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212 PROCEEDINGS OF THE IEEE, VOL. 70, NO , MARCH 1982

l / f Noise

MARVIN S. KESHNER

A b t m c t - l / f noise is a nonstatioaary random processsuitable ormodeling evolutionary or developmental sys tems. It combinesthe stronginfluence of past events on he future and, hence somewhat predictablebehavior, with the influence of random events. Nwstnt ionny autocol-

ton functions for l/f nobe a e developed to demonstmte that itspresent behavior is equrlly CorreLted with both the Rcent md distantpast. he minimtun amount of memory for a system that exhibits l/fnoise is shown to be one state w t b l e per decade o f frequency. 'Lhesystem condeneerr ts pasthistory into the pnsent dues of its state d

ables, one of which npreaa ts an average over the most recent l unitoftime,oneforthelwtlOtimeunits,100units,1000,10000,andsoon. E a d ~uch s t a t e variable has m eq influence on pnsent behavior.

IINTRODUCTION

HE 1 f noise is a random process[ 1 defined in termsofthe shape of its power spectral densityS ( f . The poweror the squa re of some variable associated with the ran-

dom process,measured n anarrowbandwidth, is roughlyproportional to reciprocal frequency (Fig.1):

constantS ( f ) = , f , r where: 0 < Y < 2

and Y is usually close to 1.

1 f was noticed f m t as an excess low-frequency noise in ac-cuum tubes and then, m uch later, in semiconductors. Modelsof l/f noise, based on detailed physical mechanism s, were de-veloped by Bernamont [21 in 193 7 for vacuum tubes and byMcWhorter [ 3 in1955forsem iconducto rs. Since the mid-fifties, 1f noise has been observedas fluctuations in the param-eters of many systems. Many are completely unrelatedo eithertubes or semiconductors and their 1f noise cannot be explainedwith either model. For example, l/f noise has been observedas fluctuations in[ 4 ] , [ 51

-the voltages or currents of:vacuum tubesdiodestransistors

-the resistance of:carbon microphonessemiconductorsmetallic thin-filmsaqueous ionic solutions

-the frequency of quartz crystal oscillators-average seasonal temperature-annual am oun t of rainfall-rate of traffic flow

work is based entirely on the Doctoral Dissertation of M . Keshna,Manuscript received April 7, 19 81 ; revised December 22 1981. This

supervised by W. M. Siebert and R. B. Adler and submitted ( 1979) tothe Electrical Engineering and Computer Science Department of theMassachusetts Institute of Technology.

The author s with HewIett-Packard Laboratories, Palo Alto, CA 94303.

f

Fig. 1. The power spectral density of I f noise. When compared withwhite noise which has equal power at all frequencies, l/fnoise has anabundance at low frequencies.

-thevoltageacrossnervemembranesand synthetic mem-

-the rate of insulin upt ake by diabetics[61-economic data [ 71-the loudness and pitchof music [ 81.

The presence of 1/f noise in such a diverse groupof systems(plus others not mentioned) has led researchers to speculatethat there exists some profound law of nature that applies toall nonequilibrium system s and results in1/f noise. Numerousspecific models have been propos ed, but not one can accountfor the presenceof 1 f noise in even mos t of the systems listed.Perhaps the only similarity am ong these systemsis the mathe-matical description that leads o 1f noise.

The applicability of l /f noise to the description of the fluc-tuations of pitch and loudnessn man y types of music stronglysuggests that 1f noise is somehow ess random than otheroise,that there is a relationship or correlation between events thatis lacking n other noise. But, to date, hat relationship hasremained a vague notion. Over what ime scales areeventsrelated? A system for which present events are influenced bythe system's past history must have some mechanism for in-formation storage, i.e., for memory. What kind of memory isrequired for 1/f noise? How muc h and for how long does thesystem remember? This paper wi address these questions andwill demonstrate that l/f noiseis appropriate for the description of fluctuations in system s that evolve with time.

