Analyst, October 1996, Vol. 12I (1359-1365) 1359 Resolution of Partially Overlapped Signals by Fourier Analysis. Application to Different ial-pulse Polarog rap h ic Responses Davide Allegri“, Giovanni Moria and Renato Seeberb* Fisirw, Uniiw-Jitu di Pal-niu, Vide delle Scienze, 431 00 Pal-ma, Italy Risoi-giniento, 4 , 40136 Bologna, Ituly Dipartiniento di Cliiniica Geneiule e Inoi,gunic.a, Chimica Analitica e Chimica Dipai-timento di Chiniica Fisicu e Inorgunicu, Universitd di Bologna, Vide del An analysis of the first harmonic components of the sequence resulting from a suitable deconvolution operation on a response composed of single partially overlapped signals allows the resolution of the over-all response into the individual signals. Ideally, it is possible to reduce the original response to single impulses with suitable height and location. If the number, n, of single signals is known, the first n - 1 harmonic components of the spectrum of the deconvoluted response, plus the continuous term, need only be computed.On the other hand, if the over-all response includes an unknown number of single signals, a method is suggested that requires the calculation of only a few harmonic components in order to build up a suitable form of the whole spectrum of the deconvoluted signal. The effectiveness of the proposed procedures, which are suitable for the treatment of responses from different analytical techniques, was tested on simulated responses, both in the absence and presence of noise, and on experimental data, viz., differential-pulse polarographic responses recorded on solutions containing two metals that were reduced at similar potentials.Keywords: Signal processing; digitul ,filtei*ing; i-csolutioiz of signals; Forri-iei- aiialysis; decon\vlution; se~-~e~~oii~oliiti(jri; diffei-ential-pulse polarogi-aphy Introduction The problem of obtaining the maximum possible information from composite signals, i.e., of defining the single components with respect to position, height and shape, has been the subject of many studies in the field of electrochemistry, spectroscopy, chromatography, and any type of detection procedure that can be thought of as a multicomponent analysis.1-22 Different approaches have been followed, the most popular and effective of which appear to be self-deconvolution based on Fourier transform, direct curve fitting, or a combination of the two techniques.It has been emphasized, however, that Fourier self- deconvolution, in spite of being, in principle, the procedure of choice, presents the serious drawback of involving divisions between very small, non-significant numbers, which leads to instability of the system: small amplitudes are common to the medium-high frequency portion of the signals usually encoun- tered in an analytical context and, consequently, of the corresponding deconvoluting sequences; hence, spurious fre- quencies often pollute the signal resulting from deconvolution. On the other hand, a simple boxcar truncation, i.e., the application of a rectangular frequency low-pass filter, causes * To whom coirespondence should be addressed. spurious oscillations due to the Gibbs phenomenon.Z3 In order to damp the oscillations as much as possible, variou\ apodiza- tion functions have been studied and proposed;4-”l2J4 however, the problem has still to be solved satisfactorily. In view of the difficulties encountered in following classical approaches to deconvolution of partially overlapped signals, we turned our attention to a more accurate analysis of the spectrum of the deconvoluted signal.On the basis of the properties of the first harmonic components of the spectra obtained from both simple and composite signals, two different procedures have been established. The first can be followed when the number of individual signals comprising the over-all response is not known; the second can be applied when the number of individual signals is known in advance.As a benchmark for the proposed algorithms, this paper considers simulated and experimental responses in differential-pulse polarography (DPP). It is well known that in this electroanalytical tech- nique,2s.26 the sensitivity of the measurement increases on increasing the amplitude of the potential step; on the other hand, this may lead to poor resolution with the presence of more than one single peak in a narrow potential range. Experimental Experimental DPP responses were recorded on aqueous solutions prepared by dilution of commercial standards. The potentiostat was a Princeton Applied Research (PAR) (Prin- ceton, NJ, USA) Model 174A equipped with a PAR Model 303 stand. The working, auxiliary and reference electrodes were a 0.5 mm diameter mercury drop, a platinum