Savitzky (1964)

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Interferences. Methylene ch lor ideor chloroform-soluble carboxylic acids(acetylsal icyl ic acid, sa l icyl ic acid,e tc . ) which inh ib i t ed 4 - , UP responsewere read i ly ex t rac ted f rom thesephases us ing 0.ln’ sod ium hydrox ide .

Span-type excipients , which gave aslight response to the 4-AAP reagent ,were eliminated b y virtue of the ir insol-ub i l i ty in 10% aqueous sodium chloride;while interfering Tween-type excipientswere precipitated ( 3 ) using the Tweenreagent . Propylene glycol , when formu-l a t e d a t a leve l of 5.25%, contributed ca .2 to 370 to the observed absorbance.This was minimized by a more favorabledistribution of th e interference withinthe aqueous phases used in the pro-cedure.

P lacebo analyses indicated that no

interference was obtained with suchcommon excipients as stearic acid,stearyl alcohol, cetyl alcohol, petrola-tum, methyl or propyl-p-hydroxy-benzoates, sesame oil, or thimerosal.Lipotropic agents (beta ine or chol ine),a l l the common vi tamins , and neomycinsulfate failed to interfere.

Scope of Reaction. I n a d d i t i o n t othe o the r typ es of s t e ro ids repor tedt o r e a c t w i t h t h e 4 - A A P r e a g e n t , t h efo l lowing s te ro ids reac ted a t e leva tedtemperature (boiling point of methanol)and a t increased concentra t ion (2.0 mg.per 10.0 ml. of reage nt): 2a-hydroxy-methyl - 17p - hydroxy - 17a - methyl-5a-androst-%one (306 mp) ; a n d 2 a -hydroxymethyl - 17p - hydroxy - 501-

androst-3-one (306 mp ). Steroids with-o u t a keto group fa iled to give any

response. Th e qua ntit ativ e aspects ofthe above responses to the 4-AAPreagent were not investigated.

Correlations of Chromopho re Wave-

length with Structure. T h e a d d i t i v e

effe ct of variou s r ing A a n d B s u b -s t i tuen t s on the chrom ophore o f thepa ren t s a tu ra ted-3-ke to s t e ro id i sp r e s e nt e d i n T a b l e 11.

LITERATURE CITED

1) Cohen, H., Bates, R. W . , J . Am.Pharm. Assoc., Sci . E d . 40, 35 (1951).

(2 ) Hii t tenrauch, R . , Z. Physiol. Chem.326,166 (1961).

(3 ) P i t t e r , P . , Chem. Ind. 42, 1832(1962).

RECEIVEDor review September 4, 1963.’Accepted March 13, 1964.

Smoothing and Differentiation of Data

b y Simplified Least Squares Pro,cedures

ABRAHAM SAVITZKY and MARCEL J. E. GOLAY

The Perkin-Elmer Corp., Norwalk, Conn.

b In attempting to analyze, on

dig a I computers, data f rom basica IIycontinuous physical experiments,

numerical methods of performing fa-

miliar operations must be developed.

The operations of differentiation and

filtering are especially important both

as an end in themselves, and as a pre-

lude to further treatment of the data.Numerical counterparts of analog de-

vices that perform these operations,

such as RC filters, are often considered.

However, the method of least squares

may b e used without additi onal com-

putational complexity and with con-

siderable improvement in the informo-

tion obt ained . The least squares cal-

culations may b e carried out in the

computer by convolution of the dat a

points with properly chosen sets ofintegers. These sets of integers and

their normalizing factors are described

and their use i s illustrated in spectro-

scopic applications. The computer

programs required are relatively sim-

pl e. Two examples are presented as

subroutines in the FORTRAN language.

H E P R I M A R Y OUTPUT of any experi-T m e n t in w hich q ua nt it at iv einformation is to be extracted is infor-mation which measures the phenomenonunde r observat ion. Superimposed uponand indis t inguishable from th is informa-t ion are r ando m errors which, regardlessof their source, are characteristicallydescribed as noise. Of fund am enta limportance t o the esperimenter is the

rem ova l of as mu ch of th is noise aspossible without, at the s am e t im e ,unduly degrading the underlying in-formation.

In much experimental w ork, the infor-mation m ay be obtained in the form ofa two-column table of numbers, A u s B.Such a table is typical ly the resul t of

digitizing a spec t rum or digitizing otherkinds of results obtained du ring thecourse of an exp eriment. If plo t t ed ,this table of nu mber s would give thefamiliar graphs of TOTus. wavelength,p H us. volume of titrant, polarographiccurrent u s applied voltage, S M R orESR spec t rum , or chromatographicelution curve, etc. Th is paper is con-cerned with co mpu tat ional methods forthe removal of the random noise fromsuch information, and with the s impleevaluation of the first few derivativesof th e information with respect to thegraph abscissa.

The bases for the methods to be dis-

cussed have been reported previously,most ly in the mathematical l i tera ture

(4 , 6, 8 , 9). The object ive here is topresent specific methods for handlingcurrent problems in the processing ofsuch tables of analyt ical dat a . Themethods apply as well to the deskcalculator, or to s imple paper and penci loperat ions for small amou nts of da ta , asthey do to the d ig i t a l com pute r fo rlarge amou nts of d ata , since their majorut i l i ty is to s implify and speed up theprocessing of da ta.

There are two important res tr ic t ionson the way in which the points in the

table may be obtained. F irs t , the pointsm us t be a t a fixed, uniform interval inthe chosen abscissa. If the independentvariable is t ime, as in chromatographyor N M R spectra with linear time sweep,each da ta po in t m us t be ob ta ined a t thesame t ime inte rval from each precedingpoint. If it is a spectrum, the intervals

may be every drum divis ion or every0.1 wavenumber, e tc . Second, thecurves formed by graphing the pointsmust be cont inuous and more or lesssmooth-as in the various exampleslisted above.

ALTERNATIVE METHOD S

One of th e simplest ways to smoothfluctuat ing data is by a moving average.In th i s p rocedure one t akes a fisednumber of points , adds their ordinatestogether, and divides by the number ofpoin ts t o ob ta in the ave rage o rd ina te a tthe center abscissa of the group. S e s t ,the point a t one end of th e group isd ropped , th e nex t po in t a t the o the r endadded , and the process is repeated.

Figure 1 i l lus tra tes how the movingaverage might be obtained. Whilethere is a much s impler way to com putethe moving average than the part icularone described, the following descriptionis correct and can be extended to moresophisticated methods as will be seenshort ly . This descript ion is based onthe concept of a convolute and of aconvolution function. Th e set ofnum bers a t the r igh t a re the da ta o rordinate values , those a t the left , theabscissa information. Th e out l ined

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1800.0 705

1799.8 712

1799.6 7 7

1799.4 718

1799.2 721

1799.0 /< 722

1798.8 x ~ - 2 ‘12 ’ 725

1798.6 I xo-l c-l I 730

1798.4 xo co I 735

1798.2 xo+l C1 736

1798.0 x0 2c 2 I 741

AXO

1 11797.8 746

- -- --‘e

17637.6 750

.xo

Figure 1 Convolution operation

Ab sc i ss a p o i nt s a t l e f t , t a b u l a r d a t a a t r i g h t .

In box a re a the convo lu t ion in tegers , c z . O p e r a -

t ion i s the mu l t ip l i ca t ion o f t h e d a t a p o i nt s b y t h e

cor respond ing C,, summat ion o f the resu lt ing

p r o d u c t s , a n d d i v is i o n b y a norma l ize r , resu l ting

in a s ing le convo lu te a t the po in t Xo The boxi s then moved down one l ine , and the p rocess

r e p e a t e d

block in the center may be consideredto be a separate piece of p ape r on whichare wri tten a new set of abscissanumbers , ranging from - 2 thru zero to

+ 2 . The C’s a t the r igh t represen t th econvoluting integers. For the movingaverage each C is numerically equal toone. To perform a convolution of theordinate numbers in the table of datawith a set of convoluting integers, C,,each number in the block is multipliedby the corresponding number in thetable of d ata , the resul t ing products are

added and this sum is divided by five.The set of ones is the convolutingfunct io n, and t he number by which wedivide, in t his case, 5 is the normalizingfactor. To get the next point in themoving average , the c enter block is sliddown one line and the process repeated.

The concept of convolution can be

generalized beyond the simple movingaverage. In the general case the C’srepresent any set of convolutingintegers. Th ere is an associatednormalizing or scaling factor. Th e pro-

cedure is to mult iply C--? t imes thenumber opposi te i t , then C-1 by i ts

num ber, e tc . , sum the results , divide by

the no rmalizing factor, if ap propriate ,and the result is the desired functionevaluated a t the point indicated byCo. For th e next point , we move the setof convoluting integers down and repeat ,e tc . The mathematical descript ion ofthis process is:

The index j represents the runningindex of th e ordinate d at a in the originalda ta t ab le .

For the moving average, each C, isequal to one and N is the number ofconvo luting integers. How ever, formany types of data the set of all l’s,which yields the average, is notparticularly useful. For example, ongoing through a sharp peak, the averagewould tend to degrade th e end of th epeak. There are other types of smooth-ing functions which might be used, anda few of these a re indicated in Fig ure 2 .

Figure 2 A illustrates th e set where allvalues have the same weight over theinterval-essentially th e mov ingaverage.

The funct ion in F igure 2B is anexponential set which simulates thefamiliar RC analog time constant-Le. , the most recent point is given thegreatest weight, and each precedingpoint gets a lesser weight determin ed bythe law of ex ponential dec ay. Future

points have no influence. Such a

funct ion t reats f u tu r e a n d past pointsdifferently and so will obviously intro-duce a unidirect ional dis tort ion in to the

numerical results, as does the RCf il ter in an actu al ins trum ent .Wh en dealing with sets of nu mbers in

hand , and no t an ac tua l run on anins trument where the data is emergingin serial order, it is possible to lookahead as well as behind. Th en we canconvolute with a func t ion tha t t rea t spast and f u t u r e on an equal basis, suchas the funct ion in F igure 2 0 . Here themost weight is given to the centralpoint, an d points on either side of t hecenter are symmetrically weighedexponentially. Th is function acts likean idealized lead-lag network, which isnot pract ical to make with res is tors ,capaci tors , and so on.

