detc2007 34228fosp overview

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AN OVERVIEW OF FRACTIONAL ORDER SIGNAL PROCESSING (FOSP)TECHNIQUES

YangQuan Chen Center for Self-Organizing and Intelligent Systems (CSOIS)

Department of Electrical and Computer EngineeringUtah State University

4160 Old Main Hill, Logan, Utah 84322, USAEmail: [email protected]

Rongtao SunPhase Dynamics, Inc.

1251 Columbia Dr.Richardson, TX 75081 USA

[email protected]

Anhong ZhouDepartment of Biological and Irrigational Engineering,

Utah State University4105 Old Main Hill, Logan, Utah 84322-4105, USA

Email: [email protected]

ABSTRACTThis paper presents a brief overview of some existing frac-tional order signal processing (FOSP) techniques where the de-

velopments in the mathematical communities are introduced; re-

lationship between the fractional operator and long-range depen-

dence is demonstrated, and fundamental properties of each tech-

nique and some of its applications are summarized. Specifically,

we presented a tutorial on 1) fractional order linear systems; 2)

autoregressive fractional integrated moving average (ARFIMA);

3) 1/f noise; 4) Hurst parameter estimation; 5) fractional orderFourier transformation (FrFT); 6) fractional order linear trans-

forms (Hartley, Sine, Cosine); 7) fractal; 8) fractional order

splines; 9) fractional lower order moments (FLOM) and 10) frac-

tional delay filter. Whenever possible, we indicate the connec-

tions between these FOSP techniques.

Keywords: Fractional order calculus, fractional order signal pro-

cessing, long range dependence, ARFIMA, 1/f noise, Hurstparameter, fractional order Fourier transformation, fractional or-

Corresponding author. Center for Self-Organizing and Intelligent Systems(CSOIS), UMC 4160, College of Engineering, Utah State University, Logan,

Utah 84322-4160, USA. Tel. 1(435)797-0148; Fax: 1(435)797-3054. URL:

http://www.csois.usu.edu/

der linear transforms, fractals, fractional order splines, fractionallower order moments (FLOM), fractional lower order statistics

(FLOS), symmetrical -stable process (SS), fractional delay.

1 Introduction

In recent years, fractional order signal processing (FOSP) is

becoming an active research area, due to the demand on analysis

of long-range dependence/ self-similarity in time series, such as

financial data, communications networks data and biocorrosion

noise [1]. We will show that FOSP is essentially based on the

idea of fractional order calculus (FOC).

Fractional order calculus is a generalization of the differ-

ential and integral operators [2]. It is the root of the fractionasystems described by fractional order differential equations. The

simplest fractional order dynamic systems include the fractional

order integrators and fractional order differentiators. The autore

gressive fractional integrated moving average (ARFIMA) mode

is a typical fractional order system. It is a generalization of au

toregressive moving average (ARMA) model [3]. The traditiona

models can only capture short-range dependence; for example

Poisson processes, Markov processes, autoregressive (AR), mov

1 Copyright c 2007 by ASME

Proceedings of the ASME 2007 International Design Engineering Technical Conferences & Computers andInformation in Engineering Conference

IDETC/CIE 2007September 4-7, 2007, Las Vegas, Nevada, USA

DETC2007-34228


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ing average (MA), autoregressive moving average (ARMA) and

autoregressive integrated moving average (ARIMA) processes

[4]. For time series which possesses long-range dependence,

ARFIMA models give a good fit. In LRD (long range dependent)

processes, there is a strong coupling between values at different

times [5]. This indicates that the decay of the autocorrelationfunction is hyperbolic and decays slower than exponential de-

cay, and that the area under the function curve is infinite. We

can also say that their autocorrelation functions are power-law

distributed. 1/f noise is a signal that possesses long-range de-pendence. The power or square of some variable associated with

the random process, measured in a narrow bandwidth, is roughly

proportional to reciprocal frequency. In this paper, we will illus-

trate that the 1/f noise could be the output of a fractional ordersystem with input of white noise. The degree of LRD in time se-

ries is analyzed by estimating their Hurst parameter. Fractional

Fourier transform based estimator has been proved by this re-

search to have a better performance than the other existing esti-

mators for many time series including fractional Gaussian noise,biocorrosion processes data and Great Salt Lake water-surface-

elevation data. According to the concept of fractional Fourier

transform, some other fractional Linear transforms have been de-

veloped in the literature such as fractional Hartley transform [6],

fractional Sine transform [7] and fractional Cosine transform [7].

A brief explanation of fractals and fractional splines is pre-

sented. Finally, a concise introduction to fractional lower order

moments (FLOM) or fractional lower order statistics (FLOS) and

fractional delay is presented to make the FOSP techniques more

inclusive. The purpose of this paper is to present a brief overview

of various fractional order signal processing techniques.

2 Fractional Derivative and Integral

The historical developments culminated in two definitions

which are based on the work of Riemann and Liouville (RL) on

one hand and on the work of Grunwald and Letnikov (GL) on

the other hand [8]. Before explaining these two definitions, it

is necessary to introduce three related mathematical functions:

Gamma function, Beta function, and Mittag-Leffler function [2].

2.1 Three Special Functions

The gamma function extends the factorial function to non-

integer numbers. It is used in the definition of fractional calculus.

(z) =

0

euuz1du, for all z R (1)

Note that z could also be complex. Then, if the real part of the

complex number z is positive, the integral in (1) converges abso-

lutely. Using integration by parts,

(z + 1) = z(z). (2

The beta function has a form similar to the fractional integral/derivative of many functions. It also called the Euler integra

of the first kind defined by

B(p, q) :=

10

(1u)p1uq1du, p, q R+. (3

Its solution can be defined in terms of gamma functions,

B(p, q) =(p)(q)

(p + q)= B(q,p), p, q R+. (4

The Mittag-Leffler function is an important function that finds

widespread use in the world of fractional calculus [2]. It arise

out of the solution of non-integer order differential equations

The definition of the Mittag-Leffler function is

E(z) =

k=0

zk

(k+ 1), > 0. (5

The exponential function corresponds to the special case ofE(z)with = 1.

