Nuclear Instruments and Methods, Vol. 161 pp. 163-165

Consideration of the Real Modulation in Correlation Technique

Laszlo J. Naszodi
Institute of Isotopes of the Hungarian Academy of Sciences

Abstract: The consideration of real modulation requires to replace the equations’ system of the correlation TOF technique. The solution and the loss of information coming from the non-ideality are discussed.

In the correlation time-of-flight technique ^1-6) the continuous function T(t) describing the transmission (modulation) is generally represented by an idealized sequence {a_j} of N elements. a_j = 1 if the chopper is open and a_j = 0 if the chopper is closed in the interval jΘ ≤ t < (j+1)Θ (j=0,1,2,…,N-1). Here Θ is the elementary period of the modulation. The matrix equation for the spectrum {S_i} (i=0,1,2,…,N-1) and the measured counts {Z_k} (k=0,1,2,…,N-1) is

Z = AS + b                                                  (1)

Where

Z is the vector of the counts;

A is an NxN circular matrix composed from {a_j}, its elements a_ki = a_{k-i mod N} (k,i=0,1,2,…,N-1);

S is the vector of the conventional spectrum elements to be determined;

b is the vector of the background counts assumed to be known and equal in every channel, i.e., b_k = b (k=0,1,2,…,N-1).

The solution of eq. (1) is generally

S = A^-1(Z – b).                                                  (2)

Using a pseudo-random binary sequence (PRBS) for chopping, the solution is more simple, since the inversion of A is its transposition, regardless of a multiplying factor and an additive constant:

S = [m(1-c)] ^-1 (A^T – C) Z – b/m,                                        (3)

where m =Σ a_j , the number of open elements in the PRBS;

c = (m-1)/(N-1), the duty cycle of the PRBS;

C is an NxN matrix, full with c.

The i th element of eq. (3) is composed by the cross-correlation between { a_k-i} and {Z_k} and that is why this technique is called correlation technique. Nevertheless, correlation comes about as an inversion in this special case.

The expressions (1), (2) and (3) would be exact only if the pseudo-random modulating function T(t) were rectangular (Fig. 1a), i.e. when the rise and fall time were infinitely short. The physically producible modulations^2-4) do not meet this criterion. The most wide-spread modulation applies a rotating disk that has transparent and blocking sections in accordance with the PRBS. This type of chopper yields a modulating function of triangular or trapezoidal shape (Fig. 1b). It can also be represented by a rectangular sequence of N elements, but the jth element a_j must be equal to the average of the transmission over the jth period, rather than 0 or 1 (Fig. 1c):

ā_j = 1/Θ ∫_jΘ ^(j+1)Θ  T(t) dt          (j=0,1,2,…,N-1)                    (4)

 

	Fig. 1. The beginning of a modulating function used in Dubna. (a) ideal; (b) real; (c) rectangular representation of the real modulation {aj}. Fig. 2. The inverse sequence { rj } of { āj } (drawn in reverse order for easier comparison with Fig. 1 a and c).

 

(4) can be expressed simply by linear combination of some elements a_j. For example, if the elementary slits of the chopper are congruent with the diaphragm slit, then

ā_j = (a_j-1 ± 6a_j+a_j+1)/8,                                                  (5)

(Fig. 1c). If the incident intensity does not change rapidly inside the jth period, the outcoming intensity is ā _j times the incident intensity and we have to write in eqs. (1) and (2) a circular matrix Ā instead of A, with the elements Ā _ki = ā_k-i(modN). Nevertheless, in eq. (3) the matrix A^T should not be substituted by Ā^T, because for real modulation the S_i is not proportional to the cross-correlation of {ā _k-i} and {Z_k}. (In other words, the inversion of Ā is not proportional to its transposition anymore.) The solution is similar to the general form of (2):

S = R (Z-b), where

R =  Ā^-1 ≠ [m(1-c)] ^-1 (A^T – C)                                        (6)

So, the elements r_j of the circular matrix R are to be determined first to obtain S.

Pelizzari and Postol published a method for obtaining the new decorrelating (inverse) sequence {r_j} empirically⁷⁾: If one takes a well-known spectrum S, its correlation measurement Z (and the known background), then eq. (6) can be regarded as the system of equations for { r_j }. After solving this system they used the obtained inverse sequence for evaluating other measurements. They found that the fluctuations of {S_i} decreased relatively to those obtained by the inversion of the ideal sequence. In principle, this method corrects other systematic imperfections of the chopper, e.g. the unequal transmissions of the open elements, too. Nevertheless, the system (6) is ill conditioned for { r_j }, i.e. the result will be highly erroneous even in the case of small errors of S and of Z.

