The document provides an overview of fundamental mathematical concepts, including Fourier transform, Euler’s theorem, complex numbers, and the center of mass, emphasizing their significance in engineering and science. It explains how Fourier transform decomposes time-based functions into their frequency components and illustrates connections with trigonometry to represent AC waveforms. Additionally, the document discusses the application of these concepts in practical scenarios, highlighting the importance of understanding complex number systems in various scientific fields.

2.  INCLUDES (FUNDAMENTAL + BASIC):
oFundamental Concept:
1) Fourier Transform (What does it mean? How did it originate? Advantages)
2) Euler’s Theorem - Beautiful Maths concept. ( ⅇⅈ𝜋
+ 1 = 0 )
3) Complex number and Imaginary number concept. (Re + Imz)
4) Center of mass (cm) & Center of gravity (cg).
5) Behaviour of even & odd functions and Fourier Series.
6) What is natural log e (Why e = 2.718 and 𝜋 = 3.14 ? ).
7) Why ⅈ
2
= -1? Rules of Vector multiplication in complex plane.
8) Sinc function, Fourier series Expansion?
o Basic Concept:
1) Trigonometry Function with their definitions.
2) Why AC wave is represented by only sine / cosine function?
3) What is the actual meaning of sine, cos and tan function in Maths?
4) What is the actual meaning of multiplication, division, log and exponential
function?
5) Why the multiplication of two negative number is +Ve? what is the logic?
6) Why √−1 is not possible? what is root function?
7) What is the maximum possible growth when rate is 100%?

3.  Fourier Transform :
Fourier Transform is one of the most important ideas evolved from Maths. It covers
many important topics of Engineering and Science like Wireless communication, Radar,
Signal processing, Modulation and Demodulation techniques. Today, when IOT, Machine
learning and Image processing are the rising branches of engineering, Fourier and
calculus play the base role in establishing the platform for it.
But … Stop, what is Fourier Transform???
The Fourier transform decomposes a function of time into its constituent frequencies.
Yes.. But what is the actual meaning of the above sentence ??? and what is the process
behind function conversion?? Let’s start the journey, for one of the most beautiful
concepts in maths.
Now suppose I have one audio signal with 400 Hz frequency, which means the wave
oscillates 400 times up & down within one second.
“What we know is a drop, what we don’t know is an Ocean”: Sir Issac Newton (1642 -
1726).

4. This is pure sine wave and you can represent it with a sinusoidal function / equation.
Now suppose we have another signal with a higher frequency of 500 Hz, which means it has
the same structure but different frequency.
Above graph represents a 500Hz audio signal.
- “Obvious – is the most dangerous word in mathematics”: E.T Bell (1883 – 1960).

5. If I play both the sound at the same time, sound produced due to combined effect is
shown in below figure.
As shown in figure, the O/P is not a pure sine wave, we get maximum value at point P2
where amplitude of both the signal match with each other, and minimum value at point
P1 where amplitude of both signal cancel out each other.
Similar things happen in daily life, the original signal is distorted due to noise signal and
we get output signal due to their combined effect. But here only two signals are
considered, but what happens when more and more signals add up to each other?
- “Mathematicians stand on each other’s shoulders”: Carl Gauss (1777 – 1855)

6. Output signal may sometimes be too different in shape when compared to the original input
signal.
Oh.. it’s very complicated and you cannot recognize your original sound frequency.
Microphone records any sound, it just picks up sound pressure at many different points, and it
considers only final sum and records it.
Now the question is: How can one recompose the original signal from these types of non-sine
wave signals??? And the answer is “FOURIER TRANSFORM”.
But, Wait … Why should we represent AC wave using sine / cosine function? What is the
relation of wave function with trigonometry?
“The only way to learn mathematics, is to do mathematics”: Paul Halmos (1916-2006).

7. Let’s discuss the basic Trigonometry concept: Directly or indirectly trigonometry is connected
with a triangle.
Let’s consider a triangle ABC where C = 90 degree.
For sin angle 𝑠ⅈ𝑛θ = Opposite / hypotenuse = a / h
Here is the sine table ….(θ values in degree)
Figure: A
Value of sin θ continuously increases from 0 to 1 with increase in the value of θ.Now let us
consider the same triangle in a unit circle.
As shown in the below figure, there is one triangle inside unit circle.
OA = radius of the circle which is 1.
So from trigonometry sin 𝛉 = (AB / OA) = AB & COS 𝛉 = (OB / OA) = OB (Since OA = 1).
“The laws of nature are but the mathematical thoughts of God”: - Euclid (Mid 4th
BC)
𝛉 0 30 45 60 90
Val. 0 1/2 1/√2 √3/2 1
0 0.5 0.707 0.860 1

8. So, we can now represent AB as sinθ and OB as cosθ.
From Pythagoras theorem:𝑂𝐵2
+ 𝐴𝐵2
= 𝑂𝐴2
.
But OA = 1 and put the value of OB= cosθ and AB= sinθ ,
We get : sin2
𝜃 + cos2
𝜃 = 1 (Identity of sin and cos Fn)
We get the same thing if we put the value of x and y
coordinate inside unit circle’s equation: 𝒙 𝟐
+ 𝒚 𝟐
= 𝟏.
But the question is: Why should we represent AC wave using sine / cosine function? What is
the relation of wave function with trigonometry? For the answer you must understand motion
of point A on circle and note the changes in value of sin𝜃 and cos𝜃.

