COLOURS OF NOISE FRACTALS AND APPLICATIONS

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1 G. Rosenhouse, Int. J. o Design & Nature and Ecodynamics. Vol. 9, No. 4 (204) OLOURS OF NOISE FRATALS AND APPLIATIONS G. ROSENOUSE Swantech Ltd. aia, Israel. Technion, aia, Israel (Retired). ABSTRAT Due to uncertainty in it, noise prevents exact prediction o the uture rom the past. Noise is generally described by spectral densities o certain unctional dependence on requency. Years o research revealed relations between natural phenomena and noise spectral distributions o either man-made or natural sources o dierent spectral density signal content. owever, since many random unctions o noise appear in nature and in technology in power spectra and power law relations, certain categories o noise spectral density distributions are generally described as powers o requency being grouped in such a way that each one represents certain speciic natural and man-made phenomena. On the other hand, most o the natural phenomena have ractal dimensions that combine together spectral behaviour that occurs in reality as can be seen by measurements results. The paper shows these unctional descriptions o noise in terms o colours and their combination with ractals theories, which enable development o advanced technologies Keywords: Frequency dependent noise, stochastic ractals and noise, applications. I ascribe to nature neither beauty, deormity, order, nor conusion. It is only rom the view point o our imagination that we say that things are beautiul or unsightly, orderly or chaotic. Baruch Spinoza, 665 DETERMINISM, UNERTAINTY AND LIFE Noise is a basic entity in the essence o lie as explained in []. Deterministic lie may become impossible or human beings, since it means clear knowledge o the uture, including unavoidable disasters, with no hope or a change. On the other hand, uncertainty encourages going on with lie, trying to improve and maintain current states, accomplish targets and keep an amount o optimism. Noise appears in all areas o science and technology, and it is involved with everything in lie. It can be described by suitable mathematical ormulation that illustrates dierent topics in an abstract way and leads to general conclusions. An important eature is that there are many kinds o noise which dier by their dependence on requency. Each such dependence leads to dierent phenomena in nature and dierent applications. 2. Ambient noise in the sea 2 EXAMPLES OF FREQUENY DEPENDENE OF NOISE While signals are desirable sounds, in the case o ambient sounds, noise is undesired. Dierent kinds o noise in the sea, with dierent spectra, have been ormulated in the past as a consequence o a huge amount o research. This ormulation includes unctional relations such as white noise, pink noise and tonal components. A pioneering research was done by Knudsen et al. [2]; see Fig.. They obtained the ollowing relation or sea surace agitation at 00 kz: 204 WIT Press, ISSN: (paper ormat), ISSN: (online), DOI: /DNE-V9-N

2 256 G. Rosenhouse, Int. J. o Design & Nature and Ecodynamics. Vol. 9, No. 4 (204) Figure : Knudsen et al. [2] curves o underwater noise. SPL sur SPL or < 000z = SPL k 7 lg k 0 or 000 > 000z () SPL k is a constant that depends on the sea state. requency (z) Mellen [3] obtained through classical statistical mechanics, theoretical ormulae or the ambient ocean noise at the ultrasonic band > 50 kz, which is due to thermal noise. The result or the thermal noise is: NL 5 20 lg ; db re m Pa; The requency in kz. (2) = + ( ) 0 In general, as Wenz [4] has shown, the ambient noise consists o at least three constituents: turbulent pressure luctuations ( 00 z), wind dependent noise due to bubbles and spray that result rom surace agitation (50 z to 20 kz) and density-dependent ocean traic (00 00 z). See Fig. 2. Using the graph, the relevant components can be added. Yet, the overall dependence o noise on requency is clearly observed. The rule o ives o Wenz considers both requency and wind speed on noise level is: U NL(, U) = 25 5 [lg ( ) lg ( )] (3) It is very important to note that the amalgamated observation o the ambient noise reveals a similarity structure, in both the acoustical and wind dependency, as Kerman [5] has shown. This topic is reviewed, or example, by Urick [6] and arey & Richard [7]. 2.2 Illustration o the eect o requency shit o noise spectra on annoyance; see Rosenhouse [8] Annoyance by noise depends strongly on its inormative, spectral contents and individual eect on people. Yet, standards dictate certain ormal limitations, ignoring such details. In practice it happens in many cases o recreational areas, industrial premises and other kinds o activities, that even when

