Cosine amplitude modulation

Figure Ila shows how an ideal cosine amplitude modulation of the RF carrier wave could be approximated by a rectangular RF pulse scheme, which is much easier to implement. Such a scheme comprises of pulses with alternating phases of 0° and 180° and is referred to as FAM. As was already mentioned, the modulation frequency should be tuned such that Vm matches tq, at least during part of the excitation. Due to the sample spinning, the quadrupolar splitting of many crystallites will pass through the v n value. It was shown that mismatches between and the powder i/qS do not create large phase distortions and simultaneous adiabatic and direct coherence transfer processes result in relatively pure MQ SQ transfers. By pure we mean that no significant phase dispersions are observed when looking at the transfer of each crystallite separately.

A very effective signal sensitivity enhancement scheme for MQMAS as well as for MAS is the DFS introduced by the group of Kentgens.The underlying principles of DFS and FAM are similar and they have been exphcitly dealt with already. In a nutshell, a cosine amplitude modulated RF carrier wave irradiates the sample at two frequencies, namely, ioq -F w, and aio — Larmor frequency of the spins and is the frequency of modulation. This means that if satellite transitions of a static spins-1 system will be simultaneously... 

As for ID data, fi quadrature detection requires two data sets to be collected which differ in phase by 90 , thus providing the necessary sine and cosine amplitude-modulated data. Since the fi dimension is generated artificially, there is strictly no reference rf to define signal phases so it is the phase of the pulses that bracket ti that dictate the phase of the detected signal. Thus, for each tj increment two data sets are collected, one with a 90 preparation pulse (ti sine modulation), the other with 90j, (ti cosine modulation), and both stored separately (Fig. 5.17). These two sets are then equivalent to the... 

ESEEM is a pulsed EPR technique which is complementary to both conventional EPR and ENDOR spectroscopy(74.75). In the ESEEM experiment, one selects a field (effective g value) in the EPR spectrum and through a sequence of microwave pulses generates a spin echo whose intensity is monitored as a function of the delay time between the pulses. This resulting echo envelope decay pattern is amplitude modulated due to the magnetic interaction of nuclear spins that are coupled to the electron spin. Cosine Fourier transformation of this envelope yields an ENDOR-like spectrum from which nuclear hyperfine and quadrupole splittings can be determined. 

Equation 10.5 indicates how amplitude modulation may be converted to phase modulation, which provides considerable flexibility in data processing. In order to obtain both sine and cosine components of amplitude modulation, we usually... 

Whereas the on-resonance nutation spectra are amplitude-modulated and can be properly phased in the FI dimension, the off-resonance nutation spectra are phase-modulated and can no longer be phased properly. Therefore, a magnitude calculation is performed after Fourier transformation. Furthermore, because the magnetisation evolves during t around the effective field, the sine and cosine components of the modulation are different and the amplitudes of positive and negative FI signals are unequal. 

Fig. 11. (a) Cosine modulation and amplitude-modulated rectangular pulses approximating the... 

Symmetrically shifted pulses have been proposed as a means of solvent suppression. Symmetrically shifted pulses are symmetrically shifted laminar pulses that contain equal numbers of rectangular pulse components of the same phase at an offset frequency. The basis of the symmetrically shifted pulse family is the SS pulse which is conceptually equivalent to applying simultaneous ir/2 rectangular pulses with two separate, but in-phase, transmitters at offset frequency from the water. On a practical basis an SS pulse is obtained by a complete Itt cosine modulation of a single transmitter (see Fig. 15). An S pulse is half of an SS pulse (i.e. a half-cycle tt pulse) which results in a narrower null and a 180° phase inversion at the transmitter frequency. They are also the soft, continuous equivalent of binomial sequences. The SS and S pulses have broader excitation maxima than the sinusoidal profile of the JR sequence. The method has maximal excitation at an offeet frequency of second-order U-shaped water suppression. The exdtation profile is related to the maximum amplitude modulation and can be determined by numerical evaluation of the Bloch equations. Hence a new pulse shape must be used for each excitation window. The SS pulses give better water suppression than the JR sequence, but at the expense of poorer excitation of resonances closer to the water. Also, there is no phase inversion at zero frequency. The S pulse gives better excitation near the water frequency but with less water suppression. 

Prior to the advent of the above methods that allowed the presentation of phase-sensitive displays, 2D data sets were collected that were phase-modulated as a function of ti rather than amplitude-modulated. Phase-modulation arises when the sine and cosine modulated data sets collected for each ti increment are combined (added or subtracted) by the steps of the phase cycle, meaning each FID per tj increment contains a mixture of both parts. Here it is the sense of phase precession that allows the differentiation of positive and negative frequencies. This method is inferior to the phase-sensitive approach because of the unavoidable mixing of absorptive and dispersive lineshapes, so is generally only suitable for routine, low-resolution work. 

