The Fourier Series

This module introduces the fundamental concepts of the Fourier Series, emphasizing the significance of periodicity in both time and space. Students will explore:

• The relationship between analysis and synthesis in Fourier Series.
• How periodic phenomena can be expressed using sine and cosine functions.
• The reciprocal relationship between frequency and wavelength.

Prior knowledge in Matlab is recommended to help students engage with the practical aspects of the Fourier Series.

Course Lectures

• The Fourier Series

  Brad G. Osgood

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  This module introduces the fundamental concepts of the Fourier Series, emphasizing the significance of periodicity in both time and space. Students will explore:

  • The relationship between analysis and synthesis in Fourier Series.
  • How periodic phenomena can be expressed using sine and cosine functions.
  • The reciprocal relationship between frequency and wavelength.

  Prior knowledge in Matlab is recommended to help students engage with the practical aspects of the Fourier Series.

• Periodicity; How Sine and Cosine Can be Used to Model More Complex Functions

  Brad G. Osgood

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  This module covers how sine and cosine functions can effectively model complex periodic signals. Key topics include:

  1. Understanding periodicity and its applications in signal modeling.
  2. Examples of periodizing a signal using sinusoidal functions.
  3. The concept of "one period, many frequencies" in the context of modeling.
  4. Complex exponential notation and its significance in Fourier Series.
  5. The symmetry property of complex coefficients in Fourier Series.

  This module aims to deepen the students' understanding of the generality of Fourier Series representations for modeling various periodic functions.

• Analyzing General Periodic Phenomena as a Sum of Simple Periodic Phenomena

  Brad G. Osgood

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  This module analyzes general periodic phenomena by representing them as sums of simpler periodic components. Key areas of focus include:

  • Understanding Fourier coefficients and their role in signal representation.
  • Exploring the generality of the Fourier Series with examples of discontinuous signals.
  • Addressing convergence issues related to Fourier series and their implications.
  • Examining different types of convergence: continuous, smooth, jump discontinuity, and general cases.

  Students will gain insights into the effectiveness of Fourier Series in approximating complex periodic signals.

• Wrapping Up Fourier Series; Making Sense of Infinite Sums and Convergence

  Brad G. Osgood

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  This module wraps up the discussion on Fourier Series and emphasizes the importance of understanding infinite sums and convergence. Key topics include:

  1. The integrability of functions and its implications for Fourier coefficients.
  2. The orthogonality of complex exponentials and its significance in Fourier analysis.
  3. Understanding the inner product and its relationship with the norm of functions.
  4. Application of Fourier Series in practical scenarios, such as heat flow.

  Students will solidify their understanding of how Fourier Series can be used to solve real-world problems.

• Continued Discussion of Fourier Series and the Heat Equation

  Brad G. Osgood

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  This module continues the discussion of Fourier Series, focusing on the transition from Fourier Series to Fourier Transforms. Key elements covered include:

  1. The relation between Fourier Series analysis and synthesis and the Fourier Transform.
  2. Understanding the spectrum picture for Fourier Series with periodicity.
  3. Effects of changing the period T on Fourier coefficients.
  4. Complications encountered while deriving Fourier Transforms as T approaches infinity.

  The aim is to help students grasp the significance of Fourier Transforms in analyzing non-periodic phenomena.

• Correction to Heat Equation Discussion

  Brad G. Osgood

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  This module addresses corrections to the heat equation discussion and sets the stage for deriving the Fourier Transform from Fourier Series. Topics include:

  • Setup for Fourier Transform derivation from Fourier Series.
  • Results of the derivation, including the definitions of the Fourier Transform and Inverse Fourier Transform.
  • The concept that every signal has a spectrum that uniquely determines it.
  • Practical examples such as the rect function and triangle function.

  Students will learn how to derive and apply the Fourier Transform effectively.