foil and an Ag- AgC1-KCl(sat.) electrode, respectively.Pulse heights of 5 (cadmium and lead solutions) or 10 mV (lead + thallium mixture) were superimposed on the potential, which was varied linearly at a scan rate of 10 (former case) or 5 mV s-1 (latter case). Solutions of 0.1 moll-’ KC1 or 1 mol 1 - I NaF were used as supporting electrolytes. Basic Theory Let us consider two finite sequences, x{ n } and z{ n ) , with equal length. The convolution sequence, y{ n }, can be conveniently computed by taking advantage of the convolution theo- rem:27.28 y { n ) = x ( n ) * z { n } = F - ’ ( F [ x { n } ] x F [ z ( n } ] ) ( I ) where the symbol * indicates the convolution operation; F and F- I indicate direct and inverse Fourier transform, respec- tively.Conversely, z is defined as the deconvolution sequence of y by x: (2) z { n } = y { n } * - ’ x In} = F-1(F[y(n}]/Flx(ii)])1360 Analyst, October 1996, VoE. 121 where the symbol *- 1 indicates the deconvolution operation. If x coincides with y in shape, but not necessarily in position, deconvolution becomes a self-deconvolution. z is, in this case, a unit impulse 6{ n ) , equal to zero over the whole interval of definition, but in a single point:29 ( 3 ) 6 ( n * a ) = x(n * a } *-I x{n) a being a constant, either equal to, or different from, zero. The unit impulse function, 6, is defined as: (4) 0 when n f a 1 when 11 = a 6 { n + a } = By considering a normalized sequence, x, basis of a vector space, a linear combination generates the generic vector v( n } : v{n) = c x{n+o,} x (‘, ( 5 ) so that deconvolution of v by x can be considered as the transformation of v into a vector space generated by 6: Hence, if the signal can be represented by deconvolution by x leads to a linear combination of 6 sequences with the same values for the coefficients a, and ‘;.The discrete Fourier transform operation on a finite sequence x consisting of N points is defined as: x{n*a,) X C17 c (7) X{ k ) = x{ x e-J(?xlwkn 0 5 k < N x ,1=0 where k is the index of a given harmonic component ( = 0 for the continuous component). In particular, the Fourier transform of 6 leads to: I . P . , Deconvolution Procedures Procedure 1 Fig. l(h) and ( c ) shows the vectorial representation of the different harmonic components relative to an impulse function, 6 [Fig.l(a)]. It consists of a sequence of unitary modulus vectors with phase angles given by: (10) Eqn. (10) can be utilized in order to establish an effective procedure to describe the signal using only a small portion of the relevant spectrum. Since the result of eyn. (8) can be viewed as Ok = (2n/N) X ka; 0 < k < N a rotation, defined by the 01, sequence, of every point of 6, a key point in the analysis of the spectrum consists in ascertaining whether the N-element sequence of the angles between two subsequent vectors is composed of equal sub-sets, each consisting of M elements. A further point is to establish if M is small enough with respect to N . An analysis of a large number of different situations allowed us to confirm that an affirmative answer can be given to both questions, M being defined by the following equation: @ A ( L ) .4 ( k + M ) = 2nM/N ( 1 1) which is valid for 0 I k < N - M ; <p is the angle between two vectors that are M indices far from each other. The advan- tageous conclusion is that M harmonic components are suitable to describe the signal completely. By properly ‘copying’ (see the procedure reported below) the first M-element part of the spectrum over the whole range of frequencies, the whole spectrum of the signal is constructed. Through this operation the signal retains fully the information content at low frequencies and discards that at higher frequencies, without the inconven- ience of the usual filtering operations.In particular, the impulse in Fig. 1 (a), located at the 200th point of a I024 point sequence, is described by the first four harmonic components reported in Fig. l(c) (M = 4). A different index for the location of the impulse leads to different situations with respect to the position of subsequent vectors on the imaginary wrsus real component plot on the vectorial plane and, consequently, to the value of M . Once a vector A { h } (0 I h I M ) has been computed, the whole spectrum A{ k } can be re-constructed as follows: A{ k } = A{ h } for 0 I h I Mi. e., for the first M + I indices k A(k} = A{m + p x M } = A{m + ( p - 1) x M}e-lv; l < m < M ; l < p < i n t [ - for k > M (12) where cp G @ in eqn. (1 1). Of course, additional A terms with respect to A(N - 1) should be disregarded.The original sequence, 6, is re-obtained by: 6 ( n } = F ’ (A{k)) (13) For two generic impulses, the sequence to be transformed can be written as: = 0 when n # aI and # ar = p when n = u1 = ‘c when n = a2 (14) d{n} = p6, { n k u1 1 + z6, { n k a 2 ) I Taking advantage of the linearity property of Fourier trans- form27 one obtains: A( k ) = F(pa1 { n ) + -&( n ) ) = p e-I(2n/W~~l + ‘c e-/(*Jr/WLu2 (15) i.e., A(N - 1) = p e-/(2n/N)(N - 1 )a1 + e- /(2x/N)(& - 1 )~2 (16) It is evident that a simple normalization on eqn. (14) can lead either p or ‘c to be unitary, so that only one of the two modulus in eqns. (14)-( 16) assumes a non-unitary value. More complex patterns for the vector representation of the spectrum of the signal may be encountered.Apart from theAnalyst, Octobei- 1996, Vol. 121 1361 1 - 0.9 0.8 0.7 > 0.6 0.5 ' 0.4 c. .