0.25****c1=a1-jb1=1-j0.3**–>a1=1 b1=+0.3**c2=a2-jb2=0.6+j0.4**–>a2=0.6 b2=-0.4**c3=a3-jb3=0.4-0.2j**–>a3=0.4 b3=+0.2**c.** Formula for the constant component **c0=a0** of the periodic function **f(t)**.**d.** Formula for complex coefficients **cn** of the periodic function **f(t)**, in other words for complex amplitudes** cn** for **nth** harmonics.

This is the doubled centroid **scn** of a rotating **trajectory** with speed** n*ω0**.

You can take it on your word of honor, but they should convince you

– **chapter 7** theory

– **chapter 11**“Checking patterns…”

I advise you to read the commentary to **Fig. 12-1c**.**e.** real component, complex amplitude** an**, also known as** cosine****f.** the imaginary component of the complex amplitude **bn**, also **sine**

Note that** ω0**, i.e. the fundamental pulsation of the function **f(t)**, appears only in formulas** a** and **b**. However, they do not appear in formulas **d, e, f**. The coefficients **an, bn,** e.g. of a **square wave**, depend only on its amplitude and duty cycle. However, they do not depend on its pulsation, frequency or period. I wrote about it in **chapter 7.6**.

**Chapter 12.7 Trigonometric Fourier Series with Positive Pulsations**

**Fig. 12-7****Trigonometric Fourier Series** with **positive** pulsations**a**. **Fourier Series** with **positive** pulsations.

It follows directly from the formulas

– **Fig. 12-6b** where **f(t)** is the **real** part of the **complex** function in braces** Re{…}**

In other words, the **complex** function in the **braces** is the sum of **rotating** vectors **(an-jbn)** and the **projection** of this **sum** is the function **f(t)**.**b.** Pulsation of the first harmonic **ω0** where **T** is the period of the function **f(t)****c.** Constant component **a0****d.** **nth** cosine component **an****e.** **nth** sinusoidal component** bn**

**The**

Chapter 12.8 Complex Fourier Series with Positive and Negative Pulsations

Chapter 12.8 Complex Fourier Series with Positive and Negative Pulsations

**Fourier Series**is based on the centroids

**scn**of trajectories

**F(njω0t)**rotating at speeds

**n*ω0**. They are the complex amplitudes for these pulsations. Previously, i.e. in

**chapters 12.6**and

**12.7**were doubled amplitudes.

**Section 12.8.1 Complex Fourier Series with positive and negative pulsations for the test function f(t).**

If in the formula

**Fig. 12-3e**and in the animation

**Fig. 12-4**we replace

**each**individual

**rotating**vector with a

**pair**of rotating vectors, we will obtain the animation below. For example, the single rotating vector of

**Fig. 12-5.1**has been replaced by a pair of rotating vectors of

**Fig. 12-5.2**.

**Fig. 12-8f(t)=(0.2+0.1j)*exp(-3jt)+(0.3-0.2j)*exp(-2jt)+(0.5+0.15j)*exp(-1jt)+0.25+(0.5-0.15j)*exp(+1jt)+(0.3+0.2j)*exp(+2jt)+(0.2-0.1j)*exp(-1jt).**The

**3**left vectors rotates

**opposite**to the

**3**right ones and form

**3**pairs of

**oppositely**rotating

**vectors**. The

**central**vector

**+0.25**does not rotate and is the

**constant**component

**a0**of the function

**f(t)**. The function

**f(t)**is the

**sum**of the above

**rotating**vectors. Unlike

**Fig. 12-4**(with the same

**f(t)**!), there is no need to use the

**real part**dereference operation of

**f(t)**. Instead of

**f(t)=Re{…}**we simply write

**f(t)=…**

**Someone**may wonder. On the

**right**side of the

**equation**are

**rotating**vectors, and on the

**left**is the real function

**f(t)**. As if there were pears on the right and cows on the left. So let’s

**sum**the right vectors (including the constant vector

**a0=+0.25**, more precisely

**a0=(0,+0.25)**. We will get the

**left**pulsating vector with a constant component

**a0=+0.25**, which is

**f(t)**!

**Fig. 12-9**

The pulsating vector as **f(t)** as the sum of the rotating vectors of **Fig. 12-8****a.** **A vector** pulsating on **Re z** from the **real axis** with a **constant** component** a0****b.** Function** f(t)** describing the motion of the pulsating vector on the **Re z** axis

This is exactly the function of **Fig. 12-2**. On the **left** side of the equation **Fig. 12-4** there is also the same function **f(t)**, but on the right there is the projection of the sum of rotating vectors, i.e. **Re of {…}**.