In Section 11, a linear system that produce s l/f noise is ana-lyzed and used to derive an exact nonstationa ry autocorr elationfunction. If the time over which the processis observedis shortcompared with the time elapsed since the proc ess began, thentheexactnonstationaryautocorrelationfunctioncan be ap-proximated. The result is an autocorrelation function that isalmost stationary and fromwhich we can describe the memoryof systems exhibiting1 f noise. In Sec tion 111, th e pow er spec-

branes

0018-9219/82/0300-0212 00.75 982 IEEE

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KESHNER: l/f NOISE 213

Fig. 2. The path of a particle undergoing Brownian motion.

tral density corresponding t o the approx imate autoco rrelationfunction is derived. Th e resultis consistent with experimentalobservations of l/f noise in tha t the power spectral densityisstationary except for its apparent steady value which dependslo ithmically on the timelapsed since th e process was start ed.In Section IV, we consider anoth er linear system and, fromit,arrive at limits fo r how much memory the system must have.Although th e derivations are based on physical systems, thesame ideas are applicable to strictly informational systems.This will be discussed in SectionV.

Ii. l / f N o ~ s ~ : NONSTATIONARY ANDOM PROCESSThe integral of th e power spectral density fdrl / f noise with

1 < < 2 (the most commo n case)is infinite. If 1 f noise werea statio nary random process, on e would conclude that its vari-ance is infinite and, therefore, that the values of the processoften wiU be extreme ly large. One way t o avoidth s problemhas been t o pos tulat e the existence of a lowest frequency, be-low which the shape of the power spectral density changedsothat the integral would converge.

Experimenters studying 1f noise have searched carefully f orany evidence of a changein the shape ofS f ) a t very low fre-quencies, The lowe st frequency at which they can accuratelydetermine the power spectral density is limited by the long-time stability of their equipm ent and by the length of timeover which they are willing to observe the random process.In spite of these limitations, one group of experime nters mea-sured the l / f noise in MOSFET's down to 10-6.3Hz or lcycle in 3 w eeks 191 No change in shape was observed. Usinggeological techniques, the l/f noise in weather data has beencomputed down to lo- Hz or 1 cycle in 300 years [ 101.Again, n o change was observed. Amongall the experimentalobservations reporte d, only in wo cases has a change beenseen: in th e resistance flu ctu atio ns of thin-films of tin at thetemperature of the superconducting transition 1 1 and in the

voltage flu ctuatio ns across nerve membranes[ 121An alternate solution to th e problem of infinite variance was

proposed by Mandelbrot [ 131. He suggested tha t l /f noise(1 < < 2) should be treated as a nonstationary random pro-cess. As such, its variance and also its power spectral densitywould be time dependent. Observing the process for a finitetime w ould alwa ys yield a f init e variance. But, only b y know-ing the exact s ta tef th e process at some prior time, could on emake any statements about resent behavior.

To illustrate this idea, consider another well-known non-stationa ry ando m process: aparticleundergoing Brownianmotion (Fig. 2). One can make statem ents abou t the presentposition of the particle only if its position at some prior time

was known. For example, assume that the particle was in thecenter at t ime t = 0. At some later ime, t = T, t is l ikely t o be

RC LINE

Fig. 3. A lumped representation of a coqtinuous resistor-capacitor rans-mission line excited by a white noise current source.

within a sphe re at adius R centered aroun dits starting position.At a s t i l l later time, the radius will be larger. Th e mean posi-tion of the particle is a constant, equal t oits starting position.But the variance in the position of the particle, correspondingto the square of the radius of the sphere in whichit is likelyt o be foun d, increases linearly with ime. If one insisted ondescribing this process as stationa ry and collected data fromall time, one would be forced t o conclude that the variancewas infin ite; the particle could be anywhere with equal prob-ability. Bu t, describing the process as nonstationary andknowing where the particle was at some prior time, one canstate where it will be later as a random variable w ith finitevariance.

In apparent conflict with Mandelbrot's suggestionis the factthat experimenters measure the power spectral density of l /fnoise, m ostly withou t difficulty, without assuming that heprocess is nonstationary, and without knowledge of its initialstate. If th e process were nonsta tionary, would not th e powerspectral density changeas a function of time and initial condi-tions?Experimenters ometimes find theamplitu de of thepower spectral density t o vary among identical systems mea-sured a t diffe rent times, bu t the shap e and,in particular, thevalue of the expo nent 7 are quite consistent. (Fo r example,see the data presented by Hooge[ 141.)

