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?

MÃ¼ller, 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

- 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)

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