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 .
A more classical example is the synthesis of pink noise from white noise, which requires a filter rolling off dB per octave. Pink noise is also called `` 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 . Many natural processes have been found to be well modeled by 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 .12
The ideal filter for synthesizing noise from white noise has transfer function
corresponding to 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 (hence the name `` filters''), thereby obtaining a distribution of poles and zeros yielding a frequency response proportional to for frequencies in some finite range . For audio, we ideally choose Hz and 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