We begin by considering in detail a simple system tha t gives

l/f noise exactly, with = 1, and deriving its nonstationaryautocorrelation unction. This procedurecan be generalizedt o derive the nonstatio nary autocorrelationfunctionsfor all1/f noise 0 < 7 < 2 [ 15 The simple system to be consideredis illustrated in Fig. 3.

The system consists of a noise current source with a whitepower spectral density of magnitudeZ driving the input of aone-dimensional con tinu ous resistor-capacitor transmission line(RC line) of infinite length. The impedance of an infinite R Cline is

z ( f ) = [ q z2nfC where: R resistance of the line perunit length

C capacitance of the line

j = f i .

per unit length

The power spectral density of the voltage at the inp ut of theline is

S ( f ) = Pn f C .R

Provided that the lineis infinite in ength,S ( f ) is exactly pro-portional to l / f down t o zero frequency. However,if the lineis finite (and terminatedin a fMte resistance), thenS(f) will

have a lowest frequ ency below whichS( f s white. The valueof the lowest frequency is determined by the length of the

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line 2 )

PROCEEDINGS O F THE IEEE, VOL. 70, NO. 3, MARCH 1982

II

I

III

I

1A==.

To derive the nonstationa ry autocorr elation function, wewillallow the line to be infinite andwill construct the system suchthat at t ime = 0 the voltage on the ineis zero everywhere.

The white noise current source will be constructed in a waytha t greatly facilitates calculations, but actuallyis a qu ite gen-eral form of sh ot noise with an average value of zero. Considera process th at once during the interval of timeT produces aunit impulse of currentwhich is either positive or negativewith equal probability, and which can occur anywherewithinthat interval with uniform probability.

I = u o t - t o ) where:

1p d f ( t o ) =

TO < t o < T

u o t - t o ) denotes animpulse at timet = t o ;

pdf( to) denotes theprobability density func-tion of to ;

= 0, otherwise.

Summing aarge number N ) f indepen dent, identical rocesseswould create a shot noise current, with a white power spectraldensity of magnitude

2 NT

S(f)=-- - Ii.

For our computation, an increase in the intervalT wi be ac-companied by a corresponding increase in the num berof inde-pendent processes, so that the magnitude of S( f ) will remainconstant. This is exactlyequivalent to requiringa constantarrival rate of a singlePoisson process.

A current impulse at t = t o excites the RC line to give a volt-age response of

where

1,or t > t o

0, for t < o .u-1(t - t o ) =

The autocorr elation function of the voltage resulting from asingle impulse is

R ( t 1 , t z ) = E { u ( t l ) v ( t z ) }where E { } denotes expected value

' u - l ( t 2 - t o ) p d f ( t o ) d t o .

Noting that both terms are nonzero onlyf the impulse occursbefore t l and t 2 , and assuming t 2 > l we obtain

Fig. 4. A sample of l/f noise where the time of observation s shortwhen compared with the time elapsed since the process began.

For N identical and indepe ndent processes, the frnal resultisjust N tim es the above, which is

Imagine th at th e R C line was constructed a very long timeago, i.e., t p is a big numb er, and that wewill observe the pro-cess for a comparatively shor t ime Tabs << t 2 . Both imest l and t 2 will be within t he observation interval, but th e start-ing time, t = 0, will not be. As a result, the ratio of t l and t 2will be nearly equal to one (see Fig.4). Thus we can approxi-ma te the previous result by

This proced ure can be easily generalized to derive the auto-correlation functions for 1 f noises for which Y 1, but isanywhere in the range 0 < 7 < 2. The details are presentedelsewhere [ 151,butasummary of the results is presentedbelow. For the purp ose of comparing the nonstationary withstationary autocorrelation functions, the quantity t 2 - t l ) h asbeen replaced by 7 ) .

R(T) : 1 7 I r - l for T > O o < y < l

R t 2 , T ) a l n 4 t z - In 171 O < T < < t 2 , y = 1

R ( t z , T ) a t 2 7 - 1 - ( c o n s t a n t s ) ( 7 ( r - 1 1 < 7 < 2 .

