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Wornell, G.W. “Fractal Signals” Digital Signal Processing Handbook Ed. Vijay K. Madisetti and Douglas B. Williams Boca Raton: CRC Press LLC, 1999 c  1999byCRCPressLLC 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. c  1999 by CRC Press LLC 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, c  1999 by CRC Press LLC 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. c  1999 by CRC Press LLC 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) c  1999 by CRC Press LLC 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) c  1999 by CRC Press LLC 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 =  m , −∞ <m<∞, (73.7) for some 1 <<∞, the limiting spectrum S x (ω) =  m  (2−γ)m ω 2 +  2m (73.8) satisfies σ 2 L |ω| γ ≤ S x (ω) ≤ σ 2 U |ω| γ (73.9) for some 0 <σ 2 L ≤ σ 2 U < ∞, and has exponenentially spaced ripple such that for all integers k |ω| γ S x (ω) =| k ω| γ S x ( k ω). (73.10) As  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) = ∞  m=−∞  s +  m+γ/2 s +  m  , (73.11) the output has power spectrum S x (ω) ∝ ∞  m=−∞  ω 2 +  2m+γ ω 2 +  2m  . (73.12) 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 insightintotherelationshipbetween1/f behaviorandthelimiting processescorrespondingto γ → 0 c  1999 by CRC Press LLC and γ → 2. On a logarithmic scale, the poles and zeros of (73.11) are each spaced uniformly along the negative real axis in the s-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 of γ , pole-zero cancellation takes place. In particular, as γ → 2, the zero pattern shifts left canceling all poles except the limiting pole at s = 0. The resulting system is therefore an integrator, characterized by a single state variable, and generates a Wiener process as anticipated. By contrast, as γ → 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 log 10 2 ≈ 6 dB/octave. However, a 1/f synthesis filter must fall off at 10γ log 10 2 ≈ 3γ dB/octave, where 0 <γ <2 is generally not an integer. With the extendedARMA 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 constitutevery good models for 1/f -type behavior. For example, whena 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) =  m  n x m n ψ m n (t), where the x m n are a collection of mutually uncorrelated, zero-mean random variables with the geo- metric scale-to-scale variance progression var x m n = σ 2 2 −γm , (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, if x(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 m n =  +∞ −∞ x(t)ψ m n (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]. c  1999 by CRC Press LLC 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. Signals x(t) of this type satisfying the deterministic scale-invariance property x(t) = a −H x(at) (73.14) 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)onlyfor values of a 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. c  1999 by CRC Press LLC 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 =  +∞ −∞ x(t)ψ m n (t) = β −m/2 q[n], where q[n] is termed a generating sequence and β = 2 2H+1 = 2 γ . 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 for q[n] to be the information bearing sequence such as a symbol stream to be transmitted, and the corresponding modulation x(t) =  m  n x m n ψ m n (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 interval T , during which they have some unknown bandwidth W 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 c  1999 by CRC Press LLC [...]... Maximum likelihood estimation of the parameters of discrete fractionally differenced Gaussian noise process, IEEE Trans Signal Processing, 41, 2977–2989, Oct 1993 [15] Deriche, M and Tewfik, A.H., Signal modeling with filtered discrete fractional noise processes, IEEE Trans Signal Processing, 41, 2839–2849, Sept 1993 [16] Granger, C.W and Joyeux, R., An introduction to long memory time series models... Fundamental 1/f noise sources, Proc IEEE, 233–258, Mar 1988 [10] Champeney, D.C., A Handbook of Fourier Theorems, Cambridge University Press, Cambridge, England, 1987 [11] Wornell, G.W., Signal Processing with Fractals: A Wavelet-Based Approach, Prentice-Hall, Upper Saddle River, NJ, 1996 [12] Barton, R.J and Poor, V.H., Signal detection in fractional Gaussian noise, IEEE Trans Inform Theory, IT-34, 943–959,... Oppenheim, A.V., Estimation of fractal signals from noisy measurements using wavelets, IEEE Trans Signal Processing, 40, 611–623, Mar 1992 c 1999 by CRC Press LLC [31] Gel’fand, I.M., Shilov, G.E., Vilenkin, N.Y., and Graev, M.I.,Generalized Functions, Academic Press, New York, 1964 [32] Wornell, G.W and Oppenheim, A.V., Wavelet-based representations for a class of self-similar signals with application to fractal... fractal point processes, IEEE Trans Signal Processing, 43, 2606–2617, Nov 1995 [38] Mandelbrot, B.B., Self-similar error clusters in communication systems and the concept of conditional stationarity, IEEE Trans Commun Technol., COM-13, 71–90, Mar 1965 [39] Johnson, D.H and Kumar, A.R., Modeling and analyzing fractal point processes, in Proc Int Conf Acoust Speech, Signal Processing, 1990 [40] Lam, W.M and... ripple A number of efficient algorithms for exploiting such models in the development of robust signal estimation algorithms for use with fractal renewal processes are described in, e.g., [37] c 1999 by CRC Press LLC From a broader perspective, the Poisson mixtures can be viewed as a nonlinear multiresolution signal analysis framework that can be generalized to accommodate a broad class of point process... generalized to accommodate a broad class of point process phenomena As such, this framework is the point process counterpart to the linear multiresolution signal analysis framework based on wavelets that is used for a broad class of continuous-valued signals 73.4.2 Extended Markov Models An equivalent description of the discrete Poisson mixture model is in terms of an extended Markov model The associated... are described by Johnson et al [39] In addition to their use in applications requiring the synthesis of fractal point processes, these multiscale models have also proven useful in signal estimation problems For these kinds of signal analysis applications, it is frequently convenient to replace the continuum Poisson mixture with a discrete Poisson mixture Typically, a collection of constituent Poisson... point processes, in Proc Int Conf Acoust Speech, Signal Processing, 1990 [40] Lam, W.M and Wornell, G.W., Multiscale analysis of fractal point processes and queues, in Proc Int Conf Acoust Speech, Signal Processing, 1996 c 1999 by CRC Press LLC ... Terreni, P., 1/f ν noise generators, in Noise in Physical Systems and 1/f Noise, D’Amico A and Mazzetti, P., Eds., North-Holland, Amsterdam, 1986, 425–428 [19] Corsini, G and Saletti, R., Design of a digital 1/f ν noise simulator, in Noise in Physical Systems and 1/f Noise, Van Vliet, C.M., Ed., World Scientific, Singapore, 1987, 82–86 [20] Saletti, R., A comparison between two methods to generate 1/f . Wornell, G.W. “Fractal Signals” Digital Signal Processing Handbook Ed. Vijay K. Madisetti and Douglas B. Williams Boca. 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