The usual spec t rum f rom a spectro-photometer is the re sul tant of two con-volut ions of th e actual spectru m of thematerial, first with a function represent-ing the s l it funct ion of the ins trum ent ,which is much like the triangular con-volute shown in F igure 2C, and th en thisfirst convolute spectrum is furtherconvoluted with a function representingthe t ime constant of the ins trument .The t r iangular convolut ing funct ioncould in many cases yield results notsignificantly different from the sym-metrical exponential function.

Figure 3 illustrates the way in whicheach of these functions would act on atypical set of spectroscopic dat a . Curve3 A is replot ted direct ly from th e ins tru-m enta l da ta . I t is a s ingle s harp band ’

recorded u nd er conditions which yield areasonable noise level. Th e isolatedpoint jus t to the r ight of the band hasthe value of 666 on the scale of zero to1000 corresponding to approximately 0to 100% t ransm it tance. This point isintroduced to i l lus trate the eff ect onthese operatio ns of a single point whichhas a gross error . The numbers a longthe bo t tom a re the d ig i ta l va lue a t the

1

E l ..4I xB

Figure 2. Various convolute functions

A . M o v i n g a v e r a g e . B Exponen t ia l func -

tion. C. Symm etr ica l t r iangu la r func t ion , rep -

resenting ideal ized spectrometer s l i t function.

3. Symmetr ica l exponen t ia l func t ion

lowest point of th e plot, and one ma yconsider that the peak goes down to34.2 t ransm it tance . The base l ine a tthe top i s a t abou t 797,.

Curve 3B is a nine-point movingaverage of th e data . As expected, thepeak is considerably shortened by thisprocess. Especially interesting is the

s tep introduced by the isola ted error.In effect, i t has the shape of t h e b o d i k econvolute in Figure 2 d , which is exactlywhat one would expect from the con-voluting process ( 3 ) .

Curve 3C is for a triangular functionwhich obviously forces both th e peak it-self and the isolated error into atriangular mold.

C u r v e 3 0 is the result of convolutingwith the numerical equivalent of aconventional RC esponential time con-sta nt filter using only five points. Th epeak is not only shortened, but is alsoshif ted to the r ight by one data point ,or 0.002 micron and the isola ted datapoint is asymmetric in th e same mann er.The convolut ion with a symmetricallead-lag exponential, as in Figure 3E,does not dis tort the peak but does s t i l lreduce its intensity.

S o t e th at while all of these functionshave h ad the desired effect of red ucingthe noise level, they are clearly unde-sirable because of the accompanyingdegradation of the peak intensity.

M E T H O D OF LEAST SQUARES

The convoluting functions discussedso far are ra ther s imple and do not

extract as much information as is pos-sible. Th e experim enter, if presentedwith a plot of the data points , wouldtend to draw through these points a linewhich best fits the m. Xumerically,this can also be done, provided one canadequately define what is meant bybest fit. Th e most, common criterion isthat of least squares which may besimply stated as follows:

A set of poin ts is to be fitted t o somecurve-for example, the curve a3x3

a 2 x 2+ lx+ a. = y. T h e a ’ s are to beselected such that when each abscissapoint , is subs t ituted into this equat ion,

1628 ANALYTICAL CHEMISTRY

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363 35442

368 3 56 3 4 2

Figure 3. Spectral band convoluted

by the various 9 point functions

T h e n u m b er a t t h e b o t t o m o f e ac h p e a k r e f e r s

to the lowes t reco rded po in t , an d is a measure

o f t h e a b i l i t y t o r e ta i n t h e s h ap e o f t h e p e a k .

A . R a w d a t a w i t h s i n gl e i s o l a te d er r o r p o i n t.

E M o v i n g a v e r a g e . C . T r i a n g u l a r f u n c -

t ion. D. N o r m a l e x p o n e n t i a l f u n c t i o n . ESy m m e t r i c a l e x p o n e n t i a l f u n c t io n . F. Least

squares smoothing function

th e squar e of thr, differences betweenthe com puted nun ibe r r , y, and theobyerved numbers is a minimum for theto ta l of the observat ions used in deter-min ing the coefficients. A11 of th eerror is assumed t o be in the ordinateand none in t h r atiscissa.

Consider thr block of seven datapoints enclosed by the left bracket inFigure 4. If these fa l l a long a curveth at can be described appros inia te ly bythe equat ion ;.hewn, th rn the re a respecific procedures-which ar e describ edin most books on numerical analysis-to f ind the a s. One then subs t i tu te sback into the resul t ing equat ion theabscisha a t the centra l point indicatedby the circle. Th e value which isobtained by this procedure is the bes tvalue a t that point based on the leas tsquares cri terion, on th e funct ion whichwas chohrn. an d on the gro up of poin tsesam ined .

This procedure can be repeated for

apx) a1 x 2 a, x ao=yIFigure 4 .moving p olyno mial smooth

Representation of a 7-point

each group of seven points, dro ppi ng onea t th e l e ft and p ick ing up one a t theright each time. A somewhat la terb lock i s ind ica ted a t the r igh t . In theusual case, there is found a different setof coefficients for each group of sevenpoints. Ev en with a high-speed com-puter this is a tedious proposition at

best.S o te , however , tha t f ind ing the a

coefficients is required onlys a

meansfor determining the final best value atjus t one point , the centra l point of theset . A careful s tu dy of th e leas t squaresprocedure us ing these constr a ints , leadsto the derivat ion of a set of integerswhich prov ide a weighting function .K it h this se t of integers the centra lpoint can be evaluated by the con-voluting procedure discussed above.This procedure is esact ly equivalent tothe least squares . I t is not approximate .

The derivat ion is presented inAppendis I. For ei ther a cubic or a

quadrat ic funct ion, the set of integersi s the s am e , and the s e t fo r up to 25

points is shown in Ta ble I of AppendixI1 with the appropria te normaliz ingfactors. X most instructive exercise isto tabulate a s imple funct ion such asy = z3 over an y in te rva l , app ly thesesmoot ,hing convolutes and comparethese new values with the original.The answers wil l be found to be exact .

In F igure 3F this least squares con-volut ing procedure has been applied tothe da ta of F igure 3 A , using a 9-pointcubic convolute . Th e value at the peakand t he shape of th e peak are es-sent ia l ly undis torted. As always , theisola ted point assumes the sh ape of th e

convolu t ing func tion . The F OR TR AYlanguag e compu t'er progra m for per-forming this operat ion is presented inP rogram I of Appendix 111.

Going beyond s imple curve f i t t ing,one can f ind in th e l i tera tu re on numeri-cal analysis a variety of least squaresprocedures for det,erminin g the first de-rivat ive . These procedures are usual lybased on interpolat ion formulas and arefor da ta at any a rb i t ra ry in te rva l .Again, if we res tr ic t ourselves to evalu at-ing the funct ion only a t th e center pointof a set of equall y spaced observa tions,then there esist sets of convolutingintegers for the first derivative as well.(These actual ly evaluate the derivat iveof th e least squares best function.)h complete set of tables for

derivat ives up to the f i f th order forpolynomials up to t he f i f th degree, usingfrom 5 to 25 points, is presented in. ippendis 11. These a re m ore thanadequate for most work, s ince, if t h epoints are taken sufficiently close to-gether, then pract ical ly any smoothcurve will look more or less like aquadrat ic in the vic ini ty of a peak, orlike a cubic in the vicinity of a shoulder.More complete tables can be found inthe s ta t is t ical l i tera ture (b, 4 , 6, 9 ) .

S M O O T H I N G - C ~ ~ ~ . y -Ouodrotic y'-Cubic

*+. ?,-

Figure 5. 9-Point convoluting functions

orthogonal polynomial) for smoothing

and first, second, and third der ivativ e

P rogram I1 of Appendis 111 shows theuse of these tables to obtain the coef-ficients of a polynomial for finding theprecise center of a n infrared ban d.

The shapes of the 9-point con-volutes for a few of th e functions arei l lus tra ted in F igure 5 . Of specialinterest is the linear relation of t he firstderivat ive convolute for a quadrat ic .

Th is is quite unique operat ional ly be-cause in processing a ta ble of d at a, onlyone multiplication is necessary for eachconvolut ion. The remainder of thepoints are found from the set calculatedfor the previous point by s imple subtrac -t ion. In F igure 6B is shown the firstderivative of th e spect rum in Figure 6A,

obtained us ing a 9-point convolute.The derivat ives are useful in cases

such as our methods of band finding ona com pute r Y), n studies of derivativespectra , in derivat ive therniogravimet-ric analysis, derivative polarography,etc.

C O N C L U S I O N S

Wi th th e increase in th e application ofcom pute rs to the ana lys i s of digit,izeddat a , the convolut ion methods describedare certa in to gain wider usage. W iththese methods, the sole function of thecomp uter is to act as a f i lter to smooththe noise f luctuat ions and hopeful ly tointroduce no dis tort ions into the re-corded da ta ( 3 ) .

Th is problem of distorti on is difficult,to assess. In an y of the curves of Figure3, there remain small f luctuat ions in th e

Figure 6.volute

17-Point first d erivati ve con-

A . O r i g i n a l s p e c t r u m . B Fi r s t de r iva t ive

spectrum.

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Figure 7. Square root relation be-

tween number of points and d egree of

smoothing

A . R a w d o t o . 8 . 5-Po in t smooth . C. 9 -Point smooth. D. 17-Po in t smooth

da ta. Are these fluctuations real, or, asis more likely, are they just a lowfrequency component of the noise levelwhich could not be smooth ed? Th e

quest ion cannot be answered by takingjus t the data from a s ingle run . How-ever, if one were to tak e more than onerun, average these and then smooth, orsmooth and then average, the computer-plus-instrument system could decide,since even low frequency noise will notrecur in exactly the same place indifferent runs. Co mp uter time is mostefficiently used if the averaging is doneprior to smoothing.