3 G. Rosenhouse, Int. J. o Design & Nature and Ecodynamics. Vol. 9, No. 4 (204) 257 Figure 2: omposite ambient noise spectra Wenz [4]. results o measurements satisy the standards limits, complaints do not stop, oten involving threats o legal acts. The ollowing case study o the eect, based on actual acoustic measurements, shows actors that cause extreme sensitivity to certain noise patterns, even i the total amount o noise remains unchanged. The eect o noise colour dierence is enhanced i the added noise has a certain periodicity, located where the background noise has lower masking eect. Since in many cases the background noise has less annoyance eect or resembles white or pink noise, other noise sources o dierent spectra can be clearly heard, i they include local amplitudes in the requency domain that are higher than the background noise. The ollowing example is o neighbours complaints about intruding noise coming rom a complex o swimming pools. The spectra o dierent cases o noise are given in Fig. 3. The top igure in Fig. 3 is a spectrum o noise without children s shouts rom a nearby swimming pool, where lower requencies are dominant. The neighbours did not complain about that. Almost the same happens when transportation noise is included but the lower requencies became more dominant, but neighbours did not complain about that. See bottom igure in Fig. 3. The spectrum in the middle igure is or the case where children s shouts were added. Those shouts caused a dominant eect at higher requencies and this was the problem that needed to be solved.

4 258 G. Rosenhouse, Int. J. o Design & Nature and Ecodynamics. Vol. 9, No. 4 (204) Figure 3: The eect o noise spectra o a swimming pools complex on residential areas. Top part: No children s shouts and no transportation noise. Middle part: hildren s shouts added. Lower part: Transportation noise added. Acoustic solutions include means or undesirable noise reduction to levels much below the background noise, by as much as 9 decibels, to allow background noise masking o disturbing sources. Such reduction alters the intruding noise status rom being strongly heard at the privacy zone. 3 OLOURS OF NOISE The noise scales were discovered by Johnson [9], where the random process S () is given by the ollowing general dependence on requency. S( ) = ; a is a constant. (4) is the requency in ertz. Following this deinition, research has shown later that many phenomena in physics (including astrophysics and geophysics), biology, technology, speech and music, economics and psychology are involved with the noise range 05. a 5. as will be shown later. A orm o a white noise is Johnson s noise (Johnson [0]) which results rom a thermal excitation o electrical components, such as a resistor. The noise is caused by the motion o the charge o atoms rom which the resistor is built. The thermal excitation is caused by heat, and thus, when the resistor

5 G. Rosenhouse, Int. J. o Design & Nature and Ecodynamics. Vol. 9, No. 4 (204) 259 Figure 4: A scheme o common colours o noise. White noise is represented by a horizontal line. Figure 5: A typical plot o white noise. gets hotter it becomes noisier. Many tests o electrical equipment and components are using several colours: white noise, pink or brown, and researchers and laboratories have, or many decades, been applying various standard noise colours or dierent acoustic measurements. The dierent colours 0 (white which is proportional to /, pink which is proportional to /, brown which is proportional 2 to / and others) have dierent coeicients, α, as will be discussed later, and the power spectra being a plot o a square magnitude o the Fourier transorm against log requency as shown in Fig White noise White noise is proportional to one. See Fig. 5. S( ) = ; a = 0 S( ) =, is a constant (5) a 0 White noise is a random signal o no correlation at two dierence times. Its power spectral density is lat over the whole range o requency bands. Theoretically, the requency band o white noise is

