Differentiation, semidifferentiation and semi-integration of a digital signals based on Fourier transformations

JO~JRN,AL OF

:-, .

i". I

ELSEVIER

Journal of Electroanalytical Chemistry 403 (1996) 1-9

Differentiation, semidifferentiation and semi-integration of a digital signals based on Fourier transformations Jun-Sheng Yu *, Zu-Xun Zhang Department of Chemistry, Nanjing Unioersity,Nanjing 210093, People's Republic of China Received 11 January 1995; in revised form 23 August 1995

Abstract

A versatile algorithm has been developed for the integer and semi-integer differentiation and the semi-integration of a discrete digital signal. A discrete differentiation function is derived from the continuous derivative theorem of Fourier transforms. Differentiation, semidifferentiation and semi-integration of any discrete digital signal can be accomplished by fast Fourier transformation using this function. In the algorithm, the order of differentiation can be a positive integer and/or a semi-integer, or a negative integer and/or a semi-integer. When it is negative, the operation corresponds to the integral or semi-integral. In the present study, the derivatives, semiderivatives and semi-integrals of theoretical and experimental voltammograms calculated using the algorithm are demonstrated. The results show that semi-integer-order derivative and semi-integral curves of theoretical cyclic voltammograms are in agreement with semidifferentiation and semi-integral electroanalysis theory. Another advantage of the algorithm is that differentiation, semidifferentiation or semi-integration with smoothing of the discrete signal can be performed simultaneously by Fourier transformation. Keywords: Signal processing; Cyclic voltammetry

1. Introduction The differentiation of analytical signals is frequently used in analytical chemistry to improve data manipulation and analysis as more useful information can be extracted from the analytical signals. Chromatographic and spectroscopic signals can be differentiated to detect any weak shoulders that may be indicative of overlapping bands and to enhance resolution. The first derivative of a symmetrical signal is used as an aid in the exact location of peaks. For example, the combined chromatographic response of two poorly resolved components may not indicate the presence of two co-eluting components. Such hidden features can be enhanced by differentiating the signal. The derivative is sensitive to subtle features of the signal distribution and therefore is effective in detecting important details such as weak shoulders. The ability of signal differentiation to detect overlapping spectral signals was demonstrated as early as 1955 [1], and the development of signal differenti-

* Corresponding author. 0022-0728/96//$15.00 © 1996 Elsevier Science S.A. All rights reserved SSDI 0 0 2 2 - 0 7 2 8 ( 9 5 ) 0 4 3 2 8 - 4

ation has led to techniques such as derivative spectroscopy [2,3]. In electrochemistry, differentiation of the signal enhances the end-points of titration curves, and the derivative dc polarographic technique challenges many of the more sophisticated methods of polarography [4-7]. Derivative linear sweep voltammetry is another electroanalytical technique based on the current-voltage curve. This approach has been reported to be very successful in improving the analytical usefulness of linear sweep voltammetry [8-14], In addition, the semiderivative technique is often used to detect asymmetrical signals such as those occurring in linear sweep voltammetry and cyclic voltammetry, i.e. semidifferential electroanalysis [15-20]. Semidifferential and semi-integral electro-analysis [21-23] are closely related techniques. In fact, semidifferential electroanalysis is simply the derivative of the semi-integral method. Linear sweep voltammetry is a method intermediate between these two techniques. Semi-integral electroanalysis has the advantage of being completely independent of the applied signal in linear sweep voltammetry and the shape of the observed curve is similar to that of the classical dc polarogram. However, the semidifferential peaks observed for reversible electrode processes are symmetrical and sharp,