• Review of Fourier Transform (and inverse) Definitions

  Brad G. Osgood

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  This module provides a comprehensive review of the definitions and properties of the Fourier Transform and its inverse. Key highlights include:

  1. Revisiting the notation of Fourier Transform and its inverse.
  2. Reviewing the Fourier transforms of basic functions, including the rect and triangle functions.
  3. Exploring the Fourier Transform of a Gaussian function through practical examples.
  4. Understanding the duality property of the Fourier Transform and its applications.

  Through this module, students will solidify their grasp of Fourier Transform definitions and their significance in analysis.

• Effect on Fourier Transform of Shifting a Signal

  Brad G. Osgood

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  This module examines the effects of shifting a signal on its Fourier Transform. Topics covered include:

  • The resulting delay formula known as the Shift Theorem.
  • Effects of scaling the time signal on the Fourier Transform.
  • Exploring the Stretch Theorem and its interpretation in the context of Fourier analysis.
  • Understanding convolution and its role in multiplying two signals in the frequency domain.

  Students will learn how signal modifications impact their Fourier representations.

• Continuing Convolution: Review of the Formula

  Brad G. Osgood

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  This module continues the discussion on convolution, focusing on its formula and practical implications. Key elements include:

  1. A review of the convolution formula and the context in which it arises.
  2. Examples of convolution in filtering and the underlying concepts.
  3. Terminology related to convolution and its interpretation in the time domain.
  4. General properties of convolution and its significance in signal processing.
  5. The Derivative Theorem for Fourier Transforms and its application to the heat equation.

  Students will deepen their understanding of convolution and its applications in engineering and science.

• Central Limit Theorem and Convolution; Main Idea

  Brad G. Osgood

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  This module introduces the Central Limit Theorem (CLT) and its relationship with convolution. Key discussions include:

  • An introduction to the Central Limit Theorem and its importance in probability.
  • Normalization of the Gaussian and its implications in convolution.
  • A pictorial demonstration illustrating the relationship between the Gaussian and convolution.
  • The setup for the CLT, including key results and proof of the distribution of sums.
  • Assumptions needed to properly set up the Central Limit Theorem.

  This module aims to connect Fourier Transforms with probability theory through the CLT.

• Cop Story

  Brad G. Osgood

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  This module, titled 'Cop Story', focuses on generalized functions or distributions, including the delta function. Key topics include:

  1. A review of rapidly decreasing functions and their significance.
  2. Understanding generalized functions and distributions, including the delta function.
  3. Viewing the delta function as a limit versus operationally defining it.
  4. Defining distributions and exploring how they interact with ordinary functions through integration.

  Students will learn how distributions extend the concept of functions and their applications in Fourier analysis.

• Setting Up the Fourier Transform of a Distribution

  Brad G. Osgood

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  This module sets up the Fourier Transform for distributions, with a focus on practical examples. Key points include:

  • Defining the Fourier Transform of a distribution and its significance.
  • Examining the delta function as a distribution and its implications.
  • Understanding distributions induced by functions and their Fourier Transforms.
  • Exploring the class of tempered distributions and their properties.
  • Calculating Fourier Transforms using the definition involving test functions.

  Students will become proficient in applying Fourier Transforms to various types of distributions.

• Derivative of a Distribution

  Brad G. Osgood

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  This module discusses the derivative of a distribution and its applications to the Fourier Transform. Key elements include:

  • Calculating derivatives of distributions, with examples like the unit step and sign functions.
  • Applying the derivative theorem to Fourier Transforms.
  • Addressing caveats when multiplying distributions and their implications.
  • Exploring special cases involving the delta function and its role in sampling.
  • Understanding convolution in the context of distributions, particularly with delta functions.

  This module aims to connect the concept of distributions with practical applications in Fourier analysis.

• Application of the Fourier Transform: Diffraction: Setup

  Brad G. Osgood

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  This module explores the application of the Fourier Transform in diffraction phenomena. Key topics include:

  1. Setting up the representation of the electric field in diffraction contexts.
  2. Applying Huyghens' principle to understand wave propagation.
  3. Discussing phase changes associated with different paths in diffraction.
  4. Utilizing the Fraunhofer approximation to simplify diffraction analysis.
  5. Studying aperture functions and their implications for single and double slit diffraction.