2.2 Definitions and Derivations

The formulation of fractional integrals and derivatives was a

natural outgrowth of integer order integrals and derivatives.

Let us start with

d

dtf(t) f(t) = lim

h0f(t) f(th)

h. (6

For the second order derivative, we know that

d2

dt2f(t) f(t) = lim

h0f(t) f(th)

h

= limh0

1

h

f(t) f(th)

h f(th) f(t2h)

h

= lim

h0f(t)2f(th) + f(t2h)

h2. (7

Similarly, for the third order derivative, we have

d3

dt3f(t) f(t) = lim

h0f(t)3f(th) + 3f(t2h) f(t3h)

h3. (8



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Therefore, in general, we get

dn

dtnf(t) fn(t) = lim

h01

hn

n

j=0

(1)j

n

j

f(t jh), (9)

in which

nj

= n

(n1)(n2)...(nj+1)j!

= n!j!(nj)! .

The Grunwald-Letnikov definition of a fractional derivative

of order is given as

Dt f(t) = limh0

1

h

j=0

(1)j

j

f(t jh) (10)

where

j

= !

j!(j)! =(+1)

(j+1)(j+1) , Dt is a FOC operator.

A generalization of fractional derivative and integral operator is

aDt .

aDt =

d/dt R() > 0,

1 R() = 0,ta (d)

R() < 0.(11)

Similarly, the common formulation for the fractional inte-

gral can be derived directly from a traditional expression of the

repeated integration of a function. This approach is commonly

referred to as the Riemann-Liouville approach. It is shown in

(12) that the formula, usually attributed to Cauchy, for evaluat-

ing the n-th integration of the function f(t).

...

ta

nfold

f()d =1

(n)

ta

f()

(t)1n d (12)

for n N, n > 0. So,

aIt f(t) aDt f(t) =

1

()t

a

f()

(t)+1 d (13)

for a, R, < 0; andaD

t f(t) =

1

(n)dn

dtn

ta

f()

(t)n+1 d, (n1 < < n), (14)

where a and t are the limits ofaDt f(t).

It should be remarked that there are numerous definitions

of fractional differentiointegral operators such as Caputo, Rietz,

and so on. Interested readers are referred to [911] for detailed

discussions. For recent development, refer to [1215].

2.3 PropertiesThe properties of aD

t f(t) are summarized in the following

[16], [2] and [8].

1. If f(t) is an analytic function oft, the derivative aDt f(t) is

an analytic function oft and .2. The operation aDt gives the same result as the usual differ-

entiation of integer order n.

3. The operator of order = 0 is the identity operator.4. Fractional operators are linear:

aDt {a f(t) + bh(t)} = aaDt f(t) + baDt h(t). (15

5. For fractional integrations of arbitrary order > 0, >0, (Re() > 0,Re() > 0) the additive index law (semigroupproperty) holds:

aDt aDt f(z) = aD(+)t f(z) (16

6. Differintegration of the product of two functions

D[x(t)y(t)] =

k=0

k

x(k)(t)y(k)(t) =

k=0

k

y(k)(t)x(k)(t) (17

7. Laplace transform ofaDt :

From the Grunwald-Letnikov definition:

0

est0Dt f(t)dt= sF(s). (18

From the Riemann-Liouville definition:

0

est0Dt f(t)dt = sF(s)

n1

k=0

sk0Dk1t f(t)|t=0,(19

for n 1 < n, where F(s) = L{f(t)} is the normaLaplace transformation.

3 Fractional Linear Systems3.1 Transfer Function

Fractional linear system may be described by fractional dif

ferential equations [17]. The theory is supported by definitions ofractional derivative and integral. The integrators, differentiator

and constant multipliers are the simplest of these systems. The

lumped parameter linear systems are cascade, parallel or feed-

back of those simple systems [18, 19]. Assuming that the coeffi

cients of the equation are constant, the corresponding system is

called a fractional linear time-invariant (FLTI) system.

Define fractional linear time-invariant (FLTI) systems de

scribed by a differential equation with the following genera



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form:

N

n=0

anDny(t) =

M

m=0

bmDmx(t) (20)

where D is the derivation operator and the is the order of differ-entiation. According to (18), applying Laplace transform to (20),

the following transfer function can be easily obtained, provided

that Re(s) > 0 or Re(s) < 0.

H(s) =Mm=0 bms

m

Nn=0 ansn

. (21)

3.2 Impulse Response

To obtain the impulse response from the transfer function,

we start by considering the simple case:

H(s) =1

s(22)

in which could be a fraction or a real number. According toRiemann-Liouville integral, the impulse response

h(t) = t1()

, > 0. (23)

Let us proceed with a further step by considering the following

transfer function:

H(s) =1

sa . (24)

For the inverse Laplace transform of (24), using the sum of the

first q terms of a geometric sequence with = 1/q we obtain:

H(s) =1

sa =

qj=1 a

j1s1j

saq . (25)

The inverse Laplace transform of (25) is

h(t) =1

a

q

j=1

ajE1j(t, aq) (26)

where

E1j(t, aq) = eaqt

0

1 j

k

t1+j+k

(j+ k)aq

k. (27

For the transfer function (21), we can expand H(s) in partiafractions like (24) when it is in commensurate order. Invert each

partial fraction and add the different partial impulse responses to

get the full impulse response.

For numerical simulation in Matlab of FLTI systems, a tool

similar to Matlab Control Systems Toolbox is provided and illus

trated in [20, 21].

