【免费】一种经典的时域滤波器。_时域滤波

ADC采样滤波

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Interferences.

Methylene chloride

chloroform-soluble carboxylic acids

(acetylsalicylic acid, salicylic acid,

etc.) which inhibited 4-,UP response

were readily extracted from these

phases using

0.ln’

sodium hydroxide.

Span-type excipients, which gave

slight response

the 4-AAP reagent,

were eliminated by virtue of their insol-

ubility in

10%

aqueous sodium chloride;

while interfering Tween-type excipients

were precipitated

(3)

using the Tween

reagent. Propylene glycol, when formu-

lated at

level of

5.25%,

contributed ca.

2 to

370

to the observed absorbance.

This was minimized by

more favorable

distribution of the interference within

the aqueous phases used in the pro-

cedure.

Placebo analyses indicated that no

interference was obtained with such

common excipients

stearic acid,

stearyl alcohol, cetyl alcohol, petrola-

tum, methyl or propyl-p-hydroxy-

benzoates, sesame oil, or thimerosal.

Lipotropic agents (betaine or choline),

all the common vitamins, and neomycin

sulfate failed to interfere.

Scope

Reaction.

In addition to

the other types of steroids reported

to react with the 4-AAP reagent, the

following steroids reacted at elevated

temperature (boiling point of methanol)

and at increased concentration (2.0 mg.

per

10.0

ml. of reagent): 2a-hydroxy-

methyl

17p

hydroxy

17a

methyl-

5a-androst-%one (306 mp)

;

and 2a-

hydroxymethyl

17p

hydroxy

501-

androst-3-one (306 mp). Steroids with-

out

keto group failed to give any

response. The quantitative aspects of

the above responses to the 4-AAP

reagent were not investigated.

Correlations

Chromophore Wave-

length with Structure.

The additive

effect of various ring

and

sub-

stituents on the chromophore of the

parent saturated-3-keto steroid is

presented in Table

11.

LITERATURE CITED

(1)

Cohen,

H.,

Bates,

W.,

Am.

Pharm.

Assoc.,

Sci.

Ed.

40, 35 (1951).

(2) Hiittenrauch,

R.,

Physiol. Chem.

326,166 (1961).

(3) Pitter, P.,

Chem.

Ind.

42, 1832

(1962).

RECEIVED

for

review September 4, 1963.’

Accepted March 13,

1964.

Smoothing and Differentiation

Data

Simplified Least Squares Pro,cedures

ABRAHAM SAVITZKY and MARCEL

GOLAY

The Perkin-Elmer Corp., Norwalk, Conn.

In attempting to analyze, on

dig i ta

computers, data

rom basica

continuous 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

analog de-

vices that perform these operations,

such as RC filters, are often considered.

However, the method of least squares

may be used without additional com-

putational complexity and with con-

siderable improvement in the informo-

tion obtained. The least squares cal-

culations may be carried out in the

computer by convolution of the data

points with properly chosen sets

integers. These sets of integers and

their normalizing factors are described

and their use

illustrated in spectro-

scopic applications. The computer

programs required are relatively sim-

ple. Two examples are presented as

subroutines in the FORTRAN language.

PRIMARY

OUTPUT

of any experi-

Tment in which quantitative

information is to be extracted is infor-

mation which measures the phenomenon

under observation. Superimposed upon

and indistinguishable from this informa-

tion are random errors which, regardless

of their source, are characteristically

described as noise. Of fundamental

importance

the esperimenter is the

removal of as much of this noise

possible without,

the same time,

unduly degrading the underlying in-

formation.

In much experimental work, the infor-

mation may be obtained in the form of

two-column table of numbers,

us.

Such a table is typically the result

digitizing

spectrum

digitizing other

kinds of results obtained during the

course of an experiment.

plotted,

this table of numbers would give the

familiar graphs of

TOT

us.

wavelength,

us.

volume of titrant, polarographic

current

us.

applied voltage,

SMR

ESR

spectrum,

chromatographic

elution curve, etc. This paper is con-

cerned with computational methods for

the removal of the random noise from

such information, and with the simple

evaluation of the first few derivatives

of the information with respect to the

graph abscissa.

