This document summarizes a lecture on Fourier series representation of periodic signals. It discusses the historical perspective of using sums of harmonically related sines and cosines to represent periodic signals dating back to Euler and Fourier. It also covers eigenvalues and eigenfunctions in linear time-invariant systems, representing periodic signals as a linear combination of complex exponentials, how to calculate Fourier series coefficients, and conditions for the convergence of Fourier series such as being absolutely integrable over one period and having a finite number of discontinuities.
3_Transmission_of_Electric_Energy.power point presentationJoseeMusabyimana
Transmission and Distribution
Advantages of interconnected systems
Better reliability
Smaller frequency deviations
Better overall system efficiency
Facilitates power exchange
Disadvantages of interconnected systems
Limitations to interconnection transport capacity
Power losses because of energy exchange
Unwanted parallel power flows
More complex system operation
3_Transmission_of_Electric_Energy.power point presentationJoseeMusabyimana
Transmission and Distribution
Advantages of interconnected systems
Better reliability
Smaller frequency deviations
Better overall system efficiency
Facilitates power exchange
Disadvantages of interconnected systems
Limitations to interconnection transport capacity
Power losses because of energy exchange
Unwanted parallel power flows
More complex system operation
EEG - Montages, Equipment and Basic PhysicsRahul Kumar
This presentation discusses the 10-20 system of electrode placement, with its modifications. Also discussed are the Equipment Specifications, basic Physics and sources of interference
EEG - Montages, Equipment and Basic PhysicsRahul Kumar
This presentation discusses the 10-20 system of electrode placement, with its modifications. Also discussed are the Equipment Specifications, basic Physics and sources of interference
Welcome to TechSoup New Member Orientation and Q&A (May 2024).pdfTechSoup
In this webinar you will learn how your organization can access TechSoup's wide variety of product discount and donation programs. From hardware to software, we'll give you a tour of the tools available to help your nonprofit with productivity, collaboration, financial management, donor tracking, security, and more.
Unit 8 - Information and Communication Technology (Paper I).pdfThiyagu K
This slides describes the basic concepts of ICT, basics of Email, Emerging Technology and Digital Initiatives in Education. This presentations aligns with the UGC Paper I syllabus.
Students, digital devices and success - Andreas Schleicher - 27 May 2024..pptxEduSkills OECD
Andreas Schleicher presents at the OECD webinar ‘Digital devices in schools: detrimental distraction or secret to success?’ on 27 May 2024. The presentation was based on findings from PISA 2022 results and the webinar helped launch the PISA in Focus ‘Managing screen time: How to protect and equip students against distraction’ https://www.oecd-ilibrary.org/education/managing-screen-time_7c225af4-en and the OECD Education Policy Perspective ‘Students, digital devices and success’ can be found here - https://oe.cd/il/5yV
Operation “Blue Star” is the only event in the history of Independent India where the state went into war with its own people. Even after about 40 years it is not clear if it was culmination of states anger over people of the region, a political game of power or start of dictatorial chapter in the democratic setup.
The people of Punjab felt alienated from main stream due to denial of their just demands during a long democratic struggle since independence. As it happen all over the word, it led to militant struggle with great loss of lives of military, police and civilian personnel. Killing of Indira Gandhi and massacre of innocent Sikhs in Delhi and other India cities was also associated with this movement.
Instructions for Submissions thorugh G- Classroom.pptxJheel Barad
This presentation provides a briefing on how to upload submissions and documents in Google Classroom. It was prepared as part of an orientation for new Sainik School in-service teacher trainees. As a training officer, my goal is to ensure that you are comfortable and proficient with this essential tool for managing assignments and fostering student engagement.
Read| The latest issue of The Challenger is here! We are thrilled to announce that our school paper has qualified for the NATIONAL SCHOOLS PRESS CONFERENCE (NSPC) 2024. Thank you for your unwavering support and trust. Dive into the stories that made us stand out!
Ethnobotany and Ethnopharmacology:
Ethnobotany in herbal drug evaluation,
Impact of Ethnobotany in traditional medicine,
New development in herbals,
Bio-prospecting tools for drug discovery,
Role of Ethnopharmacology in drug evaluation,
Reverse Pharmacology.
The Art Pastor's Guide to Sabbath | Steve ThomasonSteve Thomason
What is the purpose of the Sabbath Law in the Torah. It is interesting to compare how the context of the law shifts from Exodus to Deuteronomy. Who gets to rest, and why?
How to Create Map Views in the Odoo 17 ERPCeline George
The map views are useful for providing a geographical representation of data. They allow users to visualize and analyze the data in a more intuitive manner.
The Roman Empire A Historical Colossus.pdfkaushalkr1407
The Roman Empire, a vast and enduring power, stands as one of history's most remarkable civilizations, leaving an indelible imprint on the world. It emerged from the Roman Republic, transitioning into an imperial powerhouse under the leadership of Augustus Caesar in 27 BCE. This transformation marked the beginning of an era defined by unprecedented territorial expansion, architectural marvels, and profound cultural influence.
