Correction for Instrumental Broadening in Size Exclusion

Chapter 17 Correction for Instrumental Broadening in Size Exclusion Chromatography Using a Stochastic Matrix Approach B a s e d o n Wiener Filtering T h e o r y

Downloaded by EAST CAROLINA UNIV on April 11, 2018 | https://pubs.acs.org Publication Date: October 2, 1987 | doi: 10.1021/bk-1987-0352.ch017

L. M. Gugliotta, D. Alba, and G. R. Meira

1

Intec (Conicet and Universidad Nacional del Litoral), (3000) Santa Fe, Argentina The correction for non-uniform instrumental broadening in SEC is solved through a non-recursive matrix stochastic technique. To this effect, Tung's equation (1) must be reformulated in matrix form, and the meas urements assumed contaminated with zero-mean noise. The proposed technique is based on an extension to time-varying systems of Wiener's optimal filtering method (1-3). The estimation of the corrected chromato gram is optimal in the sense of minimizing the estima tion error variance. A test for verifying the results is proposed, which is based on a comparison between the "innovations" sequence and its corresponding expected standard deviation. The technique is tested on both synthetic and experimental examples, and com pared with an available recursive algorithm based on the Kalman filter (4). Most methods of correction for instrumental broadening in SEC (or hydrodynamic chromatography) are based on the deterministic integral equation due to Tung (_5) : oo z(t) = j g(t,x) U(T) di (1) —OO where t,x: both represent elution time or elution volume; z(t): is the base-line corrected chromatogram; g(t,x): is the time-varying or non-uniform spreading function, which is built up by the set of unit mass impulse responses g(t) of truly monodisperse polymers with dif ferent elution times τ ; and u(t): is the corrected chromatogram. When g(t,x) is considered time-invariant, then Equation 1 re duces to a convolution integral. There are two basic problems associated to Equation 1: i) the determination of the spreading g(t,x); and Correspondence should be addressed to this author. 1

0097-6156/87/0352-0287$06.00/0 © 1987 American Chemical Society

Provder; Detection and Data Analysis in Size Exclusion Chromatography ACS Symposium Series; American Chemical Society: Washington, DC, 1987.

288

DETECTION AND DATA ANALYSIS IN SIZE EXCLUSION CHROMATOGRAPHY


i i ) t h e e s t i m a t i o n of u ( t ) , based on the knowledge o f z ( t ) and g(t,x). W i t h r e s p e c t t o t h e s p r e a d i n g c a l i b r a t i o n , s e v e r a l methods have been suggested e.g. (6-lk). Numerous t e c h n i q u e s have been proposed f o r s o l v i n g the i n v e r s e f i l t e r i n g problem r e p r e s e n t e d by E q u a t i o n 1, w i t h d i f f e r e n t degrees o f success e.g. (U,15-19)» Only r e f e r e n c e s (]0 , ( l 8 ) and (19) make no assumptions on t h e shape o f g ( t , x ) . I n t h i s work, an i n v e r s e f i l t e r i n g t e c h n i q u e based on Wiener's o p t i m a l t h e o r y ( l - 3 ) i s p r e s e n t e d . T h i s approach i s v a l i d f o r t i m e v a r y i n g systems, and i s s o l v e d i n t h e time domain i n m a t r i x form. A l s o , i t i s i n many r e s p e c t s e q u i v a l e n t t o the n u m e r i c a l l y " e f f i c i e n t " Kalman f i l t e r i n g approach d e s c r i b e d i n (k) . For t h i s r e a s o n , a comparison between the two t e c h n i q u e s w i l l be made. Theory The S p r e a d i n g Model. C o n s i d e r i n g t h e d i s c r e t e v e r s i o n o f E q u a t i o n 1, and b e a r i n g i n mind t h a t a l l i n t e r v e n i n g f u n c t i o n s are o f f i n i t e l e n g t h , then one may w r i t e : k =s I g(k,k ) k =-r 0

z(k)

=

0

(k = 0 , 1 , 2 , . . . ,

u(k ) 0

n)

(2)

0

where k,kQ*. are t h e d i s c r e t e e q u i v a l e n t s o f t and τ , r e s p e c t i v e l y ; -r,s: a r e t h e lower and upper l i m i t s o f t h e sum i n E q u a t i o n 2, w i t h non-zero v a l u e s o f the i n d i c a t e d p r o d u c t . L e t ζ and u denote column v e c t o r s such t h a t : [z(0), z ( l ) ,