2

J.-S. Yu, Z.-X. Zhang/Journal of Electroanalytical Chemistry 403 (1996) 1-9

whereas conventional current-potential peaks are asymmetrical and rather broad. The current-potential relationship in linear sweep voltammetry cannot be described by an analytic function. The semidifferential technique provides high sensitivity and better resolution than standard linear sweep voltammetry and semi-integral electroanalysis [14-17]. The semidifferential electroanalysis technique for detection has also been applied in anodic stripping voltammetry [18,19,24,25]. Similarly, 2.5-order differential electroanalysis adds the advantages of higher resolution and higher sensitivity to those of semidifferential electroanalysis [26-28]. There have been two general approaches to signal differentiation, semidifferentiation and semi-integration, one based on hardware and the other based on software. The hardware approach is the utilization of RC circuits for differentiation and RC ladder circuit networks for semidifferentiation or semi-integration [15-18,29] in analog instrumentation. The shape of the peak observed in this approach depends strongly on the RC constant of the circuit, and the potential scan rate is limited by the RC parameters of the electrochemical circuit. In the software approach numerical methods are used to compute the derivative, semiderivative or semi-integral of digital signals from computerized instruments. A least-squares polynomial is a simple and useful method of signal smoothing (differentiation for the present) [30-32]. However, the shape of the observed differential curve depends on the width of the window; the larger the window the broader is the peak observed. Although Horlick [33] has demonstrated the implementation of signal differentiation using a linear ramp function in the Fourier domain, the mathematical expression for the differentiating function was not given in the discrete Fourier domain. A numerical method of semidifferentiation in common use is to compute the semi-integral of the signal by means of an approximate numerical convolution [34,35] and then to perform a numerical differentiation of the semi-integral wave to produce the semiderivative wave. Oldham et al. [17,22] described the RL0 and the G1 algorithms for the semi-integral and the semiderivative. These methods are simple and versatile, but the RL0 method is time-consuming [22] and the error magnitude of the G1 method diminishes as the number N of sampling points increases [17]. The semidifferentiation of a signal performed using the Fourier transform deconvolution method has been reported by Smith [36]. However, the form of the convolution or deconvolution function is closely related to the electrochemical process [37]. A universal method of differentiating, semidifferentiating and semi-integrating digital signals using Fourier transformations is reported in this paper. A discrete differentiation function in the frequency domain is derived from the continuous derivative theorem of the Fourier transform. Any integer- and/or semi-integer-order derivative and semi-integral of a discrete digital signal can be calculated

on the basis of this function using a Fourier transform. In this paper the method is applied to theoretical and experimental electrochemical digital signals by utilization of the fast Fourier transform (FTT) [38]. The curves computed using the algorithm are in good agreement with those predicted by electrochemical theory. The computer time required by this method is only a quarter of that required by the RL0 algorithm [22] for computing the semi-integral for the same data array. The derivative, the semiderivative and the semi-integral with smoothing of the discrete digital signal can be obtained simultaneously using the Fourier transform.

2. Theory and method

2.1. Theory Consider the response signal y(t) from an analytical instrument which is a continuous signal in the time domain. By definition [39], the Fourier transform of y(t) is given by

Y(f) = L~y (t)

exp(-j2wfi') dt

(1)

and the inverse Fourier transform is given by oo

y(t) = f

Y(f) exp(j2~rft) d f

(2)

where t is time, f is frequency and j = Vr-K-1. The functions y(t) and Y(f) are termed a Fourier transform pair, and this relationship is indicated by the notation y ( t ) ** Y ( f )

(3)

By the derivative theorem of Fourier transforms [40] we have

f'_~y
(4)

where p is the order of differentiation. When p -- 1, L~y' (t) exp(-j27rft) dt=

(j2~rf)V(f)

(5)

The Fourier transform pairs are expressed as y(P)(t) ** (j2"rrf) V r ( f )

(6)

and

y'(t) ,~ (j2crf)Y(f)

(7)

This theorem states that if the Fourier transform of y(t) is Y(f), then the Fourier transform of the pth-order derivative of y(t) is obtained by multiplying Y(f) by (j2crf) v. In other words, the pth-order derivative of y(t) can be obtained by inverse Fourier transform of the product of Y(f) and (j2zrf) v. The function (j2zrf) here is known as the basic differentiation function under the continuous condition.

J.-S. Yu, Z.-X. Zhang /Journal o f Electroanalytical Chemistry 403 (1996) 1-9

However, a continuous response signal is converted to a discrete digital signal by sampling in the computerized instrument. Therefore Eq. (6) cannot be applied directly to discrete digital signals. Thus, let y(nT) be a discrete sampling of y(t) and let Y(ra/NT) be a discrete sampling of Y(f) where n and m are variables in the time and frequency domains respectively, T is the sampling interval and N is the number of samples of signal waveform, i.e. the period. For simplicity, we replace y(nT) by y(n) and Y(ra/NT) by Y(ra). The discrete Fourier transform of the discrete signal y(n) can be written as [39]

Thus the Fourier transform pair of the first-order derivative y'(n) is established as follows:

y,( n) .. [ exp(j2~rT/N) - l ]y( m ) Continuing the above process with y"(n),

Y(P)(m) = Y'. y(')(n)

y(n)

exp(-jE,rnm/N)

(8)

In terms of the difference ratio definition, we have y'(n) = [ y ( n + 1) - y ( n ) ] / T

