Editors' Column Fourier and Hadamard Transform Methods The following exchange of comments refers to the INSTRUMENTAT I O N feature entitled "Fourier and Hadamard Transform Methods in Spectroscopy" by Alan G. Marshall and Melvin B. Comisarow, which appeared in the April 1975 issue of ANALYTICAL CHEMISTRY (page 491 A).
tions is inconsistent and no solution is possible. If w5 = 0, then the solution set of the system of equations is:
Sir: The relative merits of Hadamard and Fourier methods compared with conventional methods in spectroscopy were recently described. A basic explanation was expressed by analogy to the best way to use an ordinary double-pan balance. The authors employ the analogy with significant effect in their paper, and the present communication comments on some details of their arguments. Conventionally, the problem of determining the weights of four unknown objects by weighing each object separately is obvious. The possible objection of this method is that each unknown is measured only once. In an attempt to improve the precision of each weight, the authors proposed two schemes. In the first scheme, which was used as an analogy of the Hadamard transform method, two unknown weights were placed on the left pan, and four linearly independent arrangements of unknowns were weighed two a t a time. The following system of equations defines the relationship between the unknown weights x i (i = 1,2,3,4) and the known observed readings wi (i = 1,2, 3,4).
where k is arbitrary. Hence, this outlined experimental procedure does not provide sufficient information to determine the unknown weights x i ( i = 1, 2, 3, 4) uniquely, since the rank of the augmented matrix [ Uw]is less than the number of unknowns. The system of equations is not linearly independent. Marshall and Comisarow are therefore not justified in using the analogy between the weighing procedure outlined and the Hadamard encodingdecoding methods employed in spectrometry. The second scheme, which is used as an analogy of the Fourier transform method, does provide unique values for the unknown weights and therefore is justified. (For reference, see L. Fox, "An Introduction to Numerical Linear Algebra,'' Oxford, England, 1964.)
Eric Rushton and Albert Sagarra The University of Manchester Institute of Science and Technology Manchester, England Drs. Marshall and Comisarow Reply:
Application of the Gauss elimination procedure to the above matrix equation gives the following equivalent system of equations: 1 1 0 0
0 0 0 0
or
ux
=
I('
where w5 = w4 - w 1 + wp - w3, and the matrix U is upper triangular. If w j = 0, then the system of equa-
Rushton and Sagarra have simply pointed out that the particular four arrangements of weights in our Hadamard example are not linearly independent, since the determinant of the matrix of their coefficients vanishes. This condition is readily rectified by using the four arrangements corresponding to Equation 1below; thus, it is evident that the analogy between use of weights on a balance and Hadamard spectroscopy is indeed valid.
;]E]
1 1 0 0
[:;;
0 1 0 1
=
[]
(1)
In fact, for any N , it is always possible to find N linearly independent arrangements of N weights, where weights are placed on only one pan of the balance. In Hadamard spectroscopy it is additionally desirable that the arrangements not only be linearly independent, but also that successive arrangements differ by cyclic permutation, in order that the N arrangements may be derived by translation of a mask of (2 N - 1) slits, as discussed for Figure 4 of the article in question. I t may be shown that by choosing N = 2m - 1, m = 2,3,4, . . . , it is always possible to construct the desired array. More generally, possible values for N include N = 3, 7, 11, 15, 19, 23, 31, 35,43, 47, 63, . . . , according to the conditions [N.J.A. Sloane, T. Fine, P. G. Phillips, and M. Harwit, Appl. Opt., 8,2103 (196911:
A' = 4 tz
-
1, n = 1, 2, 3,
...
and either N = 2" - 1 , I M = 2 , 3 , 4, . . . ( 2) o r N = p , where p is a p r i m e number o r A' = p ( p + 2 ) , where p is a p r i m e number As examples, the arrays for N = 3 and N = 7 are shown below:
1 0 1 0 1 1 1 1 0
N = 3
0 1 0 1 1 1 0
1 0 1 1 1 0 0
0 1 1 1 0 0 1
1 1 1 0 0 1 0
1 1 0 0 1 0 1
1 0 0 1 0 1 1
0 0 1 0 1 1 1
'I' = 7
Of course, the scheme shown in Figure 4 should be illustrated for a value of N chosen from Equation 2 above. As an unrelated typographical correction, the signal-to-noise improvement for Hadamard spectroscopy over use of a single-slit scanning spectrometer should appear as ( m / 2 ) rather than (dlW)a t appropriate places in the text. In conclusion the analogy between weights on a balance and schematic slits in a multiplex spectrometer is valid and provides a simple and direct entry into the principles of Hadamard and Fourier transform spectroscopy.
ANALYTICAL CHEMISTRY, VOL. 47, NO. 9, AUGUST 1975
651A
