Discrete Transforms from Udemy

This course takes a detailed mathematical and also an intuitive graphical approach to learning and understanding the discrete transforms.

We start off with a recap of the continuous Fourier Transform and from this we derive the Discrete Time Fourier Transform (DTFT) then the Discrete Frequency Fourier Transform (DFFT) or as it is commonly known The Fourier Series. We then move onto deriving the Discrete Time and Frequency Transform which is commonly known as The Discrete Fourier Transform (DFT). Finally we look at the mathematics and implementation of an FFT algorithm.

You will not find another course where the mathematics of Discrete Transforms is described in as much detail but still keep a very high level of intuition. This has been achieved by the use of many 2D and 3D diagrams which are quite unique to this course. This course along with my course on Fourier and Laplace Transforms offers a unique insight to the world of mathematical transforms.

If you want a deep mathematical as well as an intuitive grasp of Discrete Transforms then this is the course for you. This is a continuation of the course Fourier and Laplace Transforms. You do not need to have taken the Fourier / Laplace course in order to do this but if may help.

What's inside

Learning objectives

Discrete time fourier transform
Discrete frequency fourier transform
Discrete fourier transform

Fft algorithm in excel
Mathematical rigour
Graphical intuition

Discrete time fourier transform
Discrete frequency fourier transform
Discrete fourier transform
Fft algorithm in excel
Mathematical rigour
Graphical intuition

Syllabus

Discrete Transforms

Introduction from me, Ross !

Course slides.

An intuitive introduction.

The Fourier Transform of an impulse train is another impulse train. This is not intuitively obvious. What we see in this video is an intuitive proof of something called the Poisson Summation Formulae.

Conversion from continuous to discrete time / frequency.

Continuous and Discrete Convolution.

Sampling Theorem and Aliasing

Discrete Time Fourier Transform

Discrete Frequency Fourier Transform

Discrete Time and Frequency Fourier Transform (DFT)

A full worked example of the Discrete Fourier Transform.

Derivation of radix 2 decimation in time fast Fourier transform.

Radix 2 FFT Algorithm in Excel VBA

Discrete Time Laplace Transform

The Z transform

Well done in completing the course. Good luck and goodbye !!!

Good to know

Know what's good

, what to watch for

, and possible dealbreakers

Explores the mathematical underpinnings of discrete transforms, which are essential for signal processing and data analysis in various engineering applications

Covers the FFT algorithm and its implementation in Excel VBA, which provides a practical understanding of its application in real-world scenarios

Requires a strong foundation in mathematics, which may be a barrier for learners without prior experience in these areas, potentially limiting accessibility

Offers a detailed mathematical treatment of discrete transforms, which is valuable for those seeking a rigorous understanding of the underlying principles

Employs 2D and 3D diagrams to enhance intuition, which can be beneficial for learners who benefit from visual aids in understanding complex mathematical concepts

Builds upon concepts from Fourier and Laplace Transforms, which may be helpful but not strictly necessary, suggesting some prior knowledge is beneficial

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 Discrete Transforms with these activities:

Review Continuous Fourier Transform

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Strengthen your understanding of the Continuous Fourier Transform, which is foundational for grasping the Discrete Transforms covered in this course.

Browse courses on Fourier Transform

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Review the definition and properties of the Continuous Fourier Transform.
Work through example problems involving common signals and their Fourier Transforms.
Compare and contrast the Continuous Fourier Transform with the Laplace Transform.

Review 'Signals and Systems' by Oppenheim and Willsky

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Deepen your understanding of the theoretical foundations of discrete transforms by studying a comprehensive signals and systems textbook.

View Signals and Systems (Second Edition) on Amazon

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Review the chapters on Fourier transforms, sampling, and discrete-time systems.
Work through the example problems and exercises in the book.
Relate the concepts in the book to the specific topics covered in the course.

Read 'Understanding Digital Signal Processing'

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Gain a broader understanding of digital signal processing principles and applications, complementing the mathematical focus of the course.

View Digital Signal Processing: A Practical Guide... on Amazon

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Read the chapters related to the Discrete Fourier Transform and Fast Fourier Transform.
Work through the examples provided in the book.
Relate the concepts in the book to the material covered in the course.

