Hadamard codes

Due to the fact that the first phase of manipulation of such data is usually a fast scanning of the entire collection, a highly compressed representation of uniformly coded data is essential in order to accelerate the handling. After the search reduces the collection to a smaller group in which the target object is supposed to be, the full (extended) representation of objects can be invoked if necessary for further manipulation. In the next sections we shall discuss the use of two methods, Fast Fourier Transformation (FFT) and Fast Hadamard Transformation (FHT), for the reduction of object representations and show by some examples in 1- and 2-dimensional patterns (spectra, images) how the explained procedures can be used... 

For the reverse transformation the same routines (source codes) can be used in FFT and FHT. However, for the reverse Fourier transformation the real and imaginary arrays of the coefficients (which are now input) should be divided by N (number of coefficients) and the imaginary array must be conjugated (multiplied by -1), while in the case of reverse Hadamard transformation only a division of N real coefficients by N is necessary. 

Non-dispersive multiplex spectrometers include Hadamard transformation spectrometers and Fourier transform spectrometers and are particularly useful for the case of very stable sources. In both cases the information, such as intensities at various wavelengths, is coded by a multiplex system, so that it can be recorded with a conventional detector. A suitable transformation is then used to reconstruct the wavelength dependence of the information. In Hadamard spectrometry use is made of a codation of the spectrum produced by recombining the information with the aid of a slit mask which is moved along the spectrum [66],... 

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. 

Arrays with rotational antisymmetry have attractive properties for applications such as coded aperture imaging (e.g. Cook et al. 1984). Finger and Prince (1985) showed that all antisymmetric URAs could be derived from skew-Hadamard cyclic difference sets. Since all two dimensional lattices have 180 symmetry, all skew-Hadamard cyclic difference sets can generate both hexagonal and rectangular URA s with 180 antisymmetry. A subclass of these with order v (the number of cells in the basic pattern) equal to a prime p with p — 1 mod 12 were shown to generate a hexagonal array (HURA) with an additional 60 rotational antisymmetry. 

FOURIER AND HADAMARD (MULTIPLEX) CODES DISCRETE SIGNAL SAMPLING... 

For an arbitrary code relating the y to the x, the problem of recovering the desired x-j from the observed can be difficult or worse.The very special feature of both the Hadamard and Fourier codes is that the desired inverse code can again be found trivially from the specified original code. In mathematical terms, these particular code matrices are said to be "well-conditioned".H Consider again the 4-channel experiment, but this time with the Hadamard code ... 

Equations 39 show that with the Hadamard code, each unknown element x is observed N times with the same absolute weight factor namely, the absolute value of each of the matrix elements in the code... 

If the first row and column of the Hadamard code of Equation 39 are deleted, it becomes clear that each row of the remaining array differs from the preceding row by cyclic permutation. This property carries two immediate advantages. First, it is no longer necessary to construct a separate code for each measurement--see Sloane Chapter and reference 10 for examples of Hadamard mask construction. Second, construction of the desired inverse transformation is trivial ... 

The Fourier code is based on the same properties just developed for the Hadamard case ... 

Plankey et al have reported on the application of Hadamard transform spectrometry to the ultraviolet and visible spectral regions. The Hadamard technique utilizes the dispersed radiation from a conventional spectrometer. The radiation is passed through a coded mask, recombined, and recorded. The mask is composed of N slits, and N measurements are made with the mask in different positions. After measurement, N simultaneous equations must be solved with a Hadamard matrix to obtain the spectrum. The mask must be moved mechanically and reproducibly and a computer is used to solve the simultaneous equations. 

Filler monochromators are of use only for flame photometry. They make use of interference filters, which often have a spectral bandpass of a few nanometers or less. Multiplex spectrometers include Hadamard transform spectrometers and Fourier transform spectrometers, and are especially useful where very stable sources are needed. Hadamard transform instruments make use of a coding of the spectrum produced by recombining the information with the aid of a slit mask which scans the spectrum [48]. 

Examples of coding matrices include the Hadamard matrix and the Fourier matrix. The advantage of this method is the statistical improvement gained as a result of the increase in total scanning time. 

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