Hadamard transform examples

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]. 

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... 

The Hadamard transform is an example of a generalized class of a DFT that performs an orthogonal, symmetric, involuntary linear operation on dyadic (i.e., power of two) numbers. The transform uses a special square matrix the Hadamard matrix, named after French mathematician Jacques Hadamard. Similarly to the DFT, we can express the discrete Hadamard transform (DHT) as... 

The upper indexes, 00), 01), 110), 111), are to emphasize that the corresponding operators only execute a true Hadamard gate when they act on the indicated states. The indexes 01, 12, and 23 indicate the pulse transition as indicated in Figure 4.2. However, the operators c), d) and e), g) can be implemented by a single pulse sequence if we use two-frequency pulses to excite simultaneously two transitions. For example, Uhj. = where 01-23 indicates a two-frequency selective pulse that act simultaneously on the transitions 01 and 23 see Figure 4.2, will implement a operation independently of the initial state. All the Hadamard transformations indicated in (4.2.11) are self-reversible. 

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. 

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 ... 

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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