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Fourier and Laplace Transforms

Ross McGowan

This course is an introduction to the Fourier and Laplace Transforms. I introduce this subject both geometrically to give a good intuition using Matlab simulations and an online graphical calculator and also in a more formal mathematical way. The equations are broken down into small elements and an intuitive 3 dimensional graphical method is used in order to build intuition and finally understanding of the mathematics. This 3D intuitive learning approach is also backed up by some solid mathematics. This course has been created in order to give you a core understanding of the methods of mathematical transforms. Some sections are quite advanced and some real thinking is required. Your time and dedication will be rewarded with a deeper understanding of this beautiful subject.

Read more

This course is an introduction to the Fourier and Laplace Transforms. I introduce this subject both geometrically to give a good intuition using Matlab simulations and an online graphical calculator and also in a more formal mathematical way. The equations are broken down into small elements and an intuitive 3 dimensional graphical method is used in order to build intuition and finally understanding of the mathematics. This 3D intuitive learning approach is also backed up by some solid mathematics. This course has been created in order to give you a core understanding of the methods of mathematical transforms. Some sections are quite advanced and some real thinking is required. Your time and dedication will be rewarded with a deeper understanding of this beautiful subject.

Subtitles manually updated in English and converted to 78 different languages.

I can assure you that if you are willing to take the time to comprehend the mathematics then the effort will be rewarded ten fold.  For example , Fourier Transform (FT) , Discrete Time Fourier Transform (DTFT) , Discrete Frequency Fourier Transform (DFFT) , Discrete Time and Frequency Fourier Transform (DTFFT) , the Discrete Fourier Transform (DFT), the Fast Fourier Transform (FFT) , discrete versions of the  Laplace Transform (Z-transform). How are they all related ? By the end of this course you will understand where they all come from and how they are all related to one another. This is not a course about the practical implementation of transforms. This course is about understanding the core subject at a deeper mathematical level. This course is for those who want to truly understand this subject. If you want a new shiny car just to drive then this is not the course for you. If you want to open up the bonnet and take the engine apart to see how it works then this is definitely the course for you.

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What's inside

Learning objectives

  • Gain an intuitive understanding of the fourier and laplace transforms.
  • Understand the core mathematics of transforms.
  • Build intuition using a unique 3d graphical approach.
  • Matlab simulation files.
  • A deeper mathematical understanding.
  • 196 course slides in colour and black and white for printing.
  • Subtitles manually updated in english and converted to 78 different languages.

Syllabus

Derivation of the Fourier Series

Introductory video to say hello and let you put a face to the voice. Also see the original course hand written notes in black and white and colour in the resources section. Note these have been superseded by the course slides which are in the resources section of the next video.

Read more

This video shows the entire 196 course slides. They are available in the resource section for download as colour and black and white for printing.

In this video I derive the Taylor series and also take a look at the approximation of the sin function on a graphical calculator. (New course video)

From the Taylor polynomial I derive the complex exponential. (New course video)

Errata - At 1.42 I say a vector is defined as a directed line segment. This a very limited view of a vector. In fact a vector has a very precise mathematical definition and is any mathematical entity that adheres to a well defined set of axioms.

This is the first of three videos deriving the Fourier Series.(New course video).

This is the second of three videos deriving the Fourier Series. (New course video).

This is the final of three videos deriving the Fourier Series. (New course video).

In this lecture we derive the Fourier series approximation to a square wave and look at it in the graphical calculator.(New course video).

We introduce time to the Fourier series and the complex exponential.

A look at different ways of writing Fourier Series and why we choose this particular realisation.

This video shows the complete course book, in order to give you a flavour of what is to come. You can download this pdf in colour or black and white for easy printing. The course book has been superseded by the new course slides but I have left it in as you may find it useful.

Derivation of the Complex Fourier Series

In this video I derive the complex Fourier series.

In this video I work through an example of the complex Fourier series. (New course video).

Derivation of the Fourier Transform and its Properties

In this video I derive the Fourier transform. (New course video).

In this video we take a graphical intuitive look at both the forward and inverse Fourier Transforms. (New course video).

Another look at the Fourier Transform from a graphical perspective. It is a function of only 2 variables (w) and (t) so we can look at it in 3 dimensions. (New course video).

In this video I look at the symmetries of the Fourier transform.(New course video).

In this video I look at the unit impulse.(New course video).

In this video we look at the delayed impulse and also simulate it in matlab. You can simulate the delay using the matlab file below. (New course video).

In this video we explore the unit step function. (New course video).

In this video I work through an example of the Fourier transform. (New course video).

In this video I take a look at the linearity property of the Fourier transform. (New course video).

In this video I look at time and frequency scaling of the Fourier transform. (New course video).