4 Autoregressive Fractional Integrated Moving Average

A typical fractional system is autoregressive fractional in-tegrated moving average (ARFIMA) or fractional autoregressive

integrated moving average (FARIMA) [22]. It is a generalization

of autoregressive moving average (ARMA) model.

4.1 Moving Average Model (MA)

The notation MA(q) means a moving average model with qterms. An MA(q) model can be written as

yt = xt +1xt1 + ... +qxtq (28

for some coefficients 1,...,q. A moving average model is essentially a finite impulse response (FIR) filter.

4.2 Autoregressive Model (AR)

The notation AR(p) means an autoregressive model with pterms. An AR(p) model can be written as

yt = 1yt1 + ... +pytp +xt (29

for some coefficients 1,...,p. An autoregressive model is essentially an infinite impulse response (IIR) filter.

4.3 Autoregressive Moving Average Model (ARMA)

The notation ARMA(p, q) means a model with p autoregressive terms and q moving average terms. This model subsume

the AR and MA models,

yt = 1yt1 + ... +pytp +xt +1xt1 + ... +qxtq. (30



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4.4 Autoregressive Integrated Moving Average(ARIMA)

An ARIMA model ARIMA(p, d, q) is a generalization ofARMA model. The p, d and q are integers greater than or equal

to zero and refer to the order of the autoregressive, integrated and

moving average parts of the model respectively.According to (30), an ARMA(p, q) model can be written as

(1p

i1

iLi)Xt = (1 +

q

i=1

iLi)t, (31)

where L is the lag operator such that LXt = Xt1, Xt is a giventime series and t are error terms. The error terms t are assumedto be normally distributed with zero mean such as Gaussian white

noise.

Then, the ARIMA model is generalized by adding a differ-

encing parameter d to (31).

(1pi1iLi)(1L)dXt = (1 +qi=1iLi)t, (32)(1L)d = k=0

dk

(L)k

where d is a positive integer.

A well-known special case, ARIMA(0, 1, 0) model, is givenby:

Xt = Xt1 + (33)

which is simply a random walk when is a white noise.

4.5 Autoregressive Fractional Integrated Moving Av-erage (ARFIMA)

The autoregressive fractional integrated moving average

model is generalized by permitting the degree of differencing to

take fractional values. The fractional differencing operator is de-

fined as an infinite binomial series expansion in powers of the lag

operator L. Fractionally differenced processes may exhibit long-

term memory (long-range dependence) or antipersistence (short

term memory) [22].

The usefulness of this model has been proved by numer-

ous studies. For instance, Diebold and ReRudebush applied

ARFIMA to real GNP (Gross National Product) data [23]; Baillieet al. found long memory in time series of inflation [24]; Carto

and Rothamn used an ARFIMA(0, d, 1) model for annual bondyields [25]; Bertacca et al. applied an ARFIMA based analysis

in sea SAR (Synthetic Aperture Radar) imagery [26]; Liu et al.

used ARFIMA for network traffic modeling [27].

An ARFIMA (p, d, q) process may be differenced a finiteintegral number until d lies in the interval [1

2, 1

2], and will then

be stationary and invertible [22]. This range is the most useful

set ofd.

1. When d= 12

, the ARFIMA (p, 12

, q) process is stationarybut not invertible.

2. When 12

< d < 0, the ARFIMA (p, d, q) process has a

short memory, and decay monotonically and hyperbolicallyto zero.

3. When d = 0, the ARFIMA (p, 0, q) process can be whitenoise.

4. When 0 < d< 12

, the ARFIMA (p, d, q) process is a stationary process with long memory, and is very useful in mod-

elling long-range dependence (LRD). The autocorrelation o

a LRD time series decays slowly as a power law function.

5. When d= 12

, the spectral density of the process is

s() =(ei)(ei)(ei)(ei)

{(1 ei)(1ei)}d

11 ...q11p

2

(2sin 12)2d C

(34

as 0. Thus the ARFIMA (p, 12

, q) process is a discretetime 1/f noise.

5 1/f NoiseWe may define a fractional stochastic process as the output

of a fractional dynamic system [28, 29]. The input could be sim-

ply white noise. Among various fractional random signals, the

1/f noise is of special importance. The 1/f noise is a randomprocess defined in terms of the shape of its power spectral den-

sity S(f). The power or square of some variable associated withthe random process, measured in a narrow bandwidth, is roughly

proportional to reciprocal frequency:

S(f) =constant

|f| . (35

The typical spectrum of 1/f noise is shown in Fig. 1.Models of 1/f noise were developed by Bernamont [30] in

1937 for vacuum tubes and by Mcwhorter [31] in 1955 for semi-

conductors. In [32], Keshner referred several examples of 1/

noise. From his considerations, we may conclude that those sig-

nals seem to belong to one of two types: those with spectrum ofthe form 1/f for every f and the others with that form only for fabove a given value. This means that the former can be consid-

ered as the output of a fractional integrator, while the second may

be the output of a low-pass fractional system with a pole near the

origin. For the first case, the system can be defined by the transfe

function H(s) = 1/s and its impulse response h(t) = t1/()with 0 < < 1. Let the input be a continuous-time stationarywhite noise w(t) with variance 2. The output can be written in



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0 5000 100000

500

1000

1500

2000

2500

3000

3500Natural scale

Frequency

Spectrum

1/f noise

100

102

104

100

101

102

103

104

Loglog scale

Frequency

Spectrum(

db)

1/f noise

Figure 1. 1/f noise spectrum

convolution form as follows:

y(t) =1

()

t

w()(t)1d. (36)

The autocorrelation function of the output y(t) is then,

R() = 2||21

2(2) cos. (37)

There is an important property shown in (37). The autocor-relation of the 1/f noise decays slowly as a power law function,

which indicates the long-range dependence [33].

5.1 Fractional Gaussian Noise

Fractional Gaussian noise is a kind of 1/f noise. It is long-range dependent with power law behavior in all frequencies [34].