., z(n)

[u(0), u ( l ) ,

u(n)]-

(3a) (3b)

_z has n o r m a l l y more non-zero elements than u. Even though t h e t h e o r y can be m o d i f i e d t o a l l o w f o r t h i s f a c t , we assume f o r s i m p l i c i t y t h a t u has the same number o f components as z_. T h e r e f o r e , one can w r i t e E q u a t i o n 2 i n m a t r i x form as f o l l o w s : ζ =

G u

(Ua)

with g(0,0) g(0,l)

. . . g(0,n)"

g(l,0)

. . . g(l,n)

g(l,l)

(kh)

,g(n,0) g ( n , l ) . . . g(n,n) Note the f o l l o w i n g : In g e n e r a l , the f i r s t and t h e l a s t elements of u w i l l be z e r o . Each column o f G c o n t a i n s an impulse response, w i t h t h e impulse a p p l i e d at t h e element i n the d i a g o n a l o f G. (See F i g u r e 1 f o r a 3-D r e p r e s e n t a t i o n o f a t y p i c a l G m a t r i x . )


17.

GUGLIOTTA ET A L .

Instrumental Broadening in

SEC

g(k,k )


0

F i g u r e 1: T r i d i m e n s i o n a l r e p r e s e n t a t i o n o f a t y p i c a l G m a t r i x .


289

DETECTION A N D DATA ANALYSIS IN SIZE EXCLUSION CHROMATOGRAPHY

290

Hess and K r a t z as f o l l o w s :

(6) t r i e d t o e s t i m a t e u d i r e c t l y

Û=

from E q u a t i o n s

k

(5)

G-lz


In general, t h i s operation i s numerically i l l - c o n d i t i o n e d , l e a d i n g to i n c o r r e c t r e s u l t s . (The degree o f i l l - c o n d i t i o n i n g may be measured by t h e c o n d i t i o n number, d e f i n e d by t h e r a t i o o f the modulus o f the l a r g e s t t o the s m a l l e s t e i g e n v a l u e o f G ) . A s t o c h a s t i c v e r s i o n o f E q u a t i o n ka. may be w r i t t e n : ζ =

G u + v

(6a)

ζ_ =

y + y_

(6b)

T where

y_ =

[ν(θ),..., v(n) ] : i s a zero-mean a d d i t i v e n o i s e ; y: i s the n o i s e - f r e e measured chromato gram; and _z,u,y_: w i l l be assumed zero-mean s t o c h a s t i c variables. In what f o l l o w s , we s h a l l seek a r e s t o r i n g m a t r i x H such t h a t t h e e s t i m a t e u_ i s c a l c u l a t e d t h r o u g h : u =

Η ζ

(Τ)

Λ.

Let e^ = (u-u) be the e s t i m a t i o n e r r o r a s s o c i a t e d w i t h u . The e s t i mate u i s chosen i n such a way t h a t t h e c o r r e s p o n d i n g mean square e r r o r Ε[(u-u) (u-u)] i s m i n i m i z e d . The Input E s t i m a t i o n Through a Wiener F i l t e r i n g Approach. Equation 6a r e p r e s e n t s a t i m e - v a r y i n g l i n e a r f i l t e r w i t h a measurement no i s e, and t h e s t a t i s t i c s o f such n o i s e may be c o n s i d e r e d n o n - s t a t i o n a r y . Simply s t a t e d , the o p t i m a l i n v e r s e f i l t e r i n g problem i s t h i s : assum i n g t h a t a s i g n a l i s f i r s t d i s t o r t e d t h r o u g h a l i n e a r f i l t e r o f known c h a r a c t e r i s t i c s and then contaminated w i t h an a d d i t i v e n o i s e , what l i n e a r o p e r a t i o n on the r e s u l t i n g measurement w i l l y i e l d the b e s t e s t i m a t i o n o f t h e o r i g i n a l s i g n a l ? . " B e s t " i n t h i s case means minimum mean-square e r r o r . T h i s branch o f f i l t e r i n g began w i t h N. Wiener's work i n the 19U0's ( 1_) . R.E. Kalman t h e n made an important c o n t r i b u t i o n i n the e a r l y 1960's; by p r o v i d i n g an a l t e r n a t i v e approach t o t h e same problem u s i n g s t a t e - s p a c e methods ( 2 0 - 2 1 ) . N. Wiener's s o l u t i o n was o r i g i n a l l y d e r i v e d i n the frequency domain f o r t i m e - i n v a r i a n t systems w i t h s t a t i o n a r y s t a t i s t i c s . In what f o l l o w s , a m a t r i x s o l u t i o n d e r i v e d from such approach but developed i n the time domain f o r t i m e - v a r y i n g systems and n o n - s t a t i o n a r y s t a t i s t i c s w i l l be p r e s e n t e d ( 2 2 - 2 3 ) · An e x p r e s s i o n f o r the r e q u i r e d t r a n s f o r m a t i o n H i n E q u a t i o n 7 w i l l be o b t a i n e d . In a l l t h a t f o l l o w s , we s h a l l denote w i t h u the b e s t e s t i m a t e o f u, i . e . an e s t i m a t e such that :

