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

Notebooks

Topic	Relation to [Müller, FMP, Springer 2015] & Description	HTML	IPYNB
Complex Numbers	[Section 2.2.3.1, Exercise 2.1] Absolute value; angle; polar representation; conjugation; inverse; arrow plot; polar coordinate plot	[html]	[ipynb]
Exponential Function	[Section 2.3.2.1, Exercise 2.15] Power series; Euler's formula; root of unity	[html]	[ipynb]
Discrete Fourier Transform (DFT)	[Section 2.1.3, Section 2.4] Inner product; DFT matrix; fast Fourier transform (FFT); runtime experiments	[html]	[ipynb]
DFT: Phase	[Section 2.3.2] Exponential function; polar coordinates; complex Fourier coefficient; phase; optimality property	[html]	[ipynb]
Discrete Short-Time Fourier Transform (STFT)	[Section 2.1.4, 2.5.3] Time localization; spectrogram; physical interpretation; interactive plot	[html]	[ipynb]
STFT: Influence of Window Function	[Section 2.5] Window type; window size	[html]	[ipynb]
STFT: Padding	[Section 2.1.4, Section 2.5.3] Padding variants; edge phenomena; interactive plot	[html]	[ipynb]
STFT: Frequency Grid Density	[Section 2.1.4] DFT frequency grid; zero padding; increasing density	[html]	[ipynb]
STFT: Frequency Interpolation	[Section 2.1.4, Section 3.1.1] Linear interpolation; cubic interpolation; log-frequency STFT; cents	[html]	[ipynb]
STFT: Inverse	[Section 2.4.2, Section 8.1.2.1] DFT; inverse DFT; window function; overlap–add technique; artition of unity	[html]	[ipynb]
STFT: Conventions and Implementations	[Section 2.1.4, 2.5.3] Time axis convention; centered windowing; frequency conversion; implementation issues	[html]	[ipynb]
Digital Signals: Sampling	[Section 2.2.2.1, Exercise 2.28] Equidistant sampling; aliasing; signal reconstruction; sinc function; sampling theorem	[html]	[ipynb]
Digital Signals: Quantization	[Section 2.2.2.2, Exercise 2.9] Uniform quantization; quantization error	[html]	[ipynb]
Interference and Beating	[Section 2.3.3.1, Exercise 2.19] Constructive interference; destructive interference; chirp; sweep	[html]	[ipynb]