An Overview of Fractional Order Signal Processing (FOSP) Techniques

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An Overview of Fractional Order Signal Processing
(FOSP) Techniques

• YangQuan Chen, Rongtao Sun, Anhong Zhou

Center for Self-Organizing and Intelligent
Systems (CSOIS), Department of Electrical
Computer Engineering, Utah State University
Phase Dynamics, Inc. Richardson,TX75081 Depart
ment of Biological and Irrigation Engineering,
Utah State University 3RD Int. Symposium on
Fractional Derivatives and Their Applications
(FDTA07) ASME DETC/CIE 2007, Las Vegas, NV, USA.
Sept. 4-7, 2007
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Outline

• Fractional derivative and integral
• Fractional linear system
• Autoregressive fractional integral moving average
• 1/f noise
• Hurst parameter estimation
• Fractional Fourier Transform
• Fractional Cosine, Sine and Hartley transform
• Fractals
• Fractional Splines
• Fractional Lower Order Moments (FLOM) and
  Fractional Lower Order Statistics (FLOS)

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Introduction

• The first reference to this area appeared during
  1695 in a letter from Leibnitz to LHospital.
• Only in the last three decades the application of
  FOSP deserved attention, motivated by the works
  of Mandelbrot on Fractals.
• Recently, many more fractional order signal
  processing techniques has appeared, such as
  fractional Brownian motion, fractional linear
  systems, ARFIMA (FIARMA) model, 1/f noise,
  Hurst parameter estimation, fractional Fourier
  transform, fractional linear transform,
  fractional splines and wavelets.
• It is very necessary to review them, and find out
  their relationships.

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Fractional derivative and integral

• Fractional derivative and integral, also called
  Fractional calculus are the basic idea of FOSP.
• The first notation of was
  introduced by Leibniz. In 1695, Leibniz himself
  raised a question of generalizing it to
  fractional order.
• In last 300 years the developments culminated in
  two calculi which are based on the work of
  Riemann and Liouville (RL) and Grunwald and
  Letnikov (GL).

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Some related functions

• Gamma function
• Beta function
• Mittag-Leffler function

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Operator

• A generalization of fractional derivative and
  integral operator

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Grunwald-Letnikov definition

• From integer order exponent
• where
• The GL definition is then given as
• where

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Riemann-Liouville definition

• Similarly, the common formulation for the
  fractional integral can be derived directly form
  the repeated integration of a function
• Then Riemann-Liouville fractional integral can be
  written as
• The RL differintegral is thus defined as

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Properties

• If f(t) is an analytic function of t, the
  derivative is an analytic function
  of t and .
• The operator gives the same result as the
  usual integer order n.
• The operator of order is the
  identity operator.
• Linearity
• The additive index law
• Differintegration of the product of two
  functions
  
• 
  

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Laplace transform of

• From the GL definition
• From the RL definition

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Fractional Linear System

• Consider fractional linear time-invariant (FLTI)
  systems described by a differential equation with
  the general format
• According to the Laplace transform of , the
  transfer function can be obtained as

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Impulse response

• Start from the simple transfer function
• its impulse response is
• Proceed to a further step

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• For transfer function like
• we can perform the inversion by the following
  steps
• 1,Transform from H(s) into H(z), by substitution
  of for z.
• 2,The denominator polynomial in H(z) is the
  indicial polynomial. Perform the expansion of
  H(z) in partial fractions.
• 3,Substitute back for z, to obtain the
  partial fractions in the form
• 4,Invert each partial fraction.
• 5,Add the different partial impulse responses.

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Autoregressive fractional integral moving average
(ARFIMA)

• Using a fractional differencing operator which
  defined as an infinite binomial series expansion
  in powers of the backward-shift operator, we can
  generalize ARMA model to ARFIMA model.
• where L is the lag operator, are error
  terms which are generally assumed to be sampled
  from a normal distribution with zero mean

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Properties of ARFIMA

• Fractionally differenced processes exhibit
  long-term memory (long rang dependence) or
  antipersistence (short term memory)
• An ARFIMA (p, d, q) process may be differenced a
  finite integral number until d lies in the
  interval (-½, ½), and will then be stationary and
  invertible. This range is the most useful set of
  d.
• 1, d-½. The ARFIMA (p, -½, q) process is
  stationary but not invertible.
• 2, -½ltdlt0. The ARFIMA (p, d, q) process has a
  short memory, and decay monotonically and
  hyperbolically to zero.

