Fourier Analysis
The author has provided a shop window for some of the ideas, techniques and elegant results of Fourier analysis, and for their applications. These range from number theory, numerical analysis, control theory and statistics, to earth science, astronomy, and electrical engineering. Each application is placed in perspective with a short essay. The prerequisites are few (the reader with knowledge of second or third year undergraduate mathematics should have no difficulty following the text), and the style is lively and entertaining.
Why Read This Book
You will gain a clear, mathematically rigorous yet entertaining view of Fourier techniques and why they work — not just how to compute them. Körner frames classical Fourier series and transforms through elegant proofs and short essays that reveal connections to concrete problems in signal processing, numerical analysis, and the physical sciences, helping you build intuition that transfers to practical DSP tasks.
Who Will Benefit
Mathematically literate engineers and applied scientists who want a deeper conceptual and theoretical grounding in Fourier methods to improve their understanding of spectral analysis, filter behavior, and the foundations behind many DSP algorithms.
Level: Intermediate — Prerequisites: Single-variable calculus, basic complex numbers and linear algebra; comfort with undergraduate-level real analysis (second–third year undergraduate mathematics recommended).
Key Takeaways
- Understand the foundations of Fourier series and Fourier transforms, including convergence, summability, and when expansions are valid.
- Analyze spectra and frequency content of signals using rigorous tools that clarify the behavior of filters and convolution.
- Relate Fourier techniques to numerical analysis and approximation methods important for implementing DSP algorithms reliably.
- Apply classical results (Poisson summation, Parseval/Plancherel identities) to sampling, reconstruction, and spectral estimation problems.
- Recognize connections between harmonic analysis and diverse applications (number theory, astronomy, control theory) that broaden engineering perspective.
Topics Covered
- Introduction and historical motivation
- Fourier series: definitions and basic properties
- Convergence and summability of Fourier series
- Fourier transform on R: definitions and properties
- Plancherel and Parseval theorems; energy and spectra
- Convolution, approximate identities, and filters
- Poisson summation and sampling perspectives
- Applications to numerical analysis and approximation
- Selected applications in physics, number theory, and engineering
- Distributions and generalized Fourier methods (selected topics)
- Spectral analysis techniques and practical implications
- Appendices and further reading
How It Compares
Compared with Bracewell's The Fourier Transform and Its Applications, Körner is more mathematically oriented and broader in theory (less hands-on DSP recipes); compared with Oppenheim & Willsky's Signals and Systems, Körner emphasizes analysis and proofs over engineering-system design.