In this video I look at the duality of the Fourier transform. (New course video).

In this video I look at time and frequency shifting of the Fourier transform. (New course video).

In this video I look at convolution. (New course video).

In this video I look at convolution in the time and frequency domain. (New course video).

In this video I look at differentiation in time and frequency. (New course video).

In this video I look at integration in time and frequency. (New course video).

In this video I derive the Fourier transform o f the sine and cosine functions. (New course video).

In this video I look at convolution with shifted impulses. (New course video).

In this video I review the properties of the Fourier transform. (New course video).

Derivation of the Laplace Transform and its Properties

In this video we start to look at the Laplace transform. (New course video).

In this video I derive the Laplace transform from a power series. (New course video).

In this video I use an second method of deriving the Laplace transform

In this video I derive the inverse Laplace transform. (New course video).

In this video I derive the Laplace transform of the unit step function. (New course video).

In this video I derive the Laplace transform of the real exponential. (New course video).

In this video I derive the Laplace transform of the complex exponential. (New course video).

In this video I derive the Laplace transform of the sinusoidal functions. (New course video).

In this video we shift both the sine and cosine functions in the frequency domain. (New course video).

In this video I look at the linearity of the Laplace transform. (New course video).

In this video I derive the Laplace transform of a derivative. (New course video).

In this video I derive the Laplace transform of an Integral. (New course video).

In this video I look at the initial value theorem. (New course video).

In this video I look at the final value theorem. (New course video).

In this video I look at the lower limit of integration of a Laplace transform. (New course video).

Goodbye and thank you for taking the time to complete this course.. (New course video).

Appendix Section

This is a proof of the dot product in component form. We covered this in lecture 7 Derivation of the Fourier Series Part 3

This is the proof of the dot product of 2 functions given the trigonometric functions as the orthonormal basis.

A derivation and example using integration by parts.

I am creating a follow up course covering the discrete transforms (DFT and Z transform). Here are a couple of videos from the course.

An intuitive graphical look at the various discrete transforms and how they are inter related.

Radix 2 Decimation in Time Fast Fourier Transform.

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Activities

Be better prepared before your course. Deepen your understanding during and after it. Supplement your coursework and achieve mastery of the topics covered in Fourier and Laplace Transforms with these activities:
Review Calculus Fundamentals
Strengthen your understanding of calculus concepts, which are essential for grasping the mathematical derivations in the course.
Browse courses on Calculus
Show steps
  • Review differentiation and integration techniques.
  • Practice solving problems involving limits and continuity.
  • Familiarize yourself with common calculus theorems.
Explore 'The Scientist and Engineer's Guide to Digital Signal Processing'
Gain a practical perspective on signal processing and transforms with this accessible guide.
View Digital Signal Processing: A Practical Guide... on Amazon
Show steps
  • Read the sections on Fourier analysis and applications.
  • Experiment with the examples provided in the book.
  • Relate the book's content to the course material.
Read 'Signals and Systems' by Oppenheim and Willsky
Supplement the course material with a classic text that provides a rigorous treatment of signals, systems, and transforms.
View Signals and Systems (Second Edition) on Amazon
Show steps
  • Read the chapters on Fourier and Laplace transforms.
  • Work through the example problems in the book.
  • Compare the book's approach to the course's approach.
Four other activities
Expand to see all activities and additional details
Show all seven activities
Solve Fourier Transform Problems
Reinforce your understanding of Fourier transforms by working through a variety of problems.
Show steps
  • Find a set of Fourier transform problems online.
  • Solve each problem, showing all steps.
  • Check your answers against the solutions.
Create a Visual Guide to Transform Properties
Solidify your understanding of transform properties by creating a visual guide that summarizes key concepts.
Show steps
  • Choose a format for your visual guide (e.g., infographic, presentation).
  • Summarize the key properties of Fourier and Laplace transforms.
  • Include examples to illustrate each property.
  • Share your guide with other students for feedback.
Implement a Signal Processing Algorithm
Apply your knowledge of Fourier and Laplace transforms by implementing a signal processing algorithm in MATLAB.
Show steps
  • Choose a signal processing algorithm that uses transforms.
  • Implement the algorithm in MATLAB.
  • Test the algorithm with different input signals.
  • Analyze the performance of the algorithm.
Create a Transform Cheat Sheet
Improve retention by compiling a cheat sheet of key formulas and properties related to Fourier and Laplace transforms.
Show steps
  • Gather all relevant formulas and properties from the course.
  • Organize the information in a clear and concise format.
  • Include examples to illustrate each formula or property.