Recent work ( 1 ) has shown the ut i l i tyof simple ave ragi ng of a larg e numb er ofruns in th e enhance ment, of signa-to-noise ratio s. Th e use of comb inedsmoothing and averaging can con-s iderably reduce the ins trument t imerequired, throwing the burden onto thecomputer which operates in a whollydifferent time domain. A character-istic of both procedures is t h a t the noiseis reduced approximately as the squareroot of the number of points used. Thi sis illustrated for the smoothing case byFigure 7. At the upper left is the rawda ta , a t the upper r igh t a & poin tsmooth, lower left 9 points, and lowerright 17 points. If it is desired toimprove the signal-to-noise ratio by afacto r of 10, simple avera ging wouldrequire a tot al of 100 runs . Similar

improvement could be achieved bymaking only 16 runs plus a 9-pointleast squares smooth (average of 16

runs 4 X impro veme nt , and 9-pointsmooth 3 X im provem ent ) or only4 uns plus a 25-point smooth (averageof 4 runs 2 X improvement plur 25-

point smooth 5 X im provem ent ) .The dis tr ibut ion between the num berof runs required and the number ofpoints which may be used for thesmo othi ng is a function of the experi-mental curve under esamin at ion. Theminimum distortion will occur when thepolynomial accurately describes the

analyt ical da ta , an d wil l deviate as thepolynomial departs from the t rue curve.The best results are obtained when theda ta are digitized at high densities-i.e., points very close together-and thenumber of points used in the convoluteis chosen to be small enough so that nomore th an one inflection in the observedd a t a is included in any convolutioninterval. Our results should be com-pared with those achievable using con-vent, ional instrumen t filtering. In asense, we ar e su bsti tuti ng idealizedfilters an d filter-networks for electronichardware such as resitztors, condensers,servos, etc. If one examines the timerelationships, it probably takes longerto get th e information using the digitizerand then the com pute r than wi th theanalogous electronic network. There is,very defini te ly, the advantage in thecompu ter of being able to v ary the proc-essing completely unfettered by thepractical restriction of real circuitry an dservo loops . r o t e too, tha t this proc-essing can be done after the fact of d ata

collection, and indeed several differentprocedures may be applied in order toassess the op timu m. This is a realadvan tage in i tse lf, and provides amplejustification for use of the computersolely as a noise filter. How ever, thegreat,est util ity of these m ethods comesin the pretreatment of data to befur ther processed, as in our bandfindingprocedures, in any curvefitting opera-tions, in quantitative analyses, etc.

This ty pe of da ta processing, so far ascomputers are concerned, requires arelatively small am oun t of programmingand relatively little use of th e computermemory or of the computer's processingcapability. Therefo re, even accounting-type computers , such as the IBM 1401can be used t o process da ta in this way.Furthermore, on such computers thereis generally a high-speed line printerwhich can be turned into a relativelycrude point plot ter . On each l ine an Xcan be placed at ' the app ropriateposition to 1 of value, and the actualvalue is printed at th e edge of th e paper.Th e rate can be on th e order of 10 lines,or points, per second.

A C K N O W L E D G M E N T

The author s apprecia te the ass is tanceof Harrison J . M inyua in the re-compilation and checking of th econvolute tables , of Robcrt Bernard ofthe Perkin-Elmer Scientific ComputingFacility for his advicc and assistancein program development, and of JoelStu tma n, 1-niversi t j - of l la ry la nd , forhis adr icc on . \ppendis I .

APPENDIX I

The general problem is formulated asfollows :

A set of 2m + 1 consecutive valuesare to be used in the determination of

the be\ t mean square f i t through thesevalue> of a polyn omial of degree n

(n esi than 2711 + 1). This polynomialis of the form(:

k = n

j t = b,i;ik = Ik-0

bnO bnl bn2i2 b,,i Ia

The deriv atives of this polynomial are :

& = bn l + 2bn2i+ 3bn3i2+d i

+ nbnnin- l I b

d 2 f ,2bn2+ 3 X 2b,& +

d i 2

+ (n - l)nb,,,i -? IC

I d

S o te th a t , in the coord ina te sys tembeing considered, the value of i rangesfrom -m to + m , a nd t h at i = 0 a t

the central point of the qet of 2m +1values. Hence, the value of the 8thde r iTa t i t e a t tha t po in t is given by:

where

dfo= bnl = aril

d iI I b

IIC

The leas t squarer cri terion requires tha tthe sum of th e sq u ar e of th e differencesbetween the observed values, y andthe calculated, f t > e a minimum oT-erthe interval being considered.

Minimizing with respect to brio we have

+ ( b d - y.1'1 = I I I a

2 2 bna b , , it = - m

+ b,,,i - = 0

and with respect to b,,,, we have

J 2 3 - m

+ b,,P - y,) i = 0 I I I b


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and with respect to the general b,,, weobta in

I I I c

or

i = m k = n i - m

C b n k i k = i i r I y a

I = - m k = O i- -?n

Where r is the index represent ing theequat ion number which runs from 0 ton ( there are n + 1 equa t ions ) . Thesummation indeses on the left s ide maybe interchanged-i.e. ,

a=m k = n k = n 2-m

b,kik = b,kik3 m k = O k = O 2 - - m

IT b

and finally, since b,k is independent of i

Va

or

where

i - m

a n d

N o t e t h a t S r + k = 0 for odd values ofr + k . Sine e ST kxists for evenvalues of r k only, the set of n + 1equat ions can 1 )e separated into tw o sets,one for even v: du es of k and one for oddvalues . Thus , for a 5th degree poly-nomial, where n = 5

Sobso )12b52 S4b54 = Fo

SIbw j S4bb2+ &bj4 = F 2 V I a

S4ho S~bsz SA4 = Fa

which can b e used to solve for baa b62,

and bS4, whi le:

Snbal f S4b53 Sob56 F1

S4bs1 -t . S6bj3 + Ssb5j Fa V I b

Ssba 4 - Ssbs3 + S ~ O ~ S SF

b60, b42 = b j 2 , arid b41 = bj4 while theset VIb has the same form for n=6, sot h a t bS1 = be l b S 3= b63, and b s s = be6.In o the r words , b,, = b,+l,, for n and s

bot h even or for n and s both odd . F o rexample, to determine the third deriva-tive for the best fit to a curve of third(or fourth) order, we would have:

Szb31 S4b33 = F i

S4b31 S6b33 = F3

SzF3 - S4Fi

SZSS - S4

from w hich b33=

When, for instance, rn = 4 (2m+ 1 = 9points) , we have from Vc t h a t

Sz = 6 0 , 84 = 708, 86 = 9780

and

F3 - 7F1-~OF3 - 708F1 -7128

33 ___

60 (9780) - (708)2

which reduces to:

which cai 1 be used to solve for b51, bas, The coefficients of yl. cons t i tu te theand bss. ' The set of equations in VI a has convo luting integers (T able T'III) forthe same form for n = 4, so tha t b40 = the third derivative of a cubic poly-

CONV OL UT E S S M O O T H I N G

P O I N T S 2 5 2 3

- 1 2- 1 1-10

- 0 9-08

- 0 7- 0 6-05

- 0 4- 0 3-02

-0 100

01

0 20 30 40 5

0 60 70 8

0 9

1 01112

- 2 5 3- 1 3 8

- 3 3

6 21 4 72 2 22 8 73 2 23 8 74 2 24 4 74 6 24 6 74 6 24 4 74 2 23 8 73 2 22 8 72 2 21 4 7

6 2- 3 3

- 1 3 8- 2 5 3

- 4 2- 2 1

- 215304 35 46 37 0

7 57 87 97 8

7 57 06 35 44 330

1 5- 2

- 2 1

- 4 2

N O R M 5 1 7 5 8 0 5 9

TA B LE I

Q U A D R A T I C C U B I C

ZL 1 9 17 15

- 1 7 1-76 I

C )

8 B14 920 4

2 4 9

2e 43c 9

32 43; 93; 431 392 8 42 4 92 . 0 41 . 4 9

9- 7 6

- 1 7 1

a 4

- 1 3 6- 5 1 - 2 2

2 4 -6

8 9 71 4 4 181 8 9 2 72 2 4 34

2 4 9 3 92 6 4 4 22 6 9 4 32 6 4 4 22 4 9 3 92 2 4 34

1 8 9 , 2 71 4 4 1 8

8 9 72 4 - 6

- 5 1 -2 1- 1 3 6

- 78-1 3428 7

1.221 4 71 6 21 6 71 6 21 4 71 2 2

8742

- 1 3- 7 8

:3059 2 2 6 1 323 1105

~ ~~

A 2 0

13

-1 10

9

16

2 12 42 52 4

2 11 6

90

- 1 1

143

A30

11 9 7 5

-3 6

9 - 2 14 4 1 4 - 26 9 39 3 - 38 4 5 4 6 1 28 9 59 7 178 4 5 4 6 1 2

6 9 3 9 3 - 34 4 1 4 - 2

9 - 2 1- 3 6

429 231 21 3 5

VOL. 36, NO. 8 ULY 1964 1631

8/13/2019 Savitzky (1964)


~T A B L E 1 ’ 1

C O N V O L U T E S S M O O T H I N G Q U A R T IC Q U I N T I C A40 A 5 0

P O I N T S 2 5 2 3 21 1 9 1 7 15 13 11 9 7 5

- 1 2-1 1

- 1 0

- 0 9- 0 8- 0 7- 0 6-0 5- 0 4- 0 3- 0 2- 0 1

00

0 1

0 20 30 40 5

0 60 70 809

10

11

1 2

1 2 6 5- 3 4 5

- 1 1 2 2

- 1 2 5 5-915- 2 5 5

5 9 01 5 0 32 3 8 53 1 5 53 7 5 04 1 2 54 2 5 34 1 2 53 7 5 03 1 5 52 3 8 51 5 0 3

