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Importance in Audio Signal Processing

In audio signal processing, we often need a spectral shaping filter having a particular roll-off, usually specified in decibels (dB) per octave over the audio band. For example, it can be desirable to arbitrarily set the slope of the log-magnitude response versus log frequency between the two transition frequencies of a shelf filter [4].

A more classical example is the synthesis of pink noise from white noise, which requires a filter rolling off $ -3$ dB per octave. Pink noise is also called ``$ 1/f$ noise'', referring to the roll-off of the power spectral density of the noise, which requires a filter for white-noise having a magnitude response proportional to $ 1/\sqrt{f}$ . Many natural processes have been found to be well modeled by $ 1/f$ noise, such as amplitude fluctuations in classical music, sun spots, the distribution of galaxies, transistor flicker noise, flood levels of the river Nile, and more [5].12

The ideal filter for synthesizing $ 1/f$ noise from white noise has transfer function

$\displaystyle H_{-\frac{1}{2}}(s) = \frac{1}{\sqrt{s}} \;\;\stackrel{\longrightarrow}{{\scriptscriptstyle s=j\omega}}\;\; \frac{1}{\sqrt{j}\,\sqrt{\omega}} \eqsp e^{-j\frac{\pi}{4}}\, \omega^{-\frac{1}{2}},
$

corresponding to $ \alpha =-1/2$ in Eq.(3). Since the filter phase is arbitrary when filtering white noise, the filter-design problem can be formulated to match only the power frequency response $ \vert H_{-1/2}(j\omega)\vert^2 = 1/\omega$ (hence the name ``$ 1/f$ filters''), thereby obtaining a distribution of poles and zeros yielding a frequency response proportional to $ 1/\sqrt{\omega}$ for frequencies $ \omega = 2\pi f$ in some finite range $ f\in[f_{\mbox{min}},f_{\mbox{max}}]$ . For audio, we ideally choose $ f_{\mbox{min}}\approx 20$ Hz and $ f_{\mbox{max}}\approx 20$ kHz. Such designs can be found on the Web13and in the FAUST distribution.14 There are also interesting ``Voss-McCartney algorithms'' which are essentially sums of white-noise processes that are sampled-and-held at various rates.15


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``Spectral Tilt Filters'', by Julius O. Smith IIIand Harrison F. Smith, adapted from arXiv.org publication arXiv:1606.06154 [cs.CE].
Copyright © 2021-06-18 by Julius O. Smith IIIand Harrison F. Smith
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University
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