Tài liệu Digital Signal Processing Handbook P73 docx

Wornell, G.W. “Fractal Signals”
Digital Signal Processing Handbook
Ed. Vijay K. Madisetti and Douglas B. Williams
Boca Raton: CRC Press LLC, 1999
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73
Fractal Signals
Gregory W. Wornell
Massachusetts Institute of Technology
73.1 Introduction
73.2 Fractal Random Processes
Models and Representations for
1/f
Processes
73.3 Deterministic Fractal Signals
73.4 Fractal Point Processes
Multiscale Models
•
Extended Markov Models
References
73.1 Introduction
Fractal signal models are important in a wide range of signal processing applications. For example,
they are often well-suited to analyzing and processing various forms of natural and man-made phe-
nomena. Likewise, the synthesis of such signals plays an important role in a variety of electronic
systems for simulating physical environments. In addition, the generation, detection, and manipu-
lation of signals with fractal characteristics has become of increasing interest in communication and
remote-sensing applications.
A defining characteristic of a fractal signal is its invariance to time- or space-dilation. In general,
such signals may be one-dimensional (e.g., fractal time series) or multidimensional (e.g., fractal

natural terrain models). Moreover, they may be continuous-time or discrete-time in nature, and
may be continuous or discrete in amplitude.
73.2 Fractal Random Processes
Most generally, fractal signals are signals having detail or structure on all temporal or spatial scales.
The fractal signals of most interest in applications are those in which the structure at different scales
is similar. Formally, a zero-mean random process x(t) defined on −∞ <t<∞ is statistically
self-similar if its statistics are invariant to dilations and compressions of the waveform in time. More
specifically, a random process x(t) is statistically self-similar with parameter H if for any real a>0
it obeys the scaling relation x(t)
P
= a
−H
x(at),where
P
= denotes equality in a statistical sense. For
strict-sense self-similar processes,this equality is in the sense of all finite-dimensional joint probability
distributions. For wide-sense self-similar processes, the equality is interpreted in the sense of second-
order statistics, i.e., the
R
x
(t, s)

= E
[
x(t)x(s)
]
= a
−2H
R
x

(at, as)
A sample path of a self-similar process is depicted in Fig. 73.1.
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FIGURE 73.1: A sample waveform from a statistically scale-invariant random process, depicted on
three different scales.
While regular self-similar random processes cannot be stationary, many physical processes ex-
hibiting self-similarity possess some stationary attributes. An important class of models for such
phenomena are referred to as “1/f processes”. The 1/f family of statistically self-similar random
processes are empirically defined as processes having measured power spectra obeying a power law
relationship of the form
S
x
(ω) ∼
σ
2
x
|ω|
γ
(73.1)
for some spectral parameter γ related to H according to γ = 2H + 1.
Generally, the power law relationship (73.1) extends over several decades of frequency. While data
lengthtypicallylimitsaccesstospectral information atlowerfrequencies, and data resolution typically
limits access to spectral content at higher frequencies, there are many examples of phenomena for
which arbitrarily large data records justify a 1/f spectrum of the form (73.1) over all accessible
frequencies. However, (73.1) is not integrable and hence, strictly speaking, does not constitute
a valid power spectrum in the theory of stationary random processes. Nevertheless, a variety of
interpretations of such spectra have been developed based on notions of generalized spectra [1, 2, 3].
As a consequence of their inherent self-similarity, the sample paths of 1/f processes are typically

fractals [4]. Thegraphs ofsample paths of random processesare one-dimensional curves in the plane;
this is their “topological dimension”. However, fractal random processes have sample paths that are so
irregular that their graphs have an “effective” dimension that exceeds their topological dimension of
unity. It is this effective dimension that is usually referred to as the “fractal” dimension of the graph.
However, it is important to note that the notion of fractal dimension is not uniquely defined. There
are several different definitions of fractal dimension from which to choose for a given application—
each with subtle but significant differences [5]. Nevertheless, regardless of the particular definition,
the fractal dimension D of the graph of a fractal function typically ranges between D = 1 and D = 2.
Larger values of D correspond to functions whose graphs are increasingly rough in appearance and,
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in an appropriate sense, fill the plane in which the graph resides to a greater extent. For1/f processes,
there is an inverse relationship between the fractal dimension D and the self-similarity parameter H
of the process: an increase in the parameter H yields a decrease in the dimension D, and vice-versa.
This is intuitively reasonable, since an increase in H corresponds to an increase in γ , which, in turn,
reflects a redistribution of power from high to low frequencies and leads to sample functions that are
increasingly smooth in appearance.
A truly enormous and tremendously varied collection of natural phenomena exhibit 1/f -type
spectral behavior over manydecadesof frequency. A partial list includes(see, e.g.,[4, 6, 7,8, 9] and the
references therein): geophysical, economic, physiological, and biological time series; electromagnetic
and resistance fluctuations in media; electronic device noises; frequency variation in clocks and
oscillators; variations in music and vehicular traffic; spatial variation in terrestrial features and
clouds; and error behavior and traffic patterns in communication networks.
While γ ≈ 1 in many of these examples, more generally 0 ≤ γ ≤ 2. However, there are many
examples of phenomena in which γ lies well outside this range. For γ ≥ 1, the lack of integrability of
(73.1) in a neighborhood of the spectral origin reflects the preponderance of low-frequency energy in
the correspondingprocesses. This phenomenon is termed the infraredcatastrophe. Formanyphysical
phenomena, measurements corresponding to very small frequencies show no low-frequency roll off,
which is usually understood to reveal an inherent nonstationarity in the underlying process. Such is