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• 3, d0. The ARFIMA (p, 0, q) process can be white
  noise.
• 4, 0ltdlt½. The ARFIMA (p, d, q) process is a
  stationary process with long memory, and is very
  useful in modelling long-range dependence (LRD).
  The autocorrelation of LRD time series decays
  slowly as a power law function.
• 5, d½. The spectral density of the process is
• as . Thus the ARFIMA (p, ½, q) process
  is a discrete-time 1/f noise.

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1/f noise

• Models of 1/f noise were developed by Bernamont
  in 1937
• where C is a constant, S(f) is the power
  spectral density.
• 1/f noise is a typical process that has long
  memory, also known as pink noise and flicker
  noise.
• It appears in widely different systems such as
  radioactive decay, chemical systems, biology,
  fluid dynamics, astronomy, electronic devices,
  optical systems, network traffic and economics

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1/f noise spectrum
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• We may define 1/f noise as the output of a
  fractional system as discussed before. The input
  could be white noise.
• Also, we can consider 1/f noise as the output of
  a fractional integrator. The system can be
  defined by the transfer function
• with impulse
  response
• Therefore, the autocorrelation function of the
  output is

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Fractional Gaussian Noise (FGN)

• FGN is a kind of 1/f noise.
• FGN can be seen as the unique Gaussian process
  that is the stationary increment of a
  self-similar process, called fractional Brownian
  motion (FBM).
• The FBM plays a fundamental role in modeling
  long-range dependence.
• The increments time series
• of the FBM process BH are called FGN.

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Relationship between fractional order dynamic
systems, long range dependence and power law
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Long-range dependence

• History The first model for long range
  dependence was introduced by Mandelbrot and Van
  Ness (1968)
• Value financial data
• communications networks data
• video traffic
• biocorrosion data

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Long-range dependence

• Consider a second order stationary time series
• Y Y (k) with mean zero. The time series Y
  is said to be long-range dependent if

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Hurst parameter

• The Hurst parameter H characterizes the degree of
  long-range dependence in stationary time series.
• A process is said to have long range dependence
  when
• Relationships
• 1,
• 2, d is the differencing parameter of ARFIMA

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Models with Hurst phenomenon

• Fractional Gaussian noise (FGN) models
  (Mandelbrot, 1965 Mandelbrot and Wallis, 1969a,
  b, c)
• Fast fractional Gaussian noise models
  (Mandelbrot, 1971)
• Broken line models (Ditlevsen, 1971 Mejia et
  al., 1972)
• ARFIMA/FIARMA models (Hosking, 1981, 1984)
• Symmetric moving average models based on a
  generalised autocovariance structure
  (Koutsoyiannis, 2000)

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Hurst parameter estimation methods

• R/S Analysis
• Aggregated Variance Method
• Dispersional Analysis Method
• Absolute Value Method
• Variance of Residuals Method
• Local Whittle Method
• Periodogram Method
• Wavelet-based
• Fractional Fourier Transform (FrFT) based

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Comparison of some important Hurst parameter
estimation methods, tested with 100 FGN of known
Hurst parameters from 0.01 to 1.00
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Fractional Fourier Transform (FrFT)

• Rotation concept of Fourier Transform
• which is the rotating angle

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Rotation concept of FrFT

• The FrFT rotates over an arbitrary angle
  , when a1 it correspond to Fourier
  transform.
• From ,we can
  define FrFT for the angle by
• Any function f can be expanded in terms of these
  eigenfunctions
  
• , with
• where Hn(x) is an Hermite polynomial

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Definition of FrFT

• By applying the operator , and use of
  Mehlers formula, as well as possible choice for
  the eigenfunctions of F we can get the definition
  for FrFT in linear integral form as

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Properties of the Kernel Function of FrFT

• If is the kernel of the FrFT,
  then

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Convolution of FrFT

• Concerns the convolution of two functions in the
  domain of the FrFT
• If
• Then its Fourier transform becomes
• Thus

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FrFT of a Delta Function
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FrFT of a Sine Function
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Fractional Linear Transform

• Generalize the FrFT method to linear transform.
  Given linear
• transform T, the procedure to find its fractional
  transform is
• Find the eigenfunctions and
  enginvalues of T
• The kernel functionis defined by
• The fractional transform T is then given by

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Fractional Hartley transform

• Hartley transform
• According to the linear fractional transform
  method, the fractional Hartley transform can be
  given by

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Relations between FrHT and FrFT
where is the fractional Hartley
transform and is the fractional
Fourier transform.
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Fractional Cosine and Sine transform

• Cosine and Sine transforms
• A.W. Lohmann, et al, in 1996, have derived the
  fractional Cosine/Sine transforms by taking the
  real/imaginary parts of the kernel of FrFT.