9. As per discussion of sine table and as shown in the above figure, with motion of point A in
anti-clockwise direction (From 0 Degree to 90 Degree), value of sine continuously increases
from 0 to 1. We can represent this change using linear graph ( Amplitude V/s Theta θ ).
As shown in the above figure, If I take the value of sin𝜃 at different angles and draw the graph
then I get maximum value at angle of 90 which is 1. Similarly, if we take sin𝜃 value for 900
to
1800 i.e. 2nd quadrant and draw the samples value we will get same
as figure2.
After connecting all sample points, you will get AC waveform/sinusoidal
graph as shown in right figure.
“Geometry is the knowledge of eternally existent”: Pythagoras

10. That’s why we use sine or cosine function to represent continuous changing graph like AC
waveforms.
Consider same figure again, <B = 90 – 𝜃
So, COS(90 − 𝜃) =
𝑎
ℎ
= sin𝜃
That Cause, COS(𝟗𝟎 − 𝜽) = sin𝜽
We have just changed the point of view on triangle.
So, it’s all about sine and cosine function, but wait… what about tan𝜽???
tan𝜃 word comes from tangent, but how ??? let’s see.
As shown in figure, let us make one tangent at point A.
OA is the radius of unit circle which is 1
So, tan𝜃 =
𝐴𝑄
𝑂𝐴
, But OA = 1 so AQ = tan𝜃 , and What is the AQ?
“Mathematics is the language in which the GOD talk to people”: Plato (429 Athens Greece)

11. AQ is the length of the tangent to the circle, that’s why the function is known as tan𝜽.
Sine & cosine are periodic function, both are repeated, their value is repeated after rotation of
360o
or 2𝜋 (of complete circle.).
So, it’s all about basic trigonometry, Let’s go back to Fourier Transform.
We were at “How you recompose the original signal from these types of non-sine wave
signals???”
Suppose we mix many different types of
frequency signals. Is there anyway, to get
back all signals in their original form as a
output signal?
Answer is yes: Using Fourier Transform.
As shown in figure output signal is in
additive form of five different signals.
“An Equation means nothing to me unless it express the thought of God”
Srinivasa Ramanujan (India : 1887-1920).

12. But How? For the perfect understanding let us consider the below 3Hz signal & concentrate on
a finite portion of the signal graph. Signal frequency is 3Hz that
means it oscillates 3 times up and down within time
one second.
We have already proved that we can represent
sinusoidal signal in circle using vector motion.
So now we will take the graph and try to wrap it within circle.
Instantaneous value of the signal represented by rotating vector.
Imagine a rotating vector where each point in a time, its
length is equal to height of the graph for that time.
Important Note: We can choose frequency of rotating
vector. So, there are two different frequency 1) Frequency of
our original signal graph 2) Rotating vector frequency.
“Two truth cannot contradict one another”: Galileo Galilei.

13. We can adjust this second frequency however we want, maybe we can wrap it around faster
or maybe we may wrap it slower. And the choice of winding frequency determines what the
wound-up graph looks like.
As shown in above figures, we tried to wound up 3Hz signals with
different rotating frequency. For example, as shown in figure 4,
rotating frequency is 0.5 Cycles/second, that means rotating vector
requires 2 second to complete one circle for representing 3 Hz
sinusoidal signal.
“All birds need to fly are right shaped wings, the right pressure and
right angle, God is great Mathematician” – Daniel Bernoulli

14. Something special will happen when winding frequency matches with frequency of our signal
3Hz. All the high points of the graph happen on the right side of the circle because of the
match of rotational winding vector frequency with frequency of our signal. There are no any
other /extra shape in the graph. You can represent entire time domain signal with the same
single image.
We did not do anything, just matched the rotational vector frequency with original signal
frequency. Means sin (2𝜋𝑓𝑡) = sin𝜃 (Here I have considered unit amplitude sine signal).
Ok, now we have completed almost 50% concept of Fourier Transform, but for further
discussion you will have to learn another important concept : Center of mass.
 Centre of mass :
Let’s prefer definition first. Centre of mass is one unique point where weighted relative
position of the distributed mass sums to zero. Suppose
you have a square notebook and you want to balance it
using single finger. you can find only one point where you
can balance: It is the centre of gravity (cg).
“If Quantum mechanics hasn’t profoundly shocked
You, you haven’t understood yet” Niels Bohr (1885-1962).

