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Mathematics of the DFT
    Logarithms and Decibels
       Logarithms
          Logarithms of Negative and Imaginary Numbers

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Logarithms of Negative and Imaginary Numbers

By Euler's identity, $ e^{j\pi} = -1$, so that

$\displaystyle \ln(-1) = j\pi $

from which it follows that for any $ x<0$, $ \ln(x) = j\pi + \ln(\vert x\vert)$.

Similarly, $ e^{j\pi/2} = j$, so that

$\displaystyle \ln(j) = j\frac{\pi}{2} $

and for any imaginary number $ z = jy$, $ \ln(z) = j\pi/2 + \ln(y)$, where $ y$ is real.

Finally, from the polar representation $ z=r e^{j\theta}$ for complex numbers,

$\displaystyle \ln(z) \isdef \ln(r e^{j\theta}) = \ln(r) + j\theta $

where $ r>0$ and $ \theta$ 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\theta }$ extracts its phase (times $ j$).

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Next: Decibels
written by Julius Orion Smith III
Julius Smith's background is in electrical engineering (BS Rice 1975, PhD Stanford 1983). He is presently Professor of Music and Associate Professor (by courtesy) of Electrical Engineering at Stanford's Center for Computer Research in Music and Acoustics (CCRMA), teaching courses and pursuing research related to signal processing applied to music and audio systems. See http://ccrma.stanford.edu/~jos/ for details.


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