Hadamard transformed spectrum

Another way to reduce spectral data is to reduce the number of coefficients of the Hadamard transformed spectrum. The reduction of the Hadamard coefficients is performed by setting some of them to zero. After reverse Hadamard transformation, the spectrum is restored. The resolution of the reduced spectrum is determined by the number of the coefficients that were not set to zero. Figure 3 shows different states of data reduced spectra versus the number of Hadamard coefficients. ... 

Hadamard transform [17], For example the IR spectrum (512 data points) shown in Fig. 40.31a is reconstructed by the first 2, 4, 8,. .. 256 Hadamard coefficients (Fig. 40.38). In analogy to spectrometers which directly measure in the Fourier domain, there are also spectrometers which directly measure in the Hadamard domain. Fourier and Hadamard spectrometers are called non-dispersive. The advantage of these spectrometers is that all radiation reaches the detector whereas in dispersive instruments (using a monochromator) radiation of a certain wavelength (and thus with a lower intensity) sequentially reaches the detector. 

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]. 

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],... 

FIGURE 6.5 The infrared spectrum of a query compound compressed by Hadamard transform for the prediction of benzene derivatives by a CPG neural networks. The spectrum exhibits some typical bands for aromatic systems and chlorine atoms. 

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. 

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]. 

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...

The original representation of infrared spectrum in this example is a set of 512 equidistant intensity values (ref. 6). In order to show the reduction of the information content, the reduction of Fourier and Hadamard coefficients (FCs and HCs) in the transform is carried out to the extreme. In Figure 5.3 the spectrum is reproduced from reduced number of coefficients obtained with the FFT and FHT of the 512-intensity-point curve. 

Data Reduction and Background Correction Reduction of data points is important if, for example, further processing of a spectrum is only feasible if the number of data points is decreased. For reduction of measurements in the original data vector, the data are transformed by means of FT or HT. After that, back transformation is performed on the basis of a limited number of Fourier or Hadamard coefficients. For back transformation, the coefficients are sorted according to importance, and the effect of less important coefficients is thus eUminated (cf. Zupan, Section 3.3). Practically, the number of coefficients is not changed, but unimportant coefficients are set to zero. 

In virtually all types of experiments in which a response is analyzed as a function of frequency (e.g., a spectrum), transform techniques can significantly improve data acquisition and/or data reduction. Research-level nuclear magnetic resonance and infra-red spectra are already obtained almost exclusively by Fourier transform methods, because Fourier transform NMR and IR spectrometers have been commercially available since the late 1960 s. Similar transform techniques are equally valuable (but less well-known) for a wide range of other chemical applications for which commercial instruments are only now becoming available for example, the first commercial Fourier transform mass spectrometer was introduced this year (1981) by Nicolet Instrument Corporation. The purpose of this volume is to acquaint practicing chemists with the basis, advantages, and applications of Fourier, Hadamard, and Hilbert transforms in chemistry. For almost all chapters, the author is the investigator who was the first to apply such methods in that field. 

In a second approach, the spectral data are expressed in terms of a vector - for example, using Hadamard or Fourier transform coefficients of IR spectra - each element of which is treated as a coordinate in multidimensional space. Each spectrum occupies a point in hyperspace and the similarity between an unknown and a reference entry is measured by the distance between the two points. Once again, the output is a rank-ordered list of structures corresponding to the spectra producing the smallest distance to the query. 

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