15. Center of gravity (cg) is a point through which the resultant of all the particles
of the body act. One important thing is centre of mass (cm) may be inside
or outside the object and depends on the geometry shape. We will take
this topics in detail but right now because of the position of ‘centre of mass’ either
inside or outside the object we will have to use complex number system.
But in maths; suppose you have one particle body, consider point P has
mass m1 and we located it in space using r1. There are many particles in
the body like Pi (with relative mass mi and vector location Ri).
∑ 𝑚ⅈ ∗ 𝑅ⅈ𝑛
ⅈ=1 = R (co-ordinates of R must be equal to the sum of all points)
Above equation represent location of cm at point R.
A body's center of gravity is the point around which the resultant torque due to gravity forces
vanishes. When gravity field is uniform, the mass-center and the center-of-gravity will be the
same.
“The scientist only imposes two things, namely truth and sincerity, imposes them upon himself
and upon other scientists.”: Erwin Schrodinger (1887-1961)

16. Let’s go back to Fourier Transform:
For all the above graph, the ‘center of mass’ location is approximately near origin or zero.
Opposite side of each ring cancels the vector location coordinates’ position of each other but
for the graph shown below centre of mass (cm) to its maximum peak value towards the right
side of the axis.
“Science cannot solve the ultimate mystery of nature. And that is because, in the last analysis,
we ourselves are a part of the mystery that we are trying to solve”: Max Plank (1858-1947)

17. So, center of mass gets its peak value in graph when wrapping frequency matches with
original signal frequency 3Hz. Now I can plot ‘centre of mass’ value of graph vs different
winding frequency.
As shown in above graph, at winding frequency 3Hz you get peak value of centre of mass.
Let’s rewind the our journey before going further in Fourier:
1) Take the Original Signal
2) Convert it into rotational vector pattern
3) Find centre of mass at different frequency
4) Find peak value of centre of mass (You will get it when frequency match.)
“All science is either physics or stamp collecting”: Ernest Rutherford (1871-1937).

18. This graph is almost Fourier Transform. We have considered only one signal 3Hz, but suppose
we have composite signals of frequency 2Hz and 3Hz and we want to wind up in circular form.
You will get proper and less pattern at 2Hz and 3Hz rotational vector frequency only where
center of mass of graph is at its peak value as shown in below figure:
“Nothing is too wonderful to be true if it be consistent with the laws of nature”: M.Faraday

19. If you individually find the Fourier Transform for 2Hz and 3Hz signal and then try to add it, the
final graph is same as shown in above.
So, F.T(2Hz) + F.T(3Hz) = F.T(5Hz).
Same way you can pull-out all-important frequency from mixed composite signal. Before
entering into Frequency domain we will have to understand Euler’s Theorem :
𝒆𝒊𝝅
+ 1 = 0: This equation known as most Beautiful identity in maths: The Euler’s Eqn.
All the numbers mentioned in above equation are constant.
π = 3.14 (constant irrational number)
e = 2.71828 (constant irrational number)
i = √−1 (Imaginary number- constant)
1 & 0 = Real numbers (constant)
So that means, above equation represent relationship between constants!!!
But What does it mean?? For answer let us go back to very basic formulas of Maths, Let us
begin.
“Anyone who thinks Sky is the limit, has limited Imagination” : J.C.Maxwell (1831-1879)

20. Q: 1) What is multiplication of two numbers?
As we know it is a form of addition process.
Ex: 3 * 5 = 15
As shown in figure you will add five times the
numbers 3 into zero.
Q: 2) Why Multiplication of one –VE and one +VE number is negative?
Ex: (-3) * 5 = -15
You are adding (-3) into zero five times.
That’s why result is negative.
Q:3) Why multiplication of two negative number is always +VE?
Ex: (-3) * (-3) = 9, In previous examples you have added
the quantity because no. of times digit (05) is +ve.
But now you must subtract from zero, because
digit 5 is negative now.
“Maths teach us always be careful with the signs”: Anonymous

21. Same things happen in process of division, but now you must subtract instead of addition. In
other words, division is a special form of subtraction.
Here is the example:
Suppose you divide 20 by number 5, That means 20/5.
For division process, we will have to subtract 5 from 20 until you get zero.
20 – 5 – 5 -5 -5 = 0 (I will use this definition in last segment FT of Pulse signal).
That means if subtract four times 05 from 20 & you get zero. So, answer is 04.

22. Now,you know from the answer of the question 3, that multiplication of two negative
numbers is always positive.
That means, there is no value which produce –ve number after self-multiplication.
So, what about √−1 ?
Suppose we have a parabola with equation 𝑥2
+ 1 = 0. Maximum power 2 indicates that the
equation has two roots / answers.
So 𝑥2
= - 1. At present we don’t have any value which represents or solves this equation that’s
why we have to enter in one of the complicated world of Mathematics: The Complex Number
System (Real numbers and Imaginary numbers).
* Complex Number System (Re + Imz):
Complex plan was discovered by famous mathematician Gauss. We cannot find all the
answers using real number system in mathematics. Complex numbers play an important role
in Engineering, Communication, Physics and many complicated problems. During discussion of
the ‘center of mass’, we learnt that depending on geometry shape, the location of cm is
outside from body for some irregular shape. So, for the representation of ‘centre of mass’ we
will have to consider complex number system.
“Be alone , that is secret of Invention ” : Nikola Tesla(1856-1943).