- m w computation of M through a simple algorithm written to find the solution of eqn. ( 1 I ) , the possibility of identifying sub-sets composed of M elements inside the whole N-element frequency sequence implies that notable geometric properties are charac- teristic of the sequence of vectors. It follows that the whole N- component spectrum is obtained by a 2x rotation of a polygon defined by just M vectors. An example is given in Fig. 2. Two impulses at the 76th and 300th points of a 1024 point sequence, respectively, with relevant heights of 2.5 : 1 [Fig. 2(u)] lead to an N-point spectrum [Fig. 2(h)] that can be obtained by subsequent rotations of the first M ( = 14) harmonic components.Fig. 2(d) shows the result of two subsequent rotations of the elemental polygon in Fig. 2(c). The procedure described can be extended to consider complex responses consisting of more than two individual signals. Fig. 3 shows the results obtained with a system of three impulses: the original sequence (u) is reconstructed on the basis of the first ten harmonic components (h). When dealing with an analytical response, cleaned of any noise, described by a sequence s { I I } , the previous discussions for x can be extended to s. Notationally: s ( n ) = x ( n + a ) (17) s { n } k ' X { I 1 ) = b { n + a ) (18) In view of eqn. (3) one can write: If the response consists of two individual signals, eventually partial 1 y over1 apped: (4 - - - - - - 0.3 0.2 0.1 - - - 0 200 400 600 800 1000 1200 index E 0.2 8 .e -0.2 (d OI L 0.4 -0.6 1 -0.8 1 so that: s { n } *-' x(n) = & ( n + a , } x ('I +6{n+a*} x ('2 (20) The result of the deconvolution by x should hence consist of two impulses with heights equal to c I and c2 and locations at a l and u2, respectively.This pattern can be correctly built up by calculating a deconvolution sequence A{ h } limited to the first M harmonic components, as discussed above. Defining: S { k } = F ( s { n } ) (21) one can compute: and then re-construct the whole spectrum, A, as described in eqn. (1 2). Finally: d{ / I } = F-'(A{ k ) ) (23) Procedure 2 The method reported above does not require the a priori knowledge of the number of impulses in the sequence, because the evaluation of M through eqn.(1 1) is directly performed on the spectrum of the signal. On the other hand, in those cases in which the number of individual component signals is known, a m a -0.6 - 0 8 -1 Real component L a, C 0.4 - 0 Q - 8 L. 0 - - k - 0 4 - c a - 1 - m -1 -05 0 0.5 Real component Fig. 1 Unit impulse ( ( I ) with thc vectorial representation of the relevant spectrum (h). The first four harmonic components are shown in ( c ) \ 1I362 Analvst, Omher 1996, Vol. I21 further simplification of the procedure is possible. Considering the sequence expressed by eqn. (14), which consists of two generic impulses, the continuous and the first harmonic components can be obtained by Fourier transform [see eqns.( I 6)]. Expressing A( 1) directly in the form of a single vector resulting from the vectorial sum of two p and 'c modulus vectors, respcctively, one obtains: where 0 and (2n/N)I7 represent modulus and phase, respec- tively: where 1111 = imaginary and Re = real. A( I), a s well a s A(0), are explicitly computed: N O ) = S(O)/X(O) A(1) = S(I)/X(l) (5 and h. as well as the sum p + T, become known quantities from which p, t, a l and 0 2 have to be computed. By normalizing the response with respect to the height of one of the two impulses, e.g., by setting p to 1, -c can be computed as A(0) - p. The 0 + c - 0.4 0.3 0.2 0.1 0 200 400 600 800 1000 1200 Index .. -1 0.5 0 0.5 1 1.5 Real component locations of the two impulses, given by u1 and u2, respectively, are hence the only unknowns: a system of two independent equations, representing the projection and the Carnot theorems, respectively, can be written: Solving the equation system (28) leads to: N 2n (29) u, = h - - arcos a2 = h + - arcos [; - N 0 0 2 - t 2 + p 2 2 n 20-c Hence, starting from a two-peak response s ( k ) , it is only necessary to calculate S(0) and S( I), as well as X(0) and X( I ) of the deconvoluting function [see eqn. (27)].Any additional individual signal only requires the consider- ation of one additional harmonic component, i.e., of two additional unknowns and of two additional equations in the system. It follows that very complicated responses can be analysed by considering in any case a small number of harmonic components. 