Three other activities

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Implement DFT and FFT Algorithms

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Solidify your understanding of the DFT and FFT algorithms by implementing them in a programming language of your choice.

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Implement the Discrete Fourier Transform (DFT) algorithm from scratch.
Implement the Radix-2 Fast Fourier Transform (FFT) algorithm.
Test your implementations with various input signals and compare the results.

Create a Visual Explanation of Aliasing

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Reinforce your understanding of the sampling theorem and aliasing by creating a visual explanation, such as a video or interactive simulation.

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Research and gather information about the sampling theorem and aliasing.
Create a visual representation of the aliasing effect using diagrams or animations.
Record a video explaining the concepts and demonstrating the visual representation.

Analyze Audio Signals Using DFT/FFT

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Apply your knowledge of DFT/FFT to analyze real-world audio signals and extract meaningful information.

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Record or obtain audio samples of various sounds or music.
Implement the DFT/FFT algorithm to analyze the frequency content of the audio signals.
Visualize the frequency spectra and identify dominant frequencies or patterns.
Write a report summarizing your findings and insights.

Career center

Learners who complete Discrete Transforms will develop knowledge and skills that may be useful to these careers:

Research Scientist

Research Scientists conduct research and development in various fields. A strong foundation in mathematical transforms, particularly Discrete Transforms, may be necessary in many areas of analysis and modeling. The course's focus on both mathematical rigor and intuitive understanding will provide a solid basis for conducting research involving signal processing or data analysis. Research scientists, especially those using transforms, may benefit.

See salaries and explore the career path for Research Scientist

Signal Processing Engineer

A Signal Processing Engineer uses mathematical techniques to analyze and manipulate signals, such as audio, images, and data. This course on Discrete Transforms helps build the foundation for understanding the mathematical underpinnings of these techniques. The course's in-depth exploration of Discrete Time Fourier Transform, Discrete Frequency Fourier Transform, and Discrete Fourier Transform directly applies to signal processing tasks. The emphasis on both mathematical rigor and graphical intuition, coupled with the coverage of the FFT algorithm, equips aspiring Signal Processing Engineers with the tools for success. The insights into discrete transforms and their practical implementation are directly applicable to real-world signal processing challenges. Someone wishing to become a Signal Processing Engineer may find this course beneficial.

See salaries and explore the career path for Signal Processing Engineer

Image Processing Specialist

An Image Processing Specialist works with algorithms to enhance, analyze, and manipulate digital images. This course on Discrete Transforms provides a detailed mathematical and intuitive understanding of concepts crucial for image processing. Topics such as the Discrete Time Fourier Transform, Discrete Frequency Fourier Transform, and Discrete Fourier Transform are fundamental to image analysis and manipulation. The course's unique approach, using 2D and 3D diagrams to explain complex concepts, will aid in visualizing and understanding image processing techniques. An Image Processing Specialist will benefit from the course's coverage of FFT algorithms and their implementation. Learners wishing to specialize in image analysis may find this course especially tailored to their needs.

See salaries and explore the career path for Image Processing Specialist

Audio Engineer

Audio Engineers record, manipulate, mix, and master sound. Sound is often represented by signals, and an understanding of the Discrete Fourier Transform is critical for analyzing and processing audio signals. This course on Discrete Transforms provides an in-depth understanding of the mathematical foundations of digital audio processing. The detailed exploration of transforms and the Fast Fourier Transform algorithm equip aspiring Audio Engineers with powerful tools for audio analysis and manipulation. The course's blend of mathematical rigor and intuitive graphical explanations greatly benefits those seeking a deeper understanding of how digital audio works. Someone who aspires to be successful as an Audio Engineer may find that this course will provide the knowledge they seek.

See salaries and explore the career path for Audio Engineer

Data Scientist

A Data Scientist analyzes large datasets to extract meaningful insights and build predictive models. Discrete Transforms, particularly the Discrete Fourier Transform, can be applied to time series data for feature extraction and pattern recognition. This course on Discrete Transforms provides the mathematical foundation for applying these techniques. The exploration of transforms and the FFT algorithm equips Data Scientists with tools for analyzing time series data and uncovering hidden patterns. The blend of mathematical detail and graphical intuition helps in understanding the underlying principles and applying them effectively. This may be a useful course for Data Scientists seeking a deeper understanding of time series analysis techniques.