Career center

Learners who complete Fourier and Laplace Transforms will develop knowledge and skills that may be useful to these careers:
Signal Processing Engineer
A signal processing engineer analyzes, designs, and develops signal processing systems. This often involves manipulating signals to remove noise, enhance features, or extract information. This course on Fourier and Laplace Transforms helps build the foundation for understanding the mathematical principles behind signal processing techniques. The course focuses on giving a core understanding of the mathematics of transforms which is critical when designing and implementing signal processing algorithms. The course's Matlab simulations are especially relevant, as they offer hands-on experience with signal processing concepts and their implementation.
See salaries and explore the career path for Signal Processing Engineer
Acoustic Engineer
An acoustic engineer analyzes and designs systems to control and manipulate sound. This often involves understanding how sound propagates, interacts with materials, and is perceived by humans. As an acoustic engineer, a course on Fourier and Laplace Transforms helps build a foundation for analyzing sound waves in the frequency domain. Understanding the Fourier Transform is particularly relevant, as it is used extensively in acoustic analysis and design. The course's emphasis on both intuitive understanding and solid mathematics offers a valuable skill set for acoustic engineers.
See salaries and explore the career path for Acoustic Engineer
Control Systems Engineer
A control systems engineer designs and implements systems that regulate the behavior of dynamic systems, found in robotics, aerospace, and manufacturing, among other fields. The role involves analyzing system stability and performance, and designing controllers to meet specific requirements. A deep comprehension of transforms, as emphasized in this course, helps build a crucial foundation for modeling and analyzing dynamic systems in the frequency domain. Understanding the Laplace transform, which this course covers, is particularly crucial, as it is a primary tool for analyzing system stability. This course's approach, which combines intuition with solid mathematics, is highly beneficial for a control systems engineer.
See salaries and explore the career path for Control Systems Engineer
Radar Systems Engineer
A radar systems engineer designs, develops, and tests radar systems for various applications, such as weather forecasting, air traffic control, and defense. Radar signal processing relies heavily on transforms for tasks such as signal detection, target tracking, and image formation. This course on Fourier and Laplace Transforms helps build a strong mathematical foundation for understanding radar signal processing techniques. The course focuses on giving a core understanding of the mathematics of transforms which is critical when designing and implementing signal processing algorithms. The course's Matlab simulations may be especially relevant, as they offer hands-on experience with signal processing concepts and their implementation.
See salaries and explore the career path for Radar Systems Engineer
Telecommunications Engineer
A telecommunications engineer designs and maintains communication systems, including wired and wireless networks. Transforms are fundamental to many aspects of telecommunications, such as signal modulation, channel equalization, and error correction. This course on Fourier and Laplace Transforms helps deepen one's understanding of the mathematical principles behind communication systems. The course focuses on giving a core understanding of the mathematics of transforms which is critical when designing and implementing signal processing algorithms. The course's Matlab simulations help illustrate the behavior of communication systems.
See salaries and explore the career path for Telecommunications Engineer
Research Scientist
A research scientist conducts experiments, analyzes data, and publishes findings in a specific scientific field. In many areas of science and engineering, transforms are essential tools for analyzing data and modeling physical phenomena. This course on Fourier and Laplace Transforms can be quite useful in helping build a strong mathematical understanding of transforms, which is crucial for many research applications. The course helps one understand the core mathematics of transforms, which can then be applied to various research problems. The Matlab simulations also enable a research scientist to explore and visualize transform behavior.
See salaries and explore the career path for Research Scientist
Robotics Engineer
A robotics engineer designs, builds, and programs robots for various applications. Control systems and signal processing are essential components of robotics, making transforms highly relevant to this field. This course on Fourier and Laplace Transforms helps build a solid foundation for understanding these concepts. The course's emphasis on the Laplace transform is particularly important, as it is used extensively in control systems design. The course's intuitive 3D graphical approach, coupled with the mathematics, offers invaluable insight into the behavior of dynamic systems.
See salaries and explore the career path for Robotics Engineer
Seismologist
A seismologist studies earthquakes and seismic waves to understand the Earth's structure and dynamics. The role involves analyzing seismic signals, locating earthquakes, and assessing seismic hazards. This course on Fourier and Laplace Transforms is useful for an aspiring seismologist, because they may develop deeper understanding of the mathematical principles behind seismic signal processing. The course focuses on giving a core understanding of the mathematics of transforms which is critical when designing and implementing signal processing algorithms for seismic data. The course's Matlab simulations are especially relevant, as they offer hands-on experience with signal processing concepts and their implementation.
See salaries and explore the career path for Seismologist
Image Processing Engineer
