Nuclear Instuments and Methods, Vol. 161 pp. 137140
On Digital Filtration in Correlation TimeofFlight Spectrometry
Laszlo J. Naszodi
Abstract: A new method of filtering the correlation timeofflight spectrum is proposed, which fits well to the direct evaluation of the spectrum. It makes additional mechanical and electronical devices unnecessary and the loss in intensity is less than that of the usual filtering technique.
1. Introduction
In the correlation technique a pseudorandom binary sequence (PRBS) is used for chopping the incident beam ^{1, 2)}. As its definition, a PRBS {a_{j } } containing m ones and Nm zeros has the autocorrelation function
C_{aa}(v) = Σ a_{j} a_{j+v} =  {  m if v=0 (mod N) h = mc if v≠0 (mod N) 
(1) 
where c=(m1)/(N1) is called the duty cycle of the PRBS. This technique yields the timeofflight (TOF) spectrum with the counts
Z_{k} = Σ a_{ki} S_{i} + b_{k} , k=0,1,2,…,N1, (2)
where
a_{ki} = a_{ki(mod N)}, the (k, i) element of the circular matrix A;
S_{i} = the ith element of the conventional TOF spectrum;
b_{k} = the background in the kth timechannel, its expected value <b_{k}>=b.
(For the sake of simplicity both the number of the elements of the PRBS and the number of the timechannels are chosen as N, i.e. the indices k, i = 0, 1, …, N1.)
Using eq. (1), the spectrum {S_{i }} can be reconstructed from eq. (2):
S_{i} = [m(1c)]^{ 1} Σ a_{ki} Z_{k} – c[m(1c)]^{ 1} Σ Z_{k} – b/m (3)
Expression (3) means that the inverse transformation of the correlation (2) is its transposition, regardless of an additive constant and of a factor. A great advantage of the correlation method using PRBS is that the solution of the linear system (2) can be obtained without a difficult inversion procedure. This is true until we do not want a further elaboration of the data. From the mathematicalstatistical viewpoint PRBSs yield the minimal total error for the spectrum {S_{i}} ^{3, 4)}.
In the theoretical research of the correlation TOF method the gain factor arises as a fundamental category. Its square
g_{i}^{2} = <Δ^{2}S_{i}^{conv}>/<Δ^{2}S_{i}^{corr}>, (4)
the ratio of the variances of the ith point of the spectrum obtained by the conventional and the correlation methods and it expresses the superiority of the correlation technique over the conventional one. With the help of the gain factor one can optimize the correlation measurement through its controllable parameters. It has been performed in one and twodimensional cases^{58)} and for filtration^{9, 10)}.
In the use of the gain factor (4) all the elements S_{i} are supposed to be reconstructable by the transformation (3). It has been shown ^{5, 10)} that the gain for S_{i} increases due to the increase of the relative signal height. This means that the identification of a small peak is difficult, if the spectrum contains a large nuisance peak increasing the “background of ignorance” ^{6)}, even in the case of optimally chosen parameters of the PRBS. Here a successful filtering of the large peak increases the relative signal height of the low intensity portions, improving the gain for the points of the interesting small peak.
The most widespread method deviced for filtration is the use of an additional chopper and of gating electronics following the filtering chopper ^{9, 10)}. In the generally accepted opinion a chopper that has the same PRBS as the correlation one should not be used for filtration ^{11)}. One goal of this paper is to prove that one chopper can fulfill both the task of correlation and that of filtration.
2. Filtration and correlation with one chopper
In an arrangement an electronic gate is set up along with the correlation chopper so that the count of an event, if it may be involved in the unwanted S_{i}, say in S_{e}, is prevented by the gate. Mathematically, the counts will be the following:
Z_{k} =(1 a_{ki} )Σ a_{ki} S_{i} + b_{k} , k=0,1,2,…,N1 (5)
Amadori and Hossfeld tried to construct the S_{i}‘s by a procedure similar to the transformation (3) and they established that the construction would be possible only if the third correlation function
C_{aaa}(u,v) = Σ a_{j} a_{j+u} a_{j+v} (6)
of the PRBS had only three levels^{10, 11)}. They declared that among the known PRBS, provided N < 1024, they did not find such a sequence. Incidentally, I found one in their earlier cited paper ^{5)} that meets the abovementioned criterion. However, it is easy to prove that every PRBS with h = 1 meets this criterion.