the case for the Wiener process (regular Brownian motion), for which γ = 2.Forγ ≤ 1, the lack
of integrability in the tails of the spectrum reflects a preponderance of high-frequency energy and is
termed the ultraviolet catastrophe. Such behavior is familiar for generalized Gaussian processes such
as stationary white Gaussian noise (γ = 0) and its usual derivatives. When γ = 1, both catastrophes
are experienced. This process is referred to as “pink” noise, particularly in the audio applications
where such noises are often synthesized for use in room equalization.
An important property of 1/f processes is their persistent statistical dependence. Indeed, the
generalized Fourier pair [10]
|τ|
γ−1
2(γ )cos(γ π/2)
F
←→
1
|ω|
γ
(73.2)
valid for γ>0 but γ = 1, 2, 3, ... , reflects that the autocorrelation R
x
(τ ) associated with the
spectrum (73.1) for 0 <γ <1 is characterized by slow decay of the form R
x
(τ ) ∼|τ|
γ−1
.
This power law decay in correlation structure distinguishes 1/f processes from many traditional
models for time series analysis. For example, the well-studied family of autoregressive moving-
average (ARMA) models have a correlation structure invariably characterized by exponential decay.
As a consequence, ARMA models are generally inadequate for capturing long-term dependence in
data.

One conceptually important characterization for 1/f processes is that based on the effects of
bandpass filtering on such processes [11]. This characterization is strongly tied to empirical char-
acterizations of 1/f processes, and is particularly useful for engineering applications. With this
characterization, a 1/f process is formally defined as a wide-sense statistically self-similar random
process having the property that when filtered by some arbitrary ideal bandpass filter (where ω = 0
and ω =±∞are strictly not in the passband), the resulting process is wide-sense stationary and has
finite variance.
Amongavariety ofimplicationsof this definition, it followsthat suchaprocessalso has theproperty
that when filtered by any ideal bandpass filter (again such that ω = 0 and ω =±∞are strictly not
in the passband), the result is a wide-sense stationary process with a spectrum that is σ
2
x
/|ω|
γ
within
the passband of the filter.
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73.2.1 Models and Representations for
1/f
Processes
A variety of exact and approximate mathematical models for 1/f processes are useful in signal
processing applications. These include fractional Brownian motion, generalized autoregressive-
moving-average, and wavelet-based models.
Fractional Brownian Motion and Fractional Gaussian Noise
Fractional Brownian motion and fractional Gaussian noise have proven to be useful mathe-
matical models for Gaussian 1/f behavior. In particular, the fractional Brownian motion framework
provides a useful construction for models of 1/f -type spectral behavior corresponding to spectral
exponents in the range −1 <γ <1 and 1 <γ <3; see, e.g., [4, 7]. In addition, it has proven useful

for addressing certain classes of signal processing problems; see, e.g., [12, 13, 14, 15].
FractionalBrownian motionis a nonstationaryGaussianself-similar process x(t)with the property
that its corresponding self-similar increment process
x(t; ε)

=
x(t + ε)− x(t)
ε
is stationary for every ε>0.
A convenient though specialized definition of fractional Brownian motion is given by Barton and
Poor [12]:
x(t)

=
1
(H + 1/2)