Alternatively, the FGN process can be defined as a process Xi, i =1, 2... satisfying the condition for any timescales kand l [34]

(Z(k)i k) d= (

k

l)H(Z

(l)j l) (38)

where Zi is denoting the aggregated process of Xi on the

timescale k, Zj is denoting the aggregated process of Xi on the

timescale l, is the mean value of Xi, the symbold= stands for

equality in distribution and H is a positive constant (0 < H < 1)known as the Hurst parameter. Hurst parameter will be discussed

in detail in the next section.

It is shown in [34] that, the autocovariance function r() of

FGN Xi is given by

r() =1

2[(+ 1)2H + (1)2H]2H, > 0 (39

Apart from small , this function is very well approximated by

r = H(2H1)2H2 (40

which shows that autocorrelation is a power function of lag .

Considering a continuous time process Y(t) with autocorrelation Cov(Y(t),Y(t+ )) = c2H2, c = H(2H 1), we canalso obtain (39) by making Y(t) discrete using time intervals oany length and taking Xi as the average of Y(t) in the interval [i, (i + 1)]. This enables an approximate calculation of thepower spectrum of the process as

s(k)() 4

0c2H2 cos(2)d c12H (41

which is a power law of the frequency .

Since FGN is a very fundamental signal, many FGN generat

ing methods have been proposed. For example, FGN can be used

as a benchmark signal to validate the accuracy of the proposed

Hurst parameter estimators [35]. The relationship between frac-

tional order dynamic systems, long range dependence and power

law has been clearly illustrated in Fig. 2 [13].

.

-

: . . . ~ .

.

-

Fractional Calculus, LRD, Power Law,

ssH

1)( =

White Noise

)()(

1

=

t

th

cos)2(2)(

12

2

=

Ryy

u(t) y(t)

2/1 f

Power laws in

Signal/Systems

Probability distributionRandom processes (correlation functions)

noise (signal) generation via fractional dynamic system

y(t) is a Brownian motion when =1, i.e., process.2/1 f

Figure 2. Relationship between fractional order dynamic systems, long

range dependence and power law [13]

Some widely used FGN generation methods are briefly re

viewed below. Other related papers include [3639]



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5.1.1 A Fast Fractional Gaussian Noise Genera-tor [40] This method is based on an algorithm to compute anapproximation to discrete fractional Gaussian noise. The algo-

rithm computes fast fractional Gaussian noise (FFGN) [40] as a

sum of a low frequency term and a high frequency term. The high

frequency term is a Markov-Gauss process and the low frequencyterm is a weighted sum of Markov-Gauss processes. Then, the

FFGN is given by

Xi = X(L)i +X

(H)i , (42)

where X(L)i and X

(H)i denote low and high frequency terms re-

spectively. Writing M(n)t for the nth Markov-Gauss process with

variance 1 and weight factor W12

n , we define

X(L)i =N

n=0

W1

2n M(n)t , (43)

Wn =H(2H1)(32H) (B

1HBH1)B2n(H1). (44)

The high frequency term X(H)i is used to compensate the defi-

ciency due to the approximation leading to (44). Finally, there

remains only the choice of B and N. Mandelbrot found 2,3 or 4

convenient for B [40], and Chi et al. (1973) recommend about

15-20 for N [41].

5.1.2 A Multiple Timescale Fluctuation Approach[34] It is shown in [34] that the autocorrelation function of theFGN process at the basic timescale could be approximated by

the weighted sum of exponential functions of the time lag. This

observation can lead to an algorithm to generate FGN. The best

(in terms of mean square error) approximation of (39) is given

by the following equation

r= 1.52(H0.5)1.32. (45)

It should be mentioned that this algorithm is based on the same

principle with the FFGN algorithm [40].

5.1.3 A Disaggregation Approach [34] [5] The dis-aggregation approach is a mathematical induction technique

which is made possible by the expressions of the statistics of

the aggregated FGN process [5]. The first step is sufficient to

describe the method application. Assume that the generation

is completed at the timescale k n and the time series will begenerated at the next timescale k/2. In the generation step, the

higher-level amount Z(k)i (1 < i < n/k) is disaggregated into two

lower-level amounts Z(k/2)2i1 and Z

(k/2)2i such that

Z(k/2)2i1 +Z

(k/2)2i = Z

(k)i . (46

Therefore, (46) can be used to generate Z(k/2)2i1 and then obtain

Z(k/2)2i . At this generation step, both the values of previous lower-

level time steps, i.e., Z(k/2)1 , ..., Z

(k/2)2i2 and the values of nex

higher-level time steps, i.e., Z(k)i+1, ..., Z

(k)n/k have become avail

able. To simplify the method, the correlations ofZ(k/2)2i1 with only

one higher-level time step behind and one ahead are considered

Thus, Z(k/2)2i1 can be generated from the linear relationship

Z(k/2)2i1 = a2Z

(k/2)2i3 + a1Z

(k/2)2i2 + b0Z

(k)i + b1Z

(k)i+1 +V (47

where a2, a1, b0 and b1 are parameters to be estimated and V isinnovation whose variance has to be estimated. The unknown

parameters can be estimated according to the correlations in the

form of Cor[Z(k/2)2iC1 ,Z

(k/2)2iC1+j] = rj , where rj is given by (39).