6 260 G. Rosenhouse, Int. J. o Design & Nature and Ecodynamics. Vol. 9, No. 4 (204) ininite and as a result carries an ininite power which, also, does not depend on time. Acoustic white noise is analogous to optical white light, which also consists o a lat power spectrum that has equal quantities o intensity at all the requencies within the visible light. Analogously, acoustic white noise contains equal quantities o intensity at all the audible requencies. In this case, each amplitude has the same chance to be at a certain point as any other amplitude has. Thus, it is not diicult to create white noise by using a program that generates stochastic signals o the size 0.5 to 0.5. White light includes all the visible colours equally, and as a result it is practically a broad band noise. White noise does not necessarily represent each phenomenon that occurs in nature. Frequently, there is interest in Gaussian distribution o the acoustic power, where most o its values are close to zero. Frequently, there is also interest in values that range between 0.5 and 0.5, as the Gaussian distribution gives. While a Gaussian process is a stochastic process, or which any inite combination o samples will be normally distributed, many other kinds o possible probability density distributions that are available suit other applications, such as Laplace and auchy probability density unctions. A major dierence between distributions is the thickness o their tails. There is a chance that a thicker tail will be responsible or more extreme events (Rosenhouse []) and the larger the Gaussian bell or the same average o noise distribution (e.g. the noise o aircrats take o measured at a control point near an airport) the noisier the extreme cases. Gaussian White Noise (GWN) The Gaussian distribution has a mean value (m) o a certain value that is statistically calculated, as well as the standard deviation (σ). In the case o Gaussian white noise the mean or the expected value is zero: m= E( x) = x p( x) dx = 0, and the mean 2 2 standard deviation, σ o the pseudo random sequence becomes: s = Ex ( m) = Ex ( ), or m=0. x is a random variable deined on a probability space. p (x) is the probability density unction (pd). I x is a discrete random variable, then a probability mass unction (pm), (x) is used and: 2 2 Ex ( ) = xpx ( ) dx= 0 ; s = Ex ( m) = Ex ( ), or m = 0. (6) i GWN provides simulation o real situations in our world. It is oten used as a source o numbers generator due to its independent statistical eatures. An additive white Gaussian noise (AWGN) channel is regularly used in communication. The input signal s (t) is contaminated by noise, n (t), that can be AWGN, during its transmission rom the source location to the receiver, and it becomes distorted in amplitude and phase, including a varying time delay. The result is the output: y() t = s() t + n() t (7) 3.2 Pink noise Pink noise depends on the ratio /. Its bandwidth ranges up to 20 kz. See Fig. 6: S( ) = ; a = a S( ) = is a constant (8) Examples in nature include brain EEG, brain MEG, human heart sound, human time estimation, squid giant axon, vacuum tube and semiconductor noise, music and natural sounds.

7 G. Rosenhouse, Int. J. o Design & Nature and Ecodynamics. Vol. 9, No. 4 (204) 26 As a result o this dependence, pink noise has a uniorm distribution along the logarithmic requency scale, which means a lat spectral density as a unction o the percentage o the band width or an equal power or each octave band. It means that pink noise is a requency spectrum with an intensity that decays approximately at the rate o 3 db per octave (or 0 db per each decade). Practically, passing white noise through a ilter o 3 db per octave intensity decay yields pink noise. Pink noise has applications in sound and audio systems and tests. Since pink noise is perceived as more natural or the human ear, it is very popular in investigation o tests in building and environmental acoustics. This application has advantages, since /3 rd octave bands suit humans ability to discriminate irregularities o the requency response, and also since measurement o /3 rd octave bands smoothens many very narrow peaks. The licker noise, which is / noise, is very close to pink noise. The dierence is that the licker noise is limited to 2 kz. It is used in solid state physics and it describes noise radiated rom deects along the wave-guide channel, as a result o base currents in conductors (Barnes and Allan [2]). 3.3 Brown or Brownian noise Figure 6: A typical plot o pink noise. Brown noise is the integral o white noise. See Fig. 7. Its deinition is: S( ) = ; a = 2 a S( ) = 2 is a constant (9) I = 0.5 then Brownian motion occurs, with independent and uncorrelated increments. > 0.5 yields smooth curves and positive correlated increments. < 0.5 results in erratic rough curves and increments with negative correlation. Each point along the route o Brownian motion depends on the value o the previous point. Each new point is moved stochastically a little bit rom the previous state due to a random walk, a term originally coined by Pearson [3]. Brown noise is generated by addition o a random number to the previous value. Equation 9 shows that brown noise expresses a drop o 20 db per decade or 6 db per octave band. That drop means more energy at the lower requencies and a roar o low requencies. It is heard also as the all o water in water cascades or heavy rain. As it is, Brownian noise can be heard i it is acoustic, but in general it carries a mathematical statistical orm o many aspects which makes it universal, including thermal luctuations that have proven the existence o molecules, lie, earth, and environmental sciences, as well as stock market behaviour, being in general responsible or dynamical lie sustaining processes under the same title o Brownian noise.