Despite of this finding, I suggest a new method which still keeps the arrangement of one chopper but changes the mathematical elaboration of the experimental data.
How can system (5) be solved, taking into account that the inverse of its matrix does not exist? (5) is an mfold underdetermined system for the S_{i}‘s, the rank of its matrix is N – m.
An evident way is to make m preliminary assumptions or constraints. For example, if we know that in a given part of the spectrum there appears no peak, we can assume that these S_{i}‘s are constant.
There is another way, which seems to be the most adequate one in the mathematicalstatistical sense and it is straightforwardly directed to the goal of the investigations. As the final result of the experiments one wants to obtain estimations for some parameters of the spectrum, such as for maximum height, the position and the standard deviation of a peak, rather than for the spectrum itself. Without filtration these few parameters can be estimated either through the usual two steps: decorrelation (3) and a fitting procedure, or through a direct fitting procedure^{12)}. In the case of the two steps filtration the decorrelation has to overcome the problem of underdetermination, which does not occur in the direct fitting. For the latter let us have an example with the assumption that the spectrum is well described by the sum of J Gaussians:
S_{i}(P) = Σ A_{r} exp[(iM_{r})^{2}/D_{r}^{2}], i=0,1,2,…,N1; r=1,2, …, J (7)
i.e. S_{i} (P) is the expected value of S_{i}. P is the vector of the 3J + 1 parameters; as the (3J + 1)st parameter, the expected value b of the background is taken.
The expected value of eq. (5):
Z_{k} = {  Σ a_{ki} S_{i} (P)+ b_{k} 0 
if a_{ke}=0 if a_{ke}=1 
(8) 
for which the parameters can be estimated by the weighted leastsquares criterion:
Σ w_{k} {Z_{k} – Σ a_{ki} [S_{i} (P)+ b/m]}^{2} => min (9)
where weighting by the reciprocals of the counts yields the best estimation for P in the sense of having the (3J + 1) dimensional dispersion ellipsoid of minimum volume. On the other hand, the filtration is also performed by the mathematical weighting, and electronic gating is unnecessary:
w_{k} = {  1/Z_{k} 0 
if a_{ke}=0 if a_{ke}=1 
(10) 
where e is the index of the spectrum element to filter out. The solution of eq. (9) with (10) can be obtained by the iterative solution of the matrix equation
G^{T}A^{T}WAGΔP = G^{T}A^{T}WΔZ,
where
G is a N x (3J + 1) matrix, its elements
G_{iu} = δS_{i}(P)/δP_{u} P = P^{f} i = 0, 1, …, N1; u = 1, 2, …, 3J + 1;
ΔP is the vector of the changes, its elements
ΔP_{u} = P_{u}P^{f}_{u} u = 1, 2, …, 3J + 1;
P^{f} is the vector of the parameters, determined in the previous step of iteration;
W is the diagonal matrix of the weights, its elements
W_{jk} = {  w_{k} if j=k 0 if j ≠ k 
ΔZ is the vector of differences between the measured and the computed counts, and the superscript T denotes transposition.
Here I mention without proof that filtration by omission of measured data generally implies a loss of information theoretically in the sense that the determinant of the information matrix M = G^{T}A^{T}WAG decreases due to leaving out symmetric dyad(s) from the positive definite matrix. Consequently, the volume of the dispersion ellipsoid of the estimated parameters increases. It is approximately equal to the expression (detM)^{1/2} . Notwithstanding this theorem, the filtration can turn out to be useful in practice. Although the correlation in eq. (2) is supposed to take place by a rectangular chopping function, with the only values of zero and one, the real modulating functions are not of this form. This inperfection results in the socalled inputtransducer error^{5)}. This effect can cause a stronger disturbance than the background of ignorance which also decreases by filtering the large uninteresting peak.
