Tài liệu 73 Fractal Signals pdf

Wornell Massachusetts Institute of Technology 73.1 Introduction 73.2 Fractal Random Processes Models and Representations for 1/fProcesses 73.3 Deterministic Fractal Signals 73.4 Fractal

Wornell, G.W “Fractal Signals”

Digital Signal Processing Handbook

Ed Vijay K Madisetti and Douglas B Williams Boca Raton: CRC Press LLC, 1999

Fractal Signals

Gregory W Wornell

Massachusetts Institute of Technology

73.1 Introduction 73.2 Fractal Random Processes Models and Representations for 1/fProcesses 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 processx(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 processx(t) is statistically self-similar with parameter H if for any real a > 0

it obeys the scaling relationx(t) = a P −H x(at), where= denotes equality in a statistical sense ForP

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)] = a4 −2H R x (at, as)

A sample path of a self-similar process is depicted in Fig.73.1

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

for some spectral parameterγ related to H according to γ = 2H + 1.

Generally, the power law relationship (73.1) extends over several decades of frequency While data length typically limits access to spectral information at lower frequencies, 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] The graphs of sample paths of random processes are 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 dimensionD of the graph of a fractal function typically ranges between D = 1 and D = 2.

Larger values ofD correspond to functions whose graphs are increasingly rough in appearance and,

in an appropriate sense, fill the plane in which the graph resides to a greater extent For 1/f processes,

there is an inverse relationship between the fractal dimensionD and the self-similarity parameter H

of the process: an increase in the parameterH yields a decrease in the dimension D, and vice-versa.

This is intuitively reasonable, since an increase inH 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 many decades of 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

Whileγ ≈ 1 in many of these examples, more generally 0 ≤ γ ≤ 2 However, there are many

examples of phenomena in whichγ 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 corresponding processes This phenomenon is termed the infrared catastrophe For many physical

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 whichγ = 2 For γ ≤ 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

20(γ ) cos(γ π/2)

F

valid forγ > 0 but γ 6= 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 (whereω = 0

andω = ±∞ are strictly not in the passband), the resulting process is wide-sense stationary and has

finite variance

Among a variety of implications of this definition, it follows that such a process also has the property

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 isσ2

x /|ω| γ within

the passband of the filter

73.2.1 Models and Representations for1/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]

Fractional Brownian motion is a nonstationary Gaussian self-similar processx(t) with the property

that its corresponding self-similar increment process

1x(t; ε)=4 x(t + ε) − x(t)

ε

is stationary for everyε > 0.

A convenient though specialized definition of fractional Brownian motion is given by Barton and Poor [12]:

x(t) =4 0(H + 1/2)1

"Z 0

−∞



|t − τ| H−1/2 − |τ| H−1/2w(τ) dτ

+

Z 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 WhenH = 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 H2

2



|s|2H + |t|2H − |t − s|2H

,

where

σ2

H = var x(1) = 0(1 − 2H )cos(πH)

πH .

The increment process leads to a conceptually useful interpretation of the derivative of fractional Brownian motion: asε → 0, fractional Brownian motion has, with H0 = H − 1, the generalized

derivative [12]

x0(t) = dt d x(t) = lim

ε→0 1x(t; ε) = 0(H01+ 1/2)

Z t

−∞|t − τ| H0−1/2 w(τ) dτ, (73.4)

which is termed fractional Gaussian noise This process is stationary and statistically self-similar with

parameterH0 Moreover, since (73.4) is equivalent to a convolution,x0(t) can be interpreted as the

output of an unstable linear time-invariant system with impulse response

υ(t) = 1 0(H − 1/2) t H−3/2 u(t)

driven byw(t) Fractional Brownian motion x(t) is recovered via

x(t) =

Z t

0

x0(t) dt.

The character of the fractional Gaussian noisex0(t) depends strongly on the value of H This

follows from the autocorrelation function for the increments of fractional Brownian motion, viz.,

R 1x (τ; ε) = E [1x(t; ε)1x(t − τ; ε)]4

= σ H2ε2H−2

2

"|τ|

ε + 1

2H

− 2

|τ|

ε

2H

+

|τ|

ε − 1

2H#

,

which at large lags (|τ|  ε) takes the form

R 1x (τ) ≈ σ2

H H (2H − 1)|τ|2H−2 (73.5) Since the right side of Eq (73.5) has the same algebraic sign asH −1/2, for 1/2 < H < 1 the process

x0(t) exhibits long-term dependence, i.e., persistent correlation structure; in this regime, fractional

Gaussian noise is stationary with autocorrelation

R x0(τ) = Ex0(t)x0(t − τ)= σ2

H (H0+ 1)(2H0+ 1)|τ|2H0

,

and the generalized Fourier pair (73.2) suggests that the corresponding power spectral density can be expressed asS x0(ω) = 1/|ω| γ0