![N~

l

r'(m)= T[n.oY(n+ l) N-,

~., y(n)

exp

(9)

exp( j2~nm)

(

j21rnm )] N

(10)

n--0

If we replace the first summation in square brackets in Eq. (10) by Y~(m) and set k = n + 1, then N

Y(( m) = exp(j2~rm/N) ~, y( k ) e x p ( - j 2 ~ k m / N ) k-I

(ll) or N-l

Y; ( m) = exp(j2zrm/N )

y(k) exp(-j2zr/an/N)

k-0

+e r

(ll')

where

er = exp(j2zrm/N)[ y(N)

e x p ( - j E r r m ) - y(0)] When N is sufficiently large, the term er can be neglected as it has no effect on the convergence and results of Eq. (11'). Substituting Eq. (11') into Eq. (10) and neglecting e r yields

exp(j2zrm/N) - 1 T N-I

× E y(n) exp(-j2zrnm/N)

(12)

n--0

(n÷)

i.e°

Y'(m) = E y'(n) exp -j2~" n=O

= [ exp(j2zrT/N) - l ]y( m).

P

Y(m)

T

where y(n) is the function y(t) evaluated at t = nT and y'(n) is the approximation of the derivative y'(t) of a function y(t) in the discrete sampling condition. When T ~ O, y'(n)~ y'(t). The Fourier transform of y'(n) can be expressed as

=

-j2vr

exp

exp(j2qrm/N) - 1

n=0

r'(,n)

y"(n) . . . . .

n=O

= ~

-

(14)

y(P)(n) yields the general equation

N-l

Y(m)

3

(13)

(15)

where Y(P)(m) is the Fourier transform of ytP)(n) and p is as defined above. Hence the Fourier transform pair of the pth-order derivative y(P)(n)is

ytP)( n)

**

[ exp(j2~rm/N) T

-

1

P

r(m)

(16)

Comparing Eqs. (15) and (16) with Eqs. (4) and (6), we can call Eq. (15) as the "derivative theorem of discrete Fourier transform" in the time domain. Eq. (15) states that if the Fourier transform of the discrete time signal y(n) is Y(m), the pth-order derivative of y(n) is obtained by the inverse Fourier transform of multiplying Y(m) by the function {[expO2~rm/N)-1]/T} p. The differentiation function [exp(j2zrm/N)- 1]/T is equivalent to the function j21rf under continuous conditions. Obviously, Eq. (15) is an approximation to Eq. (4). The accuracy of the approximation depends on that of the y(n) approximation to y(t). When N ~ o% Eqs. (4) and (15) are equivalent. Eq. (15) provides a method of obtaining the pth-order derivative of the discrete time-domain signal y(n). In general, p can be arbitrarily chosen to be a positive integer, i.e. p = 1, 2, 3 . . . . . In practice, p is set at 4 or less; otherwise, the signal-to-noise ratio of the derivative function decreases because differentiation enhances higher frequencies and attenuates lower frequencies [40]. Examples of the application of this method (with p a positive integer) to theoretical and experimental discrete signals are given in Section 4 (see Figs. 2 and 3). In Eq. (15), p can be chosen to be a semi-integer, i.e. p = 0.5, 1.5, 2.5 . . . . then we can easily obtain the 0.5th-, 1.5th-, 2.5th-,... order differentiation of signal y(n) from Eq. (15) using the FFT operation. Examples of the semi-integer derivative are shown in Figs. 4 and 5. In addition, p can be chosen to be a negative integer and/or a semi-integer. For example, the semi-integral of digital signals can be calculated from Eq. (15) with p-- - 0 . 5 , where p is a negative integer, i.e. integral operation. Examples of semiintegrals calculated using this method are also shown in Figs. 4 and 5. Thus the integer- and semi-integer-order

4

J.-S. Yu, Z.-X. Zhang/Journalof ElectroanalyticalChemistry403 (1996)1-9

derivatives and the semi-integral of the discrete signal

START )

y(n) in the time domain can be obtained from Eq. (15) at p = - 0 . 5 , 0.5, l, 1.5, 2, 2.5 . . . .

[

I IleOT p 0,o. 5, 1, •..... )1

2.2. Method

½

Although a theoretical method for differentiation, semidifferentiation and semi-integration based on the Fourier transform has been established in Eq. (15) for the discrete signal y(n) in the time domain, the technique used to implement this method is also important. Firstly, the differentiation function [exp(j27rm/N)1]/T in Eq. (15) cannot be used directly in computation and is usually converted into a triangular function. To facilitate this discussion, let Q represent the basic differentiation function, i.e.