  Students will learn how Fourier Transforms can be effectively applied to solve problems in optics.

• More on Results From Last Lecture (Diffraction Patterns and the Fourier Transform)

  Brad G. Osgood

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  This module continues the discussion on diffraction patterns and their relation to the Fourier Transform. Key highlights include:

  • Setting up the discussion for crystallography and its historical context.
  • Examining one-dimensional diffraction patterns and their properties.
  • Exploring the Fourier Transform of the Shah function and its applications.
  • Understanding the Poisson Summation Formula and its proof.
  • Applying the Fourier Transform of the Shah function to crystallography.

  Students will gain insights into how Fourier analysis aids in understanding diffraction and crystal structures.

• Review of Main Properties of the Shah Function

  Brad G. Osgood

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  This module reviews the main properties of the Shah function and its implications for interpolation problems. Key discussions include:

  1. Understanding the bandwidth assumption in signal processing.
  2. Solving the interpolation problem for bandlimited signals.
  3. Periodizing signals through convolution with the Shah function.
  4. Finding solutions to the interpolation problem and their significance.

  Students will learn how the Shah function plays a critical role in sampling and reconstructing signals.

• Review of Sampling and Interpolation Results

  Brad G. Osgood

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  This module reviews sampling and interpolation results, addressing practical issues and concepts. Key components include:

  • Terminology related to sampling, including sampling rate and Nyquist rate.
  • Identifying issues with interpolation formulas in real-world applications.
  • Understanding aliasing and its implications for signal reconstruction.
  • Providing an example of aliasing, such as the cosine function, to illustrate concepts.

  Through this module, students will gain practical insights into the challenges of sampling and interpolation in signal processing.

• Aliasing Demonstration with Music

  Brad G. Osgood

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  This module demonstrates aliasing using music, bridging the concepts of continuous and discrete signals. Key topics include:

  1. Demonstrating aliasing through practical music examples.
  2. Transitioning to discrete signals and the Discrete Fourier Transform (DFT).
  3. Creating discrete signals from continuous functions.
  4. Understanding the discrete version of the Fourier Transform and its implications.
  5. Summarizing results and culminating in the final result: the DFT.

  Students will learn how aliasing affects signal representation and the transition to discrete analysis.

• Review: Definition of the DFT

  Brad G. Osgood

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  This module reviews the definition of the Discrete Fourier Transform (DFT) and its foundational concepts. Key areas covered include:

  • The relationship between sample points and spacing in time and frequency.
  • Understanding complex exponentials in the context of the discrete DFT.
  • Exploring periodicity of inputs and outputs in the DFT.
  • Examining orthogonality of the discrete complex exponential vector.
  • Discussing the consequences of orthogonality and the Inverse DFT.

  This module aims to solidify students' understanding of the DFT and its applications in signal processing.

• Review of Basic DFT Definitions

  Brad G. Osgood

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  This module reviews basic definitions of the Discrete Fourier Transform (DFT) and highlights special cases. Key discussions include:

  1. The value of the DFT at zero and its implications.
  2. Special signals such as delta vectors and their DFTs.
  3. Understanding complex exponentials within the context of the DFT.
  4. Representing the DFT as an N x N matrix multiplication.
  5. Exploring the periodicity of input/output signals and its consequences.

  Through this module, students will gain a comprehensive understanding of the DFT and its foundational concepts.

