Introduction to Fourier Analysis
Contains 36 lectures solely on Fourier analysis and the FFT. Time and frequency domains, representation of waveforms in terms of complex exponentials and sinusoids, convolution, impulse response and the frequency transfer function, modulation and demodulation are among the topics covered. The text is linked to a complete FFT system on the accompanying disk where almost all of the exercises can be either carried out or verified. End-of-chapter exercises have been carefully constructed to serve as a development and consolidation of concepts discussed in the text.
Why Read This Book
You will get a focused, lecture-style introduction to the mathematics and practical use of Fourier analysis and the FFT, learning how to move between time and frequency domains and apply those tools to real signal-processing problems. The book pairs clear theory with a working FFT system on the accompanying disk so you can verify and experiment with almost every exercise.
Who Will Benefit
Engineers or graduate students with some mathematical background who need a compact, application-oriented treatment of Fourier methods for DSP, audio/speech, radar, and communications tasks.
Level: Intermediate — Prerequisites: Single-variable calculus, basic complex numbers and exponentials, elementary differential equations and linear systems concepts; familiarity with basic signals and sampling is helpful. Some programming experience is useful to run the disked FFT examples.
Key Takeaways
- Understand the theory of Fourier series and Fourier transforms and how they represent signals as sums of sinusoids and complex exponentials.
- Apply FFT algorithms to compute spectra, perform fast convolution, and accelerate common signal-processing operations.
- Analyze linear time-invariant systems via impulse responses and frequency transfer functions and use these to predict signal behavior.
- Perform modulation and demodulation analyses and relate time-domain operations to frequency-domain effects.
- Use windowing and spectral-analysis techniques to interpret real signals and reduce common artifacts in power spectra.
- Validate analyses with hands-on FFT code provided on the accompanying disk and complete exercises that reinforce practical workflows.
Topics Covered
- 1. Introduction: Time and Frequency Representations
- 2. Fourier Series for Periodic Waveforms
- 3. Continuous-Time Fourier Transform (CTFT)
- 4. Properties of the Fourier Transform
- 5. Discrete-Time Fourier Transform (DTFT) and Sampling
- 6. The Sampling Theorem and Aliasing
- 7. Convolution, Correlation, and Linear Systems
- 8. Impulse Response and Frequency Transfer Functions
- 9. Modulation, Demodulation, and Spectral Shifts
- 10. The Fast Fourier Transform: Algorithms and Implementation
- 11. Practical Spectral Analysis and Windowing Methods
- 12. Applications: Signal Reconstruction, Filtering, and Analysis
- 13. Exercises Linked to the Accompanying FFT System
- Appendices: Mathematical Background and Computational Notes
Languages, Platforms & Tools
How It Compares
More concentrated on Fourier theory and hands-on FFT use than Oppenheim & Willsky's broader Signals and Systems; compared with Bracewell's The Fourier Transform and Its Applications, Morrison's text is more lecture-driven with accompanying reference FFT code.