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Fractals

• The term fractal was coined in 1975 by
  Mandelbrot, from the Latin fractus, meaning
  "broken" or "fractured.
• A fractal is a geometric shape which
• is self-similar and
• has fractional (fractal) dimension.
• Fractals can be classified
• according to their self-similarity.
• 
  
  Sierpinsky Triangle

Y. Chen, \Fractional order signal processing in
biology/biomedical signal analysis," in
Fractional Order Calculus Day at Utah State
University, April 2005,http//mechatronics.ece.usu
.edu/foc/event/FOC Day_at_USU/.
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Fractal dimension estimation

• In fractal geometry, the fractal dimension is a
  statistical quantity that gives an indication of
  how completely a fractal appears to fill space,
  as one zooms down to finer and finer scales.
• Box counting dimension
• Information dimension
• Correlation dimension
• Rényi dimensions

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• Long-range dependent time series can also be
  described by a fractal dimension D which is
  related to the Hurst parameter through D
  2 - H . Here, the fractal dimension D can be
  interpreted as the number of dimensions the
  signal fills up.
• Besides, porous media model for the hydraulic
  system has fractal dimension. For example, the so
  called porous ball built by the French group
  CRONE has been used in car's hydraulic circuit.
• Also, the preparation of nanoparticles coated
  bio-electrodes is by polishing the surface with
  fractal shapes.
• In addition, the diffusion behavior of
  bioelectrochemical process will be fractional
  order dynamic, which is related with FD.

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Fractional Splines

• The fractional splines are an extension of the
  polynomial splines for all fractional degrees a gt
  -1.
• The fractional splines with one-sided power
  function can be written as
• where xk are the knots of the spline.

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Fractional B-Splines

• One constructs the corresponding fractional
  B-splines through a localization process similar
  to the classical one, replacing finite
  differences by fractional differences.
• are in L1 for all agt-1
• are in L2 for agt-1/2
• 
  Fractional
  B-Splines

http//bigwww.epfl.ch/index.html
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Properties of Fractional Splines

• If a is an integer, fractional splines are
  equivalent to the classical polynomial splines.
• 2) The fractional splines are a-Hölder continuous
  for a gt 0.
• 3) The fractional B-splines satisfy the
  convolution property and a generalized fractional
  differentiation rule. Besides, they decay at
  least like xa-2.
• 4) The fractional splines have a fractional order
  of approximation a 1.
• 5) Fractional spline wavelets essentially behave
  like fractional derivative operators.

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Fractional Lower Order Moments (FLOM) and
Fractional Lower Order Statistics (FLOS)

• The stable model can be used to characterize the
  non-Gaussian processes. including under water
  acoustic signals, low frequency atmospheric noise
  and many man-made noises.
• It has been proven that using stable model and
  fractional lower-order statistics (FLOS),
  additional benefits can be gained using this type
  of fractional order signal processing technique.
• Note that, for stable distribution the density
  function has a heavier tail than Gaussian
  distribution.
• It has been noticed that there is a natural link
  between LRD and heavy tail or thick/fat/heavy
  processes characterized by FLM/FLOS. A special
  case is the so-called SaS (symmetrical a-stable)
  process, which finds wide applications in
  engineering and non-engineering domains.

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Summary of FOSP Techniques

• Fractional derivative and integral
• Fractional linear system
• Autoregressive fractional integral moving average
• 1/f noise
• Hurst parameter estimation
• Fractional Fourier Transform
• Fractional Cosine, Sine and Hartley transform
• Fractals
• Fractional Splines
• Fractional Lower Order Moments (FLOM) and
  Fractional Lower Order Statistics (FLOS)

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Thank you!