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 |
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Complex Numbers | [Section 2.2.3.1, Exercise 2.1] Absolute value; angle; polar representation; conjugation; inverse; arrow plot; polar coordinate plot |
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Exponential Function | [Section 2.3.2.1, 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 8.1.2.1] 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 2.2.2.1, Exercise 2.28] Equidistant sampling; aliasing; signal reconstruction; sinc function; sampling theorem |
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Digital Signals: Quantization | [Section 2.2.2.2, Exercise 2.9] Uniform quantization; quantization error |
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Interference and Beating | [Section 2.3.3.1, Exercise 2.19] Constructive interference; destructive interference; chirp; sweep |
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