T

T

E[(u-Û) (u-Û)] < E [ ( u - u ) ( u - u ) ]


(8)

17.

Instrumental Broadening in

GUGLIOTTA ET AL.

291

SEC

where i s any s u b o p t i m a l e s t i m a t e of u_. The p r i n c i p l e o f o r t h o g o n a l i t y ( 2 4 - 2 6 ) , s t a t e s t h a t E q u a t i o n 8 w i l l be v e r i f i e d i f the e s t i m a t i o n e r r o r v e c t o r i s o r t h o g o n a l t o t h e measurements. I n o t h e r words, t h e f o l l o w i n g must be t r u e : E[(u-u)z ] =

T

0

and

Equation

Τ

6a

S u b s t i t u t i n g Equation a t i n g , one o b t a i n s :

T

E[uu ] G


We

T

T

+ E[uv ] =

(9) into

Equation

9 and

oper

Η E[zz_T]

(lO)

s h a l l assume t h e i n p u t u u n c o r r e l a t e d w i t h v, i . e . : E[uvT] =

E[vuT]

ο

=

( ) U

Let Z and Σ be t h e c o v a r i a n c e m a t r i c e s c o r r e s p o n d i n g t o u and z_, r e s p e c t i v e l y . (Such m a t r i c e s a r e i n g e n e r a l non-stationary")". Thus, E q u a t i o n 10 may be w r i t t e n : u

ζ

Σ

G

u

T

=

Η Σ

(12)

ζ

From: Σ

T

=

ζ

E[(Gu+v)(Gu+v) ]

and b e a r i n g i n mind E q u a t i o n 11, one Σ

=

ζ

G Σ

finds: G

u

(13)

T

+ Σ

(1*0

ν

S u b s t i t u t i n g E q u a t i o n lk i n t o E q u a t i o n 12 and arrives at: Η = In

Σ

G

η

T

[G Σ

G

α

T

+ Σ^'

o t h e r words, t h e o p t i m a l e s t i m a t e may

Û = —

Σ

u

GT

[G

Σ

G

u

T

+

o p e r a t i n g , one

1

(15)

be c a l c u l a t e d

Σ

ν

l"

finally

1

through:

(16)

ζ —

Note the f o l l o w i n g : ^ - F o r any a r b i t r a r y G , t h e e x i s t e n c e o f [ G Σ G + Σ ] ~ i s ensured by t h e i n v e r t i b i l i t y o f Σ . - A d o p t i n g Σ = q I and Σ = 0 , t h e n E q u a t i o n l 6 reduces t o E q u a t i o n 5. - W i t h E = q l and Σ = Γ Ι , E q u a t i o n 16 has a format which i s i d e n t i c a l t o t h e s o l u t i o n d e r i v e d i n (27) t h r o u g h a d e t e r m i n i s t i c minimum l e a s t squares approach f o r t i m e - i n v a r i a n t systems. T h i s i s t o be e x p e c t e d , because t h e Wiener f i l t e r i n g t e c h n i q u e may be i n f a c t i n c l u d e d as p a r t o f t h e g e n e r a l t h e o r y o f l e a s t s q u a r e s . T

