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Fractional Derivatives and Integrals

Since the $ N$ th successive time derivative/integral of $ x(t)$ Laplace-transforms to $ s^{\pm N} X(s)$ , it follows that the fractional derivative of order $ \alpha\in(0,1)$ , denoted by $ x^{(\alpha)}$ , should correspond to the Laplace transform $ s^\alpha
X(s)$ , while the fractional integral of order $ \alpha\in(0,1)$ corresponds to $ s^{-\alpha} X(s)$ . We thus let $ x^{(\alpha)}$ , $ \alpha\in\mathbb{R}$ , denote both fractional integrals ($ \alpha<0$ ) and derivatives ($ \alpha>0$ ), possibly including both an integer and fractional part. However, since integer $ \alpha=\pm N$ corresponds to a repeated ordinary derivative or integral, we will develop methods for $ \alpha\in(-1,1)$ and implement any integer part in the usual way. Expressing $ j$ as $ e^{j\pi/2}$ so that $ j^\alpha$ can be defined as $ e^{j\alpha\pi/2}$ , we obtain the corresponding Laplace and Fourier transforms for fractional integration or differentiation as

$\displaystyle \zbox{x^{(\alpha)}(t) \;\longleftrightarrow\;s^\alpha\, X(s) \;\;\stackrel{\longrightarrow}{{\scriptscriptstyle s=j\omega}}\;\; e^{j\alpha\frac{\pi}{2}}\, \omega^\alpha\, X(j\omega).} \protect$ (1)

Thus, a fractional integral or derivative of order $ \alpha$ corresponds to a phase shift by $ \alpha \pi/2$ and a spectral ``tilt'' by $ \omega^\alpha$ . For $ \alpha=1$ , we obtain the frequency-response $ j\omega$ of a differentiator, and for $ \alpha=-1$ , the integrator frequency-response $ -j/\omega$ is obtained, as required.

For integer $ \alpha=N\in\mathbb{Z}$ , $ N\ge 0$ , we have, from the convolution theorem for the unilateral Laplace transform applied to causal functions $ x(t)$ ,4

$\displaystyle s^{-N} X(s)$ $\displaystyle \longleftrightarrow$ $\displaystyle \frac{u(t)\,t^N}{(N-1)!} \ast
  $\displaystyle =$ $\displaystyle \frac{1}{(N-1)!} \int_{0}^t x(\tau)\,
(t-\tau)^{N-1} d\tau.\protect$ (2)

where `$ \ast$ ' denotes convolution, and $ u(t)$ denotes the Heaviside unit step function:

$\displaystyle u(t) \isdefs \left\{\begin{array}{ll}
1, & t\ge 0 \\ [5pt]
0, & t<0 \\
\end{array} \right.

This form was evidently developed originally as Cauchy's repeated integral formula.5

x^{(-N)}(t) &\isdef& \int_a^{t} \int_a^{\tau_1} \cdots \int_a^{\tau_{N-1}} x(\tau_N)
\,d\tau_N\,d\tau_{N-1}\,\cdots d\tau_1\\ [5pt]
&=& \frac{1}{(N-1)!}\int_a^{t} (t-\tau)^{N-1} x(\tau)\, d\tau

where $ a$ is any finite real number such that $ x(t)=0,\,\forall t<a$ .

The generalization of $ s^{-N}$ to $ s^\alpha$ for $ \alpha\in(-1,1)$ is quite natural. There is also no problem extending $ (t-\tau)^{N-1}$ to $ (t-\tau)^{-\alpha-1}$ in Eq.(2), and the lower limit of integration $ a$ can be extended (for all practical purposes) as far as needed to the left to encompass the support of $ x(t)$ . The last piece is generalizing $ (N-1)!$ to $ (-\alpha-1)!$ , which is provided by the gamma function6

$\displaystyle \Gamma(t) \isdefs \int_0^\infty \tau^{t-1} e^{-\tau} d\tau

which, for positive integers $ N$ , becomes $ (N-1)$ factorial, i.e.,

$\displaystyle \Gamma(N)=(N-1)!

for $ N=1,2,3,\ldots\,$ . We thus obtain the expression for fractional integrodifferentiation in the time domain as the convolution integral

$\displaystyle x^{(\alpha)}(t) \isdefs \frac{u(t)\,t^{-\alpha}}{\Gamma(-\alpha)} \ast x(t)
\eqsp \frac{1}{\Gamma(-\alpha)} \int_0^t \frac{x(\tau)}{(t-\tau)^{\alpha+1}}d\tau.

This is known as the Riemann-Liouville differintegral,7more commonly stated closer to the following form:

$\displaystyle x^{(\alpha)}(t) = \frac{1}{\Gamma(-\alpha)} \int_a^t \frac{x(\tau)}{(t-\tau)^{\alpha+1}}d\tau,

where $ a$ is an arbitrary fixed base point, and $ \alpha$ is any complex number with $\mbox{re\ensuremath{\left\{\alpha\right\}}}<0$ .8

The topic of fractional differentiation and integration falls within the well studied subject of fractional calculus9[1]. We will adopt the term ``differintegral'' from that literature.

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``Spectral Tilt Filters'', by Julius O. Smith IIIand Harrison F. Smith, adapted from 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