Detection in Analytical Chemistry - American Chemical Society

Chapter 7

Noise and Detection Limits in Signal-Integrating Analytical Methods 1

2

H. C. Smit and H. Steigstra

Downloaded by UNIV OF ARIZONA on January 7, 2013 | http://pubs.acs.org Publication Date: December 9, 1987 | doi: 10.1021/bk-1988-0361.ch007

1

Laboratory for Analytical Chemistry, University of Amsterdam, Nieuwe Achtergracht 166, 1018 WV Amsterdam, Netherlands Faculty of Medicine, Radboud Hospital, Nijmegen, Netherlands 2

The uncertainty in the estimation of signal parameters is discussed, particularly the 0th moment (area) of a peak determined via an integration procedure. An overview is given of the derivations of the error variance due to integrated noise, both in the frequency domain and in the time domain. As an example the uncertainty in case of some typical kinds of noise is calculated, using the derived expressions. The theory is extended with the derivation of the optimum integration interval on basis of known peak shapes and known noise character i s t i c s , assuming stationary noise without a deterministic drift component. Finally, the influence of uncorrected linear drift on the integration variance is determined, while an expression for the variance after a frequently applied drift correction is derived, using correction intervals. One o f the b a s i c problems i n a n a l y t i c a l c h e m i s t r y i s how t o c a l c u l a t e the u n c e r t a i n t y i n t h e d e t e r m i n a t i o n o f t h e parameters o f a n o i s y a n a l y t i c a l s i g n a l . Although t h i s u n c e r t a i n t y i s important, i t i s not the o n l y f a c t o r i n f l u e n c i n g t h e d e t e c t i o n l i m i t . I t must be emphas i z e d t h a t e r r o r s and u n c e r t a i n t i e s o r i g i n a t i n g from sample p r e p r o c e s s i n g , sample i n t r o d u c t i o n , l a c k o f s t a n d a r d i z a t i o n o f t h e measurement c o n d i t i o n s , e t c . , may be j u s t as i m p o r t a n t as n o i s e p e r t u r b i n g the s i g n a l . However, i t i s c e r t a i n l y u s e f u l t o c a l c u l a t e t h e c o n t r i b u t i o n of that noise to the t o t a l u n c e r t a i n t y , determining the d e t e c t i o n l i m i t , i n the a n a l y t i c a l r e s u l t . I f o n l y one measurement ( d a t a ) p o i n t i s c o n s i d e r e d , then t h e problem reduces t o a simple comparison o f the measured a m p l i t u d e w i t h the s t a n d a r d d e v i a t i o n o f t h e n o i s e , determined by r e p e a t e d measurements. O r d i n a r y s t a t i s t i c s can be a p p l i e d t o c a l c u l a t e t h e u n c e r t a i n t y . However, o f t e n dynamic s i g n a l s , l i k e peaks i n chromatography, a r e produced and s i g n a l parameters l i k e t h e 0th moment (peak area) o r h i g h e r moments a r e r e p r e s e n t a t i v e f o r t h e d e s i r e d a n a l y t i c a l i n f o r m a tion. D e t e r m i n i n g these parameters always i n c l u d e s an i n t e g r a t i o n 0097-6156/88/0361 -0126$06.75/0 © 1988 American Chemical Society

In Detection in Analytical Chemistry; Currie, L.; ACS Symposium Series; American Chemical Society: Washington, DC, 1987.