Υ

Ν

Ν

ν

u

u

Υ

The F i l t e r Adjustment. The computation o f u t h r o u g h E q u a t i o n 16 i n v o l v e s the p r e s p e c i f i c a t i o n of Σ and Σ · These m a t r i c e s a r e i n g e n e r a l symmetric; and t h e s i m p l i f i c a t i o n o f c o n s i d e r i n g b o t h v. and u w h i t e n o i s e s has been found t o p r o v i d e s a t i s f a c t o r y r e s u l t s . Thus, Σ and Σ w i l l be assumed d i a g o n a l . Η

Α

Ν

Γ


292

DETECTION A N D DATA ANALYSIS IN SIZE EXCLUSION CHROMATOGRAPHY

The s t a t i s t i c s of ν may be considered stationary with p h y s i c a l b a s i s . T h e r e f o r e , we s h a l l s i m p l y adopt: Σ

=

γ

sound

r I

(IT)

where t h e s c a l a r r may be o b t a i n e d from the sample v a r i a n c e of the chromatogram b a s e l i n e n o i s e . Note t h a t f o r any p o s i t i v e r , the i n v e r t i b i l i t y o f [G E G + Σ ] i n E q u a t i o n l 6 i s t h e o r e t i c a l l y ensured. C o n s i d e r now the e s t i m a t i o n o f the d i a g o n a l elements o f E . The f o l l o w i n g assumptions can be made: a) Take the v a r i a n c e of u(k) t o be c o n s t a n t . I n t h i s c a s e , and remem b e r i n g t h a t u(k) i s assumed o f zero mean, one may w r i t e : t

U

ν


u

Σ where t h e v a l u e o f z ( k ) as f o l l o w s :

q may

=

u

be

simply

J

q =

I k=0

=

u

estimated

[z(k)]

b) A l l o w now t h e v a r i a n c e of u(k) r e a l l i s t i c than b e f o r e ) . C a l l : Σ

(18a)

q I

from the measurement

(18b)

2

t o be t i m e - v a r y i n g . ( T h i s i s more

d i a g . [ q ( 0 ) , q ( l ) , ... q(n)]

(19)

Here, we can e s t i m a t e q(k) i n s e v e r a l ways, f o r example: q(k) =

Ci [z(k)]

2

(20)

q(k) =

C

2

(21)

or 2

[u(k)]

where C^, C a r e p o s i t i v e c o n s t a n t s , and u(k) i s any o t h e r e s t i m a t i o n of u. 2

suboptimal

The S o l u t i o n V a l i d a t i o n . Obvious c o n d i t i o n s t h a t the r e s u l t a n t s o l u t i o n u must s a t i s f y a r e : a) u must be n o n - n e g a t i v e ; b) the o p e r a t i o n Gu s h o u l d p r o v i d e a n o i s e - f r e e measured f u n c t i o n ; and c) t h e areas under t h e measured and the c o r r e c t e d chromatograms must be e q u a l . I t s h o u l d be emphasized t h a t c o n d i t i o n b) i s a n e c e s s a r y but not s u f f i c i e n t f o r good r e s u l t s . Apart from the mentioned c h e c k s , a v a l i d a t i o n procedure based on the a n a l y s i s o f t h e i n n o v a t i o n s w i l l now be presented. C o n s i d e r f i r s t the c o v a r i a n c e m a t r i x Σ~ c o r r e s p o n d i n g t o the .... . u e s t i m a t i o n e r r o r e^, i . e . : e

Σ

(22)

T

= e

E[(u-Û)(u-Û) ]

u

S u b s t i t u t i n g E q u a t i o n 6a i n t o E q u a t i o n 7 and t h e l a t t e r i n t u r n i n t o E q u a t i o n 22, one o b t a i n s : Σ

e

= u

Σ

u

- 2 H G E

u

+ H G Σ

u

G

T

H

T

+ H ϋ

v

H

T


(23)

The

293

Instrumental Broadening in SEC

GUGLIOTTA ET A L .

