Logarithms of Negative and Imaginary Numbers

GUIDE: Mathematics of the Discrete Fourier Transform (DFT). Logarithms of Negative and Imaginary Numbers

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Logarithms of Negative and Imaginary Numbers

By Euler’s formula, $e^{j\, so that

\

from which it follows that for any $x<0$, $\.

Similarly, $e^{j\, so that

\

and for any imaginary number $z = jy$, $\, where $y$ is real.

Finally, from the polar representation $z=r e^{j\ forcomplex numbers,

\

where $r>0$ 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 $e^{j\ extracts its phase (times $j$).

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 $X(\ is defined as the group delay4.1of the signal $x(t)$:

\

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 $b$.

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