23. You can represent any number in this complex plan. Please consider below graph.
Here, one axis represents all real number
(Including +ve,-ve, zero, decimal etc.) then
another represents imaginary number.
Here are some basic rules when dealing with
Complex number system.
 Addition (example):
(6+7j)+(3−5j) = (6+3)+(7−5) j= 9 + 2j …. Ans 1
 Subtraction (example) :
(12+6j) − (4+5j) = (12−4) + (6−5) j = 8+j …..Ans2
For addition and subtraction, we just need to separate real and imz parts for completing the
further process. We can represent the numbers according to points lying on the quadrant.
So, the complex number system can be defined as the algebraic extension of the ordinary
real numbers by an imaginary number i. ... A complex number whose real part is zero is said
to be purely imaginary; the points for these numbers lie on the vertical axis.
“A great deal of my work is just playing with Equation and seeing what they give” : Paul Dirac

24. One important thing about this system is that, when you multiply any number with i, the
vector rotates 90 degree in anti-clockwise direction.
We know number 1 is a complete real number, so if we multiply it with i you will get 1i
which is pure imaginary number. And if you want to represent it then need to mark it on Y
axis (IMZ axis) which is perpendicular to X axis (Real axis). That’s why whenever you multiply
any number with i you must rotate vector 90o
in anti-clockwise direction.
For more clarification, consider below graph:
“This isn't right. This isn't even wrong”:Wolfgang Pauli (1900 -1958).

25. Similarly, when we multiply 1i with i we will get ⅈ2
, due to multiplication with i you will have
to rotate the vector representation by 90 degree in anti-clockwise. After rotation you will get
pure real number -1. That’s why we consider ⅈ2
= −1 .
We can apply the same rules for multiplication of complex number. Here we have mentioned
below, some details of polar form of complex number.
x = r cos𝜃 y = r sin𝜃
As shown in figure we just need to compare polar components
with a + ib form.
a + ib = r cos𝜃 + i r sin𝜃 = r (cos𝜃 + i sin𝜃 ) = r cis𝜃
EX: You have number = 1 +2j and you want to multiply by 3j.
As shown in left figure representation
of 1 + 2j. Before multiplying with 3j, just
multiply it with 3 and your vector size increase
by 3. Result is 3 + 6j shown in right side.
“Science is about knowing, Engineering is about doing”: Henry Petroski

26. As we know on multiplication with j, vector rotates 900
in anti-clock direction. You need to
interchange X & Y. Now real part become IMZ,
and IMZ part become real number (This is just
because changing length and height of triangle).
So final result is: -6 + 3j
You can also get same things using the following method.
3j * (1 + 2j) = 3j + 3j * 2j = 3j - 6𝑗2
= -6 + 3j
So, this is all about basic things in every complex number system. Now we need to understand
some concept related to π (pi) which is another constant used in Euler’s number.
Whenever you divide circumference of a circle with its diameter you will always get one
constant number 3.14 which is pi π, it doesn’t matter how big / small the circle is.
So, π =
𝐶ⅈ𝑟𝑐𝑢𝑚𝑓𝑒𝑟𝑒𝑛𝑐𝑒
𝐷ⅈ𝑎𝑚𝑒𝑡𝑒𝑟
=
2∗𝜋∗𝑟
2∗𝑟
= 3.14.
Pi also related and plays important role in radian angle system in trigonometry.
So, what is the meaning of 1 radian? Let’s see.
“People think it’s magic, we call it Engineering”

27. Please consider the figure. Suppose the radius of the circle is r. Now we travel from one point
and stop after a distance r on the trajectory of the circle (means circular movement is r). Angle
made by these two points at the centre is known as 1 radian.
So, at 1 radian Arc length = radius length.
Why circle is 360 or 2π radian ?
As per the definition of 1 radian, arc length
covered when θ is 1 rad is r. We know circumference of
circle is 2πr. So θ required to cover entire circumference
is 2π. That’s why one complete circle is 2π radian.
Simple equation here:
Arc length Radian
r 1
2πr ? And now we know answer is 2π, the complete angle.
Area of the circle is also related with the definition of 1 radian. Let us see.
“God always takes the simplest way”: Albert Einstein

28. Consider the same figure, we need to assume that the triangle has flat base instead of arc
shape. We know the area of triangle is
1
2
* b *h. Triangle is made by 1 radian, here b=r and h=r.
So, area of triangle = 𝑟2
/ 2 (Area covered by 1 radian)
Now we know that complete circle has 2π radian.
So, total area covered by a complete circle is = 2π *
𝑟2
2
= π 𝒓 𝟐
= Area of the circle.
Now after the discussion about pi, now let see one of the beautiful number e (The Euler
number). You should know why e is 2.71828?
Euler’s number e (Natural log concept):
Suppose you have 1 INR (Rs.). A bank provides 100% interest on it after a period of one year.
That means you will get 1 extra INR after 1 year. That means you will have total 2INR. That's
cool just remember you will get it after a time period of 1 year. ------ (A)
Suppose another bank provide same scheme but instead of 100% after year, they provide 50%
interest after 6 months. So, you will get two-time interest (50% + 50%) within a year. Total
Interest remains same which is 100% after a year just we have changed only the time period.
“God used beautiful maths in creating the world” : Paul Dirac