5.1.4 A Symmetric Moving Average Approach [34[42] The symmetric moving average (SMA) approach is intro-

duced by [42], which could be used to generate stochastic pro-cess with any kind of autocorrelation structure. For an inpu

white noise Wi, the SMA filter takes the weighted average of a

number ofWi to produce the output, for example

Xi =q

j=q

a|j|Wi+j = aqWiq + ... + a1Wi1 +

a0Wi + a1Vi+1 + ... + aqVi+q (48

where aj are the weights symmetric about a center (a0) that corresponds to the variable Wi and q theoretically is infinity but in

practice can be restricted to a finite number. The autocovariance

implied by (48) is

rj =qj

l=qa|j|a|j+l|, j = 0, 1, 2... . (49

It is also shown in [42] that the discrete Fourier transform

sa() of the aj sequence is related to the power spectrum of the



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process sr() by

sa() =

2sr(). (50)

We can use (39) to approximate the inverse Fourier transform ofsa() to get aj:

aj

(22H)r032H [(j + 1)

H+0.5 + (j1)H+0.52jH+0.5], j > 0 (51)

In the end, the generation scheme (48) with coefficients aj can

lead to the SMA approach.

6 Hurst Parameter Estimation

The Hurst parameter H characterizes the degree of LRD. A

process is said to have long range dependence when 0.5


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tional form expected, typically either FGN or FARIMA. Errors

may occur if the user fails to specify the model. Local Whittle is

a semi-parametric estimation method.

6.2.2 Periodogram Method [51] The periodogram isdefined by

I() =1

2N

Nj=1

Xjei j

2

, (56)

where is the frequency and i =1. For a series with finite

variance, I() is an estimate of the spectral density of the series.A log-log plot of I() should have a slope of 12H close to theorigin.

6.2.3 Wavelet Based Method Wavelet analysis hasbeen used with success to measure the Hurst parameter in re-

cent years [52], [53], [54], [55], [56], [50], [28], [57], [58], [59]

and [60]. To decide a suitable frequency-domain Hurst param-

eter estimation method for comparison, wavelet based estimator

seems to be the right choice. Wavelets can be thought of as akin

to Fourier series but using waveforms other than sine waves. The

Hurst exponent is calculated from the wavelet spectral density by

fitting a linear regression line through a set ofxj,yj points, wherexj is the octave and yj is the logarithm of the normalized power.

The slope of this regression line is proportional to the estimate

for the Hurst exponent. A 95% confidence interval can be given.

6.2.4 FrFT Based Estimator In [35] and [61], weproposed to use a fractional Fourier transform (FrFT) based es-

timator. It uses the spectrum calculated by FrFT for estimation.

The FrFT based local estimator has proven to have better per-

formances than other existing methods in extensive validation

experiments performed in [62].

7 Fractional Fourier Transform (FrFT)

Fractional Fourier transform may be considered as a frac-

tional power of the classic Fourier transform [63]. The first ideaof fractional power of the Fourier operator appears in 1929 [64].

Like the complex exponentials are the basis functions in Fourier

analysis, the chirps (signals sweeping all frequencies in a certain

frequency interval) are the basis in fractional Fourier analysis.

Since FrFT can provide a richer picture in time-frequency anal-

ysis [65] and it is nothing more than a variation of the standard

Fourier transform [66], it can improve the performance of some

signal processing applications.

Figure 3. Rotation concept of Fourier transform.

7.1 Formulation and DerivationLet us start with interpretation of a variable along a rotated

axis system. Like the time and frequency variables in the timefrequency plane, let x be the variable along the x-axis and isthe variable along the -axis (Fig. 3). Let xa and a represent therotated variables x and , respectively. We have

xaa

=

cosF sinFsinF cosF

x

, (57

where F = a/2 is the rotating angle. It is always assumed thata = xa+1. Therefore, xa and a are always orthogonal.

If f(x) is a time signal of the variable x, it lives on thehorizontal axis. Its Fourier transform (FT) F(f(x)) = F() is afunction of the frequency , and hence it lives on the vertical axisFourier transform changes signal f(x) in time domain x to F()in frequency domain , which corresponds to a counterclockwiserotation over an angle /2 in the (x,) plane. Since applying FTtwice to f(x) results in

(F2f)(x) = (F(F f))(x) =12

F()eixd = f(x), (58

F2 is called the parity operator. Thus, the time axis rotated over

an angle for F2.Similarly, it follows that

(F3f)() = (F(F2f))() =12

f(x)eixdx = F(), (59

which corresponds to a rotation of the representation axis over

3/2. If we apply one more F , we should get another rotationover /2, which brings us back to the original time axis. There-fore,

F4(f) = f or F4 = I. (60

In conclusion, the FT operator corresponds to a rotation of the

axis over an angle /2 in the time-frequency plane.



10/18

There are six definitions of FrFT [63]. The most intuitive

way to define FrFT is by generalizing the rotation concept of

classical FT. Like FT corresponds to a rotation in the time-

frequency plane over an angle F = /2, the FrFT will corre-spond to a rotation over an arbitrary angle F = a/2 with a

R.

For a more formal definition, FrFT can be defined through

eigenfunctions. We can also start with defining the classical FT.

According to (60), the eigenvalues are all in the set {1,i,1, i}and thus FT has only four different eigenspaces. A possible

choice for the eigenfunctions of the operator F that is generally

agreed upon is given by the set of normalized Hermite-Gauss

functions [67]:

n(x) =21/4

2nn!ex

2/2Hn(x), (61)

where Hn(x) = (1)nex2Dnex2 , D= d/dx is a Hermite poly-nomial of degree n. Therefore, there is an eigenvalue n suchthat

Fn = ein/2n. (62)

Thus, the eigenvalue for n is given by n = ein/2 = n with

= i = ei/2 representing a rotation over an angle /2.

Similarly, FrFT can be defined for a rotating angle F =a/2 by

Fan = eina/2n = ann =

nan. (63)

Since any function f in the eigenspaces can be expanded in

terms of these eigenfunctions f = n=0 ann with

an = 12nn!