An image processing engineer works with digital images to enhance their quality, extract information, or prepare them for further analysis. This field relies heavily on transforms for tasks like image compression, filtering, and feature extraction. This course on Fourier and Laplace Transforms may be useful for gaining an understanding of the mathematical tools necessary for effective image manipulation. The geometric intuition provided, coupled with the underlying mathematical rigor, helps build a solid basis for understanding and applying image processing algorithms. The course's emphasis on understanding the core mathematics of transforms is highly valuable for image processing applications.
See salaries and explore the career path for Image Processing Engineer
Biomedical Engineer
A biomedical engineer applies engineering principles to solve problems in medicine and biology. Many areas of biomedical engineering, such as medical imaging and signal processing of physiological data, rely on transforms. As a biomedical engineer, a course on Fourier and Laplace Transforms helps further ones understanding of the mathematical tools necessary for these applications. The course focuses on giving a core understanding of the mathematics of transforms which is critical when designing and implementing signal processing algorithms for medical data. The course's Matlab simulations are especially relevant, as they offer hands-on experience with signal processing concepts and their implementation.
See salaries and explore the career path for Biomedical Engineer
Audio Engineer
An audio engineer records, mixes, and masters audio for music, film, and other media. This role demands a strong understanding of signal processing and frequency analysis. This course on Fourier and Laplace Transforms helps build a solid base for understanding audio signals and their manipulation. Understanding the Fourier Transform is particularly relevant, as it is used extensively in audio analysis and synthesis. The course's emphasis on understanding the relationships between different transforms, like the Discrete Fourier Transform and the Fast Fourier Transform, may be useful for an audio engineer working with digital audio.
See salaries and explore the career path for Audio Engineer
Machine Learning Engineer
A machine learning engineer develops and deploys machine learning models. While not always a core requirement, understanding transforms can be valuable in specific applications, such as feature extraction from time series data or signal processing tasks. This course on Fourier and Laplace Transforms may be useful for understanding the mathematical basis for certain machine learning techniques. The course's focus on the theoretical foundations of transforms can help a machine learning engineer understand the underlying principles and limitations of these methods. The course's exploration of Matlab simulations is helpful when implementing machine learning models that utilize transforms.
See salaries and explore the career path for Machine Learning Engineer
Data Scientist
A data scientist analyzes large datasets to extract meaningful insights and build predictive models. While data science encompasses a broad range of techniques, transforms can be valuable tools for feature engineering and signal analysis. This course on Fourier and Laplace Transforms may be useful in specialized areas of data science, such as time series analysis or signal processing applications. The focus of the course on the theoretical underpinnings of transforms can help a data scientist understand the limitations and assumptions of different techniques. The course's exploration of different transforms and their interrelationships strengthens analytical skills.
See salaries and explore the career path for Data Scientist
Software Engineer
A software engineer designs, develops, and tests software applications. While not always directly applicable, knowledge of transforms can be valuable in certain specialized areas of software development, such as signal processing libraries or scientific computing applications. This course on Fourier and Laplace Transforms may be useful should a software engineer need a deeper understanding of the mathematical principles behind these algorithms. The hands-on experience with Matlab simulations can be particularly beneficial for implementing transform-based algorithms in software.
See salaries and explore the career path for Software Engineer
Financial Analyst
A financial analyst analyzes financial data, provides investment recommendations, and manages financial risk. While not a direct application, transforms can be used in time series analysis and forecasting of financial data. This course on Fourier and Laplace Transforms may be useful if the analyst wishes to apply these advanced time series techniques. The course's focus on understanding the core mathematics of transforms can help in developing and interpreting financial models. The course's emphasis on the relationships between different transforms can strengthen analytical skills.
See salaries and explore the career path for Financial Analyst

Reading list

We've selected two books that we think will supplement your learning. Use these to develop background knowledge, enrich your coursework, and gain a deeper understanding of the topics covered in Fourier and Laplace Transforms.
Signals and Systems (Second Edition)
Save
Provides a comprehensive treatment of signals and systems, including detailed explanations of Fourier and Laplace transforms. It standard textbook used in many electrical engineering programs. Reading this book will provide a deeper understanding of the concepts covered in the course. It is particularly helpful for understanding the applications of transforms in signal processing.
Signals and Systems (Second Edition)
Paperback
Check
Signals and Systems (Second Edition)
Kindle Edition
Check
Cover image
Digital Signal Processing: A Practical Guide for...
Save
Offers a practical and intuitive approach to digital signal processing, with a strong emphasis on the Fourier transform. It valuable resource for understanding the applications of transforms in various fields. The book is particularly useful for those who prefer a less mathematical and more hands-on approach. It provides numerous examples and illustrations to aid understanding.
Digital Signal Processing: A Practical Guide for...
Paperback
Check
Digital Signal Processing: A Practical Guide for...
Kindle Edition
Check

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Effort
10 total hours
Via
Udemy
Instructor
Ross McGowan
Language
English
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