29. So as per the calculation after 6 months we will get 50% interest (half of the value).So, after 6
months you will get 0.5 INR interest on 1 INR.so now you have 1.5 INR after 6 months. Now
you will get 50% more interest on value 1.5 INR which is approx. 0.75 INR.
So, finally after the year we will get 1.5 INR + 0.75 INR =2.25 INR which is higher than 2 INR.
Amazing, same interest rate but when we change the time period we get 0.25 INR extra
benefit……………….. (B)
Now consider last case for more clarification, suppose another bank provides same interest
(means 100%) but in different time period. They will provide 33.33% after 4 months’ period.
(Means instead of 12 months, now 12/3 = 4 months period and same instead of 100% they will
provide 100/3 = 33.33%)
So, after 4 months you will get 33.33% of 1 INR =0.33 INR (Total value = 1.33 INR)
Next 4 month, you will get 33.33% on 1.33 INR = 0.44 INR (Total value = 1.777 INR)
Last 4 month you will get 33.33% on 1.777INR = 0.5922 INR (Total value = 2.37 INR)…. (C)
So, finally after a year, you will have 2.37 which is higher than 2 INR and 2.5 INR. We can
go further using 1-month time period or calculating interest per day. And we get more and
more benefit. But my question is How will we get maximum growth rate /benefit ?
“If I had just one hour left to live, I would spend it in Math class”

30. We get the same answer using following equation in the scenarios A, B and C.
Equation is (1 +
1
𝑛
) 𝑛
For the n=1, you will get 2 (as per A), same if you put 2 you will get 2.25
(as per B) and if you use n=3 then answer is 2.37(as per C). One thing is clear here above
equation is true for any value of time period.
But what maximum value can we get?
Just put n=100 and you will get 2.7048 and if you put 1000 you will get 2.7169. That means
now final answer is near 2.718, There are no much changes in this value. Now suppose you
divide time period into seconds or milli/micro seconds you cannot go beyond this value. This is
the maximum possible growth rate and this number is known as e (Natural log number or
Euler number).
In real life, any type of growth follows these rule. Suppose your current height is 5” and after
one year it may be 5.5 Ft” or 6FT.2”. But in nature growth doesn’t appear spontaneously, like
the last day of a year or month. Nature always follow the rules of continuous growth. Our
height continuously grows each day and then we will see the 1 Ft difference in our height after
a year. Nature chose this number and it represents the maximum possible output value when
the rate is 100% and time period is very small (or continuous growth every time). That’s why
this number is known as natural log number e. e = 2.71828… (It is irrational number).
“A mathematical formula should never be “owned” by anybody, Maths belong to GOD”

31. • Exponential Function and Log function: For more clarification let’s start with one example.
Consider 23
, that means number doubles itself after a certain time. Suppose there are two
fishes in a lake, each fish lays one egg after every 10 days, so after 10 days there are 4 fishes in
the lake. Similarly, each of four fishes lay 4 more eggs and after 10 days there will be 8 fishes
in lake. So finally, after 20 days’ time, total number of fishes are 8 which is 23
.
Log function allows you to see same things but in different manner, as shown in below image.
“The highest form of thought is in Mathematics”: Plato

32. We have already discussed movement of points on unit circle and their relationship with sine
and cosine function. We know that in Complex number system we represent number using
a + ib format.
We know for unit circle we can represent x coordinate using
cosθ and Y coordinate using Sinθ, but here Y axis is related
to Imaginary number. That’s why we must multiply
Y axis with Imaginary i. So, Y coordinate number is i Sinθ.
So, we can represent entire circle using:
cos θ + i sinθ equation which is also known as CIS θ
cos θ + i sin θ = ⅇⅈ𝜃
Known as Euler’s number, we have already discussed all constants used in
Euler’s equation: 𝒆𝒊𝝅
+ 1 = 0
So, it’s all about how each constant is related with Euler’s number.
Now Let’s see how above equation come.
1) Put θ =0 in this equation cos θ + i sin θ = ⅇⅈ𝜃
= cos 0 + i sin 0 = ⅇⅈ0
= 1 ( X axis point).
“Knowledge is of no value unless you put it into practice”: Anton Chekhov

33. 2) Put θ = π/2 So Equation become ⅇⅈ𝜋/2
= cos π/2 + i sin π/2 = i (Pure IMZ Y axis)
3) Put θ = π, So Equation become ⅇⅈ𝜋
= cos π + i sin π (We know sinπ = 0 & cosπ=-1)
ⅇⅈ𝜋
= −1 (LHS side there is complex No, RHS side pure real number)
ⅇⅈ𝜋
+ 1 = 0 , This equation is known as Euler’s number. This is one of the most beautiful
equation in Mathematics. (Above Equation is also known as Beauty of the Maths).
Using ⅇⅈ𝜃
we can represent above unit circle trajectory / all possible location of points on the
circle. In a unit circle we always get r=1 for any point value on the trajectory.
Ex: ei(0.15π)
= cos 0.15π + i sin 0.15π (just consider angle value = 0.15π)
= 0.89 + 0.45i So, √((0.89)^2 ) +(0.45)^2) = 1 (Unit value of r)
Let’s back to last concept of Fourier Transform:. We have already discussed how we represent
any waves using wrapping pattern in circle. Centre of mass mostly depends on geometry
shape of the circle and sometimes lies outside the body and that’s why we had to study
complex number system. Now we have the formula ⅇⅈ𝜃
, using which we can represent
rotating vector trajectory in complex plan. So, we had to do all the above derivation.
“If you aren’t confused by quantum mechanics, you haven’t understood yet”:

34. That’s what we want. If ⅇⅈ𝜃
represents trajectory of circle, then what happen if we will
multiply it with a function g(t). Yes, it represents function g(t) in circular winding pattern. So
ⅇⅈ𝜃
*g(t) provides vector length according to height/ amplitude of graph g(t). It is like sinθ
wave graph that oscillates between -1 to 1 (which is the maximum and minimum value of sine
function), but if you multiply with any amplitude like 10 then graph shape remains the same
but now wave oscillates from -10 to 10. Similarly, if you write ⅇⅈ𝜃
then it only represents a
circle with constant vector length r=1. But when ⅇⅈ𝜃
is multiplied with g(t) then vector length
changes according to the amplitude of g(t) and constant circle circumference converts into
winding shape.
For representing the entire circumference put
θ = 2π and equation becomes ⅇ2𝜋ⅈ
.
 For representing vector length at time
t, equation becomes ⅇ2𝜋ⅈ𝑡
.
 Now we need to wrap this vector at
different frequency f, so Eqn becomes ⅇ2𝜋ⅈ𝑓𝑡
.
“Math is only place where truth and beauty mean the same thing”

35. So 𝑔( 𝑡) ∗ ⅇ2𝜋ⅈ𝑓𝑡
is magic equation what we are looking for. Now let us consider motion in
clockwise direction so final equation becomes 𝑔( 𝑡) ∗ ⅇ−2𝜋ⅈ𝑓𝑡
. Using this equation, you can
convert any periodic / non-periodic function in circular winding form. But we are not
interested in finding only the shape of the graph, we have to find out ‘centre of mass’ of the
graph.
Suppose our graph is made of small particles, which are
very close to each other. So, we can consider it as one
particle body. Now, we must find average location of all
the particles which is the ‘centre of mass’ of our graph.
Let’s use equation of cm here;
Suppose there are N particles:
1
𝑁
∑ 𝑔( 𝑡) ∗𝑁
𝑘=1 ⅇ−2𝜋ⅈ𝑓𝑡 𝑘
For more accurate result consider that all the particles are very close to each other. Now we
just change particles to samples. So, if we take the samples at the different time period and
time period is very small dt then above equation converts into integration form.
Equation becomes
1
𝑡2−𝑡1
∫ 𝑔( 𝑡) ∗ ⅇ−2𝜋ⅈ𝑓𝑡𝑡2
𝑡1
𝑑𝑡 .
“Some people say it is magic, I say it is Mathematics”

36. In above equation ∫ 𝒈( 𝒕) ∗ 𝒆−𝟐𝝅𝒊𝒇𝒕𝒕𝟐
𝒕𝟏
𝒅𝒕 is Final Fourier transform equation. Using this
equation, you can decompose any mixed signal. Fourier Transform is a very powerful tool of
Mathematics, it is very useful in Audio-Video signal processing, Image processing, modern
communication and Optical fibre technology etc. Heisenberg uncertainty principle is also
related to this equation we will see it in INSIDE MATHS PART: 2.
At wrong rotating frequency of vector, average location of cm is near zero / origin. But when
rotating frequency of vector is matched, we will get center of mass at its peak value.
One important thing is our original signal is in Time domain, but you deal with rotating vector
frequency. This is the first and most important property of Fourier transform.
Here is one example of Fourier Transform to clear all the things in detail.
EX: Find Fourier Transform of following window signal:
f(t) = 1 if
−1
2
< t <
1
2
0 Otherwise
Above signal represents the square signal / pulse.
“Algebra is the Metaphysics Arithmetic”

37. F(f) = ∫ 𝑔( 𝑡) ∗ ⅇ−2𝜋ⅈ𝑓𝑡𝑡2
𝑡1
𝑑𝑡
= ∫ 1 ∗ ⅇ−2𝜋ⅈ𝑓𝑡1/2
−1/2
𝑑𝑡
=
−1
2𝜋ⅈ𝑓
[ ⅇ−2𝜋ⅈ𝑓𝑡
]
1/2
−1/2
(We know that ∫ ⅇ 𝑥
𝑑𝑥 = ⅇ 𝑥
)
=
−1
2𝜋ⅈ𝑓
[ ⅇ−𝜋ⅈ𝑓
- ⅇ 𝜋ⅈ𝑓
] (Put the value of t )
=
−1
2𝜋ⅈ𝑓
[ cos(𝜋𝑓) – i sin (𝜋𝑓) –cos(𝜋𝑓) – i sin(𝜋𝑓) ] (Put the defination of cisθ)
=
−2ⅈ sin(𝜋𝑓)
2ⅈ (𝜋𝑓)
= sinc (f) This is sinc(x) function.

38. One important thing is that our original signal gate pulse is in the time domain, but after
Fourier transform now you have signal in the frequency domain. Let’s analyze it.
We have represented two signals here; Y=X and
Y = sin(x).
If you multiply any number with sin(x), then the
instantaneous value of sin(x) depends according
to the value but the shape of oscillation remains.
There will only be changes in its maximum / minimum value.
Same things happen here when you divide sin(x) function with respect to any value, the
amplitude of sine function decreases. But here sin(x) divide by y=x graph and value of y=x
graph continuously increases, that’s why amplitude of the sin(x)/x = sinc (x) function
continuously decrease. We know that division is the form of subtraction. Deduction rate of
sin(x) signal depends on the value of x. Sinc function gets its peak value at the origin. At origin
Y=0, so there is no deduction and sinc(x) function get its maximum value at x=0.
“Anyone who claims to understand Mathematics behind Quantum theory is either lying or
crazy” : Richard Feynman