1 + c ? 0.5 E L. Q 8 0 S 0) .- -0.5 E - 1 I -1.5 -1 - 0 5 0 0.5 1 1.5 Real component Real component Fig.2 components, respectively. Two impulses ( [ I ) together with the vectorial representation of the relevant spectrum (h); (c.) and (4 show the first A4 (= 14) and 3 X A4 harmonicAnalyst, October 1996, Vol. 121 I363 Application to Signals Synthetic Signals The first problem to solve when dealing with experimental situations is the presence of superimposed noise. If the ~ ( n } sequence accounts for the deviations from the theoretical 'clean' response, s, the signal sequence, r, can be expressed by: r ( n ) = s { n } + ~ { n } (30) Deconvoluting r by x, i.e., the sequence representing a single 'ideal' response [see Fig. 5(b) as an example], leads to a sequence p { n } : p{n} = r { n ) *-I x(n) (3 1) Performing the deconvolution operation by the usual self- deconvolution algorithms, it is usually difficult to extract 6 [or 61 + .. . an] from p because of the problems discussed above. On thk other hand, according to the arguments discussed previously, even if the number of individual signals is unknown, the number, M , of harmonic components necessary to describe the proper impulse, or series of impulses, can be computed on the basis of the vectorial representation of the spectrum of the sequence resulting from deconvolution. Assuming that, as regards the first M components, the frequency content of E is negligible, it is possible to extract the spectrum of 6 [or of 6 l + 62 + . . . 6,] from that of p. The procedure is outlined in detail below for two individual signals.R(k1 = F ( r { n l ) (32) and 0.3 1 0.2} O.' 0 0 L 200 (b) 1 400 600 Index I 800 lo00 1200 -1.5 -1.5 -1 -0.5 0 0.5 1 1.5 Real component Fig. 3 components are shown in (h). Three impulses ( a ) lead to a spectrum whose first 1 1 harmonic X ( k } = F(x(nJ) (33) Defining: P,{ h } = R{ k } / X { k } limited to 0 S k d A4 (34) the remaining part of the PI( k } spectrum, which is obviously different from P ( k } = F(p{nj), is re-constructed in a way similar to eqn ( 12): P,(k} = Pl{h} P,{k}=P,(m+p x M } = P , ( n z + ( p - l ) ~ h ' } e - ' ' ~ ; for 0 5 11 I M i.e., for the first M + 1 indices k I I m I M ; l < p _ < i n t [:*)fork - > M Finally, the sequence we are seeking, relative to a single signal or to the sum of two individual signals, can be computed: ( 3 5 ) On the other hand, if it is known in advance that two signals compose the over-all response, we have only to compute the continuous and first harmonic components by relationships similar to eqn.(27). The evaluation of the quantities character- izing the deconvoluted signals is then straightforward by applying eqns. (25)-(29). Synthetic DPP Responses It is apparent that the deconvolution procedures described retain full validity independently of the measurement technique and can hence, in principle, be applied to the responses of different analytical methods. As an example of their application, let us consider a sequence representing a simulated differential-pulse polarogram relative to a single reversible uncomplicated one- electron charge transfer, the oxidized and reduced forms both being soluble in the solution phase:2s-.-70 where 6i is the measured quantity, GA and o are exponential functions of the electrode potential.E , and of the potential step, respectively, T and z' the times at which the current is sampled and the other symbols have their usual meming.25 An example of the application of the described procedures to the self- deconvolution of a simulated DPP system of two peaks representing the reduction of two species present in the solution at a relative concentration of 2.5 : 1 and with equal diffusion coefficients, is given in Fig. 4. The potential separation between the relevant formal potentials is about 120 mV. No significant differences between the results obtained with the two proce- dures are apparent.Fig. 5 illustrates the same sequence as that in Fig. 4, but this time affected by noise consisting of a linear combination of sinusoids with amplitudes proportional to the signal intensity: both deconvolution procedures give satisfactory results [Fig. 5(a) and (b) for procedures 1 and 2, respectively]. Procedure 1 preserves the location and height of the individual signals with an accuracy of about 