The bases for the methods

be dis-

cussed have been reported previously,

mostly in the mathematical literature

(4,

9).

The objective here is to

present specific methods for handling

current problems in the processing of

such tables of analytical data. The

methods apply as well to the desk

calculator,

to simple paper and pencil

operations for small amounts of data, as

they do to the digital computer for

large amounts of data, since their major

utility is to simplify and speed up the

processing of data.

There are two important restrictions

on the way in which the points in the

table may be obtained. First, the points

must be at

fixed, uniform interval in

the chosen abscissa.

the independent

variable is time, as in chromatography

NMR

spectra with linear time sweep,

each data point must be obtained at the

same time interval from each preceding

point.

spectrum, the intervals

may be every drum division

every

0.1

wavenumber, etc. Second, the

curves formed by graphing the points

must be continuous and more

less

smooth-as in the various examples

listed above.

ALTERNATIVE METHODS

One of the simplest ways to smooth

fluctuating data is by a moving average.

In this procedure one takes

fised

number of points, adds their ordinates

together, and divides by the number

points to obtain the average ordinate at

the center abscissa of the group. Sest,

the point at one end of the group is

dropped, the next point at the other end

added, and the process is repeated.

Figure

illustrates how the moving

average might be obtained. While

there is a much simpler way to compute

the moving average than the particular

one described, the following description

is correct and can be extended to more

sophisticated methods as will be seen

shortly. This description is based on

the concept of a convolute and of a

convolution function. The set of

numbers at the right are the data or

ordinate values, those at the left, the

abscissa information. The outlined

VOL.

36,

NO.

JULY

1964

1627

1800.0 705

1799.8

712

1799.6

7 17

1799.4

718

1799.2

721

1799.0

------/-

-<I

722

1798.8

x~-2

‘12

’

725

1798.6

xo-l

c-l

730

1798.4

735

1798.2

xo+l

736

1798.0 x0+2

741

1797.8

746

-*--*--‘e

17637.6 750

...4

Figure

Convolution operation

Abscissa points at left, tabular data at right.

In box area the convolution integers,

cz.

Opera-

tion

the multiplication

the data points by the

corresponding

C,,

summation of the resulting

products, and division by

normalizer, resulting

single convolute at the point

Xo.

The

box

is then moved down one line, and the process

repeated

block in the center may be considered

to be

separate piece of paper on which

are written a new set of abscissa

numbers, ranging from

-2

thru zero to

+2.

The

C’s

at the right represent the

convoluting integers.

For

the moving

average each

is numerically equal to

one. To perform

convolution of the

ordinate numbers in the table of data

with a set of convoluting integers,

C,,

each number in the block is multiplied

by the corresponding number in the

table of data, the resulting products are

added and this sum is divided by five.

The set of ones is the convoluting

function, and the number by which we

divide, in this case,

is the normalizing

factor. To get the next point in the

moving average, the center block is slid

down one line and the process repeated.

The concept of convolution can

generalized beyond the simple moving

average. In the general case the

C’s

represent any set

convoluting

integers. There is an associated

normalizing

scaling factor. The pro-

cedure is to multiply

C--?

times the

number opposite it, then

C-1

by its

number, etc., sum the results, divide by

the normalizing factor, if appropriate,

and the result is the desired function

evaluated at the point indicated by

Co.

For the next point, we move the set

convoluting integers down and repeat,

etc. The mathematical description

this process is:

The index

represents the running

index of the ordinate data in the original

data table.

For

the moving average, each

equal to one and

is the number of

convoluting integers. However, for

many types of data the set of all

l’s,

which yields the average, is not

particularly useful. For example, on

going through

sharp peak, the average

would tend to degrade the end of the

peak. There are other types of smooth-

ing functions which might be used, and

few of these are indicated in Figure

Figure

illustrates the set where all

values have the same weight over the

interval-essentially the moving

average.

The function in Figure

is an

exponential set which simulates the

familiar

analog time constant-

Le., the most recent point is given the

greatest weight, and each preceding

point gets

lesser weight determined by

the law of exponential decay.