7. SMIT AND STEIGSTRA

Noise and Detection Limits

127

p r o c e d u r e , where o f c o u r s e , t o g e t h e r w i t h t h e s i g n a l , the n o i s e i s i n t e g r a t e d as w e l l . We might f o r m u l a t e the problem as f o l l o w s : What i s the u n c e r t a i n t y i n the d e t e r m i n a t i o n o f the a n a l y t i c a l s i g n a l parameters due t o the i n f l u e n c e o f the i n t e g r a t e d n o i s e ? In t h i s paper we emphasize the d e t e r m i n a t i o n o f the peak a r e a s . P a r t i c u l a r l y i n q u a n t i t a t i v e chromatography the u n c e r t a i n t y i n t h e a r e a d e t e r m i n a t i o n i s d i r e c t l y r e l a t e d t o the d e t e c t i o n l i m i t . To e l u c i d a t e the problem f o r m u l a t i o n , a (Gaussian) peak and t h e time i n t e g r a l o f the peak i s shown i n F i g u r e 1. The r e l e v a n t i n f o r m a t i o n i s the h e i g h t I o f the i n t e g r a l w i t h r e s p e c t t o b a s e l i n e , assuming a c o n s t a n t ( f l a t ) n o i s e l e s s b a s e l i n e . I f n o i s e i s added and t h e peak i s i n t e g r a t e d a g a i n , then the f i n a l v a l u e w i l l p r o b a b l y d i f f e r from t h e t r u e v a l u e . R e p e a t i n g the same procedure w i t h a s i m i l a r peak w i t h n o i s e w i t h the same s t a t i s t i c a l p r o p e r t i e s y i e l d s a number o f s t a t i s t i c a l l y d i s t r i b u t e d data p o i n t s . I f the n o i s e i s assumed t o be s t a t i o n a r y , i . e . i f the s t a t i s t i c a l p r o p e r t i e s l i k e mean and v a r i a n c e are n o t changing w i t h time, the mean o f the mentioned data p o i n t s i s an e s t i m a t e o f the t r u e a r e a . The s t a n d a r d d e v i a t i o n or error v a r i a n c e a | determines the u n c e r t a i n t y i n t h e peak a r e a d e t e r m i n a tion. The problem o f d e t e r m i n i n g the v a r i a n c e o f i n t e g r a t e d n o i s e i s not r e s t r i c t e d t o peak parameter d e t e r m i n a t i o n . Measurement and c a l c u l a t i o n o f the average i n t e n s i t y o f a s p e c t r o s c o p i c l i n e means i n t e g r a t i n g t o o , however, the f i n a l r e s u l t i n c l u d i n g the s t a n d a r d d e v i a t i o n o f the i n t e g r a t e d n o i s e has t o be d i v i d e d by the i n t e g r a t i o n time. A l t o g e t h e r , t h i s b r i n g s us t o the d e s i r a b i l i t y t o d e r i v e an e x p r e s s i o n f o r Oj_ o r a^, r e s p e c t i v e l y , c o n t a i n i n g a l l f a c t o r s i n f l u e n c i n g the e r r o r v a r i a n c e . Of c o u r s e , t h i s e x p r e s s i o n can be used t o c a l c u l a t e the d e t e c t i o n l i m i t i n , f o r i n s t a n c e , chromatography as f a r as determined by the b a s e l i n e n o i s e . However, i t i s a l s o u s a b l e t o make an optimum c h o i c e o f parameters and c o n d i t i o n s . B e s i d e s , some r u l e s o f thumb can be g i v e n , u s a b l e i n d a i l y p r a c t i c e . One has t o keep i n mind t h a t such a d e r i v a t i o n always i m p l i e s some assumptions c o n c e r n i n g the s t a t i o n a r i t y o f the a n a l y t i c a l system and p a r t i c u l a r l y the s t a t i o n a r i t y o f the n o i s e . I n g e n e r a l , s t a t i o n a r i t y and the absence o f a d e t e r m i n i s t i c d r i f t i n g b a s e l i n e i s assumed, a l t h o u g h some d e r i v e d e x p r e s s i o n s i n the g e n e r a l form a r e v a l i d f o r n o n - s t a t i o n a r y n o i s e . However, t h e d e r i v e d t h e o r y can be used as a b a s i s f o r t h e c a l c u l a t i o n o f t h e r e m a i n i n g u n c e r t a i n t y i n the case of a c o r r e c t i o n procedure f o r d e t e r m i n i s t i c ( f o r i n s t a n c e linear) baseline d r i f t . B a s i c Theory The d e r i v a t i o n o f the e r r o r v a r i a n c e r e q u i r e s some t h e o r y from d i f f e r e n t f i e l d s . F o r the convenience o f t h e reader a v e r y s h o r t o v e r v i e w w i l l be g i v e n , i n c l u d i n g some b a s i c p r i n c i p l e s and d e f i n i t i o n s o f the r e q u i r e d t h e o r y . Another r e a s o n t o g i v e some textbook t h e o r y i s t h a t the d e f i n i t i o n o f s e v e r a l q u a n t i t i e s can d i f f e r ; l i t e r a t u r e i s n o t very c o n s i s t e n t i n that respect. A d e t a i l e d d e s c r i p t i o n of s i g n a l t h e o r y , system t h e o r y , s t o c h a s t i c p r o c e s s e s and o f course mathematics can be found i n s e v e r a l textbooks ( 1 - 5 ) . I t i s n e c e s s a r y to s o l v e t h e problem o f d e r i v i n g b o t h i n the


