Hadamard and Other Discrete Transforms in Spectroscopy

Mathematics and Statistics Research Center Bell Laboratories Murray Hill, NJ 07974 

This paper is an introduction to the use of Hadamard and other matrices for increasing the accuracy to which the spectrum of a beam of light can be measured. In the most favorable case if the spectrum has n components the mean squared error in each component is reduced by a factor of n/4. These schemes have the additional merit that the instrumentation required is relatively simple. 

If we simply weigh the objects one at a time on the balance, we make seven measurements 71, , 77. These differ from the true weights , / 7 by the (unknown) errors ei, ,e7  

On the other hand if we are clever we first put objects 1,2,3 5 on the balance and weigh them, then objects 2,3,4 6, then 3,4,5 

Again ei,--, e7 are the unknown errors in the weighings. To find the weights we solve these equations for I j, , 7 pretending that the ei s are not present. Thus we cover up ei, -,e7 and solve he equations for , 7. The results, which we call since 

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HADAMARD AND OTHER DISCRETE TRANSFORMS IN SPECTROSCOPY N. J. A. Sloane... 

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