Nyquist band

The signal band (formerly centered at, ) is thus shifted to dc. The following digital FIR filter with transmission zeros at, 2 fm, and 3 fm cancels out the dc offsets and 1/f noise of the analog front-end (now centered at fm) as well as some overtones of the original square wave abased into the Nyquist band of the signal sampled at 4 fm. 

Brief reflection on the sampling theorem (Chapter 1, Section IV.C) with the aid of the Fourier transform directory (Chapter 1, Fig. 2) leads to the conclusion that the Rayleigh distance is precisely two times the Nyquist interval. We may therefore easily specify the sample density required to recover all the information in a spectrum obtained from a band-limiting instrument with a sine-squared spread function evenly spaced samples must be selected so that four data points would cover the interval between the first zeros on either side of the spread function s central maximum. In practice, it is often advantageous to place samples somewhat closer together. 

Data were summed up to 25 /total pressure 4.5Torr, un-apodized FWHM 9.31 cm-1. Nyquist wavenumber 3950.7 cm-1, 4 shots per point, fluence 82 J cm-2. The positions of the CO v, v—1 band origins are shown together with the detector cut-off point at 1840 cm-1. Reproduced with permission from Ref. 40. 

Nyquist (45) states that the vp=0 frequencies occur at higher wavenumbers in the vapor phase than in the condensed phase. He has modified the n-constant values in (1) to better match the slightly different band positions in the gas-phase spectra. These values are shown in Table 4 together with those for the condensed phase. 

Nyquist theorem - An expression for the mean square thermal noise voltage across a resistor, given by RkTAf where R is the resistance, k the Boltzmann constant, T the temperature, and /y the frequency band within which the voltage is measured. 

The mean square value of the voltage noise produced by a resistor at the frequency v, with Av band, is known as the Nyquist-Johnson theorem [see also our previous paper ]. 

Failure to obey the Nyquist sampling theorem results in aliasing, that is, the effect of frequencies greater than the Nyquist frequency are reflected into the band of frequencies between DC and one-half the sampling rate. This situation is demonstrated in Fig. 20.70, where the samples of the high-frequency sinusoid are indistinguishable from a uniform sampling of the low-frequency sinusoid. 

Bandlimited A waveform is described as bandlimited if the frequency content of the signal is constrained to lie within a finite band of frequencies. This band is often described by an upper limit, the Nyquist frequency, assuming frequencies from DC up to this upper limit may be present. This concept can be extended to frequency bands that do not include DC. 

Nyquist frequency For a bandhmited waveform, the width of the band of frequencies contained within the waveform is described by the upper limit known as the Nyquist frequency. 

Chloroformates absorb at somewhat higher frequencies than carbonates, and the C=0 band is usually found between 1780 and 1770 cm". This is again raised by a-halogen substitution, the CCI3 derivative absorbing at 1806 cm". The bulk of the available data relates to alkyl chloroformates studied by Ory [13], Nyquist and Potts [8], and Katritzky [11]. The effect of a double bond at the ester oxygen atom has not been much studied, but it would seem that vqo is raised as in the aryl carbonates. Thus allyl chloroformate absorbs at 1799 cm". ... 

Thiol esters have been studied by Rasmussen [18], and more recently by Nyquist and Potts [82] and Baker and Harris [83]. Dialkyl thiolesters show their carbonyl bands in the narrow range 1698 to 1690cm". The reasons for this substantial fall as compared with normal esters are not fully understood and have been the subject of some controversy [80]. However, it is probably a direct consequence of the low electronegativity of sulphur. a -Con-jugation in aryl alkyl thiolesters lowers the frequency to 1665 to 1670 cm" which is a rather larger shift than is found in esters. On the other hand the rise in frequency when an aromatic ring is attached to the sulphur atom is less than in the phenyl esters, and such compounds absorb near 1710 cm". ... 

Thus Meyrick and Thompson [1] reported strong P=0 and P—H absorptions in a number of compounds of this type, and the same effects have been noted by Bellamy and Beecher [6] and by Nyquist [39]. The absence of P—OH absorptions and the fact that the P—0 absorption frequency corresponds to an unbonded phos-phoryl group confirm that not more than a very small proportion of the whole can exist as hydrogen phosphite. The basicity of the P=0 link has been measured by hydrogen-bonding methods in which the frequency shift of some standard XH donor such as CDCI3 is measured. Gramstad [40, 41] has shown that Si> can then be related to the Taft a values of the phosphorus substituents. These therefore determine both the frequency and the polarity. The intensity of the P=0 band is also related to the 6v values of proton donors and is therefore similarly determined by the polarity [41]. 

On the basis of this and other related series [6, 7], and especially of the work of Nyquist [42] who has correlated the data on over 100 compounds, the P—O—C (aromatic) absorption has been assigned to the 1200 cm" region, and it has been shown that a band... 

This band is always accompanied by a second absorption which has been attributed to either the symmetric stretch of the P—O—C system (the antisymmetric mode being that at 1200 cm ), or to a separate P—0 stretch which is not so coupled. Thomas [43] strongly favours the latter explanation which is supported by the persistence of this band in both P—O—P and P—OH compounds, and by the fact that in the latter the frequency is a linear function of the tt values of the substituents, Nyquist [42] places this band between 994 and 914 cm and notes that the band is at a lower frequency in trivalent phosphorus compounds. Thomas [43] widens this range to 905—996 cm" for pentavalent compounds and suggests 860—850 cm" for trivalent. He points out that this band is often complex and the values quoted therefore refer to the strongest band. 

---