Future

points have no influence. Such

function treats

future

and

past

points

differently and

will obviously intro-

duce

unidirectional distortion into the

numerical results,

does the

filter in an actual instrument.

When dealing with sets of numbers in

hand, and not an actual run on an

instrument where the data is emerging

in serial order, it is possible to look

ahead

well

behind. Then we can

convolute with

function that treats

past

and

future

on an equal basis, such

as the function in Figure

20.

Here the

most weight is given to the central

point, and points on either side of the

center are symmetrically weighed

exponentially. This function acts like

an idealized lead-lag network, which is

not practical to make with resistors,

capacitors, and

on.

The usual spectrum from

spectro-

photometer is the resultant of two con-

volutions of the actual spectrum

the

material, first with

function represent-

ing the slit function of the instrument,

which is much like the triangular con-

volute shown in Figure

2C,

and then this

first convolute spectrum is further

convoluted with

function representing

the time constant of the instrument.

The triangular convoluting function

could in many cases yield results not

significantly different from the sym-

metrical exponential function.

Figure

illustrates the way in which

each of these functions would act on

typical set

spectroscopic data. Curve

is replotted directly from the instru-

mental data. It is

single sharp band

’

recorded under conditions which yield

reasonable noise level. The isolated

point just to the right

the band has

the value of

666

on the scale of zero to

1000

corresponding to approximately

100%

transmittance. This point is

introduced to illustrate the effect on

these operations of

single point which

has

gross error. The numbers along

the bottom are the digital value at the

..4-

Figure

Various convolute functions

Moving average.

Exponential func-

tion.

Symmetrical triangular function, rep-

resenting idealized spectrometer slit function.

Symmetrical exponential function

lowest point of the plot, and one may

consider that the peak goes down to

34.2%

transmittance. The base line at

the top is at about

797,.

Curve

is a nine-point moving

average of the data.

expected, the

peak is considerably shortened by this

process. Especially interesting is the

step introduced by the isolated error.

In effect, it has the shape

the bodike

convolute in Figure

2d,

which is exactly

what one would expect from the con-

voluting process

(3).

Curve

for

a triangular function

which obviously forces both the peak it-

self and the isolated error into a

triangular mold.

Curve

is the result of convoluting

with the numerical equivalent of a

conventional

esponential time con-

stant filter using only five points. The

peak is not only shortened, but is also

shifted to the right by one data point,

0.002

micron and the isolated data

point is asymmetric in the same manner.

The convolution with a symmetrical

lead-lag exponential, as in Figure

3E,

does not distort the peak but does still

reduce its intensity.

Sote that while all of these functions

have had the desired effect of reducing

the noise level, they are clearly unde-

sirable because of the accompanying

degradation of the peak intensity.

METHOD

LEAST SQUARES

The convoluting functions discussed

far are rather simple and do not

extract as much information as is pos-

sible. The experimenter, if presented

with

plot

the data points, would

tend to draw through these points a line

which best fits them. Xumerically,

this can also be done, provided one can

adequately define what is meant by

best

fit.

The most, common criterion is

that of least squares which may be

simply stated as follows:

set of points is to be fitted to

some

curve-for example, the curve

a3x3

a2x2

alx

The

a’s

are to be

selected such that when each abscissa

point, is substituted into this equation,

1628

ANALYTICAL CHEMISTRY

363

354

342

368

342

Figure

Spectral band convoluted

by the various

point functions

The number at the bottom of each peak refers

to the lowest recorded point, and is a measure

of the ability to retain the shape of the peak.

Raw data with single isolated error point.

Moving average. C. Triangular func-

tion.

Normal exponential function.

Symmetrical exponential function.

Least

squares smoothing function

the square of thr, differences between

the computed nuniberr,

and the

obyerved numbers is

minimum for the

total

the observations used in deter-

mining the coefficients. A11 of the

error is assumed

be in the ordinate

and none in thr atiscissa.

Consider thr block of seven data

points enclosed by the left bracket in

Figure

these fall along a curve

that can be described approsiniately by

the equation

;.hewn,

thrn there are

specific procedures-which are described

in most books on numerical analysis-

to find the

a's.