128


DETECTION IN ANALYTICAL CHEMISTRY

Figure 1.

Integrated peak and noise.




129

time domain and i n the f r e q u e n c y domain by F o u r i e r t r a n s f o r m i n g the time s i g n a l s o r f u n c t i o n s d e r i v e d from t h e time s i g n a l s . The F o u r i e r t r a n s f o r m (FT) o f a f u n c t i o n o f the time f ( t ) i s d e f i n e d i n t h e u s u a l way: + 00 F(jco) =

f

f(t) e"

J U 3 t

dt

(1)

— CO


j

2

= -1

We w i s h t o c o n s i d e r random v a r i a b l e s i n a c o n t i n u o u s domain. T h i s can be done by u s i n g d i f f e r e n t d e s c r i p t i v e f u n c t i o n s . F o r o u r d e r i v a t i o n s we need the concepts o f p r o b a b i l i t y d e n s i t y f u n c t i o n (PDF), a u t o c o r r e l a t i o n f u n c t i o n (ACF) and power s p e c t r a l d e n s i t y (PSD). Moreo v e r , the f o l l o w i n g system f u n c t i o n s a r e used: t h e w e i g h t i n g f u n c t i o n h ( t ) and t h e complex f r e q u e n c y response H( jd)). The w e l l known p r o b a b i l i t y d e n s i t y f u n c t i o n can be c o n s i d e r e d as the l i m i t i n g v a l u e o f t h e p r o b a b i l i t y t h a t an a m p l i t u d e o f n o i s e n ( t ) l i e s i n an i n t e r v a l around a c e r t a i n v a l u e , d i v i d e d by the w i d t h o f t h a t i n t e r v a l . The shape, which i s o f t e n G a u s s i a n , and t h e w i d t h o f the PDF, e x p r e s s e d i n the s t a n d a r d d e v i a t i o n a , a r e used f o r s t a t i s t i c a l c a l c u l a t i o n s o f d e t e c t i o n l i m i t e t c . A random s i g n a l , o r i n g e n e r a l a f a m i l y o f f u n c t i o n s o f time (random p r o c e s s ) o f w h i c h t h e v a l u e s v a r y randomly even i f i t i s s t a t i o n a r y , i s n o t u n i q u e l y s p e c i f i e d by a PDF, as i s demonstrated i n F i g u r e 2. Both random s i g n a l s have the same PDF, b u t they a r e o b v i o u s l y d i f f e r e n t . An i m p o r t a n t q u a n t i t y , summarizing much i n f o r m a t i o n about a random p r o c e s s , i s the ACF. To i l l u s t r a t e the concept o f t h e ACF, F i g u r e 3 shows a f a m i l y o f s t o c h a s t i c s i g n a l s ( s i g n a l s e v o l v i n g i n time a c c o r d i n g t o p r o b a b i l i t y l a w s ) , a random ( s t o c h a s t i c ) p r o c e s s o r an ensemble. An example o f an ensemble i s a s e t o f p o s s i b l e n o i s e r e c o r d s from a chromatographic d e t e c t o r , each r e c o r d e d d u r i n g a c e r t a i n time i n t e r v a l . Now we have t o d i s t i n g u i s h ensemble s t a t i s t i c s and time s t a t i s t i c s . F o r i n s t a n c e , t h e mean v a l u e a t t h e time t i (ensemble s t a t i s t i c s ) i s d e f i n e d : n

u(

t l

)

