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?