Fast Hadamard transform

Maximum length binary sequences (MLBSs) of length N = 2l-l, where I is a positive integer, have a perfectly flat power spectrum [77]. The deconvolution in Eq. (61) can be computed very efficiently by means of a fast Hadamard transform, and they have, for example, been employed for Hadamard NMR spectroscopy [78]. 

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... 

Fast Hadamard Transform (FHT) is a mathematical transformation of signals or vectors similar to Fourier transform but is based on square wave functions rather than sines and cosines. 

Fast Hadamard Transform (FHT) Compression is a method that uses Hadamard transformation to decompose spectra into a series of Hadamard coefficients, to reduce them, and to backtransform them to achieve a compressed version of the spectrum. 

Maximum Likelihood Computations with Fast Hadamard Transform... 

It is easily verihed that the expression Eae o,i ( l) h)a is the m-order Hadamard transform of the vector v e defined by tJi(a) = Wa., where i(a) is the integer whose binary representation is a. The optimal sequence c of (8) is found by computing the fast Hadamard transform of the vector v, defined above, and then choosing the entry that yields the maximum value. The optimal sequence b of the original ML expression is now obtained by... 

HT requires only simple arithmetical operations addition and subtraction. This is in contrast to FT calculations, where complex numbers and trigonometric functions have to be processed. As a consequence, the algorithm for fast Hadamard transformation (FHT) is faster by a factor of about 3 than the FFT algorithm. 

Braun, K. L., Hapuarachchi, S., Fernandez, F. M., and AspinwaU, C. A. Fast Hadamard transform capillary electrophoresis for on-line, time-resolved chemical monitoring. Ana/. Chem. 78,1628, 2006. 

Computer methods for the reduction of measurement spaces, such as fast Fourier and fast Hadamard transforms, principal component analysis, autocorrelation, genetic algorithm, and others, can be applied successfully. 

Braun, K.L. Hapuarachchi, S. Fernandez, F.M. Aspinwall, C.A. High-sensitivity detection of biological amines using fast Hadamard transform CE coupled with photolytic optical gating. Electrophoresis 2007, 28 (17), 3115-3121. 

Figure 3 Comparison of the original IR spectrum and data reduced spectra. After fast Hadamard transformation the coefficients are truncated and transformed back. The number of remaining Hadamard coefficients determines the resolution...

In this section, we exploit the fact that the Hadamard transform can be utilized for fast calculations over linear spaces [Lem79]. We comment that the computational procedure described in this section does not uses the stationarity of the covertext sequence and the... 

A transformation related to the Fourier transformation. The Hadamard transformation calculates the periodic function as a sum of square waves, while sine waves are used in a Fourier transformation. The Hadamard transformation yields intensities and frequencies of the square waves the frequencies of the waves are harmonic. One period of the base frequency is the measured time of the periodic function. A Hadamard transformation is faster than a fast Fourier transformation, because it does not have to calculate the values of the sine function. 

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