The empire's roots lie in the city of Rome, founded, according to legend, by Romulus in 753 BCE. Over centuries, Rome evolved from a small settlement to a formidable republic, characterized by a complex political system with elected officials and checks on power. However, internal strife, class conflicts, and military ambitions paved the way for the end of the Republic. Julius Caesar’s dictatorship and subsequent assassination in 44 BCE created a power vacuum, leading to a civil war. Octavian, later Augustus, emerged victorious, heralding the Roman Empire’s birth.
Under Augustus, the empire experienced the Pax Romana, a 200-year period of relative peace and stability. Augustus reformed the military, established efficient administrative systems, and initiated grand construction projects. The empire's borders expanded, encompassing territories from Britain to Egypt and from Spain to the Euphrates. Roman legions, renowned for their discipline and engineering prowess, secured and maintained these vast territories, building roads, fortifications, and cities that facilitated control and integration.
The Roman Empire’s society was hierarchical, with a rigid class system. At the top were the patricians, wealthy elites who held significant political power. Below them were the plebeians, free citizens with limited political influence, and the vast numbers of slaves who formed the backbone of the economy. The family unit was central, governed by the paterfamilias, the male head who held absolute authority.
Culturally, the Romans were eclectic, absorbing and adapting elements from the civilizations they encountered, particularly the Greeks. Roman art, literature, and philosophy reflected this synthesis, creating a rich cultural tapestry. Latin, the Roman language, became the lingua franca of the Western world, influencing numerous modern languages.
Roman architecture and engineering achievements were monumental. They perfected the arch, vault, and dome, constructing enduring structures like the Colosseum, Pantheon, and aqueducts. These engineering marvels not only showcased Roman ingenuity but also served practical purposes, from public entertainment to water supply.
1. EEE 321
Signals and Systems
Ankara University
Faculty of Engineering
Electrical and Electronics Engineering Department
2. Fourier Series Representation of
Periodic Signals
EEE321 Signals and Systems
Lecture 8
Ankara University
Electrical and Electronics Engineering Department, EEE321
Signals and Systems
3. Agenda
• Historical Perspective
• Eigenvalues and Eigenfunctions
• Fourier Series Representation of Continuous-Time Periodic Signals
• Convergence of the Fourier Series
Ankara University
Electrical and Electronics Engineering Department, EEE321
Signals and Systems
4. Historical Perspective
• Sum of harmonically related sines and cosines to represent periodic
signals
• Euler, 1748
• Bernoulli, 1753
• Lagrange, 1759
• Fourier, 1807
• Dirichlet, 1829
• Fourier, 1822
Ankara University
Electrical and Electronics Engineering Department, EEE321
Signals and Systems
5. Eigenvalues and Eigenfunctions
• LTI Systems
• 𝑒𝑠𝑡 −−→ 𝐻(𝑠)𝑒𝑠𝑡 : continuous time
• 𝑧𝑛 −−→ 𝐻 𝑧 𝑧𝑛 : discrete time
• 𝐻 𝑠 , 𝐻(𝑧) : eigenvalues
• 𝑒𝑠𝑡, 𝑧𝑛 : eigenfunctions
Ankara University
Electrical and Electronics Engineering Department, EEE321
Signals and Systems
6. Fourier Series Representation – Continuous
Time
• To represent a periodic signal via linear combination of harmonically
related complex exponentials, 𝑒𝑗𝜔𝑡
• 𝑥 𝑡 = 𝑘=−∞
∞
𝑎𝑘𝑒𝑗𝑘𝜔0𝑡
• 𝑘: integer
• 𝑎𝑘: Fourier series coefficients
Ankara University
Electrical and Electronics Engineering Department, EEE321
Signals and Systems
7. Fourier Series Coefficients – Continuous Time
• 𝑎𝑘 =
1
𝑇 −
𝑇
2
𝑇
2
𝑥 𝑡 𝑒−𝑗𝑘𝜔0𝑡𝑑𝑡
• 𝑎𝑘:spectral coefficients
• 𝑎0 =
1
𝑇 −
𝑇
2
𝑇
2
𝑥 𝑡 𝑑𝑡 : average value of 𝑥(𝑡)
Ankara University
Electrical and Electronics Engineering Department, EEE321
Signals and Systems
8. Convergence of the Fourier Series
• Dirichlet conditions
• The periodic signal must be absolutely integrable
• Number of maxima and minima are finite during any single period
• Finite number of discontinuities
Ankara University
Electrical and Electronics Engineering Department, EEE321
Signals and Systems
9. References
• Signals and Systems, 2nd Edition, Oppenheim, Willsky, Nawab
Ankara University
Electrical and Electronics Engineering Department, EEE321
Signals and Systems
