Fourier Transform Lecture Notes

Introduction to Fourier Transform

• An integral transform that converts a function into another function expressing frequency content.
• Outputs a complex-valued function of frequency, also known as the frequency domain representation.
• Analogous to breaking down a musical chord into individual pitches.

Properties and Principles

• Uncertainty Principle: Functions localized in time have Fourier transforms spread across frequency, and vice versa.
  • Critical case: Gaussian function.
• Mathematical Definition: Can be defined as an improper Riemann integral but often requires a more sophisticated integration theory.

Generalization

• Applicable to functions of several variables in Euclidean space and functions on groups.
• Includes discrete-time Fourier transform (DTFT), discrete Fourier transform (DFT), and Fourier series.
• The Fast Fourier Transform (FFT) is an algorithm for computing the DFT.

Definition and Equations

• Fourier Transform Equation:
  • Defined for complex-valued, Lebesgue integrable functions.
  • Inversion formula provided by the Fourier inversion theorem.

Functional Spaces

• Lebesgue Integrable Functions:
  • Fourier transform is well-defined for functions in this space.
  • Important concepts include Dirac delta function and Gaussian functions.
• Square Integrable Functions:
  • Fourier transforms extend to this space via continuous extensions.
  • Automorphism of the space, enabling analysis of wave equations.

Important Properties

• Linearity: Combines two functions’ transforms linearly.
• Shift Properties: Time and frequency shifting affects the transform.
• Scaling: Affects the width and height of the transformed function.
• Convolution Theorem: Convolution in time corresponds to multiplication in frequency domain.
  • Important in linear time-invariant system theory.

Applications

• Solving differential equations, particularly heat and wave equations.
• Signal processing, including spectral analysis and filtering.
• Quantum mechanics: Relationship between position and momentum wave functions.
• Engineering fields such as NMR and MRI.

Additional Topics

• Eigenfunctions: Fourier transforms have eigenfunctions related to Hermite polynomials.
• Inversion and Periodicity: Fourier transform is four-periodic, aiding in inversion via multiple transformations.
• Generalized Forms:
  • Extensions to groups, non-abelian groups, and compact groups.

Practical Considerations

• Numerical Computation: FFT for discrete transforms, numerical integration for continuous functions.
• Units and Conventions: Importance of defining units and conventions due to non-canonical scale.

Examples

• Fourier transform of a musical waveform to identify pitches.
• Transforming heat equations to solve using spectral methods.

Conclusion

Fourier transform is a powerful mathematical tool extensively used in various fields of science and engineering, providing a bridge between time and frequency domain analyses.