5 9 0- 2 5 5- 9 1 5

- 1 2 5 5- 1 1 2 2

- 3 4 51 2 6 5

2 8 5- 1 1 4

- 2 8 5- 2 8 5- 1 6 5

3 02 6 1

4 9 57 0 58 7 09 7 5

1 0 1 19 7 58 7 07 0 54 9 52 6 1

3 0- 1 6 5- 2 8 5- 2 8 5-1 1 4

2 8 5

1 1 6 2 8

- b 4 6 0- 1 3 0 0 5-1 1 2 2 0

- 3 9 4 06 3 7 8

1 7 6 5 52 8 1 9 03 6 6 6 04 2 1 2 04 4 0 0 34 2 1 2 03 6 6 6 02 8 1 9 01 7 6 5 5

6 3 7 8

- 3 9 4 0- 1 1 2 2 0- 1 3 0 0 5

- 6 4 6 01 1 6 2 8

3 4 0- 2 5 5- 4 2 0- 2 9 0

1 84 0 57 9 0

1 1 1 01 3 2 01 3 9 31 3 2 01110

7 9 040

1 8

- 2 9 0- 4 2 0- 2 5 5

340

\ 9

-1 9 5- 2 60

-1 1 71 3 54 , 1 56 1 0

82 58 8 3

8 2 5661 0

4 1 5

135 i

- 1 1 7 ’

- 2 6 0- 1 9 5

1 9 5

2 1 4 5- 2 8 6 0- 2 9 3 7

- 1 6 53 7 5 57 5 0 0

1 0 1 2 51 1 0 5 31 0 1 2 5

7 5 0 03 7 5 5- 1 6 5

- 2 9 3 7

- 2 8 6 02 1 4 5

1 1 0-1 9 8- 1 6 0

1 1 03 9 06 0 06 7 76 0 03 9 0110

- 1 6 0- 1 9 8

1 1 0

18- 4 5 1 5-10 - 5 5 5

6 0 30 - 3 01 2 0 1 3 5 7 51 4 3 1 7 3 1 3 11 2 0 1 3 5 7 5

3 0 - 3 0- 1 0 - 5 5 5- 4 5 1 5

60

1 8

N O R M 3 0 0 1 5 6 5 5 5 2 6 0 0 1 5 7 4 2 9 4 1 9 9 4 61 89 1 2 4 3 1 4 2 9 4 2 9 2 3 1

T A B L E 1 1 1

CONVOLUTES 1 S T D E R I V A T I V E Q U A D R A T I C A 2 1

POINTS 2 5 2 3 2 1 1 9 1715 1 3 11 9

7 5

- 1 2-11-10

- 0 9- 0 8-0 7- 0 6-0 5

- 0 4- 0 3- 0 2-0 1

00

0 102

0 30 405

0 60 70 80 91 011

1 2

- 1 2-1 1

- 1 0- 9- 8- 7- 6- 5- 4- 3-2

-10

2

3

456

789

1 011

1 2

-11- 1 0-9

- 8- 7- 6- 5- 4-3-2-1

0

123

56

78

9

1 011

-10-9

- 8- 7- 6-5-4-3- 2-10

12

3456

789

10

-9

- 8- 7- 6-5- 4- 3-2-1

0

1234567

8

9

- 8- 7- 6-5-4-3- 2-1

0

1.234

567

8

7

- 6

--5- 4- 3- 2-10

123

4

567

- 6-5 - 5-4 - 4 - 4-3 -3 - 3 - 3-2 - 2 - 2 - 2 - 2-1 -1 -1 - 1 -10 0 0 0 0

1 1 1 1 12 .2 2 2 23 3 3 34 4 45 56

NORM 1300 1012 7 7 0 570 408 2 8 t 3 182 110 6 0 2 8 10


8/13/2019 Savitzky (1964)


TABLE I V

CONVOLUTES 1ST D E R I V A T I V E C U B I C Q U A R T I C A 3 1 A41

P O I N T S 2 5

- 1 2 3 0 8 6 6- 1 1 8 6 0 2- 1 0 - 8 5 2 5- 0 9 - 2 0 9 8 2- 0 8 - 2 9 2 3 6- 0 7 - 3 3 7 5 4- 0 6 - 3 5 0 0 3- 0 5 - 3 3 4 5 0- 0 4 - 2 9 5 6 2- 0 3 - 2 3 8 0 6- 0 2 - 1 6 6 4 9-0 1 - 8 5 5 8

0 0 00 1 8 5 5 80 2 1 6 6 4 90 3 2 3 8 0 60 4 2 9 5 620 5 3 3 4 5 0

0 6 3 5 0 0 30 7 3 3 7 5 40 8 2 9 2 3 60 9 2 0 9 8 21 0 8 5 2 51 1 - 8 6 0 21 2 - 3 0 8 6 6

N O R M 1 7 7 6 0 6 0

23

3 9 3 88 1 5

- 1 5 1 8- 3 1 4 0- 4 1 3 0- 4 5 6 7- 4 5 3 0- 4 0 9 8- 3 3 5 0- 2 3 6 5- 1 2 2 2

0

1 2 2 22 3 6 53 3 5 04 0 9 84 5 3 0

4 5 6 74 1 3 03 1 4 01 5 1 8- 8 1 5

- 3 9 3 8

1 9 7 3 4 0

22 1 9 1 7 15 1 3 11 9 7 5

8 4 0 7 51 0 0 3 2

- 4 3 2 8 4- 7 8 1 7 6- 9 6 9 4 7

- 1 0 1 9 0 0- 9 5 3 3 8- 7 9 5 6 4- 5 6 8 8 1- 2 9 5 9 2

02 9 5 9 25 6 8 8 17 9 5 0 4

9 5 3 3 81 0 1 9 0 0

9 6 9 4 77 8 1 7 64 3 2 8 4

- 1 0 0 3 28 4 0 7 5

6 9 3 6

6 8- 4 6 4 8- 7 4 8 1- 8 7 0 0- 8 5 7 4- 8 1 7 9- 5 3 6 3- 2 8 1 6

02 8 1 65 3 6 38 1 7 98 5 7 48 7 0 0

7 4 8 14 6 4 8- 6 8

- 6 9 3 6

748- 9 8

- 6 4 3- 9 3 0

- 1 0 0 2- 9 0 2- 6 7 3- 3 5 8

0

3 5 86 7 39 0 2

1 0 0 29 30

6 4 39 8- 7 4 8

1 2 9 2 2- 4 1 2 1

- 1 4 1 5 0- 1 8 3 3 4- 1 7 8 4 2- 1 3 8 4 3

- 7 5 0 60

7 5 0 61 3 8 4 31 7 8 4 2

1 8 3 3 41 4 1 5 0

4 1 2 1- 1 2 9 2 2

1 1 3 3- 6 6 0

- 1 5 7 8- 1 7 9 6- 1 4 8 9

- 8 3 20

8 3 21 4 8 91 7 9 61 5 7 8

6 6 0

- 1 1 3 3

300- 2 9 4 8 6- 5 3 2 - 1 4 2 2 2- 5 0 3 - 1 9 3 - 6 7 1- 2 9 6 - 1 2 6 - 5 8 -8

0 0 0 0

2 9 6 1 2 3 5 8 85 0 3 1 9 3 6 7 -15 32 1 4 2 - 2 22 9 4 - 8 6

- 3 0 0

3 6 3 4 0 9 2 2 5 5 8 1 6 2 3 2 5 6 3 3 4 1 5 2 2 4 0 2 4 5 1 4 8 1 1 8 8 2 5 2 1 2

TABLE v

CONVOLUTES S T D E R I V A T I V E Q U N T I C S E X l C A 5 1 A 6 1

P O IN T S 2 5 2 3 2 1 1 9 1 7 1 5 1 3 11 9 7 5

- 1 2- 1 1- 1 0- 0 9- 0 8- 0 7-0 6-05

- 0 4- 0 3- 0 2-0 1

00

0 10 20 30 40 50 60 70 80 91 01 11 2

- 6 3 5 6 6 2 5- 1 1 8 2 0 6 7 5- 1 5 5 9 3 1 4 1- 1 7 0 6 2 1 4 6- 1 5 8 9 6 5 1 1- 1 2 1 3 9 3 2 1

- 6 3 0 1 4 9 15 4 4 6 6 8

6 6 7 1 8 8 39 6 0 4 3 5 36 0 2 4 1 8 3

- 8 3 2 2 1 8 20

8 3 2 2 1 8 2- 6 0 2 4 1 8 3

- 9 6 0 4 3 5 3- 6 6 7 1 8 8 3

- 5 4 4 6 6 86 3 0 1 4 9 1

1 2 1 3 9 3 2 11 5 8 9 6 5 1 11 7 0 6 2 1 4 61 5 5 9 3 1 4 11 18 2 0 6 7 5

6 3 5 6 6 2 5

- 3 5 7 0 4 5- 6 5 4 6 8 7 - 1 5 9 7 7 3 6 4

- 8 4 0 9 3 7 - 2 8 7 5 4 1 5 4 - 3 3 2 6 8 4- 8 7 8 6 3 4 - 3 5 6 1 3 8 2 9 - 5 8 3 5 4 9 - 2 3 9 4 5- 7 5 2 8 5 9 - 3 4 8 0 7 9 1 4 - 6 8 6 0 9 9 - 4 0 4 8 3 - 1 7 5 1 2 5- 4 7 8 3 4 9 - 2 6 0 4 0 0 3 3 - 6 0 4 4 8 4 - 4 3 9 7 3 - 2 7 9 9 7 5 - 3 1 3 8 0- 1 0 6 9 1 1 - 1 0 9 4 9 9 4 2 - 3 4 8 8 2 3 - 3 2 3 0 6 - 2 6 6 4 0 1 - 4 5 7 4 1 - 3 0 8 4

