Linear frequency

The skeletal LN procedure is a dual timestep scheme, At, Atm, of two practical tasks (a) constructing the Hessian H in system (17) every Atm interval, and (b) solving system (17), where R is given by eq. (3), at the timestep At by procedure (23) outlined for LIN above. When a force-splitting procedure is also applied to LN, a value At > Atm is used to update the slow forces less often than the linearized model. A suitable frequency for the linearization is 1-3 fs (the smaller value is used for water systems), and the appropriate inner timestep is 0.5 fs, as in LIN. This inner timestep parallels the update frequency of the fast motions in force splitting approaches, and the linearization frequency Atm) is analogous to the medium timestep used in such three-class schemes (see below). 

Keywords continuous wave radar linear frequency modulation noise radar noise ... 

Linear Frequency Modulated Continuous Wave Radar... 

Having defined the utility of a waveform library we go on to investigate the utilities of a few libraries. Specifically, we consider libraries generated from a fixed waveform 4>o, usually an unmodulated pulse of some fixed duration, by symplectic transformations. Such transformations form a group of unitary transformations on L2(R) and include linear frequency modulation as well as the Fractional Fourier transform (FrFT) in a sense that we shall make clear. 

This section describes a new waveform design for automotive applications based on CW transmit signals which lead to an extremely short measurement time. The basic idea is a combination of linear frequency modulation (LFM) and FSK CW waveforms in an intertwined technique. Unambiguous range and velocity measurement with high resolution and accuracy can be required in this case even in multi-target situations. After introductions to FSK and LFM waveform design techniques in sections 2.1 and 2.2 the combined and intertwined waveform is described in detail in section 2.3. 

As with any other homonuclear decoupling, the transverse magnetization of the 13C , for example in the evolution time, is perturbed by the decoupling pulse, resulting in an additional precession virtually around the z axis. Since the adiabatic decoupling is applied during the entire evolution time of the 13C , a non-linear frequency shift rather than a phase shift appears in the spectrum. This is termed the Bloch-Siegert (frequency) shift in the NMR literature in honour of their discovery of the phenomenon. 

There are two common ways of expressing a frequency, namely the angular frequency co in radians per second, and the linear frequency / (with the SI unit of hertz (Hz)) in cycles per second. Note that / is often cited as cycles per second (c s ), which is another way of saying Hz. 

We will now look at the AC behaviour of these pure electrical components as a function of the frequency (o. (Remember from Chapter 7 that a linear frequency of / cycles per second corresponds to an angular frequency of 2nco radians per second.)... 

The nondimensionalized frequencies are related to linear and angular frequencies by equation 3.36. The conversion factor from linear frequencies in cm to undimension-alized frequencies is chik = 1.4387864 cm (where c is the speed of hght in vacuum). Acoustic branches for the various phases of interest may be derived from acoustic velocities through the guidelines outlined by Kieffer (1980). Vibrational modes at higher frequency may be derived by infrared (IR) and Raman spectra. Note incidentally that the tabulated values of the dispersed sine function in Kieffer (1979c) are 3 times the real ones (i.e., the listed values must be divided by 3 to obtain the appropriate value for each acoustic branch see also Kieffer, 1985). 

Fig. 9.16. Measured double piezoeiectrie response of a tube scanner. A linear frequency respon.se is observed. The responses of the two x quadrants are off by about 40%. The responses of the two y quadrants are off by about 16%. (Reproduced from Chen, 1992a, with permission.)...

As a consequence of the spectral modulation, the temporal width of the laser pulse is modified and a linear frequency sweep (chirp) is introduced. The modulated pulse duration is... 

Each frequency track has a birth and death index and dt such that bt denotes the first DFT block at which f0i is present ( active ) and d, the last (each track is then continuously active between these indices). Frequencies are expressed on a log-freuency scale, as this leads to linear estimates of the pitch curve (see [Godsill, 1993] for comparison with a linear-frequency scale formulation). The model equation for the measured log-frequency tracks fa[n is then ... 

