**Rotating Fourier Series**

**Chapter 2. The complex function exp(jωt) as a model of the real function cos(ωt)**

**2.1 Introduction**

The purpose of this article is to treat **sine/cosine** as rotating vectors. For example, a vector **1** rotating at a speed of **ω0=1/sec,** more precisely **(1,0)** corresponds to the real function **1*cos(ω0t)=1*cos(1t)**. Let me remind you that the angular velocity **ω0=1** (more precisely **ω0=1/sec**) has a period **T=2π sec≈6.28sec**. This approach will greatly facilitate the intuitive understanding of the formulas related to the **Fourier Series, Fourier Transform** and **Laplace Transform**.**Note** on **ω0** pulsation.

And why not just **ω**? Because this is an article on **Fourier Series** and** ω0** is the pulsation of the **fundamental harmonic***, and **ω=n*ω0** is about the **nth** harmonic of the periodic function **f(t)**.*****Some call the **ω0** pulsation the **ω1**–**first harmonic pulsation**.

**Chapter 2.2 Euler’s equation**

I assume you are familiar with complex numbers and the complex function **exp(jω0t)**, which are used in many fields. For me an electrician, these are rotating vectors. They show better the phase shifts** ϕ** between **curren**t and **voltage** than on ordinary time charts. If you don’t feel confident, I recommend the “**Complex Numbers**” course on the home page.**Fig. 2-1**Euler’s equation

Treat the angle

**α**generally as a good

**x**, i.e. as a

**real**number related to the equation! And

**jα**is an

**imaginary number**on the

**Im z**axis. Only this equation results in its interpretation with a circle and a right triangle. Here it is clear that the real number

**α**is the angle in

**radians**. For any

**real**number

**α**, the complex value

**z=exp(jα)**will be somewhere on a

**circle**with radius

**A=1**. At first glance, it seems as obvious as the definitions of

**sine**and

**cosine**. But what does the complex exponent function

**exp(jα)**have to do with

**trigonometry**? Mr.

**Euler**in his time could only use complex addition, subtraction, multiplication and division. What about the

**complex**function

**exp(jω0t)**? How to calculate it using only the above four operations? However,

**Euler**already knew that the

**real**function

**exp(x)**is a sum of infinitely many decreasing

**polynomials**, just like the number

**1**is an infinite sum of the series

**1/2+1/4+1/8+1/16+…**Similarly treated the complex exponential function

**exp(jx)**. i.e. calculated its value as a sum of

**polynomials**using only

**four**complex number operations. And he must have been surprised when the point

**z=exp(jx)**started rotating in a circle with radius

**A=1**. Only then did he associate

**x**with the angle

**α**. And he expected the function to go somewhere to

**infinity**. Just like the

**real**exponential function

**exp(x)**for

**x>0**! Substitute into Euler’s formula

**α=0, 0.25π, 0.5π, 0.75π …2π**in sequence. You will see the point

**z=exp(jα)**rotating in a circle.

**Chapter 2.3 Video support**

The main advantage of the article is the animation. It is much more imaginative than a simple chart.**Fig. 2-2**

Description of buttons and video indicators**– Start** clicking starts the animation and turns into a **Stop** button**– Clock** the current time of the experiment, also as a yellow bar**– Simulation time** otherwise – the duration of the experiment, here **13 sec**.**– Full screen** click to enlarge the screen, click again to reduce etc…**-Stop** Clicking stops the simulation and changes to a **Restart** button**-Restart** as the name suggests.

**Chapter 2.4 Study of various harmonic motions f(t) as their two-dimensional versions F(jω0t)**

This chapter is just an excuse to get acquainted with the complex function **Start*exp(jω0t)**. I emphasize that the parameters **Start** and** jω0t** are **complex numbers**! Now that we know how to handle video, we will examine the harmonic motions for different initial states of the rotating **Start vector** and the **ω0** pulsation. You’ll see that the time function **x(t)** is the projection of the rotating vector** Start*exp(jω0t)** onto the real axis **Re z**. So **x(t)=Re z {Start*exp(jω0t)}**.

We will examine **5** harmonic movements.**1. 1cos(1t)** as **Start*exp(jω0t)** for **Start=1** and **ω0=1/sec****2. 1sin(1t)** as **Start*exp(jω0t)** for **Start=1*exp(-jπ/2)=1*exp(-j90°)=-1j** and **ω0=1/sec****3. 1cos(1t-π/4)** as **Start*exp(jω0t)** for** Start=1*exp(-jπ/4)=1*exp(-j45°)≈0.707-j0.707** and **ω0=1/sec****4. 1cos(2t)** as **Start*exp(jω0t)** for **Start=1 and ω0=2/sec****5. 0.5cos(1t)** as **Start*exp(jω0t)** for** Start=0.5** and** ω0=1/sec**

**Chapter 2.4.1 f(t)=1*cos(1t) as F(jω0t)=1*exp(j1t) that is Start*exp(jω0t) for Start=1 and ω0=1/sec**

**The most important conclusion**:

The rotating vector

**F(jω0t)=1*exp(j1t)**is a two-dimensional version of

**f(t)= x(t)=1*cos(1t)**. Here

**ω0=1/sec**.

Later it will turn out that almost every arbitrary periodic function

**f(t)**with period

**T**corresponding to the pulsation

**ω=2π/T**has its two-dimensional version as the sum

**F(jt)=c1*exp(j1ω0t)+c2*exp(j2ω0t)+c3 *exp(j3ω0t)…**

**c1, c2, c3…**are complex numbers or vectors

**an+jbn**as initial states of rotating vectors

**cn*exp(jnω0t)**.

In the two-dimensional version

**F(jω0t)**of the function

**f(t)**, some features are more visible and intuitive than in the

**one-dimensional**one–>

**f(t)**.

**Fig. 2-3**

**Fig. 2-3a**

**x(t)=1cos(1t)**

This is

**one-dimensional**motion along the

**real**axis

**Re z**in the complex plane

**Re z, Im z**. Click the “

**Start**” button to see the

**2**periods

**T**of harmonic motion. We know the amplitude

**A=1**and the time of the

**2**periods, i.e.

**2T≈12 sec≈12.56 sec≈4πsec**shown on the video stopwatch, i.e.

**T=2πsec–>ω0=2/T=1/sec**. The absolute values of the speed are the highest in the

**middle**, and the smallest, i.e. zero, at the ends “

**at the turns**“.

**Fig. 2-3b**

The complex function

**1exp(jω0t)**as a rotating vector of length

**1**and initial state

**(1,0)**

**Note:**

**Start**

**=z=(1,0)=1+0j.**It is the complex function that has the value

**Start*exp(j1t)=(1+0j)*exp(j1t)=1*exp(j1t)**and is a rotating vector of length

**1**and initial state

**(1,0)**. It is clearly visible that the animation in

**Fig.2-3a**is a projection of the animation in

**Fig.2-3b**onto the

**real**axis

**Re z.**

Or otherwise

“The

**real**

**part**of the complex function

**1exp(j1t)**is the function

**1cos(1t)**.

Or otherwise

**Re 1exp(j1t)=1cos(1t)**

The projection of the rotating vector onto the

**real**axis

**Re z**moves as in

**Fig. 2-3a**.

Some people almost equate

**exp(jω0t)**with

**cos(ω0t)**. It’s not exact, but it’s accurate and visual.

The initial state of the circulating vector

**Start=(1+0j)**=

**+1**you can see in

**Fig. 2-3b**before the animation.

**Remember**, I won’t repeat it a second time!

Each

**Start**vector is the initial state and “stands still”. Here

**Start=z=1+0j**, but it can be e.g.

**Start=z=2-3j**.

After multiplying

**Start**by

**exp(jω0t)**, the vector

**Start*exp(jω0t)**will start to rotate at a speed of

**ω0**. You will also find out about it in the next animations.

**Fig. 2-3c**

Time chart

**x(t)=1cos(1t) (ω0=1/sec)**

**– the horizontal axis (“x”)**is time

**t**in seconds.

Keypoints

**0 sec, 0.5πsec≈1.57sec, 0.75πsec≈2.36sec, πsec≈3.14sec, 2πsec≈6.28sec**

**– the vertical axis**(

**“y”**) is

**x(t)**in units

Try to stop the simulation “more or less” at these times and compare

**Fig. 2-3a, Fig. 2-3b**and

**Fig. 2-3c**.

**Chapter 2.4.2 f(t)=1*sin(1t) as F(jt)=Start*exp(jω0t) for Start=-1j and ω0=1/sec**

**Fig. 2-4**

**Fig. 2-4a**

**x(t)=**

**x(t)=1sin(1t)**

**This is the projection of the circulating vector from**

**Fig. 2-4b**onto the real axis

**Re z**.

**Fig. 2-4b**

Complex function

**-1j*exp(j1t)**for

**ω0=1/sec**, otherwise

**exp(j1t-π/2)*exp(j1t)**because

**-1j=exp(j1t-π/2)**

**Note:**

Because the point

**Start**=

**0-1j=-1j**

This complex function has the value

**Start***

**exp(j1t)=-1j*exp(j1t)**.

You can see the initial state of the rotating vector

**Start=-1j**in

**Fig. 2-4b**before the simulation.

Compare with

**Fig. 2-3b**. You can clearly see the truth known for centuries that

**sin(ω0t)**is delayed by

**π/2 = 90º**with respect to

**cos(ω0t)**.

**Fig. 2-4c**

Time chart

**x(t)=1sin(1t)**

**Chapter 2.4.3 f(t)=1*cos(1t-π/4) as F(jt)=Start*exp(jω0t) for Start=1*exp(-jπ/4)=0.707-j0.707 and ω0=1/sec**

**Fig. 2-5****Fig. 2-5a****x(t)=1cos(1t-π/4**) or** x(t)=Acos(ω0t-ϕ)** for **A=1 ω0=1/sec** and** ϕ=-π/4 = -45º**

This is the projection of the circulating vector from **Fig. 2-4b** onto the **real** axis **Re z**.**Fig. 2-5b**

The complex function **1*exp(j1t-π/4) **as** 1*cos(1t-π/4)**.

Note that the lag** ϕ=-π/4 (ϕ=-45º)** is beautiful here. Check with Pythagoras that the amplitude **A=1**

and that **ϕ=-π/4=-45º**.

We can write the circulating vector in different ways.**Start*exp(j1t)=exp(-jπ/4)*exp(j1t)=(1/√2-j1/√2)*exp(j1t)≈(0.707-j0.707)*exp(j1t)**.

You can see the initial state of the circulating vector **Start=exp(-jπ/4) **in** Fig. 2-5b**.**Fig. 2-5c**

Time chart **x(t)=1cos(1t-π/4)**

**Section 2.4.4 f(t)=1*cos(2t) as F(jt)=Start*exp(jω0t) for Start=1 and ω0=2/sec**

We increased the speed to

**ω0=2/sec**relative to

**Fig. 2-3.**

**Fig****. 2-6**

**Rys. 2-6a**

Time chart

**2**times faster than in

**Fig. 2-3**. Besides, the initial state

**Start**is the same, i.e.

**Start=(1+0j)=+1**

Compare with the corresponding animations in

**Fig. 2-3**.

**x(t)=1cos(2t)**.

**Fig. 2-6b**

Complex function

**1exp(j2t)**as

**1cos(2t)**

**Fig. 2-6c**

Time chart

**x(t)=1cos(2t)**

**Chapter 2.4.5 f(t)=0.5cos(1t) as F(jω0t)=Start*exp(jω0t) for Start=0.5 and ω0=1/sec**

**Fig. 2-7****Fig. 2-7a**

Time amplitude **2** times lower than in **Fig. 2-3**. The initial state **Start** is the same, i.e. **Start=(0.5+0j)=+0.5**.

Compare with the corresponding animations in **Fig 2-3**.**x(t)=1cos(2t)**.**Fig. 2-7b**

Complex function **0.5*exp(j1t)** as **0.5*cos(1t)****Note:****Fig. 2-7c**

Time chart **x(t)=0.5cos(1t)**

**Chapter 2.5 Functions Start*exp(jω0t)**

Compare again the previously discussed complex functions for different parameters **Start** and **ω0**.**1- Start=1 and ω0=1/sec****2- Start=1*exp(-jπ/2)=-j1 and ω0=1/sec****3- Start=1*exp(-jπ/4) and ω0=1/sec****4- Start=1 and ω0=2/sec****5- Start=0.5 and ω0=1/sec**

**Fig. 2-8**

**5**versions

**Start*exp(jω0t)**

These functions are shown in

**Fig.2-4b…Fig.2-7b**as rotating vectors. Their ends indicate points

**z**whose coordinates are just complex functions

**Start*exp(j ω0t)**. The

**beetroot**vector is the initial state of each function. This is the

**Start**parameter that comes before

**exp(jω0t)**. Analyze each of the

**5**waveforms carefully, taking into account the

**Start**and

**ω**parameters.

Compare, for example,

**Fig. 2-8b**and

**Fig. 2-8a**. Here, it is clearer than in the time plots that

**sin(1t)–>-j1exp(j1t)**is lagged by

**90°**with respect to

**cos(1t)–>1exp(j1t).**

**Chapter 2.6 The complex function F(jt)=(a-jb)*exp(jω0t) as a two-dimensional version of f(t)=a*cos(ω0t)+b*sin(ω0t)****Chapter 2.6.1 General description**Dessert at the end. So how to build the trajectory

**F(jω0t)**for any

**sine**function

**f(t)=a*cos(ω0t)+b*sin(ω0t)=c*cos(ω0t-ϕ)**

From

**Chapter 2.4.1**we know that the complex function

**1*exp(jω0t)**corresponds to the time chart

**1*cos(ω0t)**.

Similarly, according to Chapter

**2.4.2**of the complex function

**-1j*exp(jω0t)**corresponds to

**1*sin(ω0t)**.

We’ll write it like this:

**1*exp(jω0t)<==>1*cos(ω0t)**

**–1j*exp(jω0t)<==>1*sin(ω0t)**

This is the case with amplitudes

**a=1**for cosine and

**b=1**for sine.

This works for any amplitudes a and b:

**a*exp(jω0t)<==>a*cos(ω0t)**

**-jb*exp(jω0t)<==>b*sin(ω0t)**

Instead of writing that the function corresponds to something there, you can more precisely, as below.

**Fig. 2-9**

**Re (a-jb)*exp(jω0t)=a*cos(ω0t)**+

**b*sin(ω0t)**

Vector

**(a-jb)**rotates with speed

**ω**.

The

**projection**of the rotating vector

**(a-jb)**

***exp(jω0t)**on the

**real**axis is:

**real**part

**(a-jb)**

***exp(jω0t)**

that is,

**Re**

**(a-jb)**

***exp(jω0t)**=

**a*cos(ω0t**)+

**b*sin(ω0t)**.

This is a generalization of the simplest case in

**Fig.2-3**where

**a=1**and

**b=0**.

**Fig. 2-9a**

The complex function

**(a-jb)**

***exp(jω0t)**as a vector rotating at

**ω0**

It has

**2**components:

**-real**(cosine)

**a*exp(jω0t)**

**-imaginary**(sine)

**-jb*exp(jω0t)**

What you see in

**Fig. 2-9a**is the initial state of the rotating vector, i.e. for

**t=0**.

I emphasize that “lonely”

**a**and

**b**are

**real**numbers!

**Fig. 2-9b**

The

**projection**of a rotating vector

**a*exp(jω0t)**onto the real axis is

**a*cos(ω0t)**

**Fig. 2-9c**

The projection of the rotating

**-jb*exp(jω0t)**vector onto the imaginary axis is

**b*sin(ω0t)**

**Fig. 2-9d**

The projection of the

**sum**of the rotating

**vectors**, the

**red**vector, onto the real axis is

**a*cos(ω0t)**+

**b*sin(ω0t)**=

**c*cos(ω0t-ϕ)**

It’s not very obvious, but that’s

**trigonometry**! Instead of a proof*, let’s substitute a concrete one

values e.g.

**a=0.75**and

**b=1.25**and check the animation.

* This can be done by a high school student of the mathematics class.

**Chapter 2.6.2 Concrete example F(jω0t)=(0.75-j1.25)*exp(jω0t) as a two-dimensional version of f(t)=0.75*cos(ω0t)+1.25*sin(ω0t)**

**Fig. 2-10**Rotating vector

**(0.75-j1.25)*exp(1jt)**as

**f(t)=0.75*cos(1t)+1.25*sin(1t)=1.458*cos(1t-59.04°)**

**Fig. 2-10a**

**Rotating**vector

**0.75*exp(1jt)**and its projection on the

**Re z**axis as a function of time

**0.75*cos(1t)**

**Fig. 2-10b**

**Rotating**vector

**-j1.25*exp(1jt)**and its projection on the

**Re z**axis as a function of time

**1.25*sin(1t)**

**Fig. 2-10c**

Sum of

**2**left vectors as rotating vector

**(0.75-j1.25)*exp(1jt)**

Its projection on the

**Rez**axis is a function of time

**f(t)=0.75*cos(1t)+1.25*sin(1t)=1.458*cos(1t-59.04°)**and its time chart is shown in the animation below.

**Fig. 2-11**

**1.458*cos(1t-59.04°)=0.75*cos(1t)+1.25*sin(1t)**

An angular offset o

**f 59.04° ≈1.03 radian**is shown in the time chart.