Fourier analysis comprises a set of fundamental concepts used in many scientific fields, including computer science, electrical engineering, physics, astronomy, geology and medical imaging. This book will deepen your understanding of Fourier transform and spectral analysis, making it easier to advance to more complex topics. With numerous examples, detailed explanations and plots, this book will help you understand Fourier analysis like never before.
- We start with the development of Fourier series using harmonic sinusoids to represent periodic signals in continuous and discrete-time domains.
- From here, we examine how and why the complex exponential is used to represent the Fourier series basis functions.
- Next, we describe the development of the CTFT and the DTFT for non-periodic signals.
- We show how the DTFT is modified to develop the Discrete Fourier Transform (DFT), the most used type of the Fourier transform.
- We look at the properties and limitations of the DFT and its algorithmic implementation, the Fast Fourier transform (FFT). We will also examine the use of Windows to reduce leakage due to the unavoidable truncation effect.
- We examine the applications of the DFT/FFT to random signals and the role of auto-correlation function in the development of the power spectrum. This topic is rarely covered in books and yet is the main reason why we do Fourier analysis.
- Lastly, we discuss methods of spectral power estimation. We'll focus on non-parametric power estimation of stationary random signals using the Periodogram and the Autopower. This is the most important part of this book.
