Hadamard transform advantage

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. 

The advantage of the transformed objects over the original ones in data reduction schemes lies in the order induced in the sequence of coefficients. This order is correlated with frequency while in original data the information is more or less uniformly distributed over all the sequence, in transformed object the first few low-frequency coefficients contain the information about the rough contours of the original object and the high-frequency coefficients describe the details. In both Fourier and Hadamard transforms the most important part of the information can be retained after back-transformation with the proper choice of coefficients. 

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. 

For spectral patterns Fourier-, Hadamard-, and WaIsh- transformations have been applied to generate new features C118, 154, 3353. Some success of these transformations has been reported for nuclear magnetic resonance spectra C29, 3373, and for mass spectra C122, 134, 135, 148, 154, 3193. For non-spectral data such transformations do not seem meaningful and have not been shown to be advantageous. 

To evaluate a specific attack, let us consider that Alice s and Bob s qubits are not really entangled, but Eve has sent qubits of her own choice to both of them. Eve also can listen to the classical channel. The best she can do is send a classical 0 to Alice and a Hadamard 1 to Bob. Actually, all other combinations are equivalent to, or less advantageous than, this one. Alice and Bob decide who is to measure first once they already have the qubits. With 1/2 probabihty, Alice is measuring, in which case the readings are consistent. Bob measures first, again with probability 1/2. Bob will measure a 1 or 0 with equal probability. Then Alice transforms the classical 0 with a Hadamard gate, and also reads a... 

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