Analog equipment for processing randomly fluctuating data

APPEMDI Z

Re1rcdt.-, th h,

,,ti f

the Atroneuttral Fngirteersny Qeys.., fr,e. M. 19S ,uu.

Analog Equipment for Processing Randomly

Fluctuating Data

FRANCIS B SM1TH

L oriçey

Aerar,t,cdi Lborto,, NA.CA

Errineota!

tuthec of hLeing. u!!er, and trntsphcç,r

turbulenceproduce d.ta hch ut not bIe

to nai

b

the iu procei o rneuringor '3.ating disirete valuesof rc force, pre re. or morrer.t. !ntead, 'bei rancorrit

tuctudting qudniit3e, ivcì'ed, it be,r,met nces%arv to u,r

istica1 metho-1 tt aria! u an4 decri!,r the pheoomrno',

rtng instigated The rvpe o!înformttor, uizaL

reur&

are pry,bahihlvdi ribetkris. power spectra, and cro,., pectrA

In ardu t't aod the tiror sod .zpenseinv1'ved us.ng ccii. entIoyi.iI numerica' methcid for securing he vriou, statlst&cai

a',e's.

the NACA Ix%es a Iniquie u? recordu,i d&t* on rnagl,eric tapeand pl*yio it bsktntoeectron.cs-alLganavzev which perforai the desiret! *n*iys1 andautorriaticafly pint thi

ui!,

Tht 'nagnetc tape -ecording ssternavidthe itaog analyur. are described in this paper and their appbczrioo to aunnauttoal

probern. t i}tu'sred A diaiuwortof the compa.rst.,ve accuracy and reiabi1ity o? the numerira! and analog me(hos is also

n-clud.

i INTROt,ucTrorc

Ip

_{aeronautical research and development it is possible}MeCE OP lift gXPERIMaNTAL Wi)&X undertaken iii to set up eper1rnents u that the amplitude of any quan-tity being measured remains very nearlY cotstarit dur-ing a given test run In thi, case a precise nieasure cl

the static level is sought and any uctuatiofls in the level occurring during the run are considered to be

"noise which must be eliminated by fiiter or (aired throtigh on the time-history record

Howevei. many phenomena being studied produce data in which randon amplitude uctuatiors are

in-herent and measurement o average or (aired 1eve only

is of little or no &1ue Typical examples are gurt

aim-in errors encountered in fire control systems, loads irrt-posed ori aitplanes by turbulence or bufeting, streSses

produced by ergire notse or vibration, and aircraft

landing gear hads caused by rough runways Tn thtse cises, the information sought must be secured by atial-Presented at ihr Ekuunic Aids ro the Aircrafi igd,..q!r

eoo,it Tweiy TP ted Aiii.JMeeting, AS, \e 'i't:k, Januat 2 .

.erufliutI(iI 'i,trt-h Tnstrirnent Rse.rçh

i , _itOu

ysis and stdu of thf' ra,nlj:iI itmpi:tude fluctu4t'Jris ahúut the neti '- alue.

Iht' ¡i',-t 'ractuc! wiv ti

esri1-e qu.antittte1v

these atiplitude u ariatt'r.s is to use- the st.tistcal

tr-ch-mi.lueS Which ha Ot,it dt'-ibei fir rilvzingstationar.

raridurn !im seriec fh' s'mp'lest and tr.ost faintli,&.

of these t.hn:qur

i thc probabihtv clistrihutt'i.

which, fl)r exa:nple, has beei use-d for yeaxs to de'scrtLi

atmospheric turbulence Recently,

tht

generalize'!

harrrictmc arialvss t«hniques have been developed ari are being sunecstuilv applte-d t many rand'm- tvv-data analviis priblers

The theuiretiral aspects of these techniques have beei

discussed by Wener» T.iev. and Ricr.

