**Rotating Fourier Series**

**Chapter 12. Fourier Series clasically**

**Chapter 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** center of gravity **scn** of the trajectory **F(njω0t)=f(t)*exp (njω0t)**. More precisely, it was the **complex** amplitude of the **nth** harmonic of the **periodic** function** f(t).
**Now I will present the formulas in the most common forms in the literature, i.e.

**– complex version**with

**positive**pulsations ->

**Chap. 12-6**

**– trigonometric version**with

**positive**pulsations ->

**Chap. 12-7**

**– complex version**with

**positive**and

**negative**pulsations ->

**Chap. 12-8**

**– trigonometric version**with

**positive**and

**negative**pulsations ->

**Chap. 12-9**

Most often, lectures on

**Fourier Series**start with trigonometric formulas with

**cosines**and

**sines**and then move on to the

**comple**x version. For me it is

**conversely**and probably more

**intuitive**. I start with

**F(njω0t)**with centers of gravity

**scn**, i.e. in a

**complex**version, and end in a

**classic-trigonometric**manner.

**Chapter 12.2 Once again the relationship of the center of gravity scn of the trajectory with the nth harmonic**

All **4** versions of the** Introduction** obviously result from the **nth** centers of gravity **scn** of the trajectory in **Fig. 7-2** of **Chapter 7.2**.

Let me remind you that the most important formulas that I have, are understandable. At least intuitively.

**Fig. 12-1**

Relationship of the center of gravity **scn** of the trajectory **F(njω0t)** with 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 center of gravity **scn** of the nth trajectory **F(njω0t**) for** f(t)** with any pulsation **ω0** or period **T**.

**Fig. 12-1c**

A more convenient formula on the trajectory **scn** when **ω0=1/sec** (i.e. T=2π sec). It is almost, with some reservations, a general formula -> see **Chap. 7.6**.

**Fig. 12-1d**

**cn**-the nth Fourier coefficient as a complex number in different versions

**Fig. 12-1e**

**cn**-the **nth** **Fourier** coefficient as **doubled** center of gravity **scn** of the nth trajectory **F(njω0t)**. It is shown how the coefficient is divided into **cosine** and **sinus** components. Otherwise – **real** and **imaginary**.

**Fig. 12-1f**

The formula for the** constant componen**t, i.e. the coefficient **c0=a0** of the Fourier Series.

**Fig. 12-1g**

The formula for the** an** component, or **cosine** of the Fourier Series. The red arrow shows “Origin”

**Fig. 12-1h**

The formula for the** bn** component, i.e. the **sine** component of the Fourier Series. The red arrow shows “Origin”

**Fig. 12-1i**

The nth harmonic **hn(t)** as a sum of **cosine** and **sinusoidal** components.

**Fig. 12-1j**

The **nth** harmonic **hn(t)** as** cosine** with phase shift **ϕ**. Module** |cn|** is a”Pythagoras” of **an** and **bn**.

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

We 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**).

The **cosine** and** sine** harmonics are shown **evidently**. The formulas for the **Fourier Series** should confirm them. Then we will generalize the formula for 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)**o okresie pulsacji

**T=2π sek**

**c0=a0=+0.25**constant component

**cn=an-jbn**ie.

**c1=1-j0.3**cie.

**a1=+1**and

**b1=+0.3**

c2=0.6+j0.4ie.

c2=0.6+j0.4

**a2=+0.6**and

**b2=-0.4**

c3=0.4-j0.2ie.

c3=0.4-j0.2

**a3=+0.4**and

**b3=+0.2**

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

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

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

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

**Similarly** with **chap. 2 Fig. 2-4** shows that the rotating vector **-1j*exp (1j1t)**:

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

– the projection of the rotating vector exp (-1j1t) on the imaginary axis **Im z** is the function **f(t)=1sin(1t)**

The above can be generalized, for example, to **three** vectors rotating at speeds **1/sec, 2/sec and 3/sec** from **Fig. 12-3c** 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 **projection** of rotating vectors **c1*exp (1j1t)**, **c2*exp(2j1t)** and **c3*exp(3j1t)** and constant **a0** onto 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 **a0 **are the model of the function** f(t) **from **Fig. a**.

Associate parameters **an** and **bn** with the coefficients **an** for cosines and **bn** with sine in **Fig. a**.