[

I

[

vvT ,,ubrouti.~

[

Y(m)=Y(m)×W(1)I

]

P--i0. 5, I. 5, •..... [ Y(m) --Y(m) xQ~&-"

I

I

I

,--P-o.o I

[

I 1.

[ YI'l=Yi'l×w 11 l

exp(j2zrm/N) - 1 Q=

T

(17)

Obviously, Q is a complex function. In terms of the definition of complex functions, from Eq. (17) we have

I .T l

subroutine

OUT y'-" (.)

[ [

a = ( j 2 / T ) sin(zrm/N)[cos(1rm/N) + j sin(zrm/N)]

(18) where ( 2 / T ) sin(zrm/N) is the modulus. Secondly, the computing procedure is as follows. The Fourier transform of the discrete signal y(n) in the timedomain is first performed by FFT, giving the Fourier transform Y(m). If y(n) is an experimental signal with some noise, Y(m) is multiplied by a smoothing function W(1), then by the pth-order power of differentiation function Q and then by the smoothing function W(I) again. Finally the inverse Fourier transform of the product of Y(m) and the smoothing and differentiation functions is obtained by FFT. The results are the pth-order derivative y(P)(n) of the discrete signal y(n) in the time domain. When p = - 0 . 5 , the result is the semi-integral of the digital signal y(n). The flowchart of the computer program for implementing the process is shown in Fig. 1. Computer programs based on the FTT algorithm written in BASIC, PASCALand FORTRAN are given in the literature [39,41]. The FFT program is a subroutine in the computation. In Fig. 1 FT = 1 indicates the Fourier transform of y(n) and FT = - 1 represents implementation of the inverse Fourier transform. We have used the base 2 FTT algorithm [38] here. Occasionally, although y(n) is a function about time, y(n) is not an explicit function of time or y(P)(n) may not be a derivative of y(n) with respect to time. In general, we have Q = ( j 2 / A x ) sin(7rm/N) [ c o s ( z r m / N ) + j sin(zrm/N)]

(19)

where, for example, Ax represents the potential-sampling interval AE, the time-sampling interval T, the

Fig. 1. Flowchartfor the differentiation,semidifferentiationand semi-integration subroutine. W(1) is a smoothingwindow function in the frequency domain. Other symbolsare definedin the text.

wavelength-sampling interval AA in spectrometry etc. Thus we can compute the derivative, the semiderivative and the semi-integral for various discrete signals in analytical chemistry from Eq. (15).

3. Experimental All raw electrochemical data were acquired using a microcomputerized electrochemical instrument (designed and assembled in the laboratory). The instrumentation includes an IBM P C / A T (80286-16B) microcomputer system, an Epson LQ-1600K graphic printer, a fast-response potentiostat (setting time 2/xs or less; input impedance above 10 I~ O), a high-speed data acquisition system with a 12 bit analog-digital converter and a sample-and-hold amplifier with 25 ns aperture time (69 kHz data-acquisition frequency; signal acquisition rate, 15/xs or less). The current amplifier can detect currents down to 10-Is A. Electrochemical sweep potential waveforms are generated through the potentiostat by two digital-analog (D-A) converters (a 12-bit and a 14-bit D - A converter respectively). Software for in-situ data acquisition and data processing and the sweep waveform routines were written in 80286 assembly language. The software for data-smoothing, differentiation, semidifferentiation, semi-integration and convolution by Fb-T were written in PC-BASIC.

J.-S. Yu, Z.-X. Zhang / Journal of Electroanalytical Chemistry 403 (1996) 1-9

All chemicals used were of analytical grade purity, and all experimental solutions were prepared using double-distilled water. An H-type electrochemical cell thermostated at 25 _+ 0.2°C was used. The working electrodes were ultramicrodisk platinum electrodes with r o equal to 10 or 25 /~m fabricated by heat-sealing a platinum wire into glass capillaries and polishing the surface with alumina. The auxiliary electrode was a platinum wire and the reference was a saturated calomel electrode (SCE). The experimental solutions were deaerated with nitrogen for 10-15 min to remove dissolved oxygen prior to recording the voltammogram.