• FFT Algorithm: Setup: DFT Matrix Notation

  Brad G. Osgood

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  This module dives into the intricacies of the Fast Fourier Transform (FFT) algorithm, focusing on DFT matrix notation. Students will explore:

  • The fundamental intuition behind the FFT, including how to factor matrices.
  • The approach of splitting an order N into two orders of N/2 for iterative processing.
  • New notation to keep track of powers of complex exponentials.
  • Methods for plugging new notation into the Discrete Fourier Transform (DFT).
  • Results for even and odd indices, culminating in a summary that illustrates the DFT as a combination of two half-order DFTs.
• Linear Systems: Basic Definitions

  Brad G. Osgood

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  This module covers the fundamental concepts of linear systems, presenting definitions and examples that illustrate the core principles:

  • Direct proportionality as a foundational concept.
  • Special cases of linear systems that highlight unique characteristics.
  • An introduction to eigenvectors and eigenvalues, including the spectral theorem.
  • The process of finding a basis of eigenvectors.
  • Generalization of finite-dimensional linear systems through integration against a kernel, with the Fourier transform as a key example.
• Review of Last Lecture: Discrete V. Continuous Linear Systems

  Brad G. Osgood

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  This module serves as a review of the previous lecture, focusing on the distinctions between discrete and continuous linear systems. Key points include:

  • Cascading linear systems and their relationships.
  • Derivation of impulse responses for various systems.
  • Discussion of the Schwarz kernel theorem and its relevance.
  • Special cases illustrating convolution and time invariance.
  • Summary of two main ideas: linear systems as integration against a kernel and the time-invariance of convolution-based systems.
• Review of Last Lecture: LTI Systems and Convolution

  Brad G. Osgood

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  This module continues the review of previous concepts, focusing specifically on Linear Time-Invariant (LTI) systems and convolution. Topics covered include:

  • A discussion of time-invariance in discrete systems.
  • The application of the Fourier transform to LTI systems.
  • Complex exponentials as eigenfunctions of these systems.
  • Comparison between sine/cosine functions and complex exponentials.
  • Discrete versions of these principles from a matrix perspective.
• Approaching the Higher Dimensional Fourier Transform

  Brad G. Osgood

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  This module introduces the higher dimensional Fourier transform, focusing on its notation and definitions. The content includes:

  • Understanding the definition of the higher dimensional Fourier transform.
  • Inverse Fourier transform and its significance.
  • Reciprocal relationships between spatial and frequency domains.
  • Visualizing 2-D complex exponentials and their relationships.
  • Establishing results that help in visualizing higher dimensional transforms.
• Higher Dimensional Fourier Transforms- Review

  Brad G. Osgood

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  This module reviews higher dimensional Fourier transforms, focusing on separable functions and their properties. Key aspects include:

  • Fourier transforms of separable functions, exemplified by the 2-D rectangle function.
  • Deriving the formula for the Fourier transform of separable functions.
  • Analyzing the 2-D Gaussian function and its implications.
  • Understanding radial functions and their Fourier transform properties.
  • Exploring convolution in higher dimensions and its significance in signal processing.
• Shift Theorem in Higher Dimensions

  Brad G. Osgood

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  This module discusses the shift theorem in higher dimensions, outlining its significance and applications. It covers:

  • The primary results of the shift theorem and its applications.
  • Derivation of the stretch theorem and its implications.
  • Special cases including scaling and rotation transformations.
  • Understanding the reciprocal relationships in higher dimensions.
  • Basic properties of deltas in higher dimensions and their scaling impacts.
• Shahs

  Brad G. Osgood

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  This module focuses on the concept of Shah functions, lattices, and their applications in crystallography. Important topics include:

  • An introduction to the 2-D Shah function and its properties.
  • Understanding crystals as lattices and their mathematical representations.
  • The Fourier transform of the Shah function in oblique lattices.
  • Connections between Shah functions and crystallography, including notation and key concepts.
  • Applications of these concepts in medical imaging, particularly in tomography.
• Tomography and Inverting the Radon Transform

  Brad G. Osgood

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  This module discusses tomography and the process of inverting the Radon transform, providing insights into its practical applications. Key components include:

  • Setting up the problem and introducing relevant coordinates.
  • Understanding delta functions along a line and their significance.
  • Exploring the integral of functions along a line and its implications in imaging.
  • Steps to invert the Radon transform, emphasizing practical applications in tomography.
  • Tips for effectively filling out evaluations in the context of the course.