Hadamard matrix

W. J. Diamond, Practica/ Experimenta/ Designs for Engineers and Scientists, 2nd ed.. Van Nostrand Reinhold, New York, 1989. "This book is for engineers and scientists with Httie or no statistical background who want to learn to design efficient experiments and to analyze data correctiy. .. The emphasis is on practical methods, rather than on statistical theory." The discussion is quite detailed in some areas, eg, experimental designs based on Hadamard matrices, and scanty in others. 

Table 2 shows a 12-run Plackett-Burman design. All OA(12, 2n, 2) are equivalent in the sense that they can be obtained from one another by changing signs in one or more complete columns and/or permuting rows and/or permuting columns. Hadamard matrices of higher orders, however, are not unique. 

At x t matrix H with elements 1 is called a Hadamard matrix if H H = tl. Hadamard matrices are considered to be equivalent if they can be obtained from... 

Screening design by Hadamard matrices Plackett-Burman designs... 

We see that this applies to the model matrices from factorial designs and fractional factorial designs these matrices are Hadamard matrices. It is possible to construct Hadamard matrices by other principles, and it was shown by Plackett and Burman[l] how such matrices can be obtained for zi = 4, 8, 12, 16, 20, 24, 28, 32,..., i.e. when n is a multiple of four. 

Plackett and Burman have shown the construction of all matrices for n up to 100, with one exception, (92 x 92) matrix, which they failed to construct. This matrix has been reported later.[2] The Plackett-Burman paper[l] discusses the use of Hadamard matrices to define screening experiments, in which (n —1) variables can be tested in n runs. 

The construction of a design by a Hadamard matrix is simple. Plackett and Burman have determined how the first row in the design matrix should be constructed so that the remaining rows can be obtained by cyclic permutations of the first row. An example to show this is given below. For a given n there can be many different n xn matrices which are Hadamard matrices. The Plackett-Burman matrices are by no means the unique solutions. 

A textbook largely dealing with designs based on Hadamard matrices is W.J. Diamond... 

The measured responses to the combinations of multi-frequency selective pulse excitation can be unscrambled for each volume element by transformation with a super-Hadamard matrix. The dimension of this matrix equals the product of the dimensions of the Hadamard matrices used for encoding each space axis. 

Hadamard matrices of experiments are generally used to point out the more influent qualitative and/or quantitative factors within a given experimental domain. In this method, two levels are attributed to the ctors (noted -1 and +1) as presented in Table 1. To study the seven fectors mentioned above, eight experiments are needed. The matrix of experiments is presented in Table 2 where each line corresponds to a synthesis while the columns correspond to the fectors. 

The specific values of information embedding rates appearing in these tables are due to the ECC used in these tests. The error correcting codewords we have used were the rows of the Hadamard matrices of varoius orders, together with their binary complementary sequences. The rate - R - of the code is determined from the order - h - of the Hadamard matrix by the relation R = where in our experiments h = 5,..., 12. 

Furthermore it can be shown that Hadamard matrices are also D-optimal (cf. ref. 11, Section 3.2.5 of ref. 2). Equations 11 and 12 describe a weighing design based on a Hadamard matrix of order 4. 

It is easy to prove that if a Hadamard matrix of order n exists, then n must be 1, 2, or a multiple of 4. It is generally believed that Hadamard matrices exist of every order that is a multiple of 4. 

In order for an n x n S-matrix to exist n must be 1 or a number of the form 4a + 3. Infinitely many examples can be obtained from the known constructions of Hadamard matrices. The most important examples for practical purposes are cyclic S-matrices, having the property that each row is a cyclic shift to the left (or, to the right) of the previous row. Equation 5 is an example. 

Hadamard Spectroscopy. If. instead of the exit slit, an irregular arrangement of slits of various widths is used, and an arrangement in the form of a comb-like second slit is moved in front of this, for each relative position a different summation of intensities at various wavelengths is obtained on a nonmultiplex detector. By using convolution functions. spectra can be computed rapidly for a cycle of different slit positions. Since for mathematical deconvolution so-called Hadamard matrices are used, the principle is known as Hadamard spectroscopy. It has gained some importance in waste gas analysis [50]. 

Using these matrix and the corresponding Hadamard matrices of Exercise P4.1. 

Hadamard NMR spectroscopy [3] is based on the Hadamard Transform rather than the more conventional Fourier Transform. The NMR frequencies are encoded using the Hadamard matrices and multiply-selective frequency encoding pulses. Only the frequencies of interest are used at the encoding stage. Thus vast spectral regions that contain no interesting information can be excluded and the correlation spectra can be recorded much faster. Some prior knowledge about the peak positions in the indirectly detected dimensions is required for these experiments. 

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