and

specif-applications t' aerodvnaimc prùblerns have beer

de-sctibed by Clenieritson,' Lie-pmanri,' Summers,' Press,

Chilton.' and othcrs

These authors have demnnstrated that statistical

techr.ique are extrerielv useful itt aeronautical

re-search. However, il ccnver.tional time-history records

arid ntrnencal data w&rk-up procedures are used to secure the varius reiuir,d analyses, the application o

these teduniqoes to actual samples cf experimental data s a time-consuming and expensive process. This is

es-pecially true tri th case of power and cross spectra de-terminations, ¡n addition to the digital computer

eu-pense involved tn making the thousands of required numerical calculations, several days may be required to read the necessary number of data points (rom the

tune-history r&cord. These factors often seriously

re-strict the number of statistical analyses that might be

made, or force one to limit seu-erelv the lenr..h of the

data samples and to accept the consequent poor st

tistical reliability.

To overcome these irnitations the NACA baa com-bined commercially available electronic equipment with

NACÀ-develot.4 components to prrvide as'. analog data pra-esSun Iacditv which prriduces the required

sta-tisticai anali ses i'ipidJy and ine.erisively

-cility oonshts uf toux bai.- cieITle-tltS' l) n-i

tape data storage and playback system, 1 a probahi!

analyzer, and (4) a cross pectraI drnsty analyzer. This p&per will descnbe these pieces of eqiprnent. in the order listed. aM wifl bnefly ijiuctrate the kpph cation of the varioub types of analyses to aer'riautical

TeSITh.

(Ifl

_{MAGNETIC Ta REcoRtG At PtA5CK}

SYSTPM

An essential jart ol the data proces1ng facJ*tv is (he magnetic tape ¡ecncding eqUipTntut illustrated in Fig I

This equiprntnt provides a iiwui' for storing the data in such _{way that it can be accurately r produced iii a} form most suitable for analog procecring.

Instead of recording data diretv on magnetic (ae a frequencvmodulated carnet svsteit ic Used. Thi permits the storage of very kw fre1unrv and d.'. and eliminates much oí thc 5U71OUS amplitude

_vSìi

tier, which would otherwmr be caused by tape ux1rer-fections and ttpe wea.r _{It also makes possible thr di} red recnrding of FM adit tClcflLrter dat.i.

Where the data to be processed are _{p ductd b} a stationary facility such as a wind tunnel, the _{tape rt} corder and FM rnrdu!ators are located at the fa i!it Where the data are prûdticed by an airplene. hdieoptir.

ne rirkt-propdJed rnissik, the dit _{axe tranmstter! to} a tape reeor'kt on the ground by a r1 _{te1eiuet.r li,ik} Or. if the data to be proctssed have been_{p vuuslv}

re-corded as a trace de

_{ctit,i on íihtoí'he paper}

_or

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4-A/ERAGIP4G L.. CIRGUIT R EGO R DER u-r

tilni, a tr:4ncrihilg :1i.i _{:ised jti convert t;n ,tc}

If ecl ..r- to .t fre&ii'r'i. i t.Ii_4tkd rarrtcr _{h1.'h is Ti}

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ri r r i. . ¿real rleI cf t1,hil,tv n h.niiltir: rli1. .ìnlIrks -'..q:ii'.lr-

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liv Lai tot _{.s bh as} I t. _{Iv chrigiug i}

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he rtt'irrde4 and repr io the FM s' stem t riu urr t . bi.

