Abstract
The Fourier transform is perhaps the most important transform of all, and certainly it is the most used one in audio processing.
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- 1.
Strictly speaking, for a continuous frequency variable, ω, |X(ω)|2 is a power density function and not a power function.
- 2.
Simply put, the notation \(\mathcal {O}(N^2)\) means that the complexity is upper bounded by some (possibly unknown) constant times N 2.
- 3.
Carl Friedrich Gauss famously invented the tricks that make the FFT fast already in 1805 while doing data fitting, with pen and paper, of planetary observations but never published the results himself.
- 4.
If you want to be famous in signal processing, it appears you just have to come up with a clever window!
References
A.V. Oppenheim, R.W. Schafer, Discrete-Time Signal Processing, 1st edn. (Prentice-Hall, Englewood Cliffs, 1989)
M. Frigo, A fast Fourier transform compiler, in Proceedings ACM SIGPLAN Conference on Programming Language Design and Implementation (1999)
P. Stoica, R. Moses, Spectral Analysis of Signals (Pearson Prentice Hall, Upper Saddle River, 2005)
B.G. Quinn, E.J. Hannan, The Estimation and Tracking of Frequency. Cambridge Series in Statistical and Probabilistic Mathematics (Cambridge University Press, Cambridge, 2001)
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Christensen, M.G. (2019). The Fourier Transform. In: Introduction to Audio Processing. Springer, Cham. https://doi.org/10.1007/978-3-030-11781-8_7
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DOI: https://doi.org/10.1007/978-3-030-11781-8_7
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