, whereγ0 = 2H0+ 1 In other regimes, for H = 1/2 the derivative

x0(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 used both in signal modeling and processing applications, as well as in synthesis applications

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 functionR α (τ) = e −α|τ|corresponding to a

Lorentzian spectra of the formS α (ω) = 2α/(α2+ ω2), and can be modeled as the output of a causal

LTI filter with system functionϒ α (s) = √2α/(s + α) driven by an independent stationary white

noise source The weighted superposition of a continuum of such processes has an effective spectrum

S x (ω) =

Z ∞

0

where the weightsf (α) correspond to the density of poles or, equivalently, relaxation times If an

unnormalizable, scale-invariant density of the formf (α) = α −γ is chosen for 0 < γ < 2, the

resulting spectrum (73.6) is 1/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 thes-plane The process x(t)

synthesized in this manner has a nearly-1/f spectrum in the sense that it has a 1/f characteristic

with superimposed ripple that is uniform-spaced and of uniform amplitude on a log-log frequency plot More specifically, when the poles are exponentially spaced according to

α m = 1 m , −∞ < m < ∞, (73.7) for some 1< 1 < ∞, the limiting spectrum

S x (ω) =X

m

1 (2−γ )m

satisfies

σ2

L

|ω| γ ≤ S x (ω) ≤ σ

2

U

for some 0< σ2

L ≤ σ2

U < ∞, and has exponenentially spaced ripple such that for all integers k

|ω| γ S x (ω) = |1 k ω| γ S x (1 k ω). (73.10)

As1 is chosen closer to unity, the pole spacing decreases, which results in a decrease in both the

amplitude and spacing of the spectral ripple on a log-log plot

The 1/f model that results from this discretization may be interpreted as an infinite-order ARMA

process, i.e.,x(t) may be viewed as the output of a rational LTI system with a countably infinite

number of both poles and zeros driven by a stationary white noise source This implies, among other properties, that the corresponding space descriptions of these models for long-term dependence require infinite numbers of state variables These processes have been useful in modeling physical

1/f phenomena; see, e.g., [23,24,25] And practical signal processing algorithms for them can often

be obtained by extending classical tools for processing regular ARMA processes

The above method focuses on selecting appropriate pole locations for the extended ARMA model The zero locations, by contrast, are controlled indirectly, and bear a rather complicated relationship

to the pole locations With other extended ARMA models for 1/f behavior, both pole and zero

locations are explicitly controlled, often with improved approximation characteristics [20] As an example, [6,26] describe a construction as filtered white noise where the filter structure consists of a cascade of first-order sections each with a single pole and zero With a continuum of such sections, exact 1/f behavior is obtained When a countable collection of such sections is used, nearly-1/f

behavior is obtained as before In particular, when stationary white noise is driven through an LTI system with a rational system function

ϒ(s) = Y∞

m=−∞



s + 1 m+γ /2

s + 1 m



the output has power spectrum

S x (ω) ∝ Y∞

m=−∞



ω2+ 12m+γ

ω2+ 12m



This nearly-1/f spectrum also satisfies both (73.9) and (73.10) Comparing the spectra (73.12) and (73.8) reveals that the pole placement strategy for both is identical, while the zero placement strategy is distinctly different

The system function (73.11) associated with this alternative extended ARMA model lends useful insight into the relationship between 1/f behaviorandthelimitingprocessescorrespondingtoγ → 0

andγ → 2 On a logarithmic scale, the poles and zeros of (73.11) are each spaced uniformly along the negative real axis in thes-plane, and to the left of each pole lies a matching zero, so that poles

and zeros are alternating along the half-line However, for certain values ofγ , pole-zero cancellation

takes place In particular, asγ → 2, the zero pattern shifts left canceling all poles except the limiting

pole ats = 0 The resulting system is therefore an integrator, characterized by a single state variable,

and generates a Wiener process as anticipated By contrast, asγ → 0, the zero pattern shifts right

canceling all poles The resulting system is therefore a multiple of the identity system, requires no state variables, and generates stationary white noise as anticipated

An additional interpretation is possible in terms of a Bode plot Stable, rational system functions composed of real poles and zeros are generally only capable of generating transfer functions whose Bode plots have slopes that are integer multiples of 20 log102 ≈ 6 dB/octave However, a 1/f

synthesis filter must fall off at 10γ log102 ≈ 3γ dB/octave, where 0 < γ < 2 is generally not an

integer With the extended ARMA models, a rational system function with an alternating sequence of poles and zeros is used to generate a stepped approximation to a−3γ dB/octave slope from segments

that alternate between slopes of−6 dB/octave and 0 dB/octave

Wavelet-Based Models for 1/f Behavior

Another approach to 1/f process modeling is based on the use of wavelet basis expansions.