.icd f ,

_{cli u}

lr.JTiri th (ri lt't.i

n. hfite

-trr!er aiid ici ord,n Jzre ti

The anaJ' .u;al,ui m.t}e»ds zrquirt thiit data

ph' he sa:itd j Idrgr

r: _{r q tiriii-.. b) the} ng t ut i;utnt L,rris.-tiii:ti _{t-at:h sauipk rn the trt.e}

I 'plii t-' _{! t. .1}

k ifltiilUl1L'A l.sor bruire. heitig pI.cved

1)1(1.. ii: to I in FM k -ra .ìuL i 'N j n. _{hc ao.gfv rr} lic thtte t pt-c A 4riaivjs-r _{thu uii whnle lata} c'rded t!s tai rua'1 l)e' r.ie-s-i its- a prTsfithl1It%T b;

tribuu.ri analyst .

p'tv'r

eitras,lrsitv 3fla!v:rr.

i _{51lt't'r.l deii-tt' ariai ,- 'j !eSe}

tltrt'i-udy-ier 4nd tOe vpe l _{ct.iti'ti.:aJ ul(nrull.tr,cu pr}

duçt'l by tich will tic d i.i.'.el

tri th'- f0tli v:l:g .icr

t utr.s 1 r his paper

III

PR iBAB!I ITY i )1STUi.'TlO1'

i he terri _{fuictscìn shc.ri u th top of Fig 2} its-a r-its-arvimv vits-ar) trig :1uits-antzty of the (vpe to he etnideied throi..ghc.i'. this paper 'í.)re way ti 4r !vr such a function IS ti deterrrjne _{ts 7rÛbabihV ;.} trihutrot. This :s a rueasure - _{the prtportion cf tita1} thr during which th _{mpitude of the varying quai}

tity exceeds gwen Ieves.

For example, suppose the tute functionghown at t'ri

tcip it.f the f.gure were a plot cf the g'ast velocities en

couritet-d by

n airrkae

_{virig through rough air} The intensity of the nzrhujence could be showr. by

pht-ting the probabiity that the_{gusts wcìud exceed spei.c} vekcities aganst gust veiocitv. _{The gl.ists iu the at} mosphere represent«! by the dstnbuticr. _{shown would}

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_{ARCNA'JTICi EO EfNC, EvIfA}

DATA _í.cOoro

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ANALOG QJlCMNî

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Dribtkn of .cg

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beexpecte'itnexceor4ft.persec about 5pex

't if the tiens and to xcted 2C ft per aec. only about per cent of the time.

The electronic yzer used to detercic probability distrbutiois is iJl'.istrated in Fig. 3.

Th data b..np]e

revorded cn ari endie lp of rnag'netk tape is contLo uoutly played into the analyzer, and U instazitanems amplitude is conpared to a rtferttce voltage Ea, The level of the reference voltage is deternzined by the p-Etien of the poteiitmmeter slider. If the input voltage

s larger than the erence voltage. the açlifier ll

be saturated and the outpi.lt of the bmiter will be at level B, if the tnput is lesa than the reertnce vçiitaje,

theamplifier will be cnt off and the output of th

lin-iter circnit will be at level A. The eTcentagt of tune

that the leve! is at H then represeriu the p hdity f the iriptit being above the ñzed level ¿. By ave'Mirzg the output of the limiter this probabthty nay be read

directly and recorded By slowly diai ing the level

-f the reference voltage and rivirg the paper on the re-rorder in ayncbronizaton. coriplete plot cf the prob-ability dietribution rray be obtained.

To illustrate the use of the analyzer. Fig 4 shows a

distribution of witig b.iffeting cads obtained dur-ui a gi-a4uai pull-up at high speeds. The solid line

it

the analyzer record and the circled points are values deter-mined nurizericslly.

Set-up and r.nnir

time on the

eiectronic analyzer 'wu about 2C to 30 nin. and about 7

man-hours were -ecuired to read the tie-bistc'ry

rec-ord, calculate the mean. and dete,iine nimiericaiiy the circled points.

(IV)POER SPECTRAL t)tK5iT

The probahiity dstcibitin fezr:bes the intensity

charactenstics o the data but does not dtscribe their frequency or spectral chsrcteristics.

Nor de it

fur-nub iufortnatson ufiC:erit for calculating system input-. output relationships or transer furiottors. If this sort

of intormalion is required a second tYpe nl statistical

analysis called the polcer sI.ectxal denaity anaiysi may

be ued

Power rjectxal itensity anatyss islomet,rnrs referred to as 'gtrneralired harmonic analysis" and is s3.mIar to the f aizuliar Fourier sier.es harmonic frequency analysis except in the following tespects' (l Fourier series anal

Vs

is applicable to repeutave functions while general.

izd han'nv'

s.tvsis s appfü aì'Ae to stationary ran

.i''i

'it

iuLcnon3. i!hn1 rated n Fig 5. the ipec

tn1m s Ti »re.en te-d ar a coiv in u cus -urve r ther than as d.is.rete rmsriaonkal!v-re!ated freoue-ncies, and 3;

the sçrum is pPtte-i ri terr.s of men-squared am

ttud

lier mit bandwtdm.! ouk-ale-nt to

aver,1.t-pwec per unit bindwtdth n th

electrical siem; .

stead of unule n-wae anplitude.

The .wr' spevtrux- then repeesents the distnbutic'n f energy ovei the frequency sp«tru.rni Fos example, the por ti-Lni of the tct.aJ energy induded io the frequency

band f to f in the ilure iS represented by the area

under the curve be ween f and

One extremely ucful feature çet the pcwer spectra

ccncent is the simple nput-outp-ut spectra rehtionshtp iUustra-ed in Fig. For linear systems, the power spsctra.l density of the system's output G,f ;se.qal to

the power spectral density of the mr.put G' times the

squared absolute value of the trausfer function Y(fi.

Thus, we ha'.e a re. ationship among the three nariablea su':h that if two of th three ame ru:wri. the thu-ct may

bedrterr,irtd.

1'his sort of Lnput-output.tnansier íurçti'n

rel4ti'r

ship is one whi'h is familiar to the electrical engineer

who is otten fortunate enough to be able to work with

strictly sinusvidal signa. lt should be note-J that the power- spectra c-';ncept now enables th aen-onauticiii

-meer. wh. often his rio choice ut to work with

ran-pso

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T\Y

'3D, 6 (f) -..-- r'noÑ I' O4J'rPJT

so, '

f)

I Y(f) o

.- a1(f)

i _{f Yff)}

ft

6. Lnprt oisput c-'.,a-e, spectral dresii reiaiorikuy W*N M.LJ!v*

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5 Powe' SZ,«tral ier.sity.

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INPUT

7

CONSTANT BANOWIOIN

VJNA$LE ÇLT

RECORDER

Pro. 7 Power ipoetral den,i(v &flatvzeT

domly fluctuating type nf signa}s,' to perform the same, $i*t of 'circuit analysis' as the electrical enneer

For example again consider the problem of ali air-plant flying through rough air: the airplane can be con-siderrd as a mechanism having a transfer function 1(f). the turbulence can be considered to be the input G,(f, and the att-pl.anes response can be consid. red as the

out-put G.(f).

Thus, by measuring an airplane's ransfer function and its response to atmospheric turbulence, it k pos-sible to use the plane as an instrujrtent (or measuring turbulence. as Clementson' and Surrirners' did. Or, if a specitic turbulence spectrum is assumed awl th

air-plane's tranti' function is known, it is possib!e.tn irt-dict the planes response to the turbulence.

Also, by using the techniques developed by Ricv. it is possible in many instances to use power spectra to calculate such things as the number of zero crossings and the number of times the inplítudeof a randomly vary in quantity esceeds certain le.-e!s

Power spectral '-lensity may be nu!nerleally calculated by a procedure outlined by Tukey.' Very briefly sinn-marized, this procedure is as fotlows First, rrid the time-history record point by point; second, calculate the autoc-orrelation function of the data saniple by t'vai-uattrig the mtegral

ff(t ((i +

for several d.ret.e values of time lag T and' third, de-termine the power spectral density of the data by taking the Fourier cosine transform of the autocon'elatoîi function

This process is e4uiValent to passing a tunable, c)n-stant-bandwtdth filter of known cbsracterhtics over the data and measwing the time a.'erage of the square of the alter's output. Except for squaring of the filters output and me difference in the lilt-er's characteristics. the resulta are alnost identical to those obtained from the familiar spectrum analyzers that have been used foe years by eorn,muntcation, and sound engineers

This suggests then that there are two types of dec tronic analog equipment which might be used to

de-tt'riii,nr 1sa'cr Mplctra: one type parallels the nume:-kai process 1w first determining the autocorretation functi'iii tid tht'n taking its transform to get power spectral density, the other omits the autocorrelation function entirely and measures the spectrum directly by means of a scanning electrical filter.

Since scanning filter types cf analyzers are compara-tively simple electronic devices and are commercially available, the NACA uses the direct spectrum measure-ment approach

This type of analyzer is illustrated in Fig ' and oper-ates in the following manner The data sample stored On a çontinunU Iriop of magnetic tape is applied to the bandpass filter Any frequency components in the data 'ahich tall within the filter's pass band will be passed by the ñlter. squared. averaged, and then applied to a c.liret- s-riting re--order. The filter is initially' Set at the tow end of the. freuuericy range and slowly scans ujwu-d throuh the spectrum until the entire [requencv range of interest has been covered; at the saine time the recorder purer is moving under lite atylus so that a con-tinuous iii t of power spet-tral dt':iity against Frequency is obtdirled

The scarinni sjs'etl ot the analyzer is normatIv con sers-ativetv aljust't? so that about thiee passes of the it-ata sanqile n the 1oop are made during the time re quired (or the tUter to a.uì one filter bandwidth Uiukr tht»c coniiitr',ns, the tin&e required for complete analysi, of .ì tvpn.'al record is IO to l.' min. Faster scanning speeds nav be used, hut me "smearing' 01 thi' spectrum might result

The bandwidth cf th scanning flittr 'ri the NACAs equipment cari be adjusted to values ranging froml/2 cycles per sec- to 0O cycles per sec and the true-time range of frequencies which can be analyzed is from 3

cycles per sec to l'OO() CYCICS per sec.

}y taking advantage of the pmihk changes irr tape speed preciously nit-nrtirned, it is pnrsible tu obtain equivalent filter hamivodths q4 001 cycles per sec 'ir -less and t, handle frequencies ranging f rom a fw hua-dredths of a cv.k to 50 or 60 kc. By changing tape speeds when' necessary, it is also poti.ible to handle

05 O-4 EtiCTO$iC £Id*1fZt O COP1J'TtIi POIS 0-t Q IO PtQUt5IC'Y, CRI

Pro 8 Power ipsetral disity of iriag ûirar Ios4s

AERONAJTICAL ENG'N(EiNG Q(VtEW

MAY

955

SPtCTRA.. b Cpi _o-z F1LT OUTPUT

W1MMM

/ i SQUARING AVERAGING C4RCUtT

JTh

_* SCANNING MOTOR

recd lengths ranging from a few tenths of a second to several minutes in duration.

To illustrate the use of the power spectrum analyzer,

Fig. _{shows the speetnim of the shear kadson a}

fighter-type airplane wing under buffeting conditions. _The

Continuous curve was obtained from _{a niagnetk tape} record played through the analyzes-and thecircled points

were obtained by reading the time-history record and

numerically calculating the spectrum.

The results (rom _{the two mcthxls differ by a}

maxi-mum of about 3.0 per cent at the low-frequency end of

the spectxnsn. _{This difference is due, in part at Least.}

to a large, undesired, very-low-frequency component

in the original data sample which had_{to be attenuated}

before the analysis. _{The electrical high-pass filter} used to attenuate this component _{was not identical to}

the equivalent numerical bighpasa filter used (os- the

same purpose; therefore. some difkietiers in the

spec-trum at the low-frequency end weretobe expected

Some difference between the analyzer and the

_nu

merical values also result from the fact that the _shape and bandwidth of the analyzers scanning filter were not identical to the equivalent filter resulting from the numercal process.

.etually, differences of 10 per cent are riot partic-ularly alarming. We are dealing with statistical proc-esses such that, even with errorless data _processIng

schemes, the results obtained from repeating the sante

ir'iht easily vary

as much as plus or inirwa 2. to

30 per cent. _{More will be said about this sub sect later.} Here, to read the record and determine the spectrum using an automatic digital computer required about 10

man-hours; the time required for setting up and run-¿

ning the electronic analyzer was less than 30 min.

(V) Cioss SPECTRAL DEwsir'r

It has been shown that the power spectral denstv

furnishes information regarding the frequency content

of fluctuating quantities which is not pro'ided by the probability distribution. However, where the phe-noinenon being investigated involves study of twoor

ANALOG EOLJLPMfNT FOR RANDOMLY

_{lUCT'JATlNG DATA}

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g4(f)_u.( Y(f }s. g0(f) i(f) u

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- CO-POWER - - - _{QUAD POWER} %% 00-SPECTRUM MULTIPLIER LAVERAGIP4cJ

L*v]-4 Ir Q4JAO-SPECTRUS4I SCAHHIN6 [MuLr1PuER J MOTOR DUAL RECORDER

¡11f., Io _{Crou sprtr1l drri%itvanaIvZeT}

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more related fluctuating quantities. which is usually thr case, some knowledge of the titile or phase correlation

between the two quantities might a1si be required

This infornmtìomi is not provided by the power spectra of the individual functions but niay be obtained from another statistical process called crtss spectra analysis

The cross spectrum betwccn two time functions ii a vector quantity and two spectra are required _{to fur}

nish the ('t,IH1Iett' cross spectral density. _{These arc}

illustrated in Fig. 9. _{The real part is called the}

"co-power spectrum _{and indicates the product of the}

in-phase frequency coiliporients iii the two functions The

unagunirv part is called the quadrature spectrum" and indicates the product nl ihr 90 out-of -phase_frequency components in the two functions. _{The absolute value} and phase angle of the cross sprct.rurn are determined by vectorially combining the in-phase and quadrature

.spec tra.

To gain an understanding of the physical _significance of cross spectra, consider two fluctuating time functions whose power spectra are represented_{(in Fig. 9 as PSD1} and PSI),. _{The cross spectrum between the two (uric} tions will indicate only those frequencies inPSI)1 which

as-e also contained in PSI)1 and which bear a specific.

iionrandoni phase relationship to the frequencies in PSI)1. _{For example, PSD1 and PSD, may include}

power in th same frequency band, but, if the

corre-sponding frequency components are entirely

independ-ent of one another, the phase between them _{would be} random and the cross spectral density would _{be zero} throughout the band. However, if some of _the

(re-q uency components in PSI)1 bear a definite phase _or time relationship to the corresponding components in

PSI),, this relationship will be indicated in the es-riss

spectrum.

¡I determination of the phaseresponse of a linear system One useful application of the cross spectrum is in the

subjected to a random-type input.

_{Recall that the}

power spectra of the input and output made poible calculation of the absolute amplitude of a system's

traris*er f unction but did not I urnith any knowledge of the phase response _{By u*ug cro spectra, both the}

118

ECN4uTCa ENGjNfER:r'G

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PSD(iI i cso: Et: L NijT SPC1J - SPtC1'N

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Tio Il. Crow .çrctra of wing he'din mouieet

s shown by Lee,' the transfer functhrn F(,f) equìls the input-output (4$ p ctruan G(ß divided by the power

spectrum of the tnput G11(f)

An interesting feature of this relationship rs the fact

that the equtior holds true even thougb other

nde-pendent random noises a.re present in the output. This is true since the input-output crois spectrutti wiE ignore

the presence of unc'orrelated random thctuations in

the output.

The -oss spectra has other useful, practical apl1ica-tions to systems with multiple inputs and outputs. but

these are too involved to diicuss here. Â good illus-tration of this usage, howe'rer, is included in Summer,

paper on tmopberic turbulence measurenients

The numerical proceis foi- determining c-ois spectrais

even more lengthy and expensive than that required

for determirnng power spectra. _{Howe-vet, the}

numer-ical i-nuIts e be duplicated by the analog process

Wtxstr-ated in Fig. IC). The two fluctuating data sam-ples recorded on a continuous ioop of dual -channel tape are simultaneously applied tu two synchronized filters of the same type used with the power spectral density analyzer. However, instead of squanng and averaging the outputs of the filters ai would be done to determine

power spectra, the two outputs are fed into a

multi-plies whose output is averaged anti automatically plot-ted against frequency to furnish the co-power spectral density. At the same time, one of the filtri- outputs ii

run through a 90° phase shifter and again Multiplied

by the otit put of the other filter. The product is then

averaged and plotted tri give the quadrature power

spectral density.

To illustrate the use of the aois spectrum analyzer,

Fig Il shows

me data secured from a buffeting

air-foil in a supernic wind tunnel.

The double-peaked spectrum is the power spectral density of the fluctuat-ing aerodynamic forces existfluctuat-ing at a par1,cu3ar soan-vue st*tion and the single-peaked spectrum is the power

spectral density of the fluctuating bending _mcment existing at the root of the wing. The cross spectral

densities between these two quantities _{as determined}

by the &naiyzer are shown in the center of the figure. The calculated a.mplitude and pLuie respolise showr. at the bottoni of the figure indicate _{that the wing is}

sim-llar to a spring rruss system wth low damping. This

simple example does not previde arrriynarnc data il any particular value hut w

_cht'n

tr demorstrare the use igl the analvzir and the type of priblern to whtch

is apthcable.

T-a s.i up and run thnte fçrir i,,ectra n the.detm,nr air aFçzers required ab.L1t I

h,ur

F rtad thr tnre

history records and run the sauìe finar a'alyseedigitaIlv

would mecire ui-i rstimndted -40 tü 4) mnn hc'ur - wiiih explains why nc mthcally calculated check points

are inwu in the figure.

('1) Accu*&cv aND RLIarLrrv CcinvarTç's

Electronic analog analyzers might introduce errors

in the ai-dei of piusor minus O per cent where a digital computer can pr(xesS data to lurost ai-tv drgr.n ¿f

pi-e-c-iiiun that might pritically be de-iired Cnnsitqunt1y. a given dst.a sariple can be statistically .analyìed by a

digital computer more precisely than

y an analog

machine.

However, it mgt be remembered that we are dtairr.g

with statistical qmntiùes Consequently. if an exper

iment or test is repeated several times. sizeab]e vaa tiorus in experimental results c-ecu! een if the initna merits and ..a1yzing proceases introduce n. error a

ail. For example consider again the wing ioadspowe

spectrum, previously shown irr Fig & which is fairly

typic-al of the type of problerris encountered _{Tltç sta}

tisuca! rnliabtlity mr this ease was such thai unlyone

out of three tests would be expected to 'ie)cl power

es-timates within plus or minus IO per cent ut the u-un

value. li the length of the dst& sair.plecriuld be

in-creased by a factor f 20. however, i out of 20, instead

of nut of 3. of the tests would be espected to vild re-suIts within plus nr minus I. per ..tnt of the true value.

In analysi of actual data samples, then, considera-tion e-f the statistical aspects of the problem usually

shows that improved reliability i-night be ubtairied, not by improving computational accuracy, but by analvz-Ing longer data samples.

Longer samples are not alwayi possible, especially

in airplane or niissume ighti, where steady flight con-ditions often cannot be main taand for longer than a few seconds. Where longer samples cari be secured,

however, the electronic analyzers can economically handle sai-ripies of stich length that analysis by any other process would b-e quite impractical.

