The Hadamard transform

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. 

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There is a number of alternative Raman imaging techniques these include using the Hadamard transform technique [25-27], and such as fibre-bundle image compression, which however is not yet commercially available [26-31]. However in the latter approach, the laser power on the sample could be high, since the beam is not defocused, and the possibility of sample damage increases. 

In the Fourier transform the least significant coefficient is in the middle of the series while more the coefficients are approaching both ends of the series (towards co and cn-i) the greater is their information content. On the other hand, the least important coefficient in the Hadamard transform is the last of the series. 

B.R. Kowalski, C.F. Bender, The Hadamard Transform and Spectral Analysis by Pattern Recognition, Anal.Chem., 45, (1973), 2234-2239,... 

The Hadamard transform is also called the Walsh123-Hadamard, or Hadamard-Rademacher124-Walsh, or Walsh, or Walsh-Fourier transform. 

The Hadamard transform of index m, Hm, is a 2m x 2m matrix, consisting of elements that are either 1 or — 1, which can be defined recursively ... 

A thousand points of light the Hadamard transform in chemical analysis and instrumentation, Anal Chem 61 723A-734A. 

An interesting application of the Hadamard transformation is its use for determination of the linear impulse-response function with maximum length binary sequences or m... 

Stochastic excitation with m sequences and the use of the Hadamard transformation have been investigated in NMR spectroscopy [Kail, Zie2] as well as in NMR imaging [Chal]. Processing of the nonlinear response to m sequences by Hadamard and Fourier transformation results in signal distortions reminiscent of noise, which are caused by the nonlinear parts of the response [B1U2]. 

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... 

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... 

Miyamoto, S., Tsunemi, H., and Tsuno, K. 1981, "Some Characteristics of the Hadamard Transform X-ray Telescope," preprint. 

For a description of the Hadamard transform and Hadamard tran.dorm spectroscopy, see D. K. Graff. J. Chem. duc.. 1995, 72. 304, Fourier. Hadamard. and Hilheri Transforms in Chemisirv.A. G. Marshall, cd. New York Plenum Press. 1982. 

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. 

Kaiser, R., Application of the Hadamard Transform to NMR Spectroscopy with Pseudonoise Excitation , J. Magn. Reson. 1974,15, 44—63. 

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. 

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 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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