FMP AudioLabs

Chapter 2: Fourier Analysis of Signals

The Fourier transform is perhaps the most fundamental tool in signal processing. Chapter 2 of [Müller, FMP, Springer 2015] approaches Fourier analysis from various perspectives and discusses their most important properties in the context of audio signal processing. In particular, the notion of a spectrogram, which yields a time–frequency representation of an audio signal, is introduced.

2.1 The Fourier Transform in a Nutshell
2.2 Signals and Signal Spaces
2.3 Fourier Transform
2.4 Discrete Fourier Transform (DFT)
2.5 Short-Time Fourier Transform (STFT)
2.6 Further Notes


Topic Relation to [Müller, FMP, Springer 2015] & Description HTML IPYNB
Complex Numbers [Section, Exercise 2.1]
Absolute value; angle; polar representation; conjugation; inverse; arrow plot; polar coordinate plot
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Exponential Function [Section, Exercise 2.15]
Power series; Euler's formula; root of unity
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Discrete Fourier Transform (DFT) [Section 2.1.3, Section 2.4]
Inner product; DFT matrix; fast Fourier transform (FFT); runtime experiments
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DFT: Phase [Section 2.3.2]
Exponential function; polar coordinates; complex Fourier coefficient; phase; optimality property
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Discrete Short-Time Fourier Transform (STFT) [Section 2.1.4, 2.5.3]
Time localization; spectrogram; physical interpretation; interactive plot
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STFT: Influence of Window Function [Section 2.5]
Window type; window size
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STFT: Padding [Section 2.1.4, Section 2.5.3]
Padding variants; edge phenomena; interactive plot
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STFT: Frequency Grid Density [Section 2.1.4]
DFT frequency grid; zero padding; increasing density
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STFT: Frequency Interpolation [Section 2.1.4, Section 3.1.1]
Linear interpolation; cubic interpolation; log-frequency STFT; cents
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STFT: Inverse [Section 2.4.2, Section]
DFT; inverse DFT; window function; overlap–add technique; artition of unity
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STFT: Conventions and Implementations [Section 2.1.4, 2.5.3]
Time axis convention; centered windowing; frequency conversion; implementation issues
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Digital Signals: Sampling [Section, Exercise 2.28]
Equidistant sampling; aliasing; signal reconstruction; sinc function; sampling theorem
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Digital Signals: Quantization [Section, Exercise 2.9]
Uniform quantization; quantization error
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Interference and Beating [Section, Exercise 2.19]
Constructive interference; destructive interference; chirp; sweep
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