111. THE CORRESF ONDING POWER SPECTRAL DENSITY

Assume t hat an experimenter observed th e system for a timeT O bmu ch shor ter than the time elapsed since the system wasassembled and he process begun. Fur ther assume tha t heexperimenter hought he wasobserving a stationary randomprocess and computed S( f ) based on tha t assumption. Whatwould be the result?

In the previous se ction, we derived the non station ary auto -correlation functions for 1/f noise. These functions were thenapproximated for the ase where the timeof observation to^)was mu ch shor ter than the total time (t2) elapsed since theprocessbegan. The esult was tha t eachapproximateauto-correlation function could be w rittenas the sumof two terms,one dependent only on t 2 and one dependent only onT .

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KESHNER: l / f NOISE 215

f

Fig. 5 . The power spectral density one would observe in measuring thenoise from the RC line model.

R t 2 , 7 ) = f t 2 1+ f (7).

In effect, henonstationaryautocorrelation unctionsareapproximatelystationary.Thenonstationarybehavior is en-tirely contained in the added term that depends only on t 2 .

By taking the Fourier transform with respect to the variableT

we will compu te an approximately stationaryS( f ) and evalu-ate he effect of the added erm. Since onecannot observecorrelations over times larger than the total observ ation timeTab, we will limit the value of to th e ange

0 < 171 Tabs.

For l / f noise with 1, by dividing the argum ents of bot hterms by Tab, we have

Using transform pair (1) for Erdelyi (p. 17) [ 161 and the sca l -ing properties of Fourier tran sform s, we obtain for th e secondterm

The transform of the first term with respecto ?is ust the trans-form of a constant, which yields an impulse at zero frequency.

where: uo ) denotes the unit impulse located at f = 0 . Theoverall resultis

for I I > >TObS

1

This power spectral densitys illustrated in Fig. 5 . S ) is pro-portional t o l /f down to the lowest frequency allowed by thelimited observation ime. Both he exponent and he magni-

Fig. 6. Autocorrelation functions for ( l l f ) * +7 noise. For ‘ = 1 , therecent and the distant past have almost equal correlation with hepresent.

tude of S f ) for frequencies above this lowest frequency areindependent of the value of both t 2 and T o h and are, there-fore; stationary. The apparent steady value of the process overthe intervalT h would be the integralof S ( f ) for

1< f <

Tabs T O b s

and would equal

The nonstationary autocorrelation function derived in SectionI1 corresponds to a power spectral density that appears to bestationary. Measurements of S f ) made without knowledge finitial conditions would be consistent within the range of fre-quencies allowed by the limited observ ation time . The v arianceof the process grows ogarithmically with time, but observingfor a finite intervalill always yield a finite variance.

It is comm on practice in the literature o normalizeS( f ) bydividing by the squa re of the stead y value of the process. Whenan independent constant term is not dominant, the magnitudeof S(f) would hen depend ogarithmically on t 2 . This mayexplain som e experimental observations where the value of theexponen t vanes slightly but the value of the magnitud e som e-times varies enormo usly.

IV. THE MEMORY OF l/f PROCESSES

White noise has no m emo ry of the past; current values ofthe process are nde pend ent of past values. Most processesare not white out to infinite frequency, but their mem ory islimited, often to one time constant or state variable. l / f pro-cesses do have m emor y. The questions are how muc h and forhow long are these processes influenced by their past? In thissection, we wi l l explore these questions by fmt examining theapproximate autocorrelation functions from SectionI1 t o illus

trate how long the rocess remembers and then y constructinganother linear system that yields l/f noise to illustrate howmany initial conditions the process remembers.

1/f noise is a random process with a very long mem ory. Thederived autoco rrelation functions, illustrated in Fig.6 , decayas a power of time, slower than any exponential, except or thecase of xactly equal to 1, for which the decayis logarithmicand slower than any power of time. The closer s to 1 , thegreater the influence of the distan t past when comparedwith

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216 PROCEEDINGS OF THE IEEE, VOL. 70, NO 3, MARCH 1982

log s (

SLOPE = -2

Fig. 7. Another linear system hat yields l/f noise,approximately.

remember one number.

