An Overview of Fractional Order Signal Processing (FOSP) Techniques

1. 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 #Department 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

2. 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)

3. Introduction • The first reference to this area appeared during 1695 in a letter from Leibnitz to L`Hospital. • 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.

4. 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).

5. Some related functions • Gamma function: • Beta function: • Mittag-Leffler function

6. Operator • A generalization of fractional derivative and integral operator

7. Grunwald-Letnikov definition • From integer order exponent: where The GL definition is then given as where

8. 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

9. 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:

10. Laplace transform of • From the GL definition: • From the RL definition:

11. 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:

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

13. 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.

14. 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:

15. 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, -½<d<0. The ARFIMA (p, d, q) process has a short memory, and decay monotonically and hyperbolically to zero.

16. 3, d=0. The ARFIMA (p, 0, q) process can be white noise. 4, 0<d<½. 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, d=½. The spectral density of the process is as . Thus the ARFIMA (p, ½, q) process is a discrete-time “1/f noise”.

17. 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

18. 1/f noise spectrum

19. 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

20. 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.

21. Relationship between fractional order dynamic systems, long range dependence and power law

22. 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

23. 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 dependentif

24. 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

25. 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)

26. 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

27. Comparison of some important Hurst parameter estimation methods, tested with 100 FGN of known Hurst parameters from 0.01 to 1.00

28. Fractional Fourier Transform (FrFT) • Rotation concept of Fourier Transform which is the rotating angle

29. Rotation concept of FrFT • The FrFT rotates over an arbitrary angle , when a=1it 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 •

30. Definition of FrFT • By applying the operator , and use of Mehler’s formula, as well as possible choice for the eigenfunctions of F we can get the definition for FrFT in linear integral form as:

31. Properties of the Kernel Function of FrFT • If is the kernel of the FrFT, then

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

33. FrFT of a Delta Function

34. FrFT of a Sine Function

35. 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

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

37. Relations between FrHT and FrFT where is the fractional Hartley transform and is the fractional Fourier transform.

38. 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.

39. 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@USU/.

40. 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

41. 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.

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

43. 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 a>-1 • are in L2 for a>-1/2 Fractional B-Splines* * http://bigwww.epfl.ch/index.html

44. 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 > 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.

45. 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.

46. 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)

47. Thank you!