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Chapter 2: Fourier Analysis of Signals

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\n", "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. \n", "

\n", "\n", "

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

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TopicRelation to [Müller, FMP, Springer 2015] & DescriptionHTMLIPYNB
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]
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