5%, while the results of procedure 2 are affected by errors lower than 1%. Spurious spikes are clearly detectable in the result of procedure 1 [Fig. 5(a)]. The choice of the number of harmonic components to use in order to reconstruct the whole spectrum of the deconvoluted signal (i.e., the value of M) is critical, so much so that, when dealing with signals that are very far from being 'ideal', the useful signal can be completely hidden by spurious1364 Analyst, October- 1556, Vol. I21 spikes.This is a consequence of the fact that M increases on increasing the noise. Acceptable results can, however, also be obtained with low M values in the presence of noise as high as that in Fig. 5 . The result of the elaboration of this signal is not trivial, because the application of a conventional filtering technique to a noisy signal can, on the one hand, be capable of reducing the random noise efficiently but, on the other hand, can introduce into the signal a much more subtle noise, i.e., a type of marked Index Fig. 4 result of the deconvolution performed according to procedure 1. Two partially overlapped simulated DPP signals, together with the 1.2r g o 200 400 600 800 1000 1200 0 200 400 600 800 1000 1200 Index Fig.5 Sequence shown in Fig. 4 with superimposed noise, together with the result of deconvolution by procedure I (a> and procedure 2 (h), respectively. ‘distortion’, which is the cause of the problems encountered in the usual self-deconvolution operations. Experimental Signals The procedures outlined above were tested in different experimental situations, i.e., for signals recorded under condi- tions that are far from ‘ideal’, e.g., not completely reversible charge-transfer processes, signals distorted by ‘ faradaic’ and ‘charging current’ effects,’6 and the use of relatively high values for the linear potential scan on which the pulses are superimposed. In this case, the responses do not fit the theoretical eqn.(37), resulting in a peak width different from that expected on the basis of the electronicity (the number of electrons involved in the charge transfer) of the process and asymmetric (distorted) in shape. The procedures were also tested for two signals representing processes with different electronicity. In this case, the deconvoluting function cannot be correctly evaluated by eqn. (37): the operation is no longer one of self-deconvolution. We have verified that for symmetrical peaks, satisfactory results are obtained by using appropriate values c.g., eventually non-integer numbers, for 12 in eyn. (37), provided that they lead to a particular criterion being satisfied. such as, for example, a good fit with a portion of the experimental curve.Furthermore, asymmetry is well accounted for by convolution of the peak with a suitable sequence, typically an exponential decay. These additional elaborations involve implementing the algorithms described here in more complete and flexible procedures. This aspect is currently under study in our laboratories. Two examples are given here to illustrate how a single-step-signal-treatment based on the proposed methods works. An example of the result obtained with deconvoluting functions using appropriate values for n in eqn. (37) is also provided. Fig. 6 displays the sum of DPP responses recorded for 1 ppm cadmium and lead nitrate solutions. The individual respomes have been shifted so that their maxima coincide with each other and, subsequently, the response of cadmium has been further shifted at lower abscissa indices to an extent of 40 mV; the final response has been normalized.The individual peak heights are actually in a ratio of 1 : 0.83 and appear almost equal to each other only because of the tailing of the first (cadmium) peak. The application of the second proposed procedure leads to the signals shown in Fig. 6. The results can be summarized as follows: ‘impulse’ height ratio = 0.6 (theoretical value 0.83); location of cadmium response: index 67 (theoretical value 57); 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 ’ 0.1 ’ c .- a, c-’ 0 50 100 150 200 250 0 Index Fig. 6 Elaboration, i.e., displacement and sum, of DPP responses recorded for 1 ppm cadmium(1rj and lead(ii), 1 mol I-’ NaF aqueous solutions, respectively, together with the result of deconvolution.An index difference of 1 corresponds to I mV.Analyst, October 1996, Vol. I21 1365 300 350 400 450 Index 500 550 Fig. 7 -Experimental DPP response recorded for a 9 ppm lead(r1) f 10 ppm thallium(i), 0.1 mol 1-1 KCl, aqueous solution, together with the result of deconvolution. An index difference of 1 corresponds to 1 mV. location of lead response: index 115 (theoretical value 97). 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