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Machine Learning Engineer

A Machine Learning Engineer develops and implements machine learning models. Discrete Transforms can be used for feature extraction and data preprocessing in machine learning applications. This course on Discrete Transforms provides a solid foundation in the underlying mathematical principles. The detailed coverage of transforms and the FFT algorithm will equip Machine Learning Engineers with tools for processing data and improving model performance. The course's emphasis on both mathematical rigor and graphical intuition can help in understanding what the machine learning model is actually doing. If you wish to improve the performance of machine learning models, then this course may be helpful.

See salaries and explore the career path for Machine Learning Engineer

Robotics Engineer

Robotics Engineers design, build, and program robots. Robots rely on sensor data, often processed using signal processing techniques. This course on Discrete Transforms can help future robotics engineers understand and implement these techniques. The in-depth coverage of transforms, including the Discrete Time Fourier Transform and Discrete Fourier Transform, provides a basis for analyzing and filtering sensor data. The course's focus on both mathematical understanding and intuitive visualization may significantly benefit those working with robotic sensor systems. This can be particularly helpful for analyzing sensor data and controlling robotic movement. A robotics engineer may find this course useful.

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Computer Vision Engineer

Computer Vision Engineers develop algorithms that allow computers to 'see' and interpret images. Discrete Transforms, particularly the Discrete Fourier Transform, are often used in image analysis and feature extraction. This course on Discrete Transforms may help students to better understand the mathematical underpinnings of these algorithms. The detailed exploration of transforms and the fast Fourier Transform algorithm may provide insight into the inner workings of computer vision algorithms and how they may be improved. Aspiring Computer Vision Engineers who wants to delve into the math may find this course useful.

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Financial Analyst

Financial Analysts analyze financial data and provide investment recommendations. Time series analysis, often involving Discrete Transforms, is used to identify trends and patterns in financial markets. This course may give learners greater insight into time series analysis. Concepts covered like discrete time analysis may give new insights into financial data. A financial analyst may find this course helpful.

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Software Developer

Software Developers design, develop, and test software applications. This course may be useful for software developers working on applications involving signal processing or data analysis. This course on Discrete Transforms can provide foundational knowledge for understanding and implementing algorithms related to signal processing. Topics such as the Discrete Fourier Transform and fast Fourier Transform algorithms are particularly relevant for developing software that analyzes and manipulates data. A software developer working on signal processing programs may wish to take this course.

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Network Engineer

Network Engineers design, implement, and manage computer networks. Signal processing techniques, including Fourier analysis, are used to analyze network traffic and identify anomalies. This course covers the Discrete Fourier Transform, which may enhance a network engineer's ability to analyze discrete signals. The course may be helpful for network engineers analyzing network traffic. Aspiring network engineers may find this course helpful.

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Data Analyst

Data Analysts collect, clean, and analyze data to identify trends and patterns. Fourier analysis can be a valuable tool for analyzing time series data. This course on Discrete Transforms may improve a data analyst's ability to analyze time series data and identify cyclical patterns. Concepts covered in the course may provide new tools for analyzing time-series data. A data analyst may find benefit from this course.

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Quantitative Analyst

A Quantitative Analyst, often called a 'quant,' develops and implements mathematical models for financial markets. This often includes the study of financial time series. This course on Discrete Transforms provides the mathematical grounding for understanding the behavior of financial time series. This course may find applications in building models for predicting market trends. Learners who wish to have a deep mathematical understanding of financial modeling may wish to pursue this career path.

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Econometrician

Econometricians use statistical methods to analyze economic data and test economic theories. This could involve signal processing. This course on Discrete Transforms provides theoretical knowledge useful to econometrics. The course may be helpful to econometricians.

See salaries and explore the career path for Econometrician

Statistician

Statisticians collect, analyze, and interpret data to solve real-world problems. Discrete Transforms have some use in statistical analysis, particularly in time series analysis. This course may provide a different way to see time series data. A statistician may benefit from this course.

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Discrete Transforms

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