Frequently, a flat power spectrum is not the best choice for a given problem, and it might be of advantage to focus the excitation energy into certain frequency ranges of interest. One example is a system with fast and slow relaxation processes of comparable amplitude and a large time scale separation of several decades. In such cases, a white power spectrum with a constant power density is inferior to a power spectrum, where the discrete frequencies are evenly spaced on a logarithmic instead of a linear frequency scale. 

In general, the quantities being determined by microwave measurements are complex reflection and transmission coefficients or complex impedances normalized to the impedances of the transmission lines connecting a network analyser and the device-under-test (dut). In addition to linear frequency domain measurements by means of a network analyser the determination of possible non-linear device (and thus material) properties requires more advanced measure-... 

Fig. 7.11. Four excitation-pulse waveforms achieving the highest ion yield ratio of ni29/m3i. Waveforms in a and b were obtained by the self-learning adaptive pulse control. The waveform in c was obtained by applying a linear frequency chirp at 2.2 x 10 2 ps2 by the 4/-pulse shaper. The waveform in d was obtained by applying a linear frequency chirp at 2.2 x 10 2 ps2 by adjusting the pulse compressor...

Precursors useful in the surface sol-gel process are not restricted to alkoxides. The requirements as precursors are chemisorption on surface hydroxyl groups and regeneration of the hydroxyl groups after hydrolysis. For example, TiO(acac)2 repeatedly adsorbs, when acid and alkali are added to the adsorption and hydrolysis media, respectively. Except for the case of Nb(0"Bu)5, all the compounds listed in Table 6.1 show linear frequency shifts. Adsorption conditions such as concentration, temperature, and immersion time are dependent on the solubility, reactivity, and the ease of hydrolysis of alkoxides. These conditions are varied as the structure of alkoxide units is changed. For example, Ti(01Pr)4, which exists as a monomer in solution [17], requires conditions different from Ti(0"Bu)4, which tends to form oligomer species. 

Here, z is a complex harmonic variable, r is the sampling time, whereas >( > = 27t v) and v are the complex angular and linear frequency, respectively. The sum in Eq. (317) is also known as the finite-rank Green function. In the current and the next section, the fast Pad6 transform for F(z) is represented by the two equivalent variants, FPT( ), via F(z) G (z 1), with G (z 1) being the unique polynomial quotients in their respective variables z 1 ... 

The angular Larmor frequency, in units of radians per second, can be transformed into linear frequency v (in reciprocal seconds or hertz) by division by 2k ... 

Frequency (v or co) The number of waveperiods per unit time. The linear frequency, V, is the number of cycles per unit time. The SI unit is Hz = s . For the angular frequency, the symbol (a (= 27iv) is used, with rad s as the SI unit. 

A phase difference between the carrier frequency and the pulse leads to a phase shift which is almost the same for all resonance frequencies (u)). This effect is compensated for by the so-called zero-order phase correction, which produces a linear combination of the real and imaginary parts in the above equation with p = po- The finite length of the excitation pulse and the unavoidable delay before the start of the acquisition (dead time delay) leads to a phase error varying linearly with frequency. This effect can be compensated for by the frequency-dependent, first-order phase correction p = Po + Pi((o - (Oo), where the factor p is frequency dependent. Electronic filters may also lead to phase errors which are also almost linearly frequency-dependent. 

The precessional motion of the magnetic moment around Bq occurs with angular frequency wq, called the Larmorfrequency, whose units are radians per second (rad s ). As Bq increases, so does the angular frequency that is, coq cx Bq, as is demonstrated in Appendix 1. The constant of proportionality between o>o and Bq is the gyromagnetic ratio 7, so that wq = Bq. The natural precession frequency can be expressed as linear frequency in Planck s relationship AE = Hvq or angular frequency in Planck s relationship AE = h(x)Q (coq = 2 rrvo). In this way, the energy difference between the spin states is related to the Larmor frequency by the formula... 

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