2 6 5 1 6 4 6 4 0 24 3 8 9 4 7 3 - 8 6 7 1 - 1 3 0 5 0 6 - 3 3 5 1 1 - 3 7 7 6 - 5 7 5 84 8 9 6 8 7 1 9 0 5 2 9 8 8 3 2 2 3 7 8 1 6 6 7 9 6 5 2 2 9 - 1 2 - 1 2 4 4 - 4 5 3 8 - 9 03 5 9 1 5 7 1 6 6 4 9 3 5 8 3 4 9 9 2 8 2 4 6 6 1 1 6 9 8 1 9 2 7 0 9 3 2 1 6 6 2 7 62 1 8

- 4 0 0 6 5 3 - 1 5 0 3 3 0 6 6 - 2 5 5 1 0 2 - 1 4 4 0 4 - 7 8 3 5 1 - 1 4 6 4 7 - 5 7 3 - 5 0 5 - 2

0 0 0 0 0 0 0 0 0

4 0 0 6 5 3 1 5 0 3 3 0 6 6 2 5 5 1 0 2 1 4 4 0 4 7 8 3 5 1 1 4 6 47 5 7 3 5 0 8 2- 3 5 9 1 5 7 - 1 6 6 4 9 3 5 8 - 3 4 9 9 2 8 - 2 4 6 6 1 - 1 6 9 8 1 9 - 2 7 0 9 3 - 2 1 6 6 - 2 7 6 2 - 1 8- 4 8 9 6 8 7 - 1 9 0 5 2 9 8 8 - 3 2 2 3 7 8 - 1 6 6 7 9 - 6 5 2 2 9 1 2 1 2 4 4 4 5 3 8 9 0- 2 6 5 1 6 4 - 6 4 0 2 4 3 8 - 9 4 7 3 8 6 7 1 1 3 0 5 0 6 3 3 5 1 1 3 7 7 6 5 7 5 8

1 0 6 9 1 1 1 0 9 4 9 9 4 2 3 4 8 8 2 3 3 2 3 0 6 2 6 6 4 0 1 4 5 7 4 1 3 0 8 44 7 8 3 4 9 2 6 0 4 0 0 3 3 6 0 4 4 8 4 4 3 9 7 3 2 7 9 9 7 5 3 1 3 807 5 2 8 5 9 3 4 8 0 7 9 1 4 6 8 6 0 9 9 4 0 4 8 3 1 7 5 12 58 7 8 6 3 4 3 5 6 1 3 8 29 5 8 3 5 4 9 2 3 9 4 58 4 0 9 3 7 2 8 7 5 4 1 5 4 3 3 2 6 8 46 5 4 6 8 7 1 5 9 7 7 3 6 43 5 7 0 4 5

NORM 7 1 5 3 5 7 5 3 1 2 4 5 5 5 3 1 1 7 3 5 8 1 7 1 9 4 1 9 9 0 2 0 9 9 5 2 4 3 1 1 4 3 1 4 3 1

VOL. 36, NO. 8, JULY 1964 1633

8/13/2019 Savitzky (1964)


T A B L E V I

CONVOL UT ES 2 N D D E R I V A T I V E

P O I N T S 2 5 23 2 1

- 1 2- 1 1- 1 0- 0 9

- 0 8- 0 7- 0 6- 0 5- 0 4- 0 3- 0 2- 0 1

00

0 10 20 304

0 50 60 70 80 910

1 11 2

9 26 94 82 9

1 2- 3

- 1 6- 2 7- 3 6-43

- 4 8- 5 1- 5 2- 5 1- 4 8- 4 3

- 3 6- 2 7- 1 6

- 31 22 94 86 99 2

7 75 637

2 05

- 8- 1 9- 2 8- 3 5- 4 0- 4 3- 4 4

- 4 3- 4 0- 3 5- 2 8- 1 9

- 85

2 0375 67 7

1 9 01 3 3

8 23 7- 2

- 3 5- 6 2- 8 3

- 9 8- 1 0 7- 1 1 0- 1 0 7

- 9 8- 8 3- 6 2- 3 5

- 2378 2

1 3 31 9 0

NORM 2 6 9 1 0 1 7 7 1 0 3 3 6 4 9

Q U A D R A T I C C U B I C

1 9 1 7 1 5

5 1

3 4 401 9 2 5

6 1 2- 5 1

- 1 4 - 8- 2 1 - 1 5- 2 6 - 2 0- 2 9 - 2 3- 3 0 - 2 4- 2 9 - 2 3-2 6 - 2 0- 2 1 - 1 5- 1 4 -8

- 5 16 1 2

1 9 2 53 4 40

5 1

9 1

5 21 9-8

- 2 9- 4 8- 5 3- 5 6- 5 3- 4 8- 2 9

-81 95 29 L

6 7 8 3 3 8 7 6 6 1 6 8

A 2 2

13

2 21 1

2- 5

- 1 0- 1 3- 1 4- 1 3- 1 0

- 52

1 12 2

1001

A32

It.

1 56

- 1- 6- 9

- 1 0-9

- 6- 1

61 5

42 9

9 7 5

2 8

7 5- 8 0 2

- 1 7 - 3 -L- 2 0 - 4 - 2- 1 7 - 3 -1

- 8 0 - 27 5

2 8

4 6 2 42 7.

T A B L E V I 1

CONVOL UT ES 2 N D D E R I V A T I V E Q U A R T I C Q U I N T I C A42 A52

POINTS 2 5 23 2 1 1 9 1 7 15. 13 11 9 7 5

- 1 2- 1 1- 10

- 0 9- 0 8

- 0 7- 0 6- 0 5- 04

- 0 3- 0 2-0 1

00

0 1

0 20 30 40 50 60 70 80 91 01 11 2

NOKM

- 4 2 9 5 9 43 1 1 1 9 - 3 4 6 7 3 1

2 9 8 1 5 5 6 1 8 4 5

4 1 3 4 0 9 2 8 1 9 7 94 1 4 7 8 6 3 5 8 5 3 03 3 6 2 0 1 3 3 1 6 3 52 0 7 5 7 9 2 3 6 7 0 9

5 4 8 5 5 1 0 4 4 4 5- 1 0 0 0 2 6 - 3 9 1 8 6- 2 3 9 1 0 9 - 1 7 2 9 3 5

- 3 4 8 4 2 9 - 2 8 0 2 7 5- 4 1 8 0 1 1 - 3 4 9 4 0 1- 4 4 1 8 7 0 - 3 7 3 2 3 0- 4 1 8 0 1 1 - 3 4 9 4 0 1

- 3 4 8 4 2 9 - 2 8 0 2 7 5- 7 3 9 1 0 9 - 1 7 2 9 3 5- 1 0 0 0 2 6 - 3 9 1 8 6

5 4 8 5 5 1 0 4 4 4 52 0 7 5 7 9 2 3 6 7 0 93 3 6 2 0 1 3 3 1 6 3 54 1 4 7 8 6 3 5 8 5 3 04 1 3 4 0 9 2 8 1 9 7 92 9 8 1 5 5 6 1 8 4 5

3 1 1 1 9 - 3 4 6 7 3 14 2 9 5 9 4

4 2 9 2 1 4 5 2 8 1 2 0 9 5

- 3 7 7 9 1

1 1 6 2 83 5 8 0 24 1 4 1 23 4 3 5 31 9 7 3 4

1 8 7 8- 1 5 6 7 8- 3 0 1 8 3- 3 9 . 6 7 2- 4 2 9 6 6- 3 9 6 7 2

- 3 0 1 8 3- 1 5 6 7 8

1 8 7 81 9 7 3 43 4 3 5 34 1 4 1 23 8 8 0 21 1 6 2 8

- 3 7 7 9 1

- 9 6 0 8 44 5 0 8 4 - 1 2 1 5 2 4

1 0 5 4 4 4 8 2 2 5 11 0 9 0 7 1 1 5 3 3 8 7

7 6 8 3 0 1 3 7 0 8 52 6 3 7 6 7 1 5 9 2

- 2 7 8 4 6 - 1 1 7 9 9- 7 4 6 0 1 - 8 8 7 4 9

- 1 0 5 8 6 4 - 1 4 1 8 7 3- 1 1 6 8 2 0 - 1 6 0 7 4 0- 1 0 5 8 6 4 - 14 18 7 3

- 7 4 6 0 1 - 8 8 7 4 9- 2 7 8 4 6 - 1 1 7 9 9

2 6 3 7 6 7 1 5 9 27 6 8 3 0 1 3 7 0 8 5

1 0 9 0 7 1 1 5 3 3 8 71 0 5 4 4 4 8 2 2 5 1

4 5 0 8 4 1 2 1 5 2 4- 9 6 0 8 4

- 9 3 0 9 3

1 3 3 4 8 5 9 8 0 1 0 - 1 0 5 3 08 8 8 0 3 - 7 2 9 6 3

9 5 5 6 8 1 1 5 5 3 2 2 0 3 5 8 - 4 1 5 81 9 7 3 7 5 3 2 6 2 1 7 0 8 2 1 2 2 4 3 - 1 1 7

- 5 9 2 5 3 - 3 2 0 4 3 1 1 7 4 9 8 3 6 0 3 - 3- 1 1 6 5 7 7 - 9 9 5 2 8 - 1 5 9 1 2 - 6 7 6 3 - 1 7 1 4 8- 1 3 7 3 4 0 - 1 2 4 7 4 0 - 2 2 2 3 0 - 1 2 2 1 0 - 6 3 0 - 9 0- 1 1 6 5 7 7 - 9 9 5 2 8 -1 5 9 1 2 - 6 9 6 3 - 1 7 1 4 8

- 5 9 2 5 3 - 3 2 0 4 3 1 17 4 9 8 3 6 0 3 - 31 9 7 3 7 5 3 2 6 2 1 7 0 8 2 1 2 2 4 3 - 1 1 79 5 5 6 8 1 1 5 6 3 2 2 0 3 5 8 - 4 1 5 8

8 8 8 0 3 - 7 2 9 6 31 3 3 4 8 5 9 8 0 1 0 - 1 0 5 3 0

- 9 3 0 9 3

2 4 5 1 5 7 4 9 0 3 1 4 4 7 8 6 8 6 2 7 7 1 3 4 1 6 0 4 4 6 1 6 7 3 1 4 7 1 9 9 9 3