**d.** Concrete function case where **f(t)** is from **Fig. 12-2**

**e.** Rotating vectors for a specific function from **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 the animation lasting **T=2π sec**

**+0.25** stationary vector, i.e. the constant **a0=+0.25**

**+(1-0.3j)exp(1j1t**) **first** harmonic vector rotating at the speed **1ω0=1/sec**. In time **T=2π sec** it will make **1 turn**

**+(0.6+0.4j)exp(2j1t) second** harmonic vector rotating at the speed **2ω0=2/sec**. In time **T=2π sec** it will make **2 turns**

**+(0.4-0.2j)exp(3j1t)**** third** harmonic vector rotating at the speed **3ω0=3/sec**. In time **T=2π sec** it will make **3 turns
**And what will the projections of these vectors be in time? Go back on a whlile to

**Chap. 2.6**where you will learn 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

So the

**sum**of the

**projections**of rotating vectors on the

**Re z**axis is a function of

**f(t)**in

**Fig. 12-2**!

**Or**, which is one thing

The

**projection**of the

**sum**of the

**spinning vectors**onto the

**Re**axis is a

**f(t)**function in

**Fig. 12-2**!

**Otherwise**

The

**real part**or

**Re**of the sum of

**all**rotating vectors on the

**Re**axis is the function

**f (t)**from

**Fig. 12-2**!

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

When e.g. **n=1 **and** ω0=1/se****c** then **a1=1, b1=-0.3** that is** f(t)=1*cos(1t)+0.3*sin (1t)
** This is the

**first**harmonic of the function

**f(t)**from

**Fig. 12-3d**which corresponds to the rotating vector

**(1-0.3j)*exp(1j1t)**in

**Fig. 12-4**

**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 a model of **f(t)**

**a.** Rotating vector **+(1-0.3j)*exp (1j1t)**

**b. f(t)=Re{(1-0.3j)*exp(1j1t)}**

Otherwise, the projection of a rotating **(1-0.3j)*exp(1j1t**) on the real axis **Re z** is a function **f(t)**

**Fig. 12-5.2**

Rotating** vectors pair** as a model of **f(t)**

**a.** Rotating vector **+(0.5-0.15j)*exp(1j1t**). It is **half** of the rotating vector in **Fig. 12-5.1a**

**b.** Vector **+(0.5 + 0.15j)*exp (-1j1t)** spinning in the **opposite** direction. As a c**omplex number** it is at all times a **conjugate** with respect to a spinning vector **a**.

**Note:**

**Complex numbers**, e.g. **z=5+3j** and **z*=5-3j** are called **conjugated numbers**. Note that **z*** is a mirror image of** z** when the “mirror” is the real axis **Re z**.

**c. f(t)=(0.5-0.15j)*exp(1j1t)+(0.5 +0.15j)*exp(-1j1t)**

In other words

The **sum** (complex or vector) of the opposing rotating vectors **a** and **b** is a real function** f(t)**. Indeed, vector **c** moves identically as vector **Fig. 12-5.1b**! Note that the function **f(t)** does not need to be dereferenced as a **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 **Chap 12.6 Complex Fourier Series with positive pulsations**

**3. Pairs of rotating vectors** are used in **Chap 12.8 Complex Fourier Series with positive and negative pulsations**

**Chapter 12.6. Complex Fourier series with positive pulsations**

It is a generalized formula for **single rotating vectors** from **Fig. 12-3e** and the animation **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 spinning vectors