= lim

1 1

N

E

n, (

t l

)

(2)

k r e f e r s t o s i g n a l k. The (ensemble) ACF i s d e f i n e d : N

R (ti,t!+T) n

1 = R(ti,t ) = lim - I n (t )n (ti+T) N k= 1 2

k

1

k

(3)

0 0

b e i n g the average p r o d u c t o f the v a l u e s o f t h e s t o c h a s t i c p r o c e s s a t time t i and t 2 . I f t h e r e i s no r e l a t i o n o r b e t t e r c o r r e l a t i o n between the v a l u e s a t time t i and t , then the average w i l l tend t o t h e p r o duct o f the mean v a l u e a t t i and t . I n case o f f a s t f l u c t u a t i n g n o i s e , no c o r r e l a t i o n w i l l e x i s t even a f t e r a r e l a t i v e l y s h o r t t i m e , where s l o w l y f l u c t u a t i n g n o i s e s t i l l shows an average r e l a t i o n between t h e a m p l i t u d e s . 2

2




130

Figure 2.

Fast and slowly fluctuating noise with similar PDF.





4

Figure 3.

Ensemble of noise records.


131


132

A stochastic process x^(t) i s stationary i f the two processes x(t) and x ( t + e ) have the same s t a t i s t i c s f o r any £. In other words, the values of t i and t do not influence the ensemble s t a t i s t i c s , only the difference t i - t = T i s important, R ( t i , t ) can be replaced by R ( t i ~ t ) . ^(t) °^ noise records n^ or a set of other random variables. If the expected value E[x(t)] = ]i - constant, and E [ x ( t + T ) x ( t ) ] = R ( T ) i s only dependent on T and does not vary with the time ( t i ~ t i s replaced by T ) , then the process i s weakly stationary. Ergodicity means: a l l s t a t i s t i c s can be determined from a single function x ^ ( t ) : 2

2

c

a

n

D e

2

a

s e t

2

2


T

i

f

x, (t)dt

( T , k ) = lim

1

j

u (k) = lim

(4)

and R

X X

x^t)

x^t +

(5)

'Odt

If T = 0, Equation 5 reduces to: T R(0) = l i m

i

2

2

[ x (t)dt = E[x (t)]

(6)

being the mean square value of x ( t ) , determining the average power and thus the energy of the signal. Of course, fast and slowly fluctuating noise can also be d i s tinguished i n the frequency domain. However, noise usually goes on i n d e f i n i t e l y i n time and, actually, i t s energy i s unbounded; the Fourier transform does not e x i s t . Nevertheless, FT techniques can be applied i f the average power i s bounded. The introduction of the power spectral density function (PSD), not suitable f o r a simple representation of the (stochastic) signal, allows the introduction of expressions concerning signal energy. The PSD gives the average power per unit of frequency. Of course, the ACF and the PSD are not independent; s t r i c t l y speaking, both are representing the same properties of the signal (energy, fast or slow f l u c t u a t i o n ) . The formal d e f i n i tion of the PSD i s : S(u>)

= FT

{R

X X

(T)|

(7)

( T ) i s real and even for real processes, S(OJ) i s also r e a l and:

Because R and even,

+ 00 S(OJ)

=

[

J

R

xx

(T)

COS

CUT

dx = 2

Jf

R

xx

(T)

COS

COT

dT


(8)



133

F i n a l l y , the physical r e a l i z a b l e one-sided PSD function G(to) i s defined as: oo

G(o)) = 4

j

R

X X

(T)

cos OJT dr

(9)

0 The energy of the signal can be calculated from G(co): oo

E[x (t)] = J 2

G(u>) do) = R^CO)

(10)