The second type is amplitude modulation, in which the evolution in /, is encoded as an amplitude, i.e. mathematically as sine or cosine... 

Generally, two-dimensional experiments produce amplitude modulation, indeed all of the experiments analysed in this chapter have produced either sine or cosine modulated data. Therefore most two-dimensional spectra are fundamentally not frequency discriminated in the F1 dimension. As explained above for one-dimensional spectra, the resulting confusion in the spectrum is not acceptable and steps have to be taken to introduce frequency discrimination. 

A data set from an experiment to which TPPI has been applied is simply amplitude modulated in tx and so can be processed according to the method described for cosine modulated data so as to obtain absorption mode lineshapes. As the spectrum is symmetrical about Fx = 0 it is usual to use a modified Fourier transform routine which saves effort and space by only calculating the positive frequency part of the spectrum. 

This signal is said to be amplitude modulated in t, it is so called because the evolution during gives rise, via the cosine term, to a modulation of the amplitude of the observed signal. 

With amplitude modulation the cosine and sine components may be handled in two ways to achieve quadrature detection in fl. They may be acquired in subsequent scans by either incrementing the pulse or receiver phase and the data co-added in the computer memory or they may be acquired sequentially and stored separately. With the first approach direct Fourier transformation yields frequency discrimination in fl but no absorptive lineshapes whilst with the second approach additional processing steps are necessary to achieve both, frequency discrimination and absorptive lineshapes. 

Rigorously, the above Floquet representation is valid only when the field is periodic. Imagine now that the field amplitude Eo in Eq. (12) carries a time dependence that denotes a slow modulation of the cosine field which oscillates with a frequency in the UV-Vis spectral range. Without the amplitude modulation, the dynamics under the UV-Vis field is well captured by the Floquet representation. If the amplitude modulation is slow. 

Fig. 1-11 shows five different reference patterns. Fig. 1-11 (a) is a ring pattern with phase modulation of exp[j2ncos(y/A)]. Fig. 1-11 (b) is a ring pattern with amplitude modulation of cos(y/A). Where A/2n is the period of the cosine function and is equal to 273.6pm. Fig. 1-11... 

DQPSK 3MQAM 4MQAM 12PM3 k/4. shift DQPSK with a = 0.35 raised cosine filtering Class 3 Quadrature Amplitude Modulation Class 4 Quadrature Amplitude Modulation 12 state PM with 3 bit correlation IS-54 TDMA voice and data... 

The zero crossing is independent of the amplitude of the cosine, hence effects of drift of Pin and of (varying) modulation depth M have been completely eliminated. 

The essential properties of incommensurate modulated structures can be studied within a simple one-dimensional model, the well-known Frenkel-Kontorova model . The competing interactions between the substrate potential and the lateral adatom interactions are modeled by a chain of adatoms, coupled with harmonic springs of force constant K, placed in a cosine substrate potential of amplitude V and periodicity b (see Fig. 27). The microscopic energy of this model is ... 

A double HS pulse with symmetrically placed offsets about the transmitter frequency (Acoo = 0) can be created by cosine modulation of the pulse amplitude ... 

Fig. 2.50. Generation of /-resolved two-dimensional 13C NMR spectra (a) flux diagram (b) /-modulation of Cl l doublets, CH2 triplets and CH3 quartets during evolution, vector diagrams in the x y plane and cosine curves described by the signal maxima (c) series of 13C NMR spectra of CHn groups with t1 dependent /-modulation of the signal amplitudes (d) series of /-resolved two-dimensional 13C NMR spectra formethine, methylene and methyl carbon atoms, stacked plots and contour plots (e).

Assuming that the rate of change of the optical retardation introduced by the interferometer is the same for all the input radiation frequencies (which is normally the case), each individual value of v in the broadband source output contributes a different value of /m in the a.c. component of the detector output (see Figure 3.20). These contributions are, of course, summed by the detector. The combined detector output (see Figure 3.21) arising from the simultaneous measurement of all modulated input signals, is a sum of cosine functions. Such a sum is a Fourier series, and the amplitudes of the individual... 

Signal separation in NMR spectra is not restricted to NMR parameters. Other physical properties can be used to create indirect spectral dimensions. He et al. reported 3D EP COSY experiment which utilises electrophoretic migration rates to separate resonances from individual components in mixed solutions. Additional modulation of COSY resonances is introduced in the experiment by a stepwise increase of the electric field. Due to the presence of gradient pulses in the sequence the amplitude of the resonances has a cosine dependence on the field strength with the period depending on the electrophoretic mobility. In the processed spectra, COSY spectra of individual components are observed in separate planes. 

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