Aliasing

The use of the discrete FT can lead to spectral artefacts the picket fence effect and aliasing. 

An advanced FT software package will automatically account for proper sampling, such that only the upper and lower limits of the desired spectral range need to be specified by the user. 

In summary, the sampled waveform should contain a whole number of periods to avoid leakage and the sampling should be with at least the Nyquist frequency or faster to avoid aliasing. In EIS practice, a waveform containing a predetermined number of frequencies and whole number of periods of waveforms is used and sampling is synchronized (Chap. 3.7). 

What happens if you try to image an object with periodicity outside this spatial frequency range The spatial frequency will be displaced by an integral multiple of 2n/a so as to lie in the range that the framestore can handle. There is an almost exact analogy with phonon wavevectors in a crystal lattice or, if you prefer, with why stage-coach wheels appear to go backwards in movies. The effect is illustrated in Fig. 3.8 with a feature with a periodic structure of spatial frequency Kj. It will be stored in the framestore with a spatial frequency dz JCj — 27T K/ap, with ng = 1 in the example here. This will appear as a periodic structure in the image that bears no apparent relationship to the object. 

This problem need not cause too much alarm. Most specimens do not have extended periodic structures usually a microscope is operated so that the framestore is not limiting the resolution, and the capacity of framestores is increasing, if not by the day, then at least by the year. But the problem can arise, particularly when large areas are being examined, and when it does the only remedies are to reduce the frequency or the numerical aperture of the lens, or else to reduce the area scanned. 

In summary, when the entire spectrum from zero to v ax is required at a resolution of Av, the number of points to be sampled, Ng, is given by 

If the spectral range is restricted to fall between a minimum wavenumber v maximum wavenumber Vmax, the number of points required is 

An example of a real system where these factors have been brought into consideration may be found in commercial FT-IR spectrometers that collect high-resolution 

Most near-infrared (NIR) spectra are measured from 2500 to llOOnm (4000 to 9100 cm ). To record a spectrum over this range, the interferogram must be sampled twice per wavelength of the HeNe laser interferogram to give a bandpass 

If the harmonics are outside the bandwidth allowed by the sampling frequency, they will naturally be folded into the real spectmm, thereby causing either spurious lines to appear (in the case of a line spectrum) or photometric inaccuracy (when the source is continuous). This phenomenon is explained in greater detail in Section 8.5.1. 

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The primary Cr—O bonded species is cbromium (VT) oxide, CrO, which is better known as chromic acid [1115-74-5], the commercial and common name. This compound also has the aliases chromic trioxide and chromic acid anhydride and shows some similarity to SO. The crystals consist of infinite chains of vertex-shared CrO tetrahedra and are obtained as an orange-red precipitate from the addition of sulfuric acid to the potassium or sodium dichromate(VI). Completely dry CrO is very dark red to red purple, but the compound is deflquescent and even traces of water give the normal mby red color. Cbromium (VT) oxide is a very powerful oxidi2er and contact with oxidi2able organic compounds may cause fires or explosions. 

The resolution of the analog I/O channels of the controller vaiy somewhat, with 12-bit and 14-bit conversions quite common. Sample rates for the majority of the constant sample rate controllers range from I to 10 samples/second. Hard-wired single-pole, low-pass filters are installed on the analog inputs to the controller to protect the sampler from aliasing errors. 

In addition, vibration data collected with a microprocessor-based analyzer is filtered and conditioned to eliminate non-recurring events and their associated vibration profiles. Anti-aliasing filters are incorporated into the analyzers specifically to remove spurious signals such as impacts. While the intent behind the use of anti-aliasing filters is valid, however, their use can distort a machine s vibration profile. 

Approved gene symbol Approved gene name Previous symbols and aliases Chromosomal location Sequence accession ID... 

Increasing the diameter, d, increases the number of photons in the wavefront measurement, and therefore reduces the error due to photon noise. However, increasing the diameter also increases aliasing in the wavefront sensor measurement. If the deformable mirror actuator spacing is matched to the subaperture size, then the fitting error will also depend on the subaperture diameter. There is therefore an optimum subaperture diameter which depends on the... 

Fig. 40.11. Aliasing or folding, (a) Sine of 8 Hz sampled at 16 Hz (Nyquist frequency), (b) Sine of 11 Hz sampled at 16 Hz (under-sampled), (c) A sine of 5 Hz fitted through the data points of signal (b).

As in imaging, aliasing can occur if velocities outside the FOF exist. For measurements of average velocities, the same condition holds if the total phase shift due to velocity exceeds 2 n. For velocity fields of simple structure, this problem can be accounted for by so-called unwrapping algorithms (see e.g. Chapters 2.9 and 4.2). 

Fig. 4.2.5 Unaliased (A) and aliased (B) velocity images for a 0.6% aqueous carboxy-methylcellulose solution obtained by MRI. The vertical axes represent the data obtained in the velocity encoded direction and the horizontal axis represents data obtained in the spatially encoded direction. Aliasing is achieved by setting the velocity encode gradients so large that the maximum phase evolution of a given fluid element for a pulsed gradient step exceeds Ji or 2jt...

Fig. 4.2.6 Comparison between velocity profiles obtained from aliased (circles) and unaliased (diamonds) MRI data. The data near the pipe center are shown where the difference between the two methods is most apparent.

Not only is the choice of a uniform prior-prejudice distribution not sensible it also exposes the calculation to two main sources of computational errors, both connected with the functional form of the MaxEnt distribution of scatterers, and with its numerical evaluation namely series termination ripples and aliasing errors in the numerical sampling of the exponential modulation of mix). The next two paragraphs will illustrate these issues in some detail. 

A second major source of computational difficulties associated with uniform prior-prejudice distributions is connected with the extremely fine sampling grids that are needed to avoid aliasing effects in the numerical Fourier synthesis of the modulating factor in (8). To predict the dependence of aliasing effects upon the prior prejudice, we need to examine more closely the way the MaxEnt distribution of scatterers is actually synthesised from the values of the Lagrange multipliers X. 

When the reconstruction of the density is carried out by modulation of a prior prejudice of spherical atoms, only the deformation features have to be accommodated this can be accomplished relatively easily, and the Lagrange multipliers are usually below 0.01 in modulus, or even smaller for valence-only runs. No aliasing problems occur in the synthesis of (x). 

BUSTER chooses the minimal grid necessary to avoid aliasing effects, based on the prior prejudice used and on the fall-off of the structure factor amplitudes with resolution for the 23 K L-alanine valence density reconstruction the grid was (64 144 64). The cell parameters for the crystal are a = 5.928(1)A b = 12.260(2)A c = 5.794(1) A [45], so that the grid step was shorter than 0.095 A along each axis. 

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],... 

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