8/13/2019 Savitzky (1964)


TABLE V C f . 1

C O N V O L U T E S 3R0 D E R I V A T I V E

P O I N T S 2 5 2 3 2 1

- 1 2- 1 1-10

- 0 9

- 0 8- 0 7- 0 6-0 5- 0 4-0 3- 0 2-0 100

0 1020 3

0405

0 6

07

08

0 91 01 11 2

- 5 0 6- 2 5 3

- 5 59 3

1 9 62 5 92 8 72 8 52 5 82 1 11 4 5

7 70

- 7 7- 1 4 9- 2 1 1- 2 5 8- 2 8 5

- 2 8 7- 2 5 9- 1 9 6

- 9 35 5

2 5 35 0 6

- 7 7- 3 5

- 3

2 03 54 34 54 23 52 51 3

0

- 1 3- 2 5-35- 4 2- 4 5

- 4 3- 3 5- 2 0

33 57 7

- 2 8 5-114

1 29 8

1491 7 01 6 6

1 4 21 0 3

5 40

- 5 4- 1 0 3- 1 4 2- 1 6 6-170

-149- 9 8-121142 8 5

NORM 2 9 6 0 1 0 3 2 8 9 0 8 6 5 2 6

C U B I C Q U A R T IC A 3 3

1 9 lt 1 5 13

- 2 0 4

- 6 8288 9

1 2 01 2 61 1 2

8 34 4

0

-44

- 8 3- 1 1 2- 1 2 6- 1 2 0

- 8 9- 2 66 8

2 0 4

-28-7

7

1 51 81 71 3

70

713

1718157

- 728

- 9 1- 1 3 -1 1

3 5 05 8 6

6 1 84 9 72 7 4

0 0-27 -4- 4 9 - 7- 6 1 -8- 5 8 -6

- 3 5 0

1 3 1191

4 2 6 3 6 3 8 76 7 9 5 6 5 7 2

A43

11 9 7 5

-3 06 - 1 4

2 2 7 -1

2 3 1 3 1 -11 4 9 1 2

0 0 0- 1 4 -9 ,-1 -2- 2 3 - 1 3 - 1 1

-2 2 -7 t- 6 1 430

8 5 8 198 6 2

TABLE I X

C O N V O L U T E S 3K0 D E R I V A T I V E Q U I N T IC SEXIC A53 A63

P O I N T S 2 5 2 3

- 1 2- 1 1- 1 0- 0 9-08

- 0 7- 0 6- 0 5- 0 4- 0 3

-02

-0 L

0 0

01

0 20 3

0 4

0 5

0 607

0 80 91 01 112

N O R M

1 1 8 7 4 52 1 7 6 4 02 7 9 1 0 L2 9 0 0 7 62 4 4 3 1 11 4 4 6 1 6

5 1 3 1- 1 4 6 4 0 8- 2 6 6 4 0 3- 2 9 3 1 2 8- 1 4 4 4 6 3

2 8 4 3 7 20

- 2 8 4 3 7 2

1 4 4 4 6 32 9 3 1 2 82 6 6 4 0 31 4 6 4 0 8

- 5 1 3 1- 1 4 4 6 1 6- 2 4 4 3 I 1

- 2 9 0 0 7 6- 2 7 9 1 0 1- 2 1 7 6 4 0- 1 1 8 7 4 5

5 7 2 2 8 6 0

2 3 6 9 94 2 7 0 45 2 9 5 95 1 6 8 43 8 0 1 31 3 6 3 2

- 1 6 5 8 3- 4 3 9 2 8- 5 5 2 3 3

- 3 2 2 2 44 9 1 1 5

0

- 4 9 1 1 5

3 2 2 2 45 5 2 3 34 3 9 2 81 6 5 8 3

- 1 3 6 3 2- 3 8 0 1 3- 5 1 6 8 4- 5 2 9 5 9- 4 2 7 0 4- 2 3 6 9 9

7 4 9 8 9 2

2 L 1 9 17 1 5 1 3 1 1 9 7 5

4 2 5 4 1 27 4 9 3 7 2 3 1 7 6 5 58 8 7 1 3 7 1 1 1 3 2 4 07 8 7 3 8 2 1 2 3 1 5 0 04 4 8 9 0 9 9 3 2 7 6 0- 6 2 6 4 4 2 5 9 7 4 0

- 5 9 8 0 9 4 - 5 8 9 0 8 0- 9 0 8 0 0 4 - 1 2 2 0 5 2 0

- 6 2 5 9 7 4 - 1 0 0 7 7 6 07 4 8 0 6 8 9 4 8 6 0 0

0 0

- 7 4 8 0 6 8 - 9 4 8 6 0 0

6 2 5 9 7 4 1 0 0 7 7 6 09 0 8 0 0 4 1 2 2 0 5 2 05 9 8 0 9 4 5 8 9 0 8 0

6 2 6 4 4 - 2 5 9 7 4 0- 4 4 8 9 0 9 - 9 3 2 7 6 0- 7 8 7 3 8 2 - 1 2 3 1 5 0 0- 8 8 7 1 3 7 - 1 1 1 3 2 4 0- 7 4 9 3 7 2 - 3 1 7 6 5 5- 4 2 5 4 1 2

4 9 1 58 0 2 0 9 3 1 3 5

7 9 7 5 1 4 1 3 2 04 3 8 0 1 1 3 0 6 5

- 1 7 5 5 3 8 0 0- 7 5 4 0 - 1 5 0 6 6 5- 7 7 3 5 - 2 6 0 6 8 0

5 7 2 0 - 1 6 9 2 9 50 0

- 5 7 2 0 - 1 6 9 2 9 57 7 3 5 2 6 0 6 8 07 5 4 0 1 5 0 6 6 51 7 5 5 - 3 8 0 0

- 4 3 8 0 - 1 1 3 0 6 5- 7 9 7 5 - 1 4 1 3 2 0- 8 0 2 0 - 9 3 1 3 5- 4 9 1 5

1 1 2 6 01 5 2 5 0 1 5 8 0

8 1 6 5 1 7 0 0- 6 8 7 0 - 5 5

- 1 6 3 3 5 - 2 0 1 07 1 5 0 6 4 5

0 0

- 7 1 5 0 - 6 4 51 6 3 3 5 2 0 1 0

6 8 7 0 5 5- 8 1 6 5 - 1 7 0 0

- 1 5 2 5 0 - 1 5 8 0- 1 1 2 6 0

2 2 9 51 2 8 0 6 5

- 2 2 8 5 - 4 05 0 0 5

0 0

- 5 0 0 - 52 2 8 5 4 0

- 1 2 8 0 - 6 5- 2 2 9 5

4 2 4 9 3 8 8 4 2 4 7 0 1 2 1 67 9 6 2 1 4 4 8 0 9 9 7 2 4 5 7 2 2 8 6 2

VOL. 36, NO. 8, JULY 1964 1635

8/13/2019 Savitzky (1964)


CONVOL UT ES 4T" DER1VAtIVE

P O I N T S 2 5

- 1 2 8 5 8- 1 1 8 0 3- 1 0 6 4 3- 0 9 3 9 3

- 0 8 7 8- 0 7 - 2 6 7- 0 6 - 5 9 7- 0 5 - 8 5 7- 0 4 - 9 8 2- 0 3 - 8 9 7- 0 2 - 5 1 7

- 0 1 2 5 300 1 5 1 80 1 2 5 30 2 - 5 1 70 3 - 8 9 70 4 - 9 8 20 5 - 8 5 70 6 - 5 9 7

0 7 - 2 6 708 7 809 3 9 31 0 6 4 31 1 8 0 31 2 8 5 8

NORM 1 4 3 0 7 1 5

23

8 5 87 9 3605

3 1 5- 4 2

- 4 1 7-747

- 9 5 5- 9 5 0- 6 2 7

1 3 31 4 6 3

1 3 3- 6 2 7- 9 5 0- 9 5 5- 7 4 7- 4 1 7

- 4 23 1 56 0 57 9 38 5 8

9 3 7 3 6 5

2 1

5945 4 0

3 8 51 5 0

- 1 3 0- 4 0 6- 6 1 5- 6 8 0- 5 1 0

09 6 9

0- 5 1 0- 6 8 0- 6 1 5- 4 0 6- 1 3 0

1 5 03 8 55 4 05 9 4

408595

T A B L E X

Q U A R T I C QUINTIC A44 A 5 4

19

396

3 5 22 2 7

4 2- 1 6 8- 3 5 4- 4 5 3- 3 8 8

- 6 86 1 2-6 8

- 3 8 8- 4 5 3- 3 5 4- 1 6 8

42

2 2 73 5 23 9 6

1 6 3 4 3 8

I7 1§ 1 3 11 9 7 s

363 11 7- 3

- 2 4- 3 9- 3 9- 1 3

5 2- 1 3

- 3 9- 3 9- 2 4

- 3173 1

3 6

6 2 12 5 1

- 2 4 9- 7 0 4- 8 6 9- 4 2 91 0 0 1- 4 2 9- 8 6 9- 7 0 4- 2 4 9

2 5 16 2 17 5 6

8 46 4 6

1 1 4 1 6-5 4 - 1 9 6

-9 6 - 6 - 1 1 1-66 - 6 - 2 1 -7

9 9 6 1 4 - 3- 6 6 - 6 - 2 1 7

- 9 6 -6 - 1 1 1- 5 4 -1 9 6

1 1 4 186 4 68 4

CONVOL UT ES 5TH D E R I V A T I V E

P O I N T S 2 5

- 1 2 - 2 7 5- 1 1 - 5 0 0- 1 0 - 6 3 1- 0 9 - 6 3 6- 0 8 - 5 0 1- 0 7 - 2 3 6- 0 6 1 1 9- 0 5 4 8 8- 04 7 5 3- 0 3 7 4 8- 0 2 2 5 3-0 1 - 1 0 1 200 001 1 0 1 2

