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Filter Interpretation

As derived in obtaining Eq.(1) above, every fractional differintegral corresponds to a linear time-invariant filter having frequency-response

$\displaystyle \zbox{H_\alpha(j\omega) \isdefs e^{j\alpha\frac{\pi}{2}} \, \omega^\alpha.} \protect$ (3)

Since this frequency response is not a rational polynomial in $ j\omega$ for non-integer $ \alpha$ , there is no exact realization as a finite-order filter [4]. We must therefore settle for a finite-order approximation obtained using a truncated series expansion or filter design technique [2,3]. Many filter-design methods are available in MATLAB,10 and several basic design methods, such as invfreqz, are also available in the free, open-source, GNU Octave distribution.11As far as we know, all filter-based approximations to date have been carried out along these lines.


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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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