17.

i n n o v a t i o n s sequence e^ i s d e f i n e d by:

Ëz

=

L

"1

(2*0

and t h e r e f o r e ,

£z =

(25)

ζ - Gu


because t h e best e s t i m a t e f o r ζ i s y = Gu, s i n c e Y_ i s zero mean. Sub s t i t u t i n g E q u a t i o n 6a i n t o E q u a t i o n 25 y i e l d s : e_2 =

(26)

G e^ + ν

The c o r r e s p o n d i n g c o v a r i a n c e m a t r i x i s found s u b s t i t u t i n g E q u a t i o n 23 into: Σ

= e

E[(Ge +v) u

(27)

T

(G e^ + v ) ]

z

and t h e f i n a l r e s u l t i s : Σ

e

= G Σ G u

z

+ Σ

T

- 2 G H G E

u

G

T

H

H G E G u

T

Η

Τ

G

T

+ G Η Ε }F G? + ν (28)

γ

The proposed check c o n s i s t s i n matching t h e i n n o v a t i o n s sequence o b t a i n e d from E q u a t i o n 25 w i t h t h e c o r r e s p o n d i n g expected t i m e v a r y i n g v a r i a n c e p r o v i d e d by E q u a t i o n 28. I f t h e i n n o v a t i o n s sequence i s assumed zero-mean Gaussian w h i t e , t h e n e ( k ) s h o u l d be w i t h i n t h e + σ (k) bounds f o r a p p r o x i m a t e l y two t h i r d s o f t h e t i m e . ( a (k) r e presents t h e s t a n d a r d d e v i a t i o n o f e ( k ) , found by square rooming t h e d i a g o n a l elements o f Z )· Note t h a t t h e proposed check must be perfomed a f t e r h a v i n g ob t a i n e d t h e e s t i m a t i o n o f u . I n c o n t r a s t , i n t h e Kalman f i l t e r t e c h n i q u e (k) , t h e c o r r e s p o n d i n g v a l u e s o f e^ and Σ may be r e c u r s i v e l y ^z c a l c u l a t e d along w i t h the input estimate. z

θ

e

z

e

e

Examples o f A p p l i c a t i o n In o r d e r t o compare t h e p r e s e n t t e c h n i q u e w i t h t h e method based on t h e Kalman f i l t e r (k), t h e same examples p r e s e n t e d i n t h a t p u b l i c a t i o n w i l l be attempted. The f i r s t two examples a r e s y n t h e t i c , w h i l e t h e t h i r d i s based on r e a l e x p e r i m e n t a l d a t a . A l l examples were s o l v e d by means o f a VAX 11/780 computer programmed i n FORTRAN 77. R o u t i n e s f o r m a t r i x o p e r a t i o n from t h e IMSL package (28) were u t i lized. Example 1. By p r o c e s s i n g t h e curve u(k) shown i n F i g u r e 2a t h r o u g h a t i m e - v a r y i n g f i l t e r d e f i n e d by t h e s e t o f impulse responses o f F i g u r e 1, a n o i s e - f r e e chromatogram y ( k ) i s o b t a i n e d . T h i s curve was then contaminated w i t h Gaussian w h i t e n o i s e o f a r e l a t i v e l y low v a r i a n c e (10~5) t o p r o v i d e z(k) . C l e a r l y , t h e b e s t e s t i m a t e f o r r i s 10~5 and i n t h i s case a c o n s t a n t v a l u e f o r q=5xlO*"3 v a s adopted by t r i a l and e r r o r , p r o v i d i n g an a c c e p t a b l e compromise between t h e d i f f e r e n t checks. 5

5


294



L e t ûj^(k) be t h e o p t i m a l e s t i m a t e o b t a i n e d through the Kalman a p p r o a c h , u(k) and u^(k) a r e a l s o shown i n F i g u r e 2a. The i n n o v a t i o n s c o r r e s p o n d i n g t o û(k) are r e p r e s e n t e d i n F i g u r e 2b. T h i s example was s o l v e d assuming n o i s y b a s e l i n e s e c t i o n s b e f o r e and a f t e r the peak as p a r t of the chromatogram. For t h i s r e a s o n , and because q was assumed c o n s t a n t , o s c i l l a t i o n s are observed i n u(k) and uf[(k) i n t h o s e s e c t i o n s of the c u r v e . In both t e c h n i q u e s , b e t t e r e s t i m a t i o n s a r e obt a i n e d i f q i s adopted t i m e - v a r y i n g t h r o u g h E q u a t i o n 20. In t h i s c a s e , the mentioned o s c i l l a t i o n s around t h e b a s e l i n e s e c t i o n s b e f o r e and a f t e r the peak d i s a p p e a r . Example 2. T h i s example was f i r s t suggested by Chang and Huang (29), and attemped l a t e r on by Hamielec and co-workers (19)· The problem i s i l l u s t r a t e d by F i g u r e 3, which r e p r e s e n t s the f o l l o w i n g : u ( k ) , t h e u n i f o r m s p r e a d i n g f u n c t i o n g ( k ) , t h e broadened curve z ( k ) , and t h e r e c u p e r a t e d u ( k ) by method 2 proposed i n (19)» The s o l u t i o n shown i n F i g u r e 3 i s p r a c t i c a l l y c o i n c i d e n t w i t h t h a t of (29) , and w i t h t h a t o f method 1 i n (19). C l e a r l y , t h e s e t e c h n i q u e s are unable t o approp r i a t e l y r e c o v e r the double-peaked i n p u t . T h i s problem was s o l v e d a d o p t i n g t h e same v a l u e s f o r r and q as i n (k_) , i . e . : r=0.1 and q c a l c u l a t e d t h r o u g h E q u a t i o n 20 w i t h C]_=l. The r e s u l t s are shown i n F i g u r e Ha, where the o r i g i n a l u(k) i s comp a r e d t o the e s t i m a t e s o b t a i n e d through t h e proposed t e c h n i q u e and t h r o u g h the Kalman approach ( F i g u r e 10a of 00). Figure illust r a t e s the i n n o v a t i o n s t e s t . 2

