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There are hundreds of textbooks that cover the complicated mathematics of the Fourier transform but no materials that explain its most basic principles. After many years working in signal and image processing, I have discovered that simple explanations are often overlooked. This course is targeted towards individuals who may have little experience in the area but have a desire to understand how things work.
There are hundreds of textbooks that cover the complicated mathematics of the Fourier transform but no materials that explain its most basic principles. After many years working in signal and image processing, I have discovered that simple explanations are often overlooked. This course is targeted towards individuals who may have little experience in the area but have a desire to understand how things work.
This course will provide an introduction to the Fourier transform. The first section is a review of the mathematics core to understanding Fourier integrals. We will review trigonometric functions, derivatives, integrals, and power series – both exponential and complex exponential. The course will not focus on complicated details and will instead concentrate on the basic skills required.
The second section will begin to introduce Integral Fourier transform. We will dive into the properties of Fourier transform as well as their application to engineering and communication challenges. Here, we will cover convolution, cross-correlation, modulation, demodulation, and more.
The goal of the class is to provide fundamental knowledge that can be applied to the analysis of linear systems, filtering, sampling, and some of the more advanced topics in signal processing. The course includes slides, two problem sets, and their solutions in an Adobe Acrobat file.
Discrete Fourier Transform and signal processing examples in Matlab are covered in a separate course "Discrete Fourier Transform and Spectral Analysis (MATLAB)"
Outline of course content and motivation behind creating this course
Review of basic trigonometric equations and definitions of sine and cosine functions
Review concepts of period, frequency and phase
Review of principles of calculating derivatives and integration
Review of power series - Taylor and Mc Laurin series equations
Explanation of exponential function and processes described by exponential
Explanation of complex plane, operations with complex numbers and Euler's equation
Explanation of how energy conservation law results in generation of sine wave signals
Definition and explanation of Integral Fourier transform
Calculate Fourier Transform of rectangular pulse and sinc function
Definition of delta-function and Fourier transform of delta-function. Spectrum of sine and cosine functions
Inverse Fourier Transform operator. Derivation of Fourier transform of Dirac Comb function.
Review of Fourier Transform Properties - Shift,even/odd functions, real function,derivative and integral
Fourier Transform Properties - Part 2. Convolution Theorem and Cross-Correlation
Fourier Transform Properties - Part 3. Product of Functions, Parseval's theorem and Duality
Summary review of Fourier Transform Properties and Pairs
Explanation of convolution operation
Cross - Correlation vs Convolution and signal detection in noise
Fourier transform of Periodic signals. Sampling theorem. Aliasing
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