The Fourier Transform and its Applications

Course Lectures

• The Fourier Series

  Brad G. Osgood

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  This module introduces the Fourier series, emphasizing the significance of previous knowledge in Matlab. Students will explore periodic phenomena and understand the relationship between time and space. Key concepts include:

  • Analysis versus synthesis in Fourier series.
  • Periodicity in time and space.
  • The reciprocal relationship between frequency and wavelength.
• Periodicity; How Sine and Cosine Can be Used to Model More Complex Functions

  Brad G. Osgood

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  This module discusses how sine and cosine can model more complex functions. Students will learn about:

  • Periodicity and modeling signals with sinusoids.
  • Examples of periodizing signals.
  • Modeling a signal as a sum of modified sinusoids.
  • Complex exponential notation and symmetry in Fourier coefficients.
  • The generality of the Fourier series representation for periodic functions.
• Analyzing General Periodic Phenomena as a Sum of Simple Periodic Phenomena

  Brad G. Osgood

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  This module focuses on analyzing general periodic phenomena as sums of simple periodic phenomena. Key topics include:

  • Fourier coefficients and their generality.
  • Impacts of discontinuity on Fourier series.
  • Convergence issues and different cases of convergence.
  • Detailed examples of discontinuous signals and infinite sums.
• 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, emphasizing the significance of infinite sums and convergence. Topics covered include:

  • Integrability of functions and existence of Fourier coefficients.
  • Orthogonality of complex exponentials and the inner product.
  • Fourier coefficients as projections onto complex exponentials.
  • Applications of Fourier series to heat flow analysis.
• Continued Discussion of Fourier Series and the Heat Equation

  Brad G. Osgood

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  This module continues the discussion of Fourier series and introduces the heat equation. Key concepts include:

  • Transitioning from Fourier series to Fourier transforms.
  • Fourier series analysis and synthesis in relation to Fourier transforms.
  • Understanding the spectrum picture for Fourier series.
  • Challenges in finding Fourier transforms using Fourier coefficients.
• Correction to Heat Equation Discussion

  Brad G. Osgood

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  This module corrects previous discussions on the heat equation and sets up the Fourier transform derivation from Fourier series. Key elements include:

  • Results of the derivation: Fourier transform and inverse Fourier transform.
  • Definitions of Fourier transform and inversion.
  • Understanding that every signal has a spectrum that determines it.
  • Examples of rect and triangle functions.
• Review of Fourier Transform (and inverse) Definitions

  Brad G. Osgood

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  This module reviews the definitions of the Fourier transform and its inverse, focusing on key properties. Topics include:

  • Review of rect and triangle transforms.
  • Fourier transform of a Gaussian function.
  • The duality property of the Fourier transform.
  • Application examples demonstrating the duality property.
• Effect on Fourier Transform of Shifting a Signal

  Brad G. Osgood

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  This module explores the effects of shifting a signal on its Fourier transform. Key concepts include:

  • The resulting delay formula and the shift theorem.
  • The effect of scaling time signals and interpretation of the stretch theorem.
  • Convolution as it relates to Fourier transforms.
  • Multiplication of signals in frequency and its resulting convolution formula.
• Continuing Convolution: Review of the Formula

  Brad G. Osgood

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  This module continues the discussion of convolution, focusing on its formula and application in filtering. Key topics include:

  • The context in which convolution arises.
  • Examples demonstrating convolution in filtering.
  • General properties of convolution in the time domain.
  • The derivative theorem for Fourier transforms.
  • Application of the heat equation on an infinite rod.
• Central Limit Theorem and Convolution; Main Idea

  Brad G. Osgood

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  This module covers the central limit theorem (CLT) and its relation to convolution. Key content includes:

  • Normalization of the Gaussian distribution.
  • Pictorial demonstration of convolution in probability.
  • Setup for the central limit theorem with proof.
  • Other assumptions needed for establishing the CLT.
  • Using the Fourier transform to prove the central limit theorem.
• Cop Story

  Brad G. Osgood

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  This module introduces generalized functions and distributions. Important topics include:

  • Review of rapidly decreasing functions.
  • Definitions and applications of distributions like the delta function.
  • Operational view of delta functions as limits.
  • Pairing ordinary functions with distributions through integration.
• Setting Up the Fourier Transform of a Distribution

  Brad G. Osgood

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  This module sets up the Fourier transform of a distribution, discussing essential concepts such as:

  • Examples of delta as a distribution.
  • Distributions induced by functions.
  • The Fourier transform of a distribution and tempered distributions.
  • Definition of the Fourier transform through its operation on test functions.
  • Proof of the inverse Fourier transform and calculations using distributions.
• Derivative of a Distribution

  Brad G. Osgood

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  This module covers the derivative of a distribution, including practical examples such as:

  • Derivative of a unit step function and sgn(x).
  • Applications to the Fourier transform using the derivative theorem.
  • Caveats related to multiplying distributions and their implications.
  • Special cases involving the delta function and convolution.
• Application of the Fourier Transform: Diffraction: Setup

  Brad G. Osgood

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  This module introduces the application of the Fourier transform in diffraction. Key components include:

  • Representation of the electric field in diffraction contexts.
  • Utilizing Huygens' principle to discuss phase changes.
  • Application of the Fraunhofer approximation and aperture function.
  • Results and discussions on single and double slit diffraction.
• 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 relationship with the Fourier transform. Key topics include:

  • Setup for discussing crystallography, including historical context and concepts.
  • Fourier transform of the Shah function and the Poisson summation formula.
  • Proof of the Poisson summation formula and applications to crystals.
• 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 addresses the interpolation problem. Key elements include:

  • Setup for the interpolation problem and bandwidth assumptions.
  • Exact interpolation for bandlimited signals.
  • Periodizing signals through convolution with the Shah function.
  • Solutions to the interpolation problem.
• Review of Sampling and Interpolation Results

  Brad G. Osgood

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  This module discusses sampling and interpolation results, addressing key concepts such as:

  • Terminology relating to sampling rate and Nyquist rate.
  • Issues with interpolation formulas in practice.
  • Aliasing and its effects on interpolation.
  • Detailed examples demonstrating aliasing, including a cosine example.
• Aliasing Demonstration with Music

  Brad G. Osgood

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  This module provides an aliasing demonstration using music and transitions to discrete signals. It includes:

  • Creating a discrete signal from a continuous function.
  • Formulating the discrete Fourier transform (DFT) of the sampled version.
  • Summarizing results and moving from continuous to discrete variables.
  • Final results and the DFT's significance.
• Review: Definition of the DFT

  Brad G. Osgood

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  This module reviews the definition of the discrete Fourier transform (DFT). Important topics include:

  • Sample points and their implications for time and frequency spacing.
  • Utilization of complex exponentials in the DFT.
  • Understanding periodicity of inputs and outputs in the DFT.
  • Orthogonality of discrete complex exponential vectors.
• Review of Basic DFT Definitions

  Brad G. Osgood

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  This module continues the review of basic definitions of the discrete Fourier transform (DFT). Key points include:

  • Special cases, including the DFT value at 0.
  • Handling two special signals, such as the delta vector.
  • DFT of delta functions and implications of periodicity.
  • Matrix multiplication representation of the DFT.
• FFT Algorithm: Setup: DFT Matrix Notation

  Brad G. Osgood

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  This module sets up the FFT algorithm through DFT matrix notation. Key topics include:

  • Intuition behind FFT and factoring matrices.
  • Splitting order N into two orders N/2 for efficiency.
  • Iterative process and tracking powers of complex exponentials.
  • Summary of results as a combination of half-order DFTs.
• Linear Systems: Basic Definitions

  Brad G. Osgood

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  This module introduces basic definitions of linear systems. Key concepts discussed include:

  • Direct proportionality as an example of a linear system.
  • Examples of finite-dimensional linear systems.
  • Eigenvectors and eigenvalues.
  • Matrix multiplication as a form of linear transformation.
  • Integration against a kernel to generalize matrix multiplication.
• Review of Last Lecture: Discrete V. Continuous Linear Systems

  Brad G. Osgood

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  This module reviews the previous lecture on discrete versus continuous linear systems. Key elements include:

  • Cascading linear systems and derivation of the impulse response.
  • Understanding the Schwarz kernel theorem.
  • Impulse response for the Fourier transform example.
  • Time invariance and implications of convolution.
  • Summarizing the relationship between linear systems and convolution.
• Review of Last Lecture: LTI Systems and Convolution

  Brad G. Osgood

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  This module continues the review of last lecture, emphasizing LTI systems and convolution. Key concepts include:

  • Discussion of time-invariant discrete systems and their properties.
  • Fourier transform for LTI systems as complex exponentials.
  • Comparison of sine and cosine with complex exponentials as eigenfunctions.
  • Discrete complex exponentials as eigenvectors.
• Approaching the Higher Dimensional Fourier Transform

  Brad G. Osgood

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  This module introduces the higher-dimensional Fourier transform. Key topics include:

  • Notation for thinking in terms of vectors.
  • Definition and inverse Fourier transform in higher dimensions.
  • Reciprocal relationship between spatial and frequency domains.
  • Visualizing higher-dimensional complex exponentials, particularly in 2-D.
• Higher Dimensional Fourier Transforms- Review

  Brad G. Osgood

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  This module reviews higher-dimensional Fourier transforms, focusing on separable functions. Important topics include:

  • Fourier transforms of separable functions, such as 2-D rectangular functions.
  • Formula for the Fourier transform of separable functions.
  • Examples including 2-D Gaussian and radial functions.
  • Proving that the Fourier transform of a radial function remains radial.
  • Convolution concepts in higher dimensions.
• Shift Theorem in Higher Dimensions

  Brad G. Osgood

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  This module covers the shift theorem in higher dimensions. Key components include:

  • The shift theorem and its results in higher dimensions.
  • Derivation and results of the stretch theorem.
  • Special cases including scaling and rotation effects.
  • Understanding deltas in higher dimensions and their scaling properties.
• Shahs

  Brad G. Osgood

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  This module delves into Shah functions, lattices, and their relevance to crystallography. Key topics include:

  • Understanding 2-D Shah functions and their applications.
  • Crystals represented as lattices and their Fourier transform implications.
  • Application of Shah functions in medical imaging, such as 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. Key elements include:

  • Setting up the Radon transform and its coordinates.
  • Understanding delta functions along lines.
  • Integrating functions along lines and inverting the Radon transform.
  • Practical applications and implications of the inversion process.