**plus**the constant component

**c0**.

**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**(ie

**Re {…}**) is just a function

**f t)**. The speed

**ω0**eg.

**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**

Eg. for animation from

**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 b =+0.2

**c.**Formula for the constant component

**c0 = a0**of the periodic function

**f(t)**.

**d.**Formula for the complex coefficients

**cn**of the periodic function

**f(t)**, or the complex amplitudes

**cn**for the

**nth**harmonics.

This is the

**doubled**center of gravity

**scn**of the rotating trajectory

**F(njω0t)**at speed

**n*ω0**.

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

**– chap. 7**theory

**–**

**chap. 11**“Checking formulas …”

I advise you to read the comment on

**Fig. 12-1c**.

**e.**

**an**-real component of the complex amplitude, or

**cosine**component

**f.**

**bn**-imaginary component of the complex amplitude, otherwise

**sine**component

Note that

**ω0**or the fundamental pulsation of the function

**f(t)**appears only in the formulas

**a**and

**b**. However, they are absent in the formulas

**d, e, f.**The coefficients

**an, bn,**for example, of a

**square wave**, depend only on its

**amplitude**and

**degree of filling**. However, they do not depend on its

**pulsation**,

**frequency**or

**period**. I wrote about this 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 the braces

**Re {…}**

The

**complex**function in braces is the sum of the spinning vectors

**(an-jbn)**and the projection of this sum is just 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.**The

**nth**cosine component of

**an**

**e.**The

**nth**sinusoidal component of

**bn**

**Chapter 12.8. Complex Fourier Series with positive and negative pulsations**

The **Fourier Series** is based on the centers of gravity **scn** of the trajectory **F(njω0t)** rotating with velocities **nω0**.

They are the complex **amplitude**s for these pulsations. Previously, i.e. in **chap. 12.6** and **12.7** were** double** amplitudes.

**Chapter 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 **single** rotating vector with a **pair** of rotating vectors, we will get the following animation. For example, a **single** rotating vector **Fig. 12-5.1** has been replaced with a **pair** of rotating vectors **Fig. 12-5.2**.

**Fig. 12-8**

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

The **three left** vectors spin opposite to the **right three** and form **3** pairs of counter-spinning vectors. The central vector **+0.25** does not spin and is a constant component **a0** of the function **f(t)**. The function **f(t)** is the sum of the above spinning vectors. Unlike** Fig. 12-4** (with the same** f(t)**!), You do not need to use the dereference **f(t)** real part. Instead of **f(t)=Re {…}** we just write **f(t)=…
**Someone may be wondering. On the

**righ**t side of the equation,

**rotating vectors**, and on the

**left**side the real function

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

**right vectors**(including the constant vector

**a0=+0.25**or more strictly

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

**left**pulsating

**vector**with the constant component

**a0=+0.25**being just

**f(t)**! Pears everywhere.

**Fig. 12-9**

**The pulsating vector as f(t) as the sum of the spinning vectors in Fig. 12-8**

**a.** A vector pulsating on **Re z** axis with a constant component **a0**

**b.** A function** f(t)** describing the motion of the pulsating vector on the **Re z** axis

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

Let us come back to the topic once again

**Fig. 12-10**

**a. Complex Fourier Series** with **positive** and **negative** pulsations **c(n)** coefficients for **n = 0… + ∞**.

**b.** Example of a **Complex Fourier Series** with positive and negative pulsations and with concrete **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 conjugated to **c(1)**

The coefficients **a(n)**, **b (n)** for** n** positive and negative are obtained directly from the formula **12-10b**. They did not (although can) be calculated.

**Chapter 12.8.2. Complex Fourier series with positive and negative pulsations for any function f (t).**

So the formula we get by generalizing **Fig. 12-10a**. By the way, a nice thing will come out. The **Fourier Series** with only **positive** pulsations has **complex harmonic** amplitudes, they are **double** centers of gravity **scn** of rotating trajectories **F(njω0)**. Now we know why the **scn** were **doubled** previously. Because they handled **2 times** less number of “heavier” harmonics.

**Fig. 12-11**

**a.** **Complex Fourier Series** with positive and negative pulsations with **cn** coefficients

**b.** **Complex Fourier Series** with positive and negative pulsations with more detailed **c(n)** coefficients

Note that each Fourier coefficient **c(+n)=a(n)-jb(n)** with positive **nω0** pulsations corresponds to the Fourier coefficient **c (-n)=a(-n)+jb(-n)**. These coefficients are conjugate numbers to each other **c(+n)=c(-n)***

**c.** The formula for the complex Fourier coefficient **c(n)**.

It is **2 times** smaller than the corresponding **c(n)** factor for 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 analogous ones

**c (n), a (n)**and

**b (n)**

**chap. 12.6**

**Chapter 12.9 Trigonometric Fourier Series with positive and negative pulsations
**

**Fig. 12-12**

Trigonometric Fourier 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**

**chap. 2**

**b.**Pulsation of the first harmonic

**ω0**where

**T**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 the analogous

**a(n)**and

**b(n)**

**chap. 12.7**

A