I explained Fourier analysis with a recorded video in Thunder Chasers.

1. Fourier Analysis
An Intuitive Perspective
A Basic Introductory Discussion Portraying how Fourier Series or Transformation Play Around Us

2. Axial Breakdown
v cosx
v sinx
v
x

3. Sinusoids
Input(-∝, +∝) Output [1,-1]

4. Fourier Series
𝒇 𝒙 =
𝒂 𝟎
𝟐
+
𝒏=𝟏
∞
𝒂 𝒏 𝐜𝐨𝐬
𝟐𝒏𝝅𝒙
𝒃 − 𝒂
+ 𝒃 𝒏 𝐬𝐢𝐧
𝟐𝒏𝝅𝒙
𝒃 − 𝒂
𝒂 𝟎 =
𝟐
𝒃 − 𝒂
𝒂
𝒃
𝒇 𝒙 ⅆ𝒙 𝒂 𝐧 =
𝟐
𝒃 − 𝒂
𝒂
𝒃
𝒇 𝒙 𝐜𝐨𝐬
𝟐𝒏𝝅𝒙
𝒃 − 𝒂
ⅆ𝒙 𝒃 𝐧 =
𝟐
𝒃 − 𝒂
𝒂
𝒃
𝒇 𝒙 𝐬𝐢𝐧
𝟐𝒏𝝅𝒙
𝒃 − 𝒂
ⅆ𝒙
𝒇 𝒕 =
𝒂 𝟎
𝟐
+
𝒏=𝟏
∞
𝒂 𝒏 𝐜𝐨𝐬 𝒏𝒘𝒕 + 𝒃 𝒏 𝐬𝐢𝐧 𝒏𝒘𝒕

5. 𝒚 = 𝒇 𝒙 = 𝒙 − 𝝅
𝑎0 =
2
2𝜋
0
2𝜋
𝑥 − 𝜋 𝑑𝑥
= 0
𝑏 𝑛 =
2
2𝜋
0
2𝜋
𝑥 − 𝜋 sin(
2𝜋𝑛𝑥
2𝜋
)𝑑𝑥
= −
2
𝑛
𝑎 𝑛 =
2
2𝜋
0
2𝜋
𝑥 − 𝜋 cos(
2𝜋𝑛𝑥
2𝜋
)𝑑𝑥
= 0
𝒇 𝒙 = −𝟐 𝒏=𝟏
∞ 𝟏
𝒏
𝐬𝐢𝐧 𝒏𝒙 = −𝟐 (𝒔𝒊𝒏 𝒙 +
𝟏
𝟐
𝒔𝒊𝒏 𝟐𝒙 +
𝟏
𝟑
𝒔𝒊𝒏𝟑𝒙 + ⋯ . +
𝟏
𝒏
𝒔𝒊𝒏 𝒏𝒙 ) = 𝑭(𝒙)
Fourier Series
Ch. VI : Non-Sinusoidal Wave
Example-1
Alternating Current Circuits by
RM Kerchner & GF Corcoran

6. Sawtooth Wave Transformed
Credit :Meyavuz

7. Square Wave
Credit :Mathloger

8. Straight Line with Sinusoids
Credit :Mathloger

9. Have a Look !
Credit :Faviloo

10. Fourier in Exponential Form
𝒇 𝒕 =
𝒂 𝟎
𝟐
+
𝒏=𝟏
∞
𝒂 𝒏 𝐜𝐨𝐬 𝒏𝒘𝒕 + 𝒃 𝒏 𝐬𝐢𝐧 𝒏𝒘𝒕
sin 𝑛𝑤𝑡 =
1
2𝑖
(𝑒 𝑖𝑛𝑤𝑡 − 𝑒−𝑖𝑛𝑤𝑡)
cos 𝑛𝑤𝑡 =
1
2
(𝑒 𝑖𝑛𝑤𝑡 + 𝑒−𝑖𝑛𝑤𝑡)
𝒇 𝝎 =
𝟏
𝟐𝝅 −∝
+∝
𝒇(𝒕)𝒆−𝒊𝝎𝒕 𝒅𝒕
𝑓 𝑡 =
𝑎0
2
+
𝑛=1
∞
(
𝑎 𝑛 − 𝑖𝑏 𝑛
2
𝑒 𝑖𝑛𝜔𝑡 +
𝑎 𝑛 + 𝑖𝑏 𝑛
2
𝑒−𝑖𝑛𝜔𝑡)
𝑓 𝑡 =
𝑛=−∝
∝
𝑎 𝑛 𝑒 𝑖𝑛𝜔𝑡

11. Who cares about Fourier , huh ?
Waves can be expressed as a sum of sines and cosines.

12. Signal Breakdown
Obtained Signal
Component-3
Component-2
Component-4
Component-5
Component-1

13. Who cares about Fourier , huh ?
Shifting from
Time
Domain
to
Frequenc
y Domain

14. Who cares about Fourier , huh ?
Shifting from
Time
Domain
to
Frequenc
y Domain

15. Frequency Domain
Credit: RF Wireless World
Credit: Pablo Puente Guillen

16. Applications
Image Audio
Filter
Reconstruction
Compression
Compression
Noise Exclusion
Sound Synthesize

17. References & Resources
https://www.quora.com/Why-are-Fourier-series-
important-Are-there-any-real-life-
applications-of-Fourier-series
https://www.quora.com/What-are-the-applications
-of-fourier-series-in-electrical-engineering
https://en.wikipedia.org/wiki/Fourier_series
https://www.youtube.com/watch?v=qS4H6PEcCCA
Alternating Current Circuit by
RM Kerchner & GF Corcoran
Analysis of Linear Systems by D K Cheng
https://www.youtube.com/watch?v=Mf8z3nm7prQ
https://www.youtube.com/watch?v=wNwJOBmqBsc
Special thanks to
Aninda Nandy

18. T hank You
Mohammad Iftekher Ebne Jalal (Iftu)
ID : 1802119
Department of Electrical & Electronic Engineering,
Chittagong University of Engineering & Technology
I only intended to make you feel excited about Fourier Series.
Feel free to learn more on this.