39. Here is the graph of sinc(x) Function:
Look at the graph, Graph gets its peak value at x=0. Decay rate directly depends on the value
of x. With a higher value of x, sin(x) signal oscillates with very low amplitude.Look the graph
carefully, Graph is symmetrical (Across Y axis –like a mirror). sin(x) is odd function, y=x is also
odd function, but the output sin(X) is an Even function. We get sinc(f) function as the result of
example. This graph is about Amplitude of centre of mass Vs Frequency.
“Mathematics is the door and key to the sciences.” — Roger Bacon

40. As shown in the graph, the value of ‘center of mass’ is non-zero for many different
frequencies. So you can assume that there are many different types of sine wave with
different frequency and amplitude for making square wave.
We will prove this thing later in Fourier series with its application examples. We will also prove
that “Square wave is infinite sum of different sine wave” in Fourier series example.
So, It’s all about Fourier transform and it’s core details like idea/ use /& it’s Application. Today
in modern communication technology like RADAR, Satellite communication ,Wireless
communication ,Image processing etc. mostly based on two important concept of
mathematics Fourier & Matrix.
 Even & Odd Function
Before discussing Fourier Series we need to understand concept of Even and Odd function and
their property/behaviour.
“The Power of Mathematics is often to change one thing into another ,to change Geometry
into language and equation” : Marcus du Sautoy

41. The function is evevn when f(x) = f(-x) for all x.
This is symmetry about the Y-Axis (like reflection / or mirror at Y-Axis).
Ex:
COS(x) is also Even Function ,reflection of cos graph is same across both side of Y Axis.
For Knowledge :
- Study of Trigonometry was first started in India (as part of Hellenistic Astronomy).

42. Odd Function :
The funcction is odd if f(-x) = - f(x) for all x, That means value remain same but graph is in –ve
direction of Y-Axis.
EX:
As shown in above figure value of Y is same….
But in opposite direction. That’s why we told that sine(x) is odd function.
For Knowledge :
The word hundred is derived from the word "hundrath", which actually means 120 and not 100

43. Fourier series is representation of wave shape using summation of sine/cosine function with
different frequency,phase and amplitude.
We have to understand some mathematical operation before going into Fourier series.
1) Y = sin(x) = f(x)
This one is normal sin(x) graph ,which repeat it
self after period of 2π.
Amplitude range is from -1 (min) to +1 (max).
Now let’s increase frequency and compare,
I will replace x with 2x in next graph.
2) Y= sin(2x) = f(x) :

44. In above image, blue line represented normal sin(x) graph with repating period of 2π, But look
red line graph, Graph represented function sin(2x). As shown in figure now repeatating period
is π , which is half than original.
When sin(x) graph at it’s peak value at that time graph of sin(2x) is at zero.
Now you can understand : what is the meaning of terms inside of sin() function.
3) Y= 2 sin(x) = f(x)
Here you multiply sin(x) with 2 , means period of the function remain same which is 2π , but
amplitude become double, look below , I have compared both graph sin(x) with 2sin(x).

45. Now you can understand the term ‘Amplitude’ and its relation with sin(x) function.
OK, now last term , effect of Arithmetic operation with sin(x) on result.
4) y = 1 + sin(x) = f(x)
Here you are changing level of graph. For more information,please look into below graph.
sin(x) function represented by blue line, red line represented y = 1 graph , and after addition
you can get 1 + sin(x) which represented by green line. So actually whenever you add
something into sin(x), you are clamping up or down level of the graph by that number.

46. From above discussion we can understand term/equation 𝑎0 + 𝑎 sin(𝑛𝑥) and it’s graphical
behaviour. 𝑎0 + 𝑎 sin(𝑛𝑥) is first and base term of the Fourier series.
 Fourier Series :
Suppose , you have periodic function with period p .Genrally in periodic function,during half
period time function value is increase and remaing half period of time it’s value continues
decrease. So average effect is zero .
So here , I will consider half period time L = P/2 for calculating series.
I have alrady proved that sin(x) and cos(x) both function orthogonally related to each other.
Fourier series is an expansion of periodic function f(x) in terms of infinite sums of sin(x) and
cosine function.You can genrate any periodic wave function using sine and cosine function.
Fourier series is also known as Harmonics Analysis Tool, It is basically Fourier transform of
Periodic signal.( We will see how Fourier series equation comes from Fourier transform in
next part of this book)
For knowledge:
Zero is an even number.

47. Periodic waveform f(t) , Period P = 2*L (L = Hlaf of the period), the fourier series is given by:
f(t) =
𝑎0
2
+ ∑ 𝑎 𝑛cos(
𝑛𝜋𝑡
𝐿
∞
𝑛=1 ) + ∑ 𝑏 𝑛sin(
𝑛𝜋𝑡
𝐿
∞
𝑛=1 ) (Consider range –L to L)
𝑎0 = Constant dc level (Average level of graph)
𝑎 𝑛 =
1
𝐿
∫ 𝑓( 𝑡) ∗ cos
𝑛𝜋𝑡
𝐿
𝐿
−𝐿
𝑑𝑡
𝑏 𝑛 =
1
𝐿
∫ 𝑓( 𝑡) ∗ sin
𝑛𝜋𝑡
𝐿
𝐿
−𝐿
𝑑𝑡
For odd function 𝒂 𝒏 = 𝟎 , For Even function 𝒃 𝒏 = 0 (But Why ?? )
Yes I know without derive above formula, it is very hard to accept all above equation, but 80%
things will clear after calculating following problem.
EX: Find fourier series of square pulse.
f(t) = 0 if -4 ≤ t ≤ 0
5 if 0 ≤ t ≤ 4 EQN -01
f (t) = f(t+8) EQN -02
For Knowledge:
If you folded a piece of paper in half 103 times it would be the thickness of the observable
universe.