2

Hn(x)ex2/2f(x)dx, (64)

we can get

fa := Faf = Fa[

n=0

ann] =

n=0

anFan =

n=0

aneina/2n. (65)

For computational purposes, (65) should be changed to an in-

tegral representation by replacing the an in the series by their

integral expression in (64):

fa() =

n=0

[

n(x)f(x)dx]eina/2n()

=

n=0

eina/2Hn()Hn(x)2nn!

e(x2+2)/2f(x)dx

=1

1e2iF

exp

2xeiF e2iF(2 +x2)

1 e2iF

exp

2 +x2

2

,

(66

where in the last step Mehlers formula [68] has been used.

n=0

eina/2Hn()Hn(x)2nn!

= exp 2xeiFe2iF(2+x2)1e2iF (1e2iF) (67

With the following expressions,

2xeiF1e2iF =ixcscF, (68

1

1e2iF

=e

i2 (

2 FF)

2|sinF| , (69

e2iF1e2iF +

1

2= i

2cotF, (70

where F = sgn(sinF), we can obtain a more tractable integrarepresentation ofF as (71).

fa() := (Faf)() =

ei2 (

2 FF)e

i2

2 cotF

2|sinF|

exp i x

sinF+

i

2x2 cotF

f(x)dx,

(71

where 0 < |F| < .

It is shown that defining via (71), the FrFT exists in the same

conditions as in which the FT exists [63]. Therefore, we have



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proved the definition of FrFT in linear integral form as follows.

The ath order of fractional Fourier transform of f(x) is

fa() =

Ka(,x)f(x)dx, (72)

where Ka(,x) = AF exp[i(cotF22cscFx + cotFx2)],

AF =

1 i cotF, F = a/2.

7.2 Properties of the Kernel Function Ka(,x)

By Fb = FbaFa, we have

fa() := (Faf)() =

ei2 (

2 FF)e

i2

2 cotF2|sinF|

exp i xsinF + i2x2 cotFf(x)dx.

(73)

Using the expressions in (73) and the interpretation of the

FrFT as a rotation, it is directly verified that the kernel Ka has the

following properties [67].

If Ka(,x) is the kernel of the FrFT as in (72), then

1. Ka(,x) = Ka(x,);

2. Ka(,x) = Ka(,x);3. Ka(,x) = Ka(,x);4.

Ka(, t)Kb(t,x)dt = Ka+b(,x);

5. Ka(t,)Ka(t,x)dt = (x).

7.3 Convolution of FrFT

Since FrFT is a generalization of FT, it has similar prop-

erties as FT. Therefore we only mention the convolution prop-

erty here. FT transforms a convolution in time domain into a

product in frequency domain. This property remains true when

it concerns the convolution of two functions in the domain of

the FrFT Fa: if ga(xa) = fa(xa) ha(xa), then its FT becomesga+1(a) = fa+1(a)ha+1(a) [67]. Thus,

F{F[fa(xa)ha(xa)]} = F{fa+1(xa+1)ha+1(xa+1)}= fa+2(xa+2)ha+2(xa+2). (74)

7.4 FrFT of Some Common Functions

The fractional Fourier transform of some common functions

can be derived by the use of (72).

FrFT of Delta Function [63]

FrFTF{()}=

1 i cotFexp(i2 cotF) (75

FrFTF{( )}=

1 i cotFexp[i(2 cotF2cscF +2 cotF)] (76

Here, F is rotating angle of FrFT. As can be seen from (75) and(76), a delta function is transformed into a linear chirp function

by the FrFT.

FrFT of a Sinusoid [63]

FrFTF{exp(i2)} = 1 + i tanFexp[i(2 tanF2s+2 tanF)].

Notice that similar to the delta function, the FrFT of a sinusoid

is also a linear chirp function.

FrFT of a Constant [63]

FrFTF{C}= C

1 + i tanF exp(i2 tanF) (78

Since the DC term can be considered as a sinusoid with zero

frequency, the result is a special case of the previous FrFT pair

with = 0.

FrFT of a Chirp Function [63]

FrFTF{exp(i2)}=

1 + i tanF1 + tanF

exp

i2

tanF1 + tanF

(79

The FrFT of a linear chirp function is also a linear chirp function

with a special type of sweeping rate.

FrFT of a Gaussian Function [63]

FrFTF{exp(2)}= exp(2) (80

It is interesting to notice that the FrFT of a Gaussian function is

still a Gaussian function. In fact, the result is independent of the

transform angle F.



12/18

0

0.5

1

1.5

2

0

2

4

6

80

2

4

6

8

10

Fractional order

FrFT spectrum of a Sine function

Frequency

FrFTSpectrum

Figure 4. Fractional Fourier transform of a sine function.

40 30 20 10 0 10 20 30 400

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1symmetric delta

Figure 5. A sample of symmetric delta function.

0

0.5

1

1.5

2

40

20

0

20

400

0.2

0.4

0.6

0.8

1

Fractional order

FrFT spectrum of a symmetric delta function

Frequency

F

rFTSpectrum

Figure 6. Fractional Fourier transform of the symmetric delta function.

Figure 4 is an example of fractional Fourier transform of a

sine function. The orders of FrFT vary from 0.0 to 2.0.

For a symmetric delta function shown in Fig. 5, the FrFT

spectra with orders in (0,2) are plotted in Fig. 6.

8 Fractional Hartley and Sine/Cosine Transform

This section is re-written in concise form based on [6], [7

and [69].

8.1 Fractional Linear Transform

The derivation of fractional Fourier transform can be gener

alized to fractional linear transform. Let T be a linear transform

that maps function f(x) into the function g(). Then, the corresponding fractional transform ofT can be defined by

Ta{f(x)}= ga() (81

Let n(x) be an eigenfunction of linear transform T with eigenvalue n, then, we have

T{n(x)}= nn(). (82

Similar to (66), we can show that

Ta{f(x)} =

n=0

anann()

=

n=0

[

f(x)n(x)dx]ann()

=

Ka(x,)f(x)dx (83

where the transform kernel is defined by

Ka(x,) =

n=0

ann(x)n(). (84

8.2 Fractional Hartley TransformConsider the well-known Hartley transform [6] defined by

T{f(x)}= gH() = 12

f(x)cas(x)dx (85

where cas(x) = cos(x) + sin(x).