Let’s get back to the topic again**Fig. 12-10****a. Complex Fourier Series** with **positive** and **negative** pulsations of **c(n)** coefficients for** n=0…+∞.****b.** **Example** of a **Complex Fourier Series** with **positive** and **negative** pulsations and specific complex coefficients **c(n)** for **n=0,1,2,3**.

e.g. **c(-1)=0.2+0.1j, c(0)=+0.25, c(1)=0.2-0.1j**

Note that e.g. **c(-1)** is the** conjugate** of **c(1)**

We obtained the coefficients **a(n),b(n)** for positive and negative **n** directly from formula **12-10b**. They did not need to be calculated (although they could be).

**Chapter 12.8.2 Complex Fourier Series with positive and negative pulsations for any function f(t).**This is the formula we obtain by generalizing

**Fig. 12-10a**. By the way, something cool will come out. A

**Fourier series**with only

**positive**pulsations has

**complex**harmonic amplitudes, they are

**double**the centroids of the

**scn**of the rotating trajectories

**F(njω0)**. Now we know why

**scn**were previously

**doubled**. Because they handled

**2**times

**less**harmonics.

**Fig. 12-11****a. Complex Fourier** Series with** positive** and **negative** pulsations with coefficients **cn****b. Complex Fourier** Series with **positive** and **negative** pulsations with detailed coefficients **cn**

Note that each Fourier coefficient **c(+n)=a(n)-jb(n)** at **positive** pulsations **nω0** corresponds to a Fourier coefficient **c(-n)=a(-n)+jb(-n)**. These coefficients are conjugate numbers **c(+n)=c(-n)*****c. formula** for the complex **c(n)** Fourier coefficient. It is **2** times** smaller** than the corresponding coefficient **c(n)** on the **Fourier Series** in **Fig. 12-6d****d. **formula for **a(n)=a(-n)****e.** formula for **b(n)=-b(-n)**

Note that the coefficients **c(n), a(n)** and** b(n)** are **2** times smaller than the **c(n), a(n)** and **b(n)** in **chapter 12.6**.

**Chapter 12.10 Trigonometric Fourier Series with positive and negative pulsations.**

**Fig. 12-12****Trigonometric Fourie**r Series with **positive** and **negative** pulsations**a. Fourier** series with **positive** pulsations.

It follows directly from the formulas**– Fig. 12-11b** where **f(t)** is the **real part** of the **complex** function in braces **Re{…}****– Fig. 2-9d** **chapter 2****b. Pulsation** of the **first** harmonic **ω0** where **To** is the period of the function **f(t)****c. Constant** component **a0****d. **formula for **a(n)=a(-n)****e.** formula for **b(n)=-b(-n)**

Note that the coefficients **a(n)** and **b(n)** are **2** times smaller than **a(n)** and **b(n)** in the **chapter. 12.7**

**Chapter 12.11 Trigonometric Fourier Series with positive and negative for arbitrary T.**

Often, the coefficients** an, bn** do not depend on the period** T** of the function **f(t)**. For example, for a **square wave** with **50%** duty cycle, they are the same for the period **T=1** sec and** T=3 sec**. Then we can assume that **Tπ=2π** as in **Fig. 12-12**. It’s always one parameter less and easier calculations. See **chapter 7.7.** But this is not always the case. You will find out about this in the article “Fourier Transform” **chapter. 3.3**. Then we use the formula with “general T”**Fig. 12-13****Trigonometric Fourier** series with **positive** pulsations for the interval **T/2…+T/2**.

Such a range can sometimes make our calculations easier.

**Fig. 12-14****Trigonometric Fourier** series with **positive** and **negative** pulsations for the interval **0…T** or **-T/2…+T/2**.

Such a range can sometimes make our calculations easier.