Application of the Hadamard Transform to Gas Chromatogramsof Continuously Sampled Mixtures M. Kaljurand/ E. Kfillik Institute of Chemistry, Academy of Sciences of the Estonian S.S.R., Tallinn 200026, U.S.S.R. Su m mary A method to establish a chromatogram by pseudo random injections and the Hadamard transform technique is described. It is possible to accumulate a peak from a detector output signal, superimposed by background noise. The method is tested on simulated process. Introduction During the last decade there has been increasing interest in the possibilities of continuous gas chromatography, and in particular, the application of a pseudo random binary perturbation of sample gas injections [ 1]. Hard-wired correlators are commercially available and can be used to determinate the impulse response of the system i.e. a chromatogram obtaining from a single injection but the present high cost of correlation equipment and the num- ber of points available to the user (limited resolution) are their disadvantages. It is possible to perform the necessary computations using a mini-computer which is really available in most laboratories. It requiresproper algorithms for the effective use of computer time and memory. One of these algorithms is the Hadamard transform. Recently the relationship between the maximum length binary pseudo random sequences and Hadan~ard matrices have been explored and fast algorithm for the deconvolution of the output from such input signals developed in nuclear magnetic resonance spectroscopy [2] and optics [3]. The aim of this paper is to modify this technique for gas chromatography. Theoretical The matrix form of the numerical convolution can be written as Y=Uh (1) where Y, h and U are respectively the output vector, the discretised chrornatogram and convolution matrix. The convolution matrix is formed cyclically ("foldaround") from the given pseudo random binary sequence vector. A cyclic matrix is one for which each row may be obtained from the preceding row by shifting the elements one posi- tion to the right (left); the element at the end (beginning) of a row is moved to the beginning (end) of the next row. One additional (artifical) degree of freedom is introduced in the following manner. Extended vectors Y and h may be defined by ~fo = 0 ; Yi = Yi i = 1, ..., N "ho = - ~Yi ; hi = hi i = 1 ..... N where N is the number of measurements made (the num- ber of points on the chromatogram). An extended matrix U may be defined by Uoo =Uoj= Ujo =0 Ui,k = Ui,k Eqn. (1) becomes j=l ..... N i,k= 1,...,N Y = U h = (2) Kaiser has shown [2] that the following statement holds H = P2 UP1 (3) where P1 and P2 are orthogonal matrices which have a single element "1" in each row and column, all other elements being zero. A "1" in position Pjk puts the "j"-th column into "k"-th place or the "k"-th row into "j"-th place. Thus P~ permutates the columns and P2 permutates the rows of U. The permutations depend on the particular pseudo random binary sequence used. The method for finding the permutations is exhaustively ex- plained by Kaiser [2]. A Hadamard matrix H of order N + 1 is a square matrix of elements 0 and 1 (or +1 and -1) with the property HH r =(N + 1)I (4) where the superscript indicates the transpose and I is the unity matrix of order N + 1. It is necessary for our pur- pose to construct Hadamard matrices of order N + 1 = 2 n 3 28 Chromatographia, Vol. 11, No. 6, June 1978 Originals