NOTE: THIS DOCUMENT IS OBSOLETE, PLEASE CHECK THE NEW VERSION: "Mathematics of the Discrete Fourier Transform (DFT), with Audio Applications --- Second Edition", by Julius O. Smith III, W3K Publishing, 2007, ISBN 978-0-9745607-4-8. - Copyright © 2017-09-28 by Julius O. Smith III - Center for Computer Research in Music and Acoustics (CCRMA), Stanford University
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Logarithms of Negative and Imaginary Numbers
By Euler's formula, , so that
from which it follows that for any , .
Similarly, , so that
and for any imaginary number , , where is real.
Finally, from the polar representation forcomplex numbers,
where and are real. Thus, the log of the magnitude of a complex number behaves like the log of any positive real number, while the log of its phase term extracts its phase (times ).
As an example of the use of logarithms in signal processing, note that the negative imaginary part of the derivative of the log of aspectrum is defined as the group delay4.1of the signal :
Another usage is in Homomorphic Signal Processing [8, Chapter 10] in which the multiplicative formants in vocal spectra are converted to additive low-frequency variations in the spectrum (with the harmonics being the high-frequency variation in the spectrum). Thus, the lowpass-filtered log spectrum contains only the formants, and the complementarily highpass-filtered log spectrum contains only the fine structure associated with the pitch.Exercise: Work out the definition of logarithms using a complex base .