Now let us calculate the losses of intensity, which result from the two types of filtration. The arrangement of the additional filtering chopper with parameters (N_{f}, m_{f}, c_{f}) and of the following gate transmits the portion (m / N) (m_{f }/ N_{f}) (1 – m_{f }/ N_{f}) ≈ cc_{f }(1 – c_{f}) of the incident intensity. Its maximum is c/4. In the new method of one chopper the remaining part is: (m / N)(1 m / N) ≈ c(1 – c) times the incident intensity. (Of course, the optimum is influenced not only by the maximum intensity, but by the signaltobackground rate, too.)
If the ignorable peak is so wide that it is worth to cut out more than one channel, say s channels, the remaining portion is approximately (m / N)(1 – sm / N) ≈ c(1 – sc) for the new method, which has the maximum of order ¼ s, if c = s/2. The filtration of the channels e_{1}, e_{2}, …, e_{s} is performed by the weights
W_{jk} = {  1/Z_{k} if a_{ke1}= a_{ke2} = _{… }a_{kes}=0 0 otherwise, 
instead of eq. (10). For two filtered channels the exact value of the remaining portion is (m / N) [1 – (2m – h) / N]. For more channels the parameters of the autocorrelation functions of higher order should also be considered for the exact calculation of loss in intensity. For more channels one must consider that by increasing the number of the zero weights, the information matrix M can become singular. To avoid this, the duty cycle c should be decreased. The optimization is not too difficult by computer trials and it can also be performed analytically for fixed models [e.g. for spectrums following eq. (7)]. The use of a smaller duty cycle does not mean any disadvantage for the reason that filtering is effective only for low background, and in these cases a smaller than 1/2 duty cycle can be optimal^{6)} . If c is small, the posterior filtration outlined here does not erase too many Z_{k}, so it can be extended to more channels without making the system underdetermined.
3. Droppedout points
The posterior filtration can be applied for the appearance of droppedout points, too. Because of electronic error some Z_{k} can become false (e.g. contents of some timechannels are erased). These channels can be filtered by the weights w_{k} = 0, if Z_{k} is dropped out. Here the decorrelation (3) is of no use. This is shown by computersimulated measurements. We produced a TOF spectrum by eq. (2), adding Poissondistributed errors to the Z_{k} ‘s. The parameters of the used PRBS: N = 31, m = 6, c = 1/6. The model function: S_{i }= A exp[– (i – M)^{2} / D^{2}] with A = 100, M = 6, D^{2} = 3 and with background b = 100. Half of the measured points Z_{k} were simulated as rejected points. These randomly chosen channels got the mean content Z . (One could not do better.) The spectrum {S_{i }} was reconstructed by eq. (3), the traditional decorrelation (Fig. 1), and by my method according to eq. (9) (Fig. 2).


The former resulted in an unevaluable spectrum. The latter yields nearly the exact parameters. The mean square deviations of the parameters are about √2 time the MSDs given by the simulation without droppedout points, but still they are sufficiently small (table 1). It means that the proposed evaluation of experiments with lost counts provides a good result that reflects the loss in accuracy proportionally to the loss of information. As opposed, the formerly used method results in total loss of accuracy due to some loss of information.


Table 1. Estimation for the parameters of TOF spectrum, with posterior filtration of the droppedout points. In the simulation 50% of the measured points were dropped out. A is the amplitude, M is the abscissa, D^{2} is the square width of the peak, b is the background level.
4. Summary
A new method is introduced for filtering posterior the correlation TOF spectrum. It has the following advantages over the usual decorrelation technique:
 The loss of intensity is significantly less.
 It is no use to add another chopper and gating electronics to the usual TOF correlation arrangement.
 Filtering is performed after the measurement, so it is adjustable, the channels to be filtered can be chosen a posteriori, and the unfiltered spectrum can also be reconstructed.
 It fits well to the data evaluation process, yielding the best estimation for the parameters of the spectrum.
 The droppedout points can also be filtered by this technique contrary to the usual decorrelation method.
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