According to the linear fractional transform method, the



13/18

fractional Hartley transform can be given by

gaH() =

KaH(x,)f(x)dx (86)

where the Hartley kernel function is

KaH(x,) =

1 j cot

2ej

x2+2

2 cot[cos(xcsc) +

ej(2 ) sin(xcsc)] (87)

where = a/2. It is clear that the fractional Hartley transformgaH() of real function f(x) is complex valued except = k/2for integers k. Besides, the eigenvalues of Hartley transform

2

n = 1 for n = 0, 1,...,. (88)

Thus, the period of fractional Hartley transform is 2. Moreover,

the relationship between fractional Fourier transform and frac-

tional Hartley transform can be derived as

gaH() =1 + ej

a2

2fa() +

1ej a22

fa() (89)

8.3 Fractional Cosine and Sine Transforms

The fractional cosine and sine transforms [7] are now ac-

tively used in optics and signal processing.

The Cosine transform is defined as

CT{f(x)}= gC() = 12

cos(x)f(x)dx. (90)

The Sine transform is defined as

ST{f(x)}= gS() = 12

sin(x)f(x)dx. (91)

In [7], the fractional Cosine/Sine transforms have been de-

rived by taking the real/imaginary parts of the kernel of FrFT.

gaC() =

Re[KaF(,x)]f(x)dx, (92)

gaS() =

Im[KaF(,x)]f(x)dx. (93)

0 5 10 15 20 25 30 350

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

half delta

Figure 7. A sample of right half delta function

0

0.5

1

1.5

2

0

10

20

30

400

0.2

0.4

0.6

0.8

1

Fractional order

FrCT of a half delta function

Frequency

FrCTSpectrum

Figure 8. Fractional Cosine transform of the half delta function

As illustrations, Fig. 8 is the fractional Cosine transform of

the half delta function in Fig. 7. The orders of FrCT vary from

0.0 to 2.0. Figure 10 is the fractional Sine transform of the hal

delta function in Fig. 9. The orders of FrST vary from 0.0 to 2.0

9 Fractals9.1 What Are Fractals?

A huge amount of applications of FOC are motivated by

the research of fractals. In 1967, Mandelbrot investigated self

similarity in papers such as How Long Is the Coast of Britain?

Statistical Self-Similarity and Fractional Dimension [70]. After

that, Mandelbrot introduced the word fractal to denote an ob

ject whose Hausdorff-Besicovitch dimension is greater than it



14/18

35 30 25 20 15 10 5 01

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0

half delta

Figure 9. A sample of left half delta function

0

0.51

1.5

2

40

30

20

10

00

0.5

1

1.5

Fractional order

FrCT of a half delta function

Frequency

FrCTSpectrum

Figure 10. Fractional Sine transform of the half delta function

topological dimension [71]. He illustrated fractals with images

based on recursion in [71].

A fractal is a geometric shape which is self-similar and has

fractional dimension. Self-similar has the same meaning as long-

range dependence. But for a geometric shape, a simple explana-

tion of LRD could be a small part of the shape has a similar

appearance to the full shape. Figure 11 is an example of fractals,

which is from [72].

9.2 Fractal Dimension (FD)

In fractal geometry, the fractal dimension is a statistical

quantity that gives an indication of how completely a fractal ap-

pears to fill space, as one zooms down to finer and finer scales

[73]. Fractals can have non-integer dimensions like 2.5. That is,

while we are in the 3 dimensional space, looking at a piece of

Figure 11. Sierpinsky Triangle

paper which is in 2 dimension, fractals are in between the integer

dimensions. However, fractals always have a smaller dimension

than what they are on. If you draw a Kochs Curve [74] on a

plane, the fractal cant have a dimension higher than the plane

which is 2.

It can be assumed that for any fractal object (of size P, made

up of smaller units of size p), the number of units (N) that fitsinto the larger object is equal to the size ratio (P/p) raised to thepower ofd, which is called the Hausdorff dimension [75].

D =logN

log(P/p). (94

There are some other ways to define fractional dimensions

such as box-counting dimension [76], information dimension

[77], and the correlation dimension [78].

As an example of calculating FD, consider Sierpinsky trian

gle shown in Fig. 11. It contains 3 identical triangles, each of

which requires 2X magnification to become identical to the en

tire figure. The fractal dimension is then log3/ log2, which isapproximately 1.58.

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 [79]. 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 fracta

dimension [80]. For example, the so called porous ball built by

the French group CRONE has been used in cars hydraulic cir

cuit. When the hydraulic gas passes through the porous ball

the pressure/speed shows a fractional order dynamics which is

apparently related to the FD of the ball [81]. Also, the prepa

ration of nanoparticles coated bio-electrodes is by polishing the

surface with fractal shapes. In addition, the diffusion behavio



15/18

of bioelectrochemical process will be fractional order dynamic,

which is related with FD [72].

10 Fractional Splines

10.1 SplinesThe first mathematical reference to splines was discovered

by Schoenberg in 1946 [82]. They are used for smoothing poly-

nomial approximations. A wide class of functions that are used

in applications requiring data smoothing can be referred to as

splines [83]. In [84], splines have been used to minimize the

measures of roughness.

A polynomial spline is a piecewise polynomial function

[85]. In its most general form a polynomial spline S : [a, b] Rconsists of polynomial pieces Pi : [xi,xi+1]R, where the knotssatisfy a = x0 < x1 < ... < xk2 < xk1 = b. The given k ponitsxi are called knots. The vector x = (x0,...,xk1) is called a knotvector for the spline. If the polynomial pieces on the subintervals

[xi,xi+1], i = 0,...,k2 all have a degree of at most n, the splineis said to be of degree n or of order n + 1 [86].