0 2 - 25 30 3 - 7 4 80 4 - 7 5 305 - 4 8 806 - 1 1 90 7 2 3 608 5 0 10 9 6 3 61 0 6 3 11 1 5 0 0

1 2 2 7 5

N O R M 1 3 0 0 6 5 0

2 3

- 6 5- 1 1 6- 1 4 1

- 1 3 2-87

- 1 27 7

1 5 21 7 1

7 6- 2 0 9

0

2 09

- 7 6- 1 7 1- 1 5 2

- 7 71 287

1 3 21 4 11 1 6

6 5

1 7 0 4 3 0

21

-1 4 0 4- 2 4 4 4- 2 8 1 9- 2 3 5 4- 1 0 6 3

7 8 82 6 1 83 4 6 81 9 3 8

- 3 8 7 60

3 8 7 6

- 1 9 3 8- 3 4 6 8- 2 6 1 8

- 7 8 81 0 6 32 3 5 42 8 1 92 4 4 41 4 0 4

1 9 3 15 4 0

T A B L E X I

Q U I N T I C S E X I C A55 A65

1 9

- 4 4- 7 4- 7 9- 5 4

- 3

5 89 86 8

- 1 0 20

1 0 2

- 6 8- 9 8- 5 8

35 47 97 444

2 9 7 1 6

17 1s 1 3 11 9 7 5

- 5 5

- 8 8- 8 3- 36

3 91 0 4

9 1- 1 0 4

01 0 4

- 9 1-1 0 4

- 3 93 68 38 85 5

- 6 7 5- 1 0 0 0

- 7 5 14 4

9 7 91 1 4 4

- 1 0 0 10

1 0 0 1

- 1 1 4 4- 9 7 9

- 4 47 5 1

1000

675

- 2 0- 2 6 - 4-11 - 4 -9

1 8 1 - 4 -33 3 6 11 4

- 2 2 - 3 - 4 -10 0 0 0

2 2 3 4 1

- 3 3 - 6 - 1 1 -4- 1 8 - 1 4 5

1 1 4 9

26 420

1 6 7 9 6 8 3 9 8 0 8 8 4 5 2 2 6 2

1636 e ANALYTlCP4 CHEMISTRY

8/13/2019 Savitzky (1964)


PROGRAM’ 1

* SUBROUTINE SMOOTH -9 P O I N T

c

C

C I N P U T S

C N NUMBER OF R A W D A T A P O I N T S

C NDATA ARRAY 3 F N R A W D A T A P O I N T S S T O R E D I N M A I N PROGRAM

C D U M M Y D I M E N S I O N

C O U T P U T SC M NUMBER OF SMOOTHED DATA P OI NT S = N - 8

C MDATA ARRAY 3F M S MO OT HE D P O I N T S S T O RE D I N M A I N P R OG RA M

C M AY B E SA ME R E G I O N I N M A I N P R OG RA M A S N D A T A

C D U M M Y D I M E N S I O N

C

CC I N I T I A L I Z A T I O N SEGMENT

C

S U B R O U T IN E SM OO TH ( N t N D A T A t M t M D A T A l

D I M E N S I O Y Y D A T A ~ 1 0 0 0 ~ ~ M D A T A ~ l O O O ~ ~ N P [ 9 ~

M=N-8

J = I - 1DO 10 1 ~ 2 ~ 9

10 N P ( I ) = N D A T A ( J 1

C

CC SMOOTHING L o o p

DO 2 0 0 1 r l . M

DO 11 K = l t 8

J = I +8

KA = K + 1

Tf N P ( K ) = N P ( K A )

N P ( 9 ) = V D A T A ( J )

N S U M = 5 9 * N P ( 5 ) + 5 4 * ( Y P 1 4 ) + N P ( 6 ) ~ t 3 9 * ( N P ( 3 ) t N P ( 7 ) ) t l 4 * ( N P ( 2 ) + N P ( 8 ) ) ~

1 2 1 * ( N P ( l ) + N P ( 9 ) 1M D A T A ( 1 ) = N S U M / 2 3 L

200 C O N T I N U E

C

C

* E N D

9999 R E T U R N

123

456

7

8

910

111 21 3

14

1 516

17

18

19

2 0

2 1

2 2

2 3

2 42 5

2 6

27

2 8

29

30

3 1

32

3 3

34

3 53 6

3 738

39

nomial determined from a leas t squ aresfit to 9 points. Since the value of a33is 3 b Z 3 , he denominator in the aboveexpression must be divided by 6 to ge tthe normalizer of 198 found in Tab le 111.

In a l l of the above derivat ions , i t hasbeen assumed th at th e sampling intervalis the same as the absolute abscissainterval-Le., A x = 1 . If not , thevalue of Ax must be included in thenormalizat ion procedure . Hence, toeva lua te the s th de r iva t ive at the cen t ra lpoint of a set of m values, based on a n

a th degree polynomial f i t , we musteva lua te

i = m

Note that s ince A x o = l , the interval isof no concern in th e case of smo othing.

Repeated Convolution. T h e p r oc -es s of convolu t i an can be repea tedif des i red . F or exam ple , one m igh twish to fu r the r sm ooth a set of pre-v iously sm oothed po in t s , o r t o o b t a i nt h e d e r i v a t i v e o n l y a f t e r t h e r a w

da ta has been sm oothed . Th us , ifwe convolu te us ing p poin t s the f i r s tt i m e , a n d m p o i n t s t h e s e c o n d ,

i - m i n

i - m i - n

where h = i +

Equa t ion XI shows th at one need no tgo through the convolut ion proceduretwice, but can do a s ingle convolut ion,us ing 2 ( m + p) + l po in ts , and a table ofnew integers formed by combining thec’s .

For the case where a cubic smooth isto be fol lowed by obtaining the quad-ra t i c f ir s t de riva tive us ing m = 2 andp = 2 :

C, = -2, -1, 0, 1, 2, N , = 10C’, = -3, 12, 17, 12, -3, N, = 35d- 4

= C-zC’-z = 6d - 3 = c-2C‘-1 + C-,C’-2 =

d - 2 = C-2C‘O + COC’2 +

d - 1 = C-2C‘l + C1C’-2 +

o = C-2C12 + C2CI-2 + C l C ‘ 4 -

dl = By sym m et ry =

-36 + 3 = -33

C-iC’-i = -34 - 12 = -46

c-IC‘O COC’-l =

-24 - 3 - 17 = -44

c-IC’l + COCO = 044

d2 = 46d3 = 33

d a = - 6n ’ h = 350Ax

VOL. 36 NO, 8, JULY 1964 1637

8/13/2019 Savitzky (1964)


P R O G R A M 2

u. S U B R O U T I N E C E N T E Q L 3 R E N ZC

C

C C O M P U T A T I O Y O F P R E C I S E PEA K P O S I T I O N AND I N T E N S I T Y U S I N G 9 P O I N T S

C TO A Q U A D R A T I C * Y = A 2 0 + A 2 1 X + A 2 2 X S Q * X = ( - A 2 1 / 2 A 2 2 ) o I N ORDER T O

C A P P R O X I M A T E A L O R E N T Z C O N T n U R t V A L U E S A RE C O Y V E R T E D T O ABSORBAVCE

S U BR O UT I NE C E N T E ~ ( N P O I V T I X ~ Y ~ T Q A N S * D E N S )

C A ND T HE R E C I P H O C 4 L S A RE U S ED I N D E T E R M I N I N G TH E C O E F F I C I E N T S B YC O R T H O G O N A L P O L Y N O M I A L S

CD IM E N S I O N N P O I N T ( 2 5 I * D Y S ( 9 )

60 DO 6 1 Is199

I P = I + 8

P 4 = O N S ( l ) + O N S ( 9 )

P 3 = D N S ( 2 l + D N S ( 8 )

P 2 = D N S ( 3 ) + D N S 1 7 )

P l = D N S ( 4 ) + D N S ( 6 )

6 1 D N S ( I ) = ~ , / A L O G ~ O F ( ~ ~ O O ~ / N P O I N T ( I P ) )

A 2 0 = - 2 1 e * P 4 ) + 1 4 o . P 3 ) t 3 9 o ~ P ~ ) + ~ 5 4 e t P l ) t 5 9 e * D N S 5 ) )

A 2 0 = A 2 0 / 2 3 1 o

A21=14o* DNS 91-DNSIl)I)t 3o* DNS 8)-DNS 2)))t 2.~ DNS~7)-DNS 3))

l + I D N S ( 6 ) - D Y S f 4 ) )

A 2 1 = A 2 1 / 6 0 0A 2 2 = 2 8 . . P 4 ) + 7 o * P 3 ) - 8 o ~ P 2 ) - 1 7 o P l 2 0 0 ~ D ~ S S ) ~

A 2 2 = A 2 2 / 9 2 4 .

X = ( - A 2 1 / ( 2 o O * A 2 2 ) )

Y = A 2 0 + X * ( A 2 1 + X * A 2 2 )

D E N S = l o O / YT R A N S = l O O O o / l O o O * * D E N S )

1000 RETURN

* END

2

34

56

7

89

1 0

11

1 213

1 4

1 516

1 718

19

202 12 22 32 4

2 5

262 7

2 82930

3 1

APPENDIX I1

The following eleven tables containthe convolut ing integers for smoothing(zeroth derivat ive) through the f i f thderivative for polynomials of degree twothrou gh five . The y are in the form oftables of A , , , where i is the degree of t hepolynomial and j is the order of t h ede r iva t ive . Thus , to ob ta in the th i rdderivative over 17 points, assuming afourth degree polynomial ( A d 3 ) , newould use the integers in the columnheaded 17 of Tabl e VIII .

APPENDIX 111 COMPUTER PROGRAMS

Th e programm ing of today’s high

speed digi ta l computers is s t i l l an artra the r than a science. Diffe rent pro-grammers presented with the sameproblem will, in general, write quitedifferent programs to satisfactorilyaccomplish the calculation. Two pro-grams are presented here as esamplesof th e techniques discussed in this pape r.T h e y a r e w r i t t e n i n t h e F O R T R A Slanguage, since this is one of the m os twidespread of the computer pro-gramming languages . Programmersusing other languages should be able tofollow the logic quite readily and makethe appro pria te t rans la tions . Each is

wri t ten as a subroutine for incorporation

into a larger program as required.P rogram 1 is a 9-point least squares

smooth of spectroscopic da ta. Th e rawdata has previously been s tored by themain program in the region NDXTA.