8 262 G. Rosenhouse, Int. J. o Design & Nature and Ecodynamics. Vol. 9, No. 4 (204) Figure 7: An example o Brownian noise urst parameter = 0.5 perormed by Daphne Sobolev. Figure 8: Brownian motion as measured by Perrin (909). The situation where stochastic events, such as pollen movement in luids (see Fig. 8) and their collective behaviour can be quantitatively given in terms o probability and statistics, leads to the general mathematical theory o the random walk. This theory has uses in many areas, such as molecular biology, wireless nets and changes in the stock market. The mathematical ormality o describing Brownian motion (BM) can be modelled by a simple description as ollows: The position o a particle x(t) at the time t is a result o a random process. The change o location in each step leads to a new location as ollows:. x( t+ Δt) = x( t) + vδt 05 N( 0, ) (0) v is the average speed o a particle and N(0,) is a normal randomly distributed variable. The increment x(t 2 ) x(t ) has a Gaussian distribution, with the average E and the variance = σ 2 (σ standard deviation) properties as ollows: Ext [ ( 2) xt ( ) ]= 0; Var[ xt ( 2) xt ( ) ] t2 t. ()

9 G. Rosenhouse, Int. J. o Design & Nature and Ecodynamics. Vol. 9, No. 4 (204) 263 The position o a particle x(t) is continuous, but not dierentiable. The increments: xt ( 0 rt) xt ( 0) x( t0 + t) x t + ( 0) and are statistical sel-similar. Also: I t = 0, x(t 0 ) = 0, then, r x(t) and xrt ( ) are statistically equivalent. These last assumptions allow or development o the algorithm that generates D Brownian motion. 05. r The ractional Brownian motion (Bm) is a continuous Gaussian process B (t), on [0,T]. It starts at zero and has zero expectation or all t s in this domain. In act it is an extension o the Brownian motion (Bm), which carries the orm: 2 var [ xt ( 2) xt ( ) ] t2 t (2) xt () andr xrt ( )are statistically sel-similar with respect to the urst parameter (or index), which is a real number associated with Bm. Thus, the Bm is properly rescaled by dividing the amplitudes by r. 0 determines the roughness o the curve., which is associated with Bm, is a real number named ater urst (urst, [95]), and it was introduced by Mandelbrot and van Ness [968]. The increment ΔB() t = B( t ) B() t is the ractional Brownian noise. The Bm satisies the ollowing covariance unction: 2 2 E B() t B( t) = t + t t t 2 2 (3) The Bm o the Weyl type (see in Mandelbrot and van Ness [968], is deined as B () t B () 0 = Γ ( ) 05. [( t t ) ( t ) ] db t t + ( ) 05. t t db ( t ) 0 or t > 0 and or t < 0. (4) Both Bm o the Riemanian Liouville type and the one o Weyl type are sel-similar, having the property: B ( at) a B ( t); a>0; (5) The symbol in eqn (5) is equality in the stochastic sense. ence, Weyl integral is a ractional integral o white noise, used to deine the Brownian motion process. The random walk is based on the mutual dependence between two random variables e.g. Gaussian distribution: ( ) 2 xi m p( xi, m, s) = exp (6) 2 s 2p 2s This link was very important or the uture development o tools to orecast stock market behaviour as the wave principle suggested by Elliott (87 948) in a book he published in 940.