These lead to representations for processes exhibiting 1/f -type behavior that are useful in a wide

range of signal processing applications

Orthonormal wavelet basis expansions play the role of Karhunen-Lo`eve-type expansions for 1/f

-type processes [11,27] More specifically, wavelet basis expansions in terms of uncorrelated random variables constitute very good models for 1/f -type behavior For example, when a sufficiently regular

orthonormal wavelet basis{ψ m

n (t) = 2 m/2 ψ(2 m t − n)} is used, expansions of the form x(t) =X

m

X

n

x n m ψ n m (t),

where thex m

n are a collection of mutually uncorrelated, zero-mean random variables with the

geo-metric scale-to-scale variance progression

varx n m = σ22−γ m , (73.13) lead to a nearly-1/f power spectrum of the type obtained via the extended ARMA models This

behavior holds regardless of the choice of wavelet within this class, although the detailed structure of the ripple in the nearly-1/f spectrum can be controlled by judicious choice of the particular wavelet.

More generally, wavelet decompositions of 1/f -type processes have a decorrelating property For

example, ifx(t) is a 1/f process, then the coefficients of the expansion of the process in terms of a

sufficiently regular wavelet basis, i.e., the

x n m=

Z +∞

−∞ x(t) ψ n m (t) dt

are very weakly correlated and obey the scale-to-scale variance progression (73.13) Again, the detailed correlation structure depends on the particular choice of wavelet [3,11,28,29]

This decorrelating property is exploited in many wavelet-based algorithms for processing 1/f

signals, where the residual correlation among the wavelet coefficients can usually be ignored In addition, the resulting algorithms typically have very efficient implementations based on the discrete wavelet transform Examples of robust wavelet-based detection and estimation algorithms for use with 1/f -type signals are described in [11,27,30]

73.3 Deterministic Fractal Signals

While stochastic signals with fractal characteristics are important models in a wide range of engi-neering applications, deterministic signals with such characteristics have also emerged as potentially important in engineering applications involving signal generation ranging from communications to remote sensing

Signalsx(t) of this type satisfying the deterministic scale-invariance property

for all a > 0, are generally referred to in mathematics as homogeneous functions of degree H

Strictly homogeneous functions can be parameterized with only a few constants [31], and constitute

a rather limited class of models for signal generation applications A richer class of homogeneous signal models is obtained by considering waveforms that are required to satisfy (73.14) only for values ofa that are integer powers of two, i.e., signals that satisfy the dyadic self-similarity property x(t) = 2 −kH x(2 k t) for all integers k.

Homogeneous signals have spectral characteristics analogous to those of 1/f processes, and have

fractal properties as well Specifically, although all non-trivial homogeneous signals have infinite energy and many have infinite power, there are classes of such signals with which one can associate a generalized 1/f like Fourier transform, and others with which one can associate a generalized 1/f

-like power spectrum These two classes of homogeneous signals are referred to as energy-dominated and power-dominated, respectively [11,32] An example of such a signal is depicted in Fig.73.2

FIGURE 73.2: Dilated homogeneous signal

Orthonormal wavelet basis expansions provide convenient and efficient representations for these classes of signals In particular, the wavelet coefficients of such signals are related according to

x m

n =

Z +∞

−∞ x(t)ψ m

n (t) = β −m/2 q[n],

whereq[n] is termed a generating sequence and β = 22H+1 = 2γ This relationship is depicted

in Fig.73.3, where the self-similarity inherent in these signals is immediately captured in the time-frequency portrait of such signals as represented by their wavelet coefficients More generally, wavelet expansion naturally lead to “orthonormal self-similar bases” for homogeneous signals [11,32] Fast synthesis and analysis algorithms for these signals are based on the discrete wavelet transform

FIGURE 73.3: The time-frequency portrait of a homogeneous signal

For some communications applications, the objective is to embed an information sequence into

a fractal waveform for transmission over an unreliable communication channel In this context,

it is often natural forq[n] to be the information bearing sequence such as a symbol stream to be

transmitted, and the corresponding modulation

x(t) =X

m

X

n

x n m ψ n m (t)

to be the fractal waveform to be transmitted This encoding, referred to as “fractal modulation” [32] corresponds to an efficient diversity transmission strategy for certain classes of communication channels Moreover, it can be viewed as a multirate modulation strategy in which data is transmitted simultaneously at multiple rates, and is particularly well-suited to channels having the characteristic that they are “open” for some unknown time intervalT , during which they have some unknown

bandwidthW and a particular signal-to-noise ratio (SNR) Such a channel model can be used, for

example, to capture both characteristics of the transmission medium, such as in the case of meteor-burst channels, the constraints inherent in disparate receivers in broadcast applications, and/or the effects of jamming in military applications

73.4 Fractal Point Processes

Fractal point processes correspond to event distributions in one or more dimensions having self-similar statistics, and are well-suited to modeling, among other examples, the distribution of stars