10.2 Fractional Splines

The fractional splines are a generalization of Schoenbergs

polynomial splines for all fractional orders [87, 88]. They in-

volve the fractional operator and many techniques derived from

wavelet and approximation theory. The piecewise power func-

tions of fractional degree a are the basic configuration of frac-

tional splines. Since the natural building blocks for the frac-

tional splines are Liouvilles one-sided power functions, the frac-

tional splines of degree a with the increasing sequence of knots

{xk}k Z as functions that can be written in the following form:

sa(x) = kZ

k(xxk)a+ (95)

where the xks are the knots of the spline. xa+ = max(0,x)

a is one

sided power function.

Before the fractional B-splines are discussed, a relevant def-

inition should be given as

a+f(x) = k

0

(1)k

a

k

f(x k). (96)

Similar to the classical B-splines, [87] defines the fractionalcausal B-splines by taking the (+ 1)th fractional difference ofthe one-sided power function

a+(x) =1

(a + 1)a+1+ x

a+ =

1

(a + 1) k0

(1)k

a + 1

k

(x k)a+. (97)

Figure 12 is an example of fractional B-splines with a > 0 [87].The fractional splines have the following properties [87]:

Figure 12. The fractional B-splines [87]

1) If a is an integer, fractional splines are equivalent to the

classical polynomial splines.

2) The fractional splines are a-Holder continuous for a > 03) The fractional B-splines satisfy the convolution property

and a generalized fractional differentiation rule. Besides, they

decay at least like xa2.4) The fractional splines have a fractional order of approxi-

mation a + 1.5) Fractional spline wavelets essentially behave like frac

tional derivative operators.

11 Fractional Lower Order Moments (FLOM) andFractional Lower-Order Statistics (FLOS)

We all know that for random processes, the Gaussian modeis commonly used in statistic signal processing. This Gaussian

model assumption is usually acceptable and justifiable by the

Central Limit Theorem. A generalization of this assumption i

the so-called stable model which can be used to characterize

a broad class of non-Gaussian processes [8991] including un-

der 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), additiona

benefits can be gained using this type of fractional order sig-

nal processing technique [90]. Note that, statistics of fractiona

lower order means < 2. When = 2, it is the Gaussian process which is the basis of second-order moment theory. When

> 2, it is called the high order statistics (HOS) [92,93]. When < 2, the density function has a heavier tail than Gaussian dis-tribution. Therefore, FLOS noises are more spiky. In addition

the variance or the second-order moment of a stable distribution

with < 2 does not exist. In fact, for a non-Gaussian stabledistribution with characteristic exponent , only the moments oorders less than are finite. Therefore, variance can no longer beused as a measure of dispersion and in turn, many standard sig-

nal processing techniques such as spectral analysis and all leas



16/18

squares (LS) based methods may give misleading results.

A simple test of infinite variance is to plot the running sam-

ple variance estimate Sn with respect to number of points n where

S2n = (nk=1(xk xn)2)/(n 1) and xn = nk=1xk/n. For finite

variance processes xk, Sn will converge to a constant value as n

increases. If Sn does not converge to a constant value, xk is anon-Gaussian infinite-variance process with fractional lower or-

der < 2.An excellent collection of papers on stable distributions,

processes and related topics can be found in [94] where it is

also clear 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 SS (symmetrical -stable) pro-cess [89, 95] which finds wide applications in engineering and

non-engineering domains.

A very interesting fact is that, there is a strong link between

fractional calculus and stable processes [96, 97].

Due to the space limit, we shall not expand our discussion

here. However, the papers and web links cited in this sectionshould give a good start if the readers are interested in exploring

this FOSP technique.

12 Fractional DelayIn many digital signal processing tasks in communication

systems, it is often required to design a filter that implements a

fractional delay, that is, the block of zD with D a nonintegerpositive real number and z the Z-transform symbol. An excellent

review on fractional delay can be found in [98] with a tutorial

flavor.

Fractional delay filter design based on B-spline transform is

proposed in [99]. Closed-form design of all-pass fractional delayfilters is proposed in [100].

Most interestingly, in [101], improved design of digital

fractional-order differentiators approximating s using fractional

sample delay is proposed. This shows a connection between frac-

tional calculus and fractional delay filter. Once again, details

are omitted here in the interest of space limit. Interested readers

can consult the 207 papers at ieeeXplore.ieee.org from

1990-2007 on Fractional Delay.

13 ConclusionsIn this paper, we have given an overview of some exist-

ing fractional order signal processing (FOSP) techniques wherethe developments in the mathematical communities are intro-

duced; relationship between the fractional operator and long-

range dependence is demonstrated, and fundamental properties

of each technique and some of its applications are summarized.

Specifically, we presented a tutorial on 1) fractional order linear

systems; 2) autoregressive fractional integrated moving average

(ARFIMA); 3) 1/f noise; 4) Hurst parameter estimation; 5)fractional order Fourier transformation (FrFT); 6) fractional or-

der linear transforms (Hartley, Sine, Cosine); 7) fractal; 8) frac

tional order splines; 9) fractional lower order moments (FLOM)

and 10) fractional delay filter.

We believe that FOSP techniques will keep developing and

in many realistic applications, FOSP will find more and more

unique place that conventional signal processing methods cannoeven deliver the answer. It is expected the FOSP will become a

more dedicated research field with a strong emphasis on applica-

tion driven scenarios.

ACKNOWLEDGMENTWe wish to thank Review #1 for his/her two-page long review comments

Both reviewers helped improve this paper. This research was supported in par

by USU Skunk Works Research Initiative Grant ( Fractional Order Signal Pro

cessing for Bioelectrochemical Sensors, 2005-2006).

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