10 264 G. Rosenhouse, Int. J. o Design & Nature and Ecodynamics. Vol. 9, No. 4 (204) Figure 9: An example o price changes o a stock, sel-similar and sel-aine systems. 3.4 Black noise It is an anti-phased white noise, which as such it cancels the primary white noise, by a negative overlapping the original signal shape. By deinition, it is an anti-sound, which has a speciic application o cancelling undesirable noise. This is the meaning o active noise control. Black noise is also deined as noise whose spectrum varies as: S( ) = ; a > 2 a S( ) < 2 is a constant (7) Black noise is used to describe various environmental processes as natural and unnatural catastrophes as loods, drought, bear-like inancial markets and population persistence problems because o environmental changes, all backed up by scientiic evidence. Such disasters tend to appear in groups. 4 SUMMARY Noises can be classiied in accordance with the speciic spectral shapes and speciic parameters o each topic involved. Some basic ideas about noise, out o a huge amount o topics in science, art, medicine and technologies involved with noise were presented here. Noise is colourul. It exists everywhere, carrying endless shapes. REFERENES [] Rosenhouse, G., The essence o noise in nature with reerence to acoustics. 6 th Int. onerence on Design and Nature, eds. S. ernandez &.A Brebbia, Wit Press Southampton, pp. 3 3, 202. [2] Knudsen, V.O. Alord, R.S., Emling, J.W., Underwater ambient noise. J. Mar. Res., 22, pp , 948. [3] Mellen, R.., Thermal noise limit in the detection o underwater acoustic signals. J. Acoust. Soc. Am., 24, pp , 952. doi:

11 G. Rosenhouse, Int. J. o Design & Nature and Ecodynamics. Vol. 9, No. 4 (204) 265 [4] Wenz, G.M. Acoustic ambient noise in the ocean: spectra and sources. J. Acoust. Soc. Am., 34(2), pp , 962. doi: [5] Kerman, B.R., Underwater sound generation by breaking wind waves. J. Acoust. Soc. Am., 75(), pp , 984. doi: [6] Urick R.J., Ambient Noise in the Sea, Naval Sea Systems ommands Dept. Navy, Washington D.., 984. [7] arey, W.M. & Richard, B.E., Ocean Ambient Noise Measurement and Theory, Springer, NY, p. 2, 20. doi: [8] Rosenhouse, G., The spectral eect o masking o intruding noise by environmental background noise. Acoustical Society o America, Spring 204 Meeting, Providence, Rhode Island, 5 9 May, 204. doi: [9] Johnson, J.B., The Schottky eect in low requency circuits. Phys. Rev. 26, pp. 7 85, 925. doi: [0] Johnson, J.B., Thermal agitation o electricity in conductors. Phys. Rev. 32, pp , 928. doi: [] Rosenhouse, G., Smaller samples o the same properties are less vulnerable than larger ones, as a general rule that emerges rom the tail statistics. Int. J. o Design and Nature and Ecodynamics, 3(), pp., doi: [2] Barnes, J.A. & Allan, D.W., A statistical model or licker noise. Proc. IEEE, 54(2), pp , 966. doi: [3] Pearson, K., The problem o random walk. Nature, 72(865), pp. 294, 342, 905. doi: dx.doi.org/0.038/072294b0 [4] urst,.e., Long term storage capacity o reservoirs. Trans. Am. Soc. ivil Engineers, 6, pp , 95. [5] Mandelbrot, B.B. & Van Ness, J.W., Fractional Brownian motions, ractional noises and applications. SIAM Review, 0(4), pp , 968. doi: