Lecture: Fourier Analysis of Signals

After working through the material of this lecture, you should be able to answer the following questions:

• What is the main idea of Fourier analysis?
• What is a music signal? How can one mathematically model an analog or continuous-time (CT) signal? How a discrete-time (DT) signal?
• What is equidistant sampling? What is the relation between the sampling period and the sampling rate? (See Eq. 2.18.)
• What is the difference between the Fourier transform and the Fourier representation?
• What are the mathematical definitions of the continuous Fourier transform and Fourier representation? (See Eq. 2.12 and Eq. 2.17.)
• What is the relation between the real-valued and complex-valued version of the Fourier transform? What is the interpretation of the magnitude and phase of the complex-valued Fourier coefficients?
• What is the definition of the discrete Fourier transform (DFT)? (See Eq. 2.24.)
• What is the relation between the DFT and the continuous Fourier transform?
• How can one interpret the coefficients obtained by the DFT? (See Eq. 2.25.)
• What is the fast Fourier transform (FFT) good for?
• What is the motivation for introducing the short-time Fourier transform (STFT)? What is the main idea?
• What are important properties of a window function? What is the trade-off between time and frequency resolution?
• What is the definition of the discrete STFT? (See Eq. 2.26.)
• How can one interpret the coefficients obtained by the discrete STFT? (See Eq. 2.27 and Eq. 2.28.)
• What is a spectrogram? How can it be visualized?

Reading Assignments

Müller, FMP, Springer 2021
Chapter 2: Fourier Analysis of Signals

• Introduction of Chapter 2
• Section 2.1: The Fourier Transform in a Nutshell
• Section 2.2: Signals and Signal Spaces
  • Section 2.2.1: Analog Signals
  • Section 2.2.2: Discrete Signals
• Exercises
  • Exercise 2.2
  • Exercise 2.3
  • Exercise 2.4
  • Exercise 2.5

FMP Notebooks

• Part 2: Fourier Analysis of Signals
• Digital Signals: Sampling
• Discrete Fourier Transform (DFT)
• DFT: Phase
• Discrete Short-Time Fourier Transform (STFT)

Additional Material

• Interference and Beating

Slides and Notes

• Audio Features: Slides (PDF), Handouts (6 slides per page) (PDF)
• Relation between CT-based Fourier transform and DFT: Handwritten Notes (PDF)
• Relation between complex and real version of Fourier Transform: Handwritten Notes (PDF)
• Redundancy of negative frequencies for real-valued signals: Handwritten Notes (PDF)
• Fourier transform as solution of an optimization problem: Handwritten Notes (PDF)

Videos

One can find many videos at YouTube that explain the idea of Fourier analysis in different ways. Here are some example videos. Explore YouTube videos by yourself.

• Discrete Fourier Transform – Simple Step by Step (10:34)
  Fourier series; white noise; continuous Fourier transform; correlation; complex exponential; sine and cosine; sampling; discrete Fourier transform; Euler; magnitude; phase; Nyquist limit
• The Discrete Fourier Transform (DFT) (17:36)
  DFT; FFT; discrete signal; data vector; Fourier transform vector; inverse DFT; fundamental frequency (primitive $n$-th root of unity); DFT matrix; magnitude; phase

Question & Answer Session