5

1.10

\

II

X

0.70

0.30

4. Results and discussion

The operational sequence of differentiation, semidifferentiation and semi-integration based on Fourier transforms is quite straightforward, and some steps are shown in Fig. 1. It is similar to the operations for smoothing [42] and convolution [43] using Fourier transforms. To illustrate the use of Eq. (l 5), two examples of differentiation are shown in Figs. 2 and 3. Fig. 2(A) is a theoretical linear slow-sweep voltammetric curve calculated from the theoretical equation [44] I

1

1~

exp[(E-Ei/E)nF/RT] + 1

-0.10 ......... -15.00

X = nFAco~/nFD° v/RT

, ......... 5.00

15.00

~ ......... 5.00

, 15.00

0.05

-0.05

using a computing step length of 0.1 and N = 256. The first- and third-order derivative curves of Fig. 2(A) calculated from Eq. (15) for p -- 1 and p = 3 are shown in Figs. 2(B) and 2(C) respectively. The peak width at half maximum in Fig. 2(B) is about 9 0 / n , in agreement with theoretical prediction. The third-order derivative curve in Fig. 2(C) is obtained directly by multiplying Y(m) by Q 3 (see Q in Eq. (18)), and not by multiplying Y(m) by Q three times in sequence. The two algorithms are different in the computer programs. Fig. 3(A) shows experimental staircase sweep cyclic voltammograms for 1.0 × 10 -3 mol dm -3 K4Fe(CN) 6 in 0.50 mol dm -3 KCl at an ultramicrodisk platinum electrode ( r 0 = l 0 / x m ) for a staircase potential height AE of 2 mV and a step width T of 200 ms. Current sampling was performed at the end of each potential step. The first- and second-order derivative curves of Fig. 3(A) are shown in Figs. 3(B) and 3(C) respectively. Examples of the semi-integer-order derivative and semi-integral for the theoretical and experimental signals are shown in Figs. 4 and 5 respectively. Fig. 4(A) is a theoretical cyclic sweep voltammogram obtained by digital simulation [45] (step length, 0.08; N = 512) of a reversible electrode process I

, ......... -5.00

-0.15

J -0.25 ......... -15.00

, ......... -5.00 uF (~-g

°' )/RT

0.20

?,.. "-- 0.10 X

X,

0.00

-O.IO

f

.........

-15.00

, .........

-5.00

p

, .........

5.00

,

I5.00

nF (g-~,°' )/RT Fig. 2. Theoretical dc voltammetry curve and derivative curves: (A) dc voltammogram; (B) first-order derivative curve; (C) third-order derivative curve.

6

J.-S. Yu, Z.-X. Zhang/Journal of Electroanalytical Chemistry 403 (1996) 1-9

where the symbols have their usual meanings [45]. The 0.5th-, 1.5th- and 2.5th-order derivative and semi-integral curves of Fig. 4(A) calculated from Eq. (15) for p values

of 0.5, 1.5, 2.5 and -0.5 are shown in Figs. 4(B), 4(C), 4(D) and 4(E) respectively. The pairs of curves in Figs. 4(B) and 4(D) are completely with symmetrical. The total

7.04)

k 6.00

5.00

4.00

3.00

2.00

;

0.00

. . . . . .

'

'

I

. . . . . . .

0.20

'

'

I

. . . . . . . . .

0.40

I

0.60

8

T

0.32

"-- 0.30

0.28

0.26

~

~

.........

0.00

~

-

, .........

0.20

~

, .........

0.40

,

0.60

ElY (va. SCE)

29.80 C ~,. 29.80 ,.< =L

29.40

.% 29.20

29.00

28.80

. . . . . . . . .

0.00

,

. . . . . . . . .

0.20

,

. . . . . . . . .

0.40

0.80

v./V ( va. see,) Fig. 3. Staircase sweep cyclic voltammograms at an ultramicrodisk platinum electrode (to= 10 pm; v= I0 mV s - ' ) : (A) experimental curves; (B) fwst-order differential curves; (C) second-order differential curves.

J.-S. Yu, Z.-X. Zhang / Journal of Electroanalytical Chemistry 403 (1996) 1-9

7

20.00

D

A

0.40

=_ tO.O0

d

0.20

== X c= ~.J

0.00

r.

0.00

II - 10.00

-0.20

-0.40 ................... - 15.00 -5.00

,

, ......... 5.00

-

15.00

1.50

20.00 -15.00

-5.00

5.00

, ......... -5.00

5.00

15.00

1000.00 B

=.. 500.00 X

0.50

r=

ill

0.00

-0.50 -500.00

-1.50 ......... - 15.00

= ......... -5.00

, ......... 5.00

, 15.00

-1000.00

......... -15.00

,

......... 15.00

nF~z-wo'~/xr g

0.25

i

.~ 0.15 X

i

0.05

-0.05 -15.00 ,

,

,

,

,

,

,

,

,

i

,

-5,00

,

,

,

,

,

,

,

,

i

,

5.00

,

,

r

,

,

,

,

,

I

15.00

nit (g-E °" )/RT

Fig. 4. Theoretical cyclic voltammograms and semi-integer-order differential and semi-integral curves: (A) theoretical cyclic voltammograms; and (B) 0.5th-order differential curve; (C) 1.5th-order differential curve; (D) 2.5th-order differential curve; (E) semi-integral curves.