Example 3· Curve z ( k ) i n F i g u r e 5 r e p r e s e n t s the chromatogram o f a PS s t a n d a r d o f MW=525, when f r a c t i o n a t e d through an A-802 Shodex column mounted on a S e r i e s 3-B P e r k i n Elmer l i q u i d chromatograph. The chromatogram o f pure benzene g(k) i s adopted as t h e u n i f o r m s p r e a d i n g f u n c t i o n . The polymer sample i s expected t o be i n t e g r a t e d by the f i r s t PS o l i g o m e r s , w i t h preponderance o f the pentamer. I d e a l l y , d e l t a f u n c t i o n s ought t o be r e c u p e r a t e d , w i t h the h i g h e s t peak at a m o l e c u l a r weight o f 520. Here, a v a l u e o f r=5xlO~5 was a d o p t e d , and q was c a l c u l a t e d t h r o u g h E q u a t i o n 20 w i t h C]_=0.75. In F i g u r e 5, the r e s u l t o f the p r e s e n t t e c h n i q u e i s compared t o the r e s u l t i n F i g u r e 12a o f 00. As w i t h a l l p r e v i o u s examples, t h e e s t i m a t e d n o i s e - f r e e chromatogram y ( k ) i s p r a c t i c a l l y c o i n c i d e n t w i t h the measured z ( k ) . Conclusions The proposed t e c h n i q u e i s n u m e r i c a l l y " r o b u s t " , and i t s r e s u l t s are comparable t o t h o s e o b t a i n e d t h r o u g h a r e c u r s i v e method based on the Kalman f i l t e r {k) . I t s h o u l d be noted t h a t because the p r e s e n t t e c h n i q u e u t i l i z e s a l l o f t h e i n f o r m a t i o n s i m u l t a n e o u s l y , the r e s u l t s have been compared t o those of the o p t i m a l smoother e s t i m a t e s i n 00 , w h i c h a r e " b e t t e r " than the t r u e f i l t e r e d e s t i m a t e s . The main advantage o f t h e s t o c h a s t i c m a t r i x approach i s the s i m p l i c i t y f o r i t s computer i m p l e m e n t a t i o n . E q u a t i o n 17 d i r e c t l y p r o v i d e s the d e s i r e d r e s u l t , and E q u a t i o n 28 i s the b a s i s of a v a l i d a t i o n t e s t which may or may not be performed a c c o r d i n g t o p r e v i o u s e x p e r i e n c e . In o t h e r words, the proposed method i s c o n c e p t u a l l y and p r a c t i c a l l y e a s i e r t o implement than t h e Kalman c o u n t e r p a r t . The



17.


GUGLIOTTA ET A L .