(VII) CocLeu!No Rgaxs

In condulon, the various statistical analyses

re-quired in aeronautical research a,r4 development cs.ri be secured r*pidly a.nd economically by the use of elec-tronic analog analyzers. The accuracy of the analog

artalyurs in processing i specific data sample is sorne-what inferior to that available from digital computers.

This (actor is normally outweighed. however, by the analog equipments c4pabílity for economicaIly

han-dung longer data sample.s and providing improved sta-tistical reliability.

ANALOG EQL'IPMN T FOR

_{RANDOMLY FLUCTUAI}

_{NG DA I}

fr ₁₎₉

lt appears that more

_{terve u}

_{of thr}

naiognaty-lu

t'hniqur might íj,dbtat

£urtha.r ad' _{cc.znent in}

the apç,&ation of st2tstiçaI mrthod t _aerodynamic

problems.

Wrie,, Noi1iett. Axrpu*. ¡*k so*, a4 SuiioMa

çf Sww T%?TE Sf4!J th &i(ta'wIL .

iioai.

Tb

Tthaoky Pre, Me.e.cbuiew

inailat

_{of Tuiojy.}

Cam,ridi. Mus.. a.d Jon À.ky & tozia, ln* ,

\pw ï'rk.

1.949.

I _{Tukey, Jitn W, Th Smptwg Theo'y}

..'f Pm'r 5p«waa

&i'.etes, Smposiwi u _Apptwatici

of Atrrt1tàon

t&is te PyJ Prbkr»

_{S.od o}

_{ONP.. Depart.}

men of (lie Navy). Waod5 Hole, M.ss pp ,'ur.e 314,

194g

'Rkc. S C.

't!

_{4älyiu ol'}

_{asàs !Qoi. Pt.}

E

a.od U, 13e9 Syst T&d Jour _{, Voi XXIII, o 3, p'}

-132.

July. 1944; P _{Jil .nd IV. Vol XXIV. i. p 4f}

Jaauarv. 194.

C'eihgn1t C.. Aa ¿t,,vwa

_{'f thr Ps}

sec-l.s,t Draii',

.aèr

tSe. Z) Ths

, a

4tKI 1aor, Mschuas

Iisiture of TcçhnJoy, Cssnt,rida._{Mass May. l9iO.}

'Lipm,,, H W., Oa ii

r4Sloa_{,,f i&i.tftca! C.v.r'4}

,f*Ji*g P'ìem, Joort.J ot ¿ht Arn4uU4* &etce,.

Vc. 19, No. 12, p _I92.

'5wuntrs, R A

A _{Ñ&cis:eeaj Dfsc'soq çf !Afc- csk} AimespErs Tiir.,,Àea,re, Rep. T _{(& i) Tb,eaM. ¡oit jmezoe}

no's Latorstúrv. Maia.huvtrs I bruto

_{M T-olog', C,n}

tviugt Maie. May, IG4.

Prii, deny, sod koi, Johi

C, .Seiac .4frpicshoas

'

Ge*ra4a,d !i'a'e',s,c tno.{yu i. C, Laadr oa

cl lbs Aa'onawtI &'envtl, Vol. _{, No 1, pp 173e,}

6t, January, lW I_{Clijiton, &ob«-t G.}

eo.a'C1uss of

4& Tar.

hkacs Qéiin

F's

_{'-Øiraioa Vas,, Afaard}

_{a s,}

w,e, 'ACA TN 3318, 1954

'Lst, Y 'e.'

.4 .aa's

_{of .%itziai .VIAg, la LO aa}

¿i,ìø Pv'ai, rh. Rti,.

'k _{t$i. Re,arch Lb*toey o?} 5loet'onacs. M _{aCc.e(ts L 3gtL M T,v-IooiGE.,}

_Setrmr

1,