Introduction to Fourier Series and Transform

Sep 7, 2024

ENG 860: Special Topics in Digital Image Processing - Lecture 5

Main Topic

  • Introduction to Fourier Series and Fourier Transform

Readings

  • First four sections of Chapter 4
  • Important to derive equations line by line to avoid being overwhelmed

Lecture Outline

  1. Background on Fourier Series and Transform
  2. Preliminary Concepts
  3. Fourier Series of Periodic Functions (1D)
  4. Fourier Transform of Functions (1D)
  5. Convolution Representation in Fourier Domain
  6. Sampling and Fourier Transform of Sampled Functions

Key Concepts

Fourier Series

  • Represent periodic functions as sums of sine/cosine functions.
  • Applied to functions with periodicity.
  • Any periodic function can be expressed as a sum of sine and cosine functions of different frequencies, each multiplied by different coefficients.

Fourier Transform

  • Applied to non-periodic functions with finite area under the curve.
  • Converts functions to a frequency domain representation.
  • Becomes a weighted integral of sine and cosine functions.

Computational Efficiency

  • Fourier domain operations are often computationally cheaper than spatial domain operations.
  • Example: Convolution in Fourier domain can be exponentially less complex than in spatial domain.

Complex Numbers in Fourier Analysis

  • Complex Number: Consists of real (R) and imaginary (I) parts.
  • Imaginary Unit (j): Square root of -1.
  • Representation forms:
    • Vector
    • Polar: Magnitude and angle (using Euler's Equation)

Impulse Functions

  • Continuous Impulse Function: Infinite amplitude, zero duration, unit area.
  • Discrete Impulse Function: Defined for discrete values (e.g., delta of X).
  • Impulse trains and their Fourier series representations.

Convolution in Fourier Domain

  • Convolution in time domain = Multiplication in frequency domain.

Sampling of Functions

  • Multiply continuous function with impulse train for sampling.
  • Fourier transform of sampled signals is a periodic sequence.
    • Over-sampled, Critical, and Under-sampled cases.

Sampling Theory

  • A function can be perfectly reconstructed if sampling rate > 2 * maximum frequency (Nyquist rate).

Practical Implications

  • Aliasing: Occurs when sampling rate is too low, causing different signals to appear identical.
  • Anti-aliasing: Pre-sampling filtering to avoid aliasing by smoothing the high frequencies.

Upcoming Topics

  • Discrete Fourier Transform (DFT)
  • 2D Functions and their Fourier transforms
  • Image smoothing and filtering in frequency domain

Recommendations

  • Review textbook and derive equations to fully understand the concepts, especially if only attending lectures.