Each section has one state variable (the capacitor voltage) and can

the influence of the recentpast. When Y equals 1, presentevents are approximately equally correlated with ev ents fromthe recent and the very distant past. For Y near either 0 or 2,the process is influenced by the recent past much more stronglytha n by the distant past.

Since it is clearfrom the autocorrelation functions hat a1/f process is strongly nfluenced by me past, especially for7 nearly 1, one might inquire how much information the pro-cess remembers. Assume that he processis modeledwith al inear system, and hat ts entire past history is represented

by t he present values of its state variables. How many statevariables are required for such a system when its fluctuationshave a l / f power spectral density? Or how many numbers arerequired t o summarize the influence of the past on the present?For white noise, the numberis zero. For Brownian motion, itis one ; only t he initial position is required. But for l/f noise,hundreds of numbers are required.

The continuous R C line model that yields an exact fit to a1 f noise power spectral density has an infiniteumber of statevariables. So does thesum-of-distributed-time<onstants modelintroduced by Bernamont [ 21 in 1937 to explain l/fnoise invacuum tubes. However,if we require only an approximate fitprovided by a lumped, inear system, hen only hundreds of

state variables are required and the number ncreases slowlywith the degree of precision required. Alu mpe d linear system,quitedifferentfrom the R C lineused nSection 11, will beanalyzed t o place both a lower and an upper limit on the re-quired number of state variables for a l/ f process. Itwill also

be used to illustrate the similarity among 1If processes withslightly different values of the parameter7'.

A simple linear system can be constructed to have a powerspectral density tha t differs from (l/f )' by a constant perce nt-age error which can be m ade arbitrarily small. It is illustratedin Fig. 7. It bears little relation to th e previous mod el excep tthat i t s also a linear systemdriven by a white noise source andthat S(f) of some variable Vo f ) will be l /f . Itconsists ofresistor-capacitor sections separated and isolated by unity-gainbuffer amplifiers. Each section contributes one pole and onezero to he overall response and hasone state variable, thevoltage on the capacitor. The response of a single section isshown i n Fig. 8.

Of course, the required number of sections depends on howclosely the approxim ate and exact curves must agree.An ex-act fit would require an infinite number. However, itis inter-esting t o determine how many are required for a5 percent fit,and then to use that value to discuss the nu mber of statevari-ables and the number of values needed by a l/f process tosummarize the past. A construction based on he foregoingconcept is presented n Fig. 9. The continuou s ines are hepowe r spec tral de nsities for various alues of7 plotted on a logversus log scale. Superimposed on each of these lines are ottedlines representing the asym ptotes for(f) f an approximating

POLE ZERO I fLOCATIONOCATION

Fig. 8. The frequency response of a single section of the linear systemillustrated in Fig. 7.

log f

Fig. 9. A curve fit of the power spectral densities of approximatinglinear systems to obtain l/f noire with various values of 7 . Themarks on the abscissa and the ordinate, are one decade apart.

linear system. The positions of the poles and zeroswere chosenfor a maximum m o r of 5 percent.

Theresults may be somewhatsurprising.Approximately1section (also 1state variable) is requiredperdecad e of fre-quen cy t o fit a l/f power spectral density. Slightly fewer arerequiredornd If3/ , withhe value decreasingtowards ero per decade as S(f) approacheswhite noise7 = 0) or Brownian motion 7 = 2). (Note: Brownian motion

requires a total of one state variable.) For a 1f power spectraldensity extending from 1 O- Hz t o 1 0 H z , nly a hundredsections are required for a better than 5-percen t fit. For a 1-percent fit, the num ber is probably less than 500. This meanstha t the system would have only hundreds of sta te variables

and would summarize he past with only hundredsof numbers.Even if we demand a perfect fit, but only observe the process

for a finite time, the number of state variables whose presentvalues must be treated individua lly is not infinite. The effectof a finite observation time (Tab) is to sm ooth the frequencyresponse by averaging over a bandwidth( fo of approximatelythe reciprocal of tha t time.

1f0 --.

T O b s

At most, we w ould need a pole and zero at every multiple off o ) . Thus the maximum umber of distinguishable state

variables is the ratio of the high-frequency imit (fh) to thereciprocal of the observation time.

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KESHNER: I/f NOISE 217

f hmaximum = -.fo

For purpose of comparison with the number for a 5-percentfit, choose the low-frequency limito coincide w ith th e lowestfrequency allowed by the observation time. (The effect of allstate variables whose time constants are much longer than the

observation ime can be grouped in to one present value: th eapparent steady value.) Th e resultis a range for the num berofstate variables (N ) given by

l o g [ t ] < N < - . f ho

The construction of Fig. 9 suppo rts the view that l / f noiseswith slightly different valuesof re fundam entally similar.They all have a u niform distrib ution of poles per decade of fre-quency an d over the range: 12 < < 312 have approximatelythe same numberof poles per decade. The major difference inthe system for different valuesof s in the relative weightingof the effect of each state variable on th e present valueof theoutput .

For the durationof this paper, the autho rwill use the numberof sta te variables required for a 5-percent fit as the minimumrequire d for the process. Choosing a slightly bett er or worsefit would have on ly a small effect o n th e minimum number.Combining the result for the minimum number of state vari-ables required with the characterizationof the autocorrelationfunctions developed earlier, we arrive at t he following des cri ption of the m emory of 1/f noise for ery near 1. The correla-tion of present events with those from a ime7 long ago wouldresult from the voltage on all cap acito rs whose discharge ratewas too slow to allow for an appreciable change in the timeT.

All capacitors whose poles were at low enough frequencies but

above the lowest frequencyfo would be included.1

f o < l f l < G

Since each capacitor has n equal influence on present behavior,the magnitude of the correlation for 1 is just proportionalto the total number of capacitors included.

R 7 ) : [ 1 per decade] x [number of decades]

- log - - : fo].2m

- onstant - log

Of course, t is is the same resultas before.

V. DISCUSSION

l / f noise is an evolution ary random process. Its present be-havior is strongly influenced by its entire history. Furthermore,its memory is dynamic; the influenceof recent eventsis addedto andgraduallysupersedes the nfluen ce of tant events.The influence of the distant past fades very slowly, eitherasa small power of elapsed tim e or logarithm ically, very. mu chslower than he exponen tial relaxation imes associated withlow-orderdifferentialequationscomm only used forsystemmodeling. These deas will be llustrated first by consideringl/f noise in MOSFET’s and then in informational systems.

Most physical ystemsaregraduallyapproaching hermo-

dynamic equilibrium and the influenceof their initial condi-tions decreases with time. Fo r example,a MOSFET consistsof concentrations of material, which given enough time, woulddisperse. The usual description of its dynamics presumes thatit can be divided into tw o subsystems, one tha t changes veryslowly when compa red w ithhe time of observation andanotherthat changes very fast. Its structu re changesso slowly that for

practical purposes, it is considered static. On the oth er han d,the distributionsof the electrons and holes within he transistorhave very short relaxation times and are considered to be inquasi-equilibrium.

The presence of 1 f noise in MOSFET’s, down to the owestfrequency allowed by t he limited observation ime[91 , suggeststha t t he division in to just two subsystems is inappropriate. In-stead, he ransistor mustb e c o m p a e d of subsystemswithrelaxation times comparable toall time scales of interest. Nomatter how longt is observed , some rocesseswill have reachedequilibrium and th e values of their nitial condition s wi beforgotten. A few processeswill be demonstrably changing overthe observation interval,while otherswill be changing oo slowly

for detection and will preserve their initial conditions almostcompletely. The length of time tha t one observes the systempartition s the subsystems in to three categories: fast, compar-able, and slow. Changing the observation time merely alters th ecategory in to which a particular subsystemis placed; the pro-cess is fundamen tally the same. T o characterize th e behaviorof th e transistor, one must know the present valuesof each ofthe state variables with times comparable to or faster tha n th eobservation time. In addition, o ne must know the aggregate ef-fect of all the s ta tevariables thatd o no t change and whose sepa-rate effects cannot be distinguished during the observation in-terval. ll of these slowly changingvariables arerepresented byone num ber: the appare nt steady value of th e process. Chang-

