Instantaneous frequencies

I will be using a seeond order, eommon-mode filter. The diffieulty in eonsider-ing an input eondueted EMI for this power faetor eorreetion eireuit is its variable frequency of operation. The lowest instantaneous frequency of operation occurs at the crests of the sinusoid voltage waveform. This is where the core requires the longest time to completely discharge the core. The estimated frequency of operation has been 50 kHz, so I will use this as an assumed minimum frequency. 

Figure 12. Bottom panel Theoretical distributions of instantaneous frequencies for the uncoupled (F (co)) and coupled (Pc(to)) chromophores. Top panel Inverse participation ratios R(co) and Rm(w). Both panels are for H20 at room temperature.

To investigate the relationship between the reaction driven v7 mode and the subsequent protein motions along the dissociative pathway, further modulations of the frequency of the v7 mode by the surrounding intramolecular and protein bath fluctuations were found using an instantaneous frequency (IF) analysis. The IF was derived from the data by applying a Gaussian filter around the v7 mode in the Fourier spectrum. An inverse Fourier transform produced the time trace TT(t) given by ... 

Fig. 6.2. Illustration of the frequency-time dependences of pump and Stokes pulses in three different CRS excitation pulse schemes and their corresponding spectral resolution of Raman shifts. A Using a pair of transform-limited femtosecond pulses of broad spectral and narrow temporal widths results in a broad bandwidth of Raman shifts that exceeds the line width of a single Raman resonance. B Using transform-limited picosecond pulses of broad temporal and narrow spectral width readily provides high spectral resolution matching the Raman resonance line width to be probed. Selection of a Raman resonance shifted by AQr is achieved by tuning the frequency of one of the laser beams by the same amount. C Spectral focusing of a pair of identically linear chirped pump and Stokes femtosecond pulses results in a narrow instantaneous frequency difference in the CRS process, thus also providing narrow-bandwidth CRS excitation. Selection of a Raman resonance shifted by AQr is achieved by adjusting the time delay At between the pulses. Shifted pulses in (B) and (C) are depicted hatched...

Fourier channel k. The phase is known up to a multiple of 27t(since only exp /( ),(/")) is known). Time-scale modifications also require the knowledge of the instantaneous frequency G) (f ). 0), Uua) can also be estimated from successive short-time Fourier transforms for a given value of k, computing the backward difference of the short-time Fourier transform phase yields... 

The duration T of the analysis window must be small enough so the amplitudes and instantaneous frequencies of the sinusoids can be considered constant within... 

A similar analysis can be made for quasi-periodic signals which consist of a sum of sine waves with slowly-varying amplitude and instantaneous frequency each of which is assumed to pass through a single filter. 

The phase vocoder has been useful in a number of applications4. In time-scale modification, for example, the goal is to maintain the perceptual quality of the original signal while changing its apparent rate of articulation . In performing time-scale modification with the phase vocoder, the instantaneous frequency and amplitude of each channel are interpolated or decimated to a new time scale . In one scenario, the phase of each filter output in Equation (9.9) is first unwrapped, and the channel amplitude and unwrapped phase are then time scaled. With time-scale modification by a factor p, the modified filter output is given by... 

Time-Scale Modification. In time-scale modification, the magnitude, frequency, and phase of the sine-wave components are modified to expand the time scale of a signal without changing its frequency characteristic. Consider a time-scale modification by a factor p. By time-warping the sine-wave frequency tracks, i.e., C0/(pt) = 0/(pf), the instantaneous frequency locations are preserved while modifying their rate of change in time [Quatieri and McAulay, 1986], Since d/dt[Bi(tP)/p] = C0 (P/), this modification can be represented by... 

The maximum instantaneous frequency deviation A (Umax is therefore given by A (i)max I(i>m. When the modulation index/is nonzero, side frequencies occur above and below the carrier 00c, and the number of side frequencies increases with increasing I. 

Now we will assume that the analysis window h(n) is sufficiently short so that the instantaneous frequencies and amplitudes of the sinusoids can be assumed constant over the duration of h. As a result, we have... 

Compute the short-time Fourier transform at next analysis time-instant +1 and calculate the instantaneous frequency in each channel according to Eq. (7.14). 

Note that if an analysis time-instant t" is not an integer, it can be rounded to the nearest integer prior to the calculation of the instantaneous frequency, provided that the corrected value of R(u - 1) = / - t"A is used in Eq. (7.14). It is easy to show that for a constant-amplitude, constant-frequency sinusoid, the procedure above outputs a perfect time-modified sinusoid2 provided... 

Figure 12.2c shows the temporal variation of the instantaneous frequencies for the two modes. It is interesting to observe how the frequency of the fast mode is modulated in a fairly regular manner. With about 17 modulation cycles for fjast during the 500 s of observation time, we conclude that the frequency of the fast mode is modulated by the presence of the slow mode, indicating that the two modes interact with one another. If one compares the phase of the tubular pressure variations in Fig. 12.2a with the phase of the frequency modulation in Fig. 12.2c it appears that the maximum of ffast occurs about 60° after the maximum of Pt. It is important to note, however, that the various steps of our wavelet analysis may have introduced a certain phase lag. We are presently trying to correct for such effects in order to obtain a better understanding of the instantaneous relation between the two variables. 

The t scale disappeared with the development of frequency-scan instruments and of the pulsed FT mode, which is essentially an instantaneous frequency scan. The terms upfield and downfield are now obsolete and have been replaced, respectively, by shielded (lower S, or to the right) and deshielded (higher S, or to the left). 

At the resonance w(t) = A(x), the adiabatic potentials i.e. the eigenvalues of (5.9) show avoided crossing and the population splits into the two adiabatic Floquet states. In the case of quadratically chirped pulses, the instantaneous frequency meets the resonance condition twice and near-complete excitation can be achieved due to the constructive interference. The nonadi-abatic transition matrix Ujj for the two-level problem of (5.9) is given by the ZN theory [33] as... 

An ideal FM laser is a laser which produces an output of constant amplitude but whose instantaneous frequency is sinusoidally modulated about a central carrier frequency. Thus the electric field can be described as... 

An elegant way to get 0-H vibrational frequencies from short ab initio MD trajectories is based on the monitoring ofroH(t), obtaining the instantaneous vibrational frequencies from the time delay, r, between two consecutive maxima as w = 27r/r. A vibrational density of states can be constmcted from the histogram of the instantaneous frequencies, giving some pieces of information about the expected width of the corresponding IR band. The OH frequency, about 3550 cm , obtained by this procedure from Car-Parrinello PAW trajectories for the sodalite framework is very close to the experimentally expected value. 

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