Fourier transform aliasing

The calculation of the thermally-smeared core fragment and the valence monopoles densities was carried out by a Fourier transform of a set of aliased structure factors computed with the program VALRAY [46] details of this calculation have been published elsewhere [49],... 

Answer The spectral width is too narrow to allow the Nyquist limit ( 1.3.1) to be satisfied for all the frequencies in the spectrum and the methyl signals are folded into the window. On spectrometers that use a different version of the Fourier transformation, the aliased data may appear at the other end of the frequency spectrum but will still be out of phase with the rest of the signals. Clearly, the spectral width needs to be increased. 

The way in which phase cycling selects a series of values of Ap which are related by a harmonic condition is closely related to the phenomenon of aliasing in Fourier transformation. Indeed, the whole process of phase cycling can be... 

What you see here is that an eight-point Fourier transform apparently cannot count beyond 4, because it clearly confuses B = 5 with B = 3, and likewise 6 with 2, and 7 with 1. This is called aliasing a cosine with B = 5 can masquerade under the alias B = 3. 

The Bayesian spectral density approach approximates the spectral density matrix estimators as Wishart distributed random matrices. This is the consequence of the special structure of the covariance matrix of the real and imaginary parts of the discrete Fourier transforms in Equation (3.53) [295]. Another approximation is made on the independency of the spectral density matrix estimators at different frequencies. These two approximations were verified to be accurate at the frequencies around the peaks of the spectmm. The spectral density estimators in the frequency range with small spectral values will become dependent since aliasing and leakage effects have a greater impact on their values. Therefore, the likelihood function is constructed to include the spectral density estimators in a limited bandwidth only. In particular, the loss of information due to the exclusion of some of the frequencies affects the estimation of the prediction-error variance but not the parameters that govern the time-frequency structure of the response, e.g., the modal frequencies or stiffness of a structure. 

Figure 17. Demonstrate of foldover aliasing, (a) Hypothetical spectrum, with peaks located at their true frequencies, (b) Discrete cosine Fourier transform of the time-domain signal corresponding to (a), with sampling and Nyquist frequencies as shown. The peaks in (b) have correct relative intensities, but are folded-back to lower apparent displayed frequencies.

There are of course problems, common to any analyses requiring numerical Fourier transformations, of aliasing, truncation, and time referencing. Happily, these can be handled quite readily for dielectric problems because the pulse forms to be transformed are relatively simple and because effects of apparatus response characteristics and unwanted reflections can be eliminated or minimized by judicious apparatus design and use. In the sections that follow, we present basic analysis of TDS measurements, useful sample cell designs and working equations, methods for evaluation of transforms involved, and representative results. 

The principal errors in numerically approximating the transform integral to infinite time by a discrete Fourier transform (DFT), or series of a finite number of values at discrete points, are from aliasing and truncation. The magnitudes of the errors for the functions G(t) usually encountered can be readily assessed and kept... 

Horlick and Yuen make extensive use of aliasing or folding of different spectral regions into the final spectrum. This topic is discussed in many references on Fourier transform spectroscopy (c.f. reference 19), and therefore will not be developed here. It is because of their use of aliasing that several spectra in Figure 9 have multiple frequency scales. 

Because computerized Fourier Transforms are performed on discrete, digital data arrays of finite length, two well-known problems arise. The discreteness of the data array leads to a phenomenon referred to as "aliasing" in which frequencies which are higher than one-half the data point acquisition frequency (the Nyquist frequency) appear at values which are lower than the true frequency. This effect is illustrated in Figure 6 for the case of a sine wave. 

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