8

J.-S. Yu, Z.-X. Zhang/Journal of Electroanalytical Chemistry 403 (1996) 1-9

overlapping of the pairs of curves in Figs. 4(C) and 4(E) clearly indicate their symmetry. These results agree with semidifferential and semi-integral electroanalysis theory [17,22] for the reversible electrode process. The results show that the derivatives, semiderivatives and semi-integrals of discrete digital signals calculated using Eq. (15) are reasonably accurate. Hence Eq. (15) can be used in electrochemical investigations as a general differentiation, semidifferentiation and semi-integration method for time or potential (see Q in Eq. (19)) when it is assumed that the waveforms of the time-damped function do not vary with the electrochemical process. Since the algorithm for the differentiation, semidifferentiation and semi-integration of the discrete digital signal is a pure mathematical method, Eq. (15) can also be applied to spectroscopy and chromatography for signal determination and waveform modification.

Fig. 5(A) shows an experimental staircase sweep cyclic voltammogram obtained at an ultramicrodisk platinum electrode (r 0 = 25 /zm; 1.00 × 10 -2 mol dm -3 K4Fe(CN) 5 in 0.5 mol dm -3 KC1; A E = 2 mV; ~'= 0.2 ms; AE/z= 10 V s-I). The 0.5th-and 1.5th-order derivative and semi-integral curves of Fig, 5(A) are shown in Figs. 5(B), 5(C) and 5(D) respectively. Since noise is always present in experimental digital signals. The Fourier transform Y(m) of the current in Fig. 5(A) is multiplied by Q05, Q15 o r Q-01S in the frequency domain, and then by a trapezoidal filter function. The curves in Figs. 5(B), 5(C) and 5(D) were obtained by performing inverse Fourier transforms of the product. The theoretical and experimental results reported above show that the algorithm provides a versatile method for determining the derivative, semiderivative and semi-integral of discrete signals. In addition, the larger the number

A/>

1.00

0.50

25.00

=L /

/

L

//'

0.00

/ \

i

5.00

t

-15.00

t

-0.50

/

-I.00 ½ . . . . . . . . 0.00

, ......... 0.20

, ........ 0.40

-35.00

-~ 0.60

........

0,00

,,

.........

0,20

, .........

,

0.40

0.60

2.50

,~

1.5o

<¢ :=L

0.50

~

~0.50

,

/

\

0.10

----~

/

~

o.os

- 1.50

0.00

-2.50

.........

0.00

, .........

0.20

, .........

0.40

E/V ( r a . SCE)

,

0.60

'',

0.00

....

, , I ' , , , , ' , , , i , , , ,

0.20

0.40

.....

i

0.60

E/V ( r s . SCD

Fig. 5. Staircase sweep cyclic voltammograms at ultramicrodisk platinum electrode ( r 0 = 25 p.m, v = 10 V s -~ ) (see text for discussion): (A) staircase sweep voltammetric curves; (B) 0.5th-order differential curves; (C) 1.5th-order differential curves; (D) semi-integral curves.

J.-S. Yu, Z.-X. Zhang / Journal of Electroanalytical Chemistry 403 (1996) 1-9

N of discrete samples, the better will the results of the method approximate to the theoretical values. High computational speed is also an advantage of this algorithm; for example, the approach takes only 4 min for the semi-integral operation of 512 original data points, whereas the RL0 algorithm [22] takes about 16 min under the same conditions (Epson 386 computer; computer program written in PC-BASIC).The flowchart in Fig. 1 also shows that the derivative, semiderivative or semi-integral with smoothing of the digital signal can be simultaneously calculated using FFT. Since smoothing a digital signal using FFT is an important method of extracting useful information from experimental signals, the method presented in this work will be a useful tool for digital signal processing in computerized instruments.

Acknowledgments Support of this work by the Natural Science Foundation of the Jiangsu Province of China is gratefully acknowledged. The suggestion by the referee to investigate the use of p = - 0 . 5 is acknowledged with thanks.

References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12]

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