ing the observation interval alters which variables are groupe dtogether and which must be treated individually.Informationalsystemsaccum ulate nformation . Withpass-

ing time, they exhibit a general increase in structu re and com-plexity, rather han a decreaseas in the case of a ransistor.Biological evolution,ultural evelopment, the growth ofgovernments, the development of economicystems,ndpersonal growth and development are good examples. In addi-tion works of art tha t evelop central themes, such asovels andsymphonies arealso good examples. Voss and Clarke [ 81 haveshown hatwhen music (which is obviously no t a andomcomposition) is treated as if it were a random process, the n thepowerspectraldensity of both ts amplitudesandpitch asfunctions of time) are l / f . The same maybe true for manyinformation accumulating, evolving systems; some parameterswhen treated as random processes may have powerspectraldensities that are 1If 1 f noise com bines the strong influenceof past events with the infl uenc e of current events. The resultis an overall conte xt or pattern and som ewh at predictable be-havior, b ut with th e possibility of new trend s developing andof occasional surprises.

A l / f process appears to extract rends and conden se datainto summaries that, in turn, determine the valuesof the sys-tem’s state variables and influence present behavior. Focusingon the minimum num ber of state variables required, we haveone with a t ime constant of say 0.1 units of time , one with 1unit,10units,100units,and so on. The value of the s ta tevariable whose relax ation (or averaging) time is 1s representsthe average behavioror trend in th e process over the past 1-s

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218

o r so. The value of the 1 00- s state variable represents a sum-mary of the most recent 100 s. Present behavioris influencedequally by each of these state variables. Each one representsa tren d in th e da ta, but over different time scales. Recent datamight dominate the valuesof the shortest time state variables,but it will have a progressively diminishing imp act on the lon gerones.Persistentnew rends will cause the process to adapt

logarithmically w ith time, as more and more of the state vari-ables reflect the new trends in the data rather than theld.The modelsdeveloped for l/f noise might be useful for

modeling someof the informationalsystems mentioned above.The idea that inform ation describing the pastis summarizedand stored in the form of trends over different time scalesisparticularly appealing. It seemsclose to the way humans re-member (and incorrectly remem ber) information, as parts ofconsistent patterns rather han as separate,unrelated pieces.The fact thatmusic has 1f noise statistics and tha t when note sarechosen at random , hey sound most musicalwhen theirpower spectral density is l/f, suggests a connection betwee nthe way hum ans perceive and remember, and the struc ture ofl / f noise.Because of thisconnectionand the influence ofhumanmem ory nd behavior on he developmentof ourinstitutions:he development of ourselves, our cono micsystem, our governm ent, and our culture may each have thestatisticsof a l /f noise random process.

As a final note, consider th e task of best predicting the futu rebehavior of a l/f process. Since present and futu re behav ior arehighly correlated with past behavior, one must understand theinfluence of the past to best predict the future. For al / f pro-cess, the past behavior can be summ arized by th e present valuesof the system’s st ate variables. Approximately o ne per decadeover the time spanof interest is the minimum number required.Any fo rm of regression analysis appliedto a I/f process wouldhave t o est imate not just the verage value of a pa rameter overthe entire extent of the dat a, but also its value averaged over 1unit of time, 10 units, 100 units, 1000, 10 000, etc., extendingfrom the sh ortest o th e ongest timesof interest. Eachof theseaverages would det erm ine the value of one sta te variable, an deach state variable will have an equal influence on the futurebehavior of the process. Fo r best predictions, one would com-bine the present value of eachof the s ta tevariables with an un-derstanding of the mechanism bywhich each of them influencespresent behavior.

PROCEEDINGS O F THE IEEE, VOL. 70, NO. 3, MARCH 1982

ACKNOWLEDGMENTThe author would ike to acknowledge thecontributions

of his thesis co-supervisors,Profs.William M. Siebert ndRichard B. Adler, to the materialpresented in thispaper.Prof. Siebert introd uced me to this topic with the idea thatl/f noisemightbeappropriate for modeling th e long-timebehavior of complexsystems. Prof. Adler’s majorcontribu-tion to the thesis work was on the topicof thermal d iffusionmodels for 1 f noise, which may be published at a later date.

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