12. Fourier Series classically

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 complex 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 component, 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+j0.4 ie. a2=+0.6 and b2=-0.4
c3=0.4-j0.2 ie. 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/sec 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 complex 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 amplitudes 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 right 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

 

 

 

 

 

 

 

 

 

 

 

 

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