0
−∞

|t − τ|
H−1/2
−|τ|
H−1/2

w(τ) dτ
+

t

0
|t − τ|
H−1/2
w(τ) dτ

(73.3)
where 0 <H <1 is the self-similarity parameter, and where w(t) is a zero-mean, stationary white
Gaussian noise process with unit spectral density. When H = 1/2,(73.3) specializes to the Wiener
process, i.e., classical Brownian motion. Sample functions of fractional Brownian motion have a
fractal dimension (in the Hausdorff-Besicovitch sense) given by [4, 5]
D = 2 − H.
Moreover, the correlation function for fractional Brownian motion is given by
R
x
(t, s) = E
[
x(t)x(s)
]
=
σ
2
H
2

|s|
2H
+|t|
2H
−|t − s|
2H


,
where
σ
2
H
= var x(1) = (1 − 2H)
cos(πH )
πH
.
The increment process leads to a conceptually useful interpretation of the derivative of fractional
Brownian motion: as ε → 0, fractional Brownian motion has, with H

= H − 1, the generalized
derivative [12]
x

(t) =
d
dt
x(t) = lim
ε→0
x(t; ε) =
1
(H

+ 1/2)

t
−∞

|t − τ|
H

−1/2
w(τ) dτ,
(73.4)
which is termed fractional Gaussian noise. This process is stationary and statistically self-similar with
parameter H

. Moreover, since (73.4)isequivalenttoaconvolution,x

(t) can be interpreted as the
output of an unstable linear time-invariant system with impulse response
υ(t) =
1
(H − 1/2)
t
H−3/2
u(t)
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driven by w(t). Fractional Brownian motion x(t) is recovered via
x(t) =

t
0
x

(t) dt.

The character of the fractional Gaussian noise x

(t) depends strongly on the value of H. This
follows from the autocorrelation function for the increments of fractional Brownian motion, viz.,
R
x
(τ; ε)

= E
[
x(t; ε)x(t − τ; ε)
]
=
σ
2
H
ε
2H−2
2


|τ|
ε
+ 1

2H
− 2

|τ|
ε


2H
+

|τ|
ε
− 1

2H

,
which at large lags (|τ|ε) takes the form
R
x
(τ ) ≈ σ
2
H
H(2H − 1)|τ|
2H−2
.
(73.5)
Since the right side of Eq. (73.5) has the same algebraic sign as H −1/2, for 1/2 <H <1 the process
x

(t) exhibits long-term dependence, i.e., persistent correlation structure; in this regime, fractional
Gaussian noise is stationary with autocorrelation
R
x

(τ ) = E


x

(t)x

(t − τ)

= σ
2
H
(H

+ 1)(2H

+ 1)|τ|
2H

,
and the generalized Fourier pair (73.2) suggests that the corresponding power spectral density can be
expressed as S
x

(ω) = 1/|ω|
γ

,whereγ

= 2H

+ 1. In other regimes, for H = 1/2 the derivative

x

(t) is the usual stationary white Gaussian noise, which has no correlation, while for 0 <H <1/2,
fractional Gaussian noise exhibits persistent anti-correlation.
A closely related discrete-time fractional Brownian motion framework for modeling 1/f behavior
has also been extensively developed based on the notion of fractional differencing [16, 17].
ARMA Models for
1/f
Behavior
Another class of models that has been used for addressing signal processing problems involving
1/f processes is based on a generalized autoregressive moving-average framework. These models
have been usedbothinsignal modelingandprocessingapplications, as wellasinsynthesisapplications
as 1/f noise generators and simulators [18, 19, 20].
One such framework is based on a “distribution of time constants” formulation [21, 22]. With
this approach, a 1/f process is modeled as the weighted superposition of an infinite number of
independent random processes, each governed by a distinct characteristic time-constant 1/α >
0. Each of these random processes has correlation function R
α
(τ ) = e
−α|τ|
corresponding to a
Lorentzian spectra of the form S
α
(ω) = 2α/(α
2
+ ω
2
), and can be modeled as the output of a causal
LTI filter with system function ϒ
α

(s) =
√
2α/(s + α) driven by an independent stationary white
noise source. The weighted superposition of a continuum of such processeshas an effective spectrum
S
x
(ω) =

∞
0
S
α
(ω) f (α) dα,
(73.6)
where the weights f(α)correspond to the density of poles or, equivalently, relaxation times. If an
unnormalizable, scale-invariant density of the form f(α) = α
−γ
is chosen for 0 <γ <2, the
resulting spectrum (73.6)is1/f , i.e., of the form (73.1).
More practically, useful approximate 1/f models result from using a countable collection of single
time-constant processes in the superposition. With this strategy, poles are uniformly distributed
along a logarithmic scale along the negative part of the real axis in the s-plane. The process x(t)
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