295

b

0

50

100

F i g u r e 2: Example 1: a) Comparison between t h e " t r u e " i n p u t u ( k ) , t h e e s t i m a t i o n o f t h a t i n p u t t h r o u g h t h e present t e c h n i q u e u ( k ) and t h e same e s t i m a t i o n t h r o u g h t h e method d e s c r i b e d i n (h) u^-(k) ; b) I n n o v a t i o n s sequence and +o (k) bounds c o r r e s p o n d i n g t o u ( k ) . e

F i g u r e 3: Example 2:

( a f t e r Hamielec and co-workers

(19))·




296

F i g u r e 5: Example 3: a) E x p e r i m e n t a l chromatogram, s p r e a d i n g f u n c t i o n and comparison o f p r e s e n t r e s u l t s w i t h those i n {k); b) V a l i d a t i o n t e s t f o r û(k).


17.

GUGLIOTTA ET A L .


297

p r i n c i p a l drawback o f t h e p r e s e n t t e c h n i q u e i s i t s r e l a t i v e l y h i g h c o m p u t a t i o n a l c o s t , both i n memory and computation t i m e . T y p i c a l l y , i n o r d e r t o s o l v e a chromatogram o f 128 p o i n t s w i t h a t i m e - v a r y i n g q(k) , a computation time o f 5 mins. was r e q u i r e d t o e s t i m a t e u(k) , and 4.5 more mins. were n e c e s s a r y f o r t h e v a l i d a t i o n t e s t . A p o i n t t h a t has not been i n v e s t i g a t e d i s t h e p o s s i b i l i t y o f c o n s i d e r i n g u ( k ) a c o l o u r e d n o i s e i n s t e a d o f white n o i s e , and t h e r e f o r e a non d i a g o n a l E . F o r example, t h e c h o i c e o f a t r i d i a g o n a l E would imply t h e assumption o f u ( k ) a random walk p r o c e s s . On t h e one hand, by imposing a c o r r e l a t i o n among s u c c e s s i v e v a l u e s o f u ( k ) , t h e f l e x i b i l i t y o f t h e output i s reduced, and f o r example a d e l t a f u n c t i o n c o u l d not be r e c u p e r a t e d . On t h e o t h e r hand, smoother o u t p u t s and b e t t e r s o l u t i o n s c o u l d be o b t a i n e d i f good "a p r i o r i " e s t i m a t i o n s o f t h e r e a l a u t o c o r r e l a t i o n s o f u ( k ) c o u l d be p r o v i d e d . F i n a l l y , i t s h o u l d be noted t h a t a p a r t from i t s use i n chromato g r a p h i c data t r e a t m e n t , i n v e r s e f i l t e r i n g t e c h n i q u e s such as t h a t de s c r i b e d i n t h i s work have a l s o p o t e n t i a l a p p l i c a t i o n s i n o t h e r areas o f p o l y m e r i z a t i o n e n g i n e e r i n g , ( s e e f o r example (30) and ( 3 l ) ) . u


u

Acknowledgments We would l i k e t o thank Mr. M. B r a n d o l i n i f o r h i s h e l p w i t h t h e e x p e r i m e n t a l work, CONICET and U.N.L. f o r t h e i r f i n a n c i a l support and Dr. J . F . Weisz f o r r e v i s i n g t h e m a n u s c r i p t .

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20. Kalman, R.E., Trans. ASME, Series D, J . Basic Eng., 1960, 82, 35. 21. Kalman, R.E. and Bucy, R.S., Trans. ASME, Series D, J. Basic Eng., 1961, 83, 95. 22. Booton, R.C., Proc. IRE, 1952, 40, 977. 23. Davis, M.C., IEEE Trans, on Automatic Control, 1963, AC-8, 196. 24. Papoulis, Α., "Probability, Random Variables and Stochastic Proc esses", McGraw-Hill 1965. 25. Srinath, M.D., Rajasekaran, P.K.; "An Introduction to Statistical Signal Processing with Applications", Wiley 1979. 26. Anderson, B.D.O. and Moore, J.B., "Optimal Filtering", Prentice Hall 1979. 27. Rosen, E.M. and Provder, T., J . Appl. Polym. Sci., 1971, 15, 1687. 28. The International Mathematical Statistical Libraries, Inc. 1980. 29. Chang, K.S. and Huang, R.Y.M., J . Appl. Polym. Sci., 1969, 13, 1459. 30. Couso, D., Alassia, L. and Meira, G.R., J. Appl. Polym. Sci., 1985, 30(8), 3249. 31. Gugliotta, L.M. and Meira, G.R., Die Makromoleculare Chemie, (in press). RECEIVED

February

26, 1987


Correction for Instrumental Broadening in Size Exclusion

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