48. We can draw the graph from above description.
Here L = P/2 = 8/2 = 04 (Half time period)
Eqn 2 tell us that the signal repeat itself after 8 sec.
Let’s find fourier series :
𝑎0 =
1
𝐿
∫ 𝑓( 𝑡) 𝑑𝑡
𝐿
−𝐿
=
1
4
∫ 𝑓( 𝑡) 𝑑𝑡
4
−4
=
1
4
[ ∫ (0) 𝑑𝑡
0
−4
+ ∫ (5) 𝑑𝑡
4
0
]
= ¼ * 5 [𝑡]0
4
=
20
4
= 5 So,
𝑎0
2
= 5/2 = 2.5
You can find same things using the formula of Area .From range -4 to 4 , there is only one
rectangle R2.
Area of R2 is = 4*5 =20 (A=L*W)
For calculating the average ,you need to divide it by L which is 04, so you will get 20 / 4 = 05.

49. Now need to find 𝑎 𝑛 :
𝑎 𝑛 =
1
𝐿
∫ 𝑓( 𝑡) ∗ cos
𝑛𝜋𝑡
𝐿
𝐿
−𝐿
𝑑𝑡
= [ ∫ 0 ∗ 𝑐𝑜𝑠
𝑛𝜋𝑡
4
0
−4
dt + ∫ 5 ∗ 𝑐𝑜𝑠
𝑛𝜋𝑡
4
4
0
dt ]
=
1
4
[ 0 + 5 (
4
𝑛𝜋
) sin(
𝑛𝜋𝑡
4
)0
4
]
=
1
4
*
4
𝑛𝜋
* 5 [ 0 – sin (𝑛𝜋) ]
= 0
From the figure , function is odd , and we know that for odd function 𝒂 𝒏 = 𝟎.
You can directly put the value 𝒂 𝒏 = 𝟎 . It is true for any type of /shape of odd function.
𝑏 𝑛 =
1
𝐿
[∫ 𝑓( 𝑡) ∗ sin
𝑛𝜋𝑡
𝐿
𝐿
−𝐿
𝑑𝑡]
=
1
4
[∫ (0) ∗ sin
𝑛𝜋𝑡
4
0
−4
𝑑𝑡 + ∫ (5) ∗ sin
𝑛𝜋𝑡
4
4
0
𝑑𝑡]
=
1
4
[ 0 + (-5) (
4
𝑛𝜋
) [𝑐𝑜𝑠
𝑛𝜋𝑡
4
]0
4
] (We know that ∫ sin( 𝑥) 𝑑𝑥 = −cos(𝑥))
=
−5
𝑛𝜋
[ cos(n𝜋) – 1]

50. Put all above values, and you will get final fourier series of square wave.
= 2.5 -
5
𝜋
∑
1
𝑛
(cos( 𝑛𝜋) − 1) sin(
𝑛𝜋𝑡
4
∞
𝑛=1 )
You can use above equation to genrate your square wave using sine and cosine function.
Let see, How ????
put the value of n.
n = 1 >>> -2sin(
𝜋𝑡
4
)
n = 2 >>> 0
n = 3 >>>
−2
3
sin(
3𝜋𝑡
4
)
n = 4 >>> 0
n = 5 >>>
−2
5
sin(
5𝜋𝑡
4
)
n = 6 >>> 0
n = 7 >>>
−2
7
sin(
7𝜋𝑡
4
) and so on…..
After adding the value from n=1 to n=3 in fourier series , you will get above graph.
If I am continuesly adding n=1 to n=7 , then I will get something like shown in below graph.

51. This is not pure sine wave , but if you will add
more and more terms , your graph shape will
turn into square wave.
After adding infinite terms you will get
perfect square wave, and that’s why we
called that “Infinite sum of sine/cosine
wave produce square wave”.
In Fourier transform example section ,We have used same square signal for calculating
transform. We got sinc(f) function as output. So now we decode it as : If you want to produce
square wave you will have to add infinite sine wave with different amplitude and frequency
(As per the sinc(f) graph , there are many points where center of mass is non zero for different
frequency level).
Finally with this example , I have stopped Part-1 here, and it’s all about basic of Fourier
Transform and Fourier series.We will see more about other core concept of mathematics,
including Fourier series ,Matrix calculation & relationship of Fourier series with Heisenberg
uncertainty principle.
END OF PART:1

52. About Author:
Jaydeep Shah (Electronics & Communication Engineer)
Ahmedabad – India
Contact Info:
E-mail : radhey04ec@gmail.com
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53. Dedicated to :
“जिनके सामने पूरा ब्रहमाांड शिि झुकाता है ,में भी उनके िरणमे नतमस्तक हूां ॥ ”
To my Lord
शिवा
& सती
Thanks :