The inverse sequence can be determined analytically, free from measured errors, too, if the real modulation deviates from ideal primarily because of the finite fall and rise times. One can show⁸⁾ that the circular matrix of the elements (5) can be diagonalized in the form

Ā = UΛU*,                                                  (7)

where U is a unitary matrix with the elements of

U_jk = ω^k_j /√N,                    j, k = 0, 1, …, N-1;

U* is the transposed conjugate of U;

ω_j = exp (i2πj/N);

Λ is the diagonal matrix of the eigenvalues λ_k of Ā;

λ_k = Σ ā_s ω^s_k      k = 0, 1, …, N-1;

The inversion of Ā:

R = UΛ^-1U*,                    (8)

with the elements

R_pq = 1/N  Σ exp[i2π(p-q)k/N]/λ_k = r_{p-q mod N}   p, q = 0, 1, …, N-1,                              (9)

i.e. R is the circular matrix of the elements

r_j = 1/N  Σ_k {exp[i2πjk/N] / Σ_s ā_s exp[i2πks /N]},   j = 0, 1, …, N-1                              (10)

of the new inverse sequence. If ā_j = a_j, we get back the ideal case: r_j ~ a_-j.

The above computation has been performed for the modulating sequence of the ANL chopper and the result is similar to the decorrelating sequence published in reference 7. One can show that some significant differences come from the statistical errors of the empirical method. Fig. 2 shows the analytically determined inverse sequence of a chopper working at the IBR-30 reactor in Dubna.

The elements differ from the ideal rectangular ones by about 15%. This deviation causes a loss of information, which can be measured by the ratio of the determinants of the information matrices M coming from the ideal and the real modulations. The square root of this ratio is equal to the ratio of the volumes of the dispersion ellipsoids of S obtained from the two modulations¹⁰⁾. In the approximation used in ref. 10 this ratio is ⁹⁾

(11)

I have to emphasize that this loss arises from the very fact of the non-ideal modulation and not from its consideration. However, if we disregard the real modulation, we disregard the real dispersion ellipsoid and the errors of the S_i‘s computed in the usual way will be false.

If the time division is shifted by ½ θ, i.e. the channel is centered not at the moment of covering the diaphragm by the slit, then, instead of eq. (5), ā_j = ½ (a_j + a_j+1) or

a _j   = ½ (a_j-1 + a_j), and a similar derivation leads to the expression

(12)

i.e., a half-channel shift of time causes not only a corrigible shift in the spectrum {S_i}, as is the case with an integer-channel shift, but a greater loss in a mathematical statistical sense.

The substitution of the old decorrelating sequence by {r_j } does not change the computation of S significantly and does not increase its time if the sequence used earlier was stored as a set of arithmetic zeros and ones and not of logic constants.

References

1. L. Pal, N. Kroo, P. Pellionisz, F. Szlavik and I. Vizi, Proc. Symp. on Neutron inelastic scattering, Copenhagen (IAEA, Vienna, 1968) vol. 2, p. 407.
2. F. Gompf, W. Reichardt, W. Glaser and K. Beckurts, ibid, p. 417.
3. F. Hossfeld, R. Amadori and R. Scherm, Proc. Symp. on Neutron inelastic scattering, (IAEA, Vienna, 1970) p. 117.
4. J. Gordon, N. Kroo, G. Orban, L. Pal, P.Pellionisz and I. Vizi, Phys. Lett. 26A (1968) 122.
5. N. Kroo and L. Cser, Fizika Elementarnych Tsjastii I Atomnogo Jadra, 8 (1977) 1412 (in Russian).
6. R. Amadori and F. Hossfeld, Proc. Symp. on Neutron inelastic scattering, Grenoble, IAEA/SM-155/F-5 (1972).
7. C. A. Pelizzari and T. A. Postol, Nucl. Instr. and Meth. 143 (1977) 139.
8. P. Rozsa, Linear algebra and its applications (Technical Publisher, Budapest, 1974) p. 254 (in Hungarian).
9. Ibid., p. 267.
10. L. Cser and L. J. Naszodi, Nucl. Instr. and Meth., this issue, p. 141.