0 2

2

E [ x ( t ) ] i s the mean square value i(/ of the signal x ( t ) . I f the mean of x ( t ) i s not zero, f o r instance i f x(t) i s a noise with a Direct Current (DC) component, then the mean (DC) value contributes to the t o t a l energy. However, we are only interested i n the stochastic variations of the signal and not i n the mean value which can be estimated and corrected. Therefore, i n the following we assume a mean value of zero, i n which case E [ x ( t ) ] becomes the variance a . Some d e f i n i t i o n s from l i n e a r system theory are required. The weighting function h(t) i s the response (output) of a l i n e a r system applied to an impulse s i g n a l , t h e o r e t i c a l l y a Dirac-delta function 6 ( t ) , or more precisely, a 6-distribution with the properties: 2

2

+ oo

6(t) dt (ID 6(t) = 0

(t + 0)

The FT of the impulse response i s the complex frequency response H(joj). Suppose x(t) with a PSD = G (OJ) i s the input signal of a l i n e a r system with a complex frequency response H(jio) and suppose y(t) i s the r e s u l t i n g output. Then i t can be proved (J_) that the PSD of y ( t ) i s given by: X

2

G (u>) = |H(jco)| G

y

x

(o>)

(12)

Variance of Integrated Noise The system functions, mentioned i n the previous paragraph, can be determined for an integrator. The response of an integrator applied to an impulse 6(t) i s the value 1 ( t > 0 ) , as follows from the d e f i n i t i o n of the 6-distribution (Equation 11). The FT can e a s i l y be calculated, r e s u l t i n g i n :

H(jO)) = FT | h ( t ) | = J

h(t) e

j U ) t

dt = J

-joot

dt =

J _


(13)


134

A c c o r d i n g t o E q u a t i o n 12 we c a n c a l c u l a t e t h e PSD o f t h e o u t p u t , assuming a s t o c h a s t i c i n p u t s i g n a l w i t h known PSD: G (0)) = |H(ja))| G (co) = - L (u>) y x ^2 x

(14)

2

G

At f i r s t s i g h t , i t appears p o s s i b l e t o c a l c u l a t e , w i t h o u t any p r o blem, t h e v a r i a n c e o f t h i s o u t p u t s i g n a l by i n t e g r a t i n g the c a l c u l a ted output PSD over the 0)-range from zero t o (see E q u a t i o n 10): 0 0

?

k V

=

°° i f - L G (co)dco J u)


(15)

x

2

However, t h i s i s n o t c o r r e c t , as a l r e a d y i s shown i n p r e v i o u s papers ( 6 , 7 ) . A s i g n a l i s never i n t e g r a t e d from -°° to+°°, b u t always d u r i n g a l i m i t e d time i n t e r v a l . In r e a l i t y , t h e impulse response h ( t ) o f an i n t e g r a t o r i s n o t 1, b u t i s g i v e n by: h(t)

= 1

(0 12. The c o n t r i b u t i o n t o t h e i n t e g r a l r e s u l t i n g from t h e i n t e r v a l between U)0 /2 ( > 1 2 ) and i n f i n i t e i s n e g l i g i b l e . The v a l u e o f t h e i n t e g r a l w i t h i n t e g r a t i o n l i m i t s 0 and i s TT/2. The f i n a l r e s u l t i s : T

00



136

k

a

2

- a

I

2

& n u)

(25)

Q

In this p a r t i c u l a r case the variance of the integrated noise i s proportional to the integration i n t e r v a l and to the variance a of the o r i g i n a l baseline noise. We note that a and COQ 22£ independent, reducing COQ means reducing o . Noise with a strong 1/f character i s much more r e a l i s t i c (7), p a r t i c u l a r l y i n chromatography. The PSD of 1/f (or 1 /to) noise i s proportional to 1/to. Because of the s i n g u l a r i t y in co=0, a s l i g h t l y modified model i s more r e a l i s t i c . Such a PSD might be: n

A R E

n


n

G(u)) = K/oj£

0 < oo < u>£ (26)

G(oj) =

K/00

< co <

Detection in Analytical Chemistry - American Chemical Society

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