Lines through 25 are evplan atoryan dhousekeeping to set up the initial con-di t ions . The 9-point array N P containsthe current set of data points to besmooth ed. Th e main loop consists oflines26thr ough 35. The inn er loop, lines28 through 30, moves th e previous set ofpoints up one position. The next pointis added b y line 31. In lines 32 and 33the convoluting integers are multipliedby the corresponding data , and theproduct s wm m ed. In line 34 , the sumis divided by the normalizing constantand the resul t ing smoothed point isstored.

Program 2 computes the precise peakposition and intensity of a set of pointswhich is known to contain a spectro-scopic peak. In order to appros im ate aLorentz contour ( 5 ) , values are con-verted t o absorbance a nd th e reciprocalsare used in deter mining the coefficientsof a polynomial (6) having the form:

Y = am u21z+a22z2

The center is a t the point where z =

- 21/2a22

The dat a points to be used a re points9 th r ou g h l 7 s t o re d i n t h e a r ra y S P O I K Tand may have any value from 30 through999. In the loop lines 12 hrough 14, eachof these points is conve rted to absorb -ance, the reciprocal taken , an d th e re-su l t s to red in the a r ray DN S .

Since for azoand u22 the convolutefunct ion is symmetric abo ut the origin,f o r m i n g h e s u m s P 4 t h r o u g hP1 in lines15 th rough 18shortens the com putat ion.The f i rs t constant , azo s found in line19, using the values of the 9-po intconvolute from Table I, and normalizedin line 20.

The value of uZ1 s computed in lines21-22 using the convolute from TableI11 (first derivative- quadratic). Xo tethat this is an ant isymmetric funct ion.Tab le V I furnished the constants for thecompu tation of in line 24. Ko tethat the normalizing factor of 924 is 2times the value given in the table. X

is computed in line 26 and the corre-sponding value of y is computed in line27 by subst i tu t ing the appropria tevalues into the polynomial . Theabsorbance or optical density (DEXS)is, of c ourse, th e recip rocal of y (line 28) .


8/13/2019 Savitzky (1964)


LITERATURE CITED 4)Guest, P. G. Numerical Methods of 9)Whittaker, E. obinson, G., “TheCurve Fitting,” p. 349 f f . University Calculus of Observations,” p. 291 f f .

1)Allen, L. c . , Johnson, L. F . ~. A m . Press, Cambridge, England, 1961, Blackie and Son, Ltd., London(5) Jones, R. N., Seshadri, K. S., Hopkins, Glasgow, 1948.hem. SOC. 5, 2668 1963).

2) Anderson, R. L., Houseman, E. F.,

ues Extended t o K = 104,”Agricultural 6)Kerawala, S M., Indian J Phys. 15, RECEIVEDor review February 10 1964.Experiment Station, Iowa State College 241 lg41). Accepted April 17, 1964. Presented inof Agriculture and Mechanical Arts, ( 7 ) SavitzkY, A., A N A L . 33, 25.4 part at the 15th Annual Sum mer Sym-Ames, Iow a, 1942. (Dec. 1961). posium on Analytical Chemistry, Univer-3)Cole, LaMont C., Science 125, 874 8)Whitaker, S., Pigford, R. L. , I n d . Eng. sity of Maryland, College Park, Md.,1957). Chem. 52, 185 1960). June 14, 1962.

“Tables of Orthogonal Polynominal Val- J ’ J Can’ Jour’ Chem. 40 334 lg6’).

Control ed -Potential Cou Iometric Ana ysis

of N-Substituted Phenothiazine Derivatives

F. HENRY MERKLE and CLARENCE A. DISCHER

College o f Pharmacy, Rutgers-The State University, Newark 4 N. 1.

b Controlled-potential electrolysis is

suitable for the coulometric deter-

mination of several pharmaceutically

impor tant N-substituted phenothi azines.

The concentration of sulfuric acid, used

as the supporting electrolyte, has a

differentiating effect on the half -wave

potentials of the compounds studied.

Polarographic measurements obtained

with a rotati ng platinum microelec-

trode established current-voltage

curves. The compounds could b e

quantitatively oxi dized to a fr ee

radic al or to a sulfoxide by selection

of suitable acid concentrations and

app li ed potentials. Electroreduction

of the free radicals occurs at appr oxi -

mately +0.25 volt vs. S.C.E. on a

platinum electrode. In the case of the

sulfoxides, a single 2-electron reduc-tion step occurred at ca. -0.95 volt

vs. S.C.E. on a mercury pool cathode.

The determination s showed g ood re-

producib ility and an accuracy of ca.

1 was obtained with sample con-

centrations of 10-3 M or greater.

HE establishm ent of an oxidationT echanism for the various N -amino- substituted pheno thiazines is asubject of considerable biologicalsignificance 2 ) . The character of thisoxidat ion has been inves t igated byusing a varie ty of analy t ical methods

(1, 4 7 , 8). However, because of thetrans ient nature of a free radical inter-mediate , re la t ively l i t t le has been doneto dem o ns t ra te the quan t i t a t iv e a spec t sof this reaction.

The application of controlled-poten t ia l e lectrolysis t o the analysis ofphenothiazine derivat ives is anextension of previous work by theauthors on the e lectro-oxidat ion ofchlorpromazine (6). Since these com-pounds are e lectrolyt ical ly act ive a tmoderate appl ied potent ia ls , con-trolled-potential coulometry offers arap id and abso lu te approach for their

quan t i t a t ive de te rm ina t ion . Moreover,this technique makes possible t he d irectdete rm ina tion of the ind ividual speciesinvolved in th e oxidat ion sequence.

The oxidation reactions for pheno-thiazine derivatives are, in general,represented by:

(1)

(2)

(3)

R: . R . + e -

a n d

R . + H z O S + 2 H + + e -

in 1 2N sulfuric acid, and

R : .R . + e -

a n d

spontaneous

2 R . + H20-- : + S + 2 H +

4)in 1 N sulfuric acid ( 2 , 6 ) , where R:represents the initial reduced form ofthe com pound , R. represents the freeradical obtained up on 1-electron oxida-t i o n , a n d S represents th e correspondingsulfoxide.

EXPERIMENTAL

Instrumentation. P ola rog ram s wereo b t a i n e d o n a S a r g e n t M o d e l XXI re-cord ing po la rog raph . An H- ty pe ce llwas used , wi th a s intered -glass disk ofm e d i u m p o r o s i t y s e p a r a t i n g t h e t w oe l e c t r o d e c o m p a r t m e n t s . A r o t a t i n g

p l a t i n u m m i c r o e le c t r o d e s e r v ed a s t h ework ing e lec t rode u s S . C . E . a s re fe r -ence. T o obt ain well defined reproduci-ble S-shaped curves it was necessaryto pretre at th e microelectrode by anodicpolarization for 10 m inutes in 1N or12N sulfuric acid at f1.0 volt v s S.C.E. ,followed immediately by a brief 2- to 3-minute e lectrolys is with the pla t inummicroelectrode as the c athod e.

The controlled-potential electrolyseswere performed with an electronic con-trol led-potent ia l coulometric t i t ra tor ,Model Q-2005 O R N L 5) . Readoutvoltages were measured with a ?Jon-Linear Systems Model 484 A digitalvol tmeter .

Electrolysis Cells and Electrodes.Tw o c losed ce ll s wi th ope ra t in g capac i -t ies of 100 an d 20 ml. , respect ively,were used fo r ox ida t ions . Th ey were

c o n s t r u c t e d t o a c c o m m o d a t e l a r g ecy l indr ica l , wi re m esh , ro ta t in g p la t i -num e lec t rodes , 2 . 5 cm . in d iam ete ra n d 5 cm. in height for the large cel l ,a n d 1 cm. in diameter and 3 cm. inheight for the small cell. Th e referenceelectrode (S .C.E .) and auxiliary plati-num cathode were isola ted from thesample compartment by s intered-glassdiaphragms a nd ag ar plugs .

The sample compartment of thereduction cell consisted of a 125-m1.wide-mouthed Erlenmeyer flask with as tanda rd-tap er neck a nd f i t ted. cover.Approximately 30 ml. of mercury wasused as the cathode pool. A glasspropel ler- type s t i rrer served to agi ta te

the mercury pool. Th e reference elec-trode (S .C.E. ) and auxil iary e lectrode(plat inu m anode) chambers mere sepa-ra ted f rom the ca thode com par tm ent byglass side-arms fitted w ith sintered-glassdiaphragms an d agar plugs .

The three cells were constructed sothat ni t rogen gas could be bubbledthrough the solut ion before and duringelectrolysis. Th e pla tinu m gauze elec-trodes and the glass s t i rrer were rota te da t 600 r .p .m. us ing a Sargent Syn-chronous Rota to r .

Materials. R E A G E N T S . h e a c i dso lu t ions were p repa red us ing su l fu r icac id , Baker ana lyzed re agen t , d i s t i ll edw a t e r , a n d e t h a n o l U . S . P . g r a d e .

PHENOTHIAZINEhRIVATIVES. T h ecompounds studied were provided inpowdered form by the suppliers in-dicated : chlorpromazine hydrochloride,chlorpromazine sulfoxide hydrochloride,prochlorperazine e thanedisulfonate , a ndtri f luoperazine dihydrochloride, Smith,Kline French Laboratories ; pro-methazine hydrochloride and promazinehydrochloride, Wy eth Laboratories ; t r i -f lupromazine hydrochloride, E R .Squib b Laboratories ; thioridazine hy-drochloride, SandDz Labor atories. Th es tructural formulas and generic namesof the compounds used in this work areshown in F igure 1.

VOL. 36, NO. 8, JULY 1964 1639

Savitzky (1964)

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