One then substitutes

back into the resulting equation the

abscisha at the central point indicated

by the circle. The value which is

obtained

this procedure is the best

value at that point based on the least

squares criterion, on the function which

was

chohrn. and on the group of points

esamined.

This procedure can be repeated for

ao=y

Figure

moving polynomial smooth

Representation

a 7-point

each group of seven points, dropping one

at the left and picking up one at the

right each time.

somewhat later

block is indicated at the right. In the

usual case, there is found a different set

of coefficients for each group of seven

points. Even with a high-speed com-

puter this is

tedious proposition

best.

Sote, however, that finding the

coefficients is required only

means

for

determining the final best value

just one point, the central point of the

set.

careful study of the least squares

procedure using these constraints, leads

to the derivation of

set of integers

which provide a weighting function.

Kith this set of integers the central

point can be evaluated

the con-

voluting procedure discussed above.

This procedure is esactly equivalent to

the least squares. It is not approximate.

The derivation is presented in

Appendis

For

either

cubic

quadratic function, the set of integers

is the same, and the set for up to

points is shown in Table

of Appendix

with the appropriate normalizing

factors.

most instructive exercise is

to tabulate a simple function such as

over any interval, apply these

smoot,hing convolutes and compare

these new values with the original.

The answers will be found to be exact.

In Figure

this least squares con-

voluting procedure has been applied to

the data of Figure

3A,

using

9-point

cubic convolute. The value

the peak

and the shape of the peak are es-

sentially undistorted.

always, the

isolated point assumes the shape of the

convoluting function. The FORTRAY

language comput'er program for per-

forming this operation is presented in

Program

of Appendix

111.

Going beyond simple curve fitting,

one can find in the literature on numeri-

cal analysis a variety of least squares

procedures for det,ermining the first de-

rivative. These procedures are usually

based on interpolation formulas and are

for data

any arbitrary interval.

Again,

we restrict ourselves to evaluat-

ing the function only at the center point

of a set of equally spaced observations,

then there esist sets of convoluting

integers for the first derivative as well.

(These actually evaluate the derivative

of the least squares best function.)

complete set of tables for

derivatives

to the fifth order for

polynomials up to the fifth degree, using

from

points, is presented in

.ippendis

11.

These are more than

adequate for most work, since,

the

points are taken sufficiently close to-

gether, then practically any smooth

curve will look more or less like a

quadratic in the vicinity of a peak,

cubic in the vicinity of

shoulder.

More complete tables can be found in

the statistical literature

(b,

9).

SMOOTHING-C~~~.

y'-Ouodrotic

y'-Cubic

?,-

Figure

9-Point convoluting functions

(orthogonal polynomial) for smoothing

and first, second, and third derivative

Program

of Appendis

111

shows the

use

these tables to obtain the coef-

ficients of

polynomial for finding the

precise center of an infrared band.

The shapes of the 9-point con-

volutes for a few of the functions are

illustrated in Figure

Of special

interest is the linear relation of the first

derivative convolute for a quadratic.

This

quite unique operationally be-

cause in processing a table of data, only

one multiplication is necessary for each

convolution. The remainder of the

points are found from the set calculated

for

the previous point by simple subtrac-

tion. In Figure

is shown the first

derivative of the spectrum in Figure

6A,

obtained using

9-point convolute.

The derivatives are useful in cases

such as our methods of band finding on

a computer

(Y),

in studies of derivative

spectra, in derivative therniogravimet-

ric analysis, derivative polarography,

etc.

CONCLUSIONS

With the increase in the application

computers to the analysis

digit,ized

data, the convolution methods described

are certain to gain wider usage. With

these methods, the sole function of the

computer is to act as a filter to smooth

the noise fluctuations and hopefully to

introduce no distortions into the re-

corded data

(3).

This problem of distortion is difficult,

to assess. In any of the curves of Figure

there remain small fluctuations in the

Figure

volute

17-Point first derivative con-

Original spectrum.

First derivative

spectrum.

VOL.

36,

NO.

JULY

1964

1629

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一种经典的时域滤波器。