Estimating Harmonic Amplitudes

When you are on the bench, you don t want lengthy calculations interfering with your insight or thinking process. So it is always valuable to have some quick rules of thumb at your disposal (provided of course they are reliable ). 

In that spirit, we now detail a quick procedure for estimating harmonic amplitudes of the square current waveform (assuming fast transitions). We will apply it to our specific lOOkHz/lA example as we go along  

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An interesting aspect of these non-linear oscillations is that they can be put to use to estimate the amplitude of the acoustic pressure both in-vivo and in-vitro. In this paper, theoretical aspects of this non-Unear resonance are first discussed. Then, experimental results showing the second harmonic using microbubbles created in the laboratory are presented. Finally, a roadmap is drawn which could eventually lead to in-vivo calibration of ultrasoimd waves intensity. 

In this work, Equ. 2 and Equ. 3 have been used in order to estimate the amplitude of the first two harmonics. 

Because of the observed asymmetry in the anti-peak, Eqs. (13) and (14) were applied over a region corresponding to 2o

determined values of AH, torsional potential was estimated including V[-, V2-, V3-terms in the Fourier series expansion. The resulting... 

Several techniques have been proposed for modifying the shape of the short-time spectrum so as to emphasize those portions deemed to be important for speech perception or to reduce the amplitude of those portions assumed to be noise. One approach is adaptive comb filtering[Lim et al., 1978], In this method, the fundamental frequency of voiced speech sounds is estimated, and a comb fdter is then constructed to pass signal power in the regions of the pitch harmonics and to suppress power in the valleys in between. Experimental results with normal-hearing subjects, however, have shown no significant improvement in intelligibility with this type of system [Perlmutter et al., 1977][Lim et al., 1978],... 

An advantage of having an analytic expression, such as Equation 14, for the potential well is that it can be used to estimate the vibrational frequency of the adsorbed molecule with respect to the surface. For vibrations of small amplitudes the system behaves as a harmonic oscillator for which the frequency is given by... 

With an estimated Tp =0.4K and rric = (4.1 0.1) me the harmonic ratio for Ml/M2 is 130. Therefore, the maximal amplitude of the second harmonic would be smaller by this factor compared to the maximal fundamental amplitude. This, however, would mean that M2 and of course also all higher harmonics are always below the noise level for the present crystal and experimental sensitivity. The fact that harmonics up to the eighth order are visible at the angular range where AF = 0 shows the invalidity of (3.6) for these angles where only one FS area like in a perfect 2D material exists. 

Sinusoidal models use a harmonie model and decompose each frame into a set of harmonics of an estimated fundamental frequency. The model parameters are the amplitudes and phases of the harmonics. With these, the value of the fundamental can be changed while keeping the same basic spectral envelope. 

For weakly nonlinear systems, the contributions of the higher harmonics and higher FRFs decrease with the increase of their order. Different harmonics of the output can be estimated directly by harmonic analysis of the output signal. On the other hand, as can be seen from Equation (11.13) to Equation (11.16), the first-order function Gi(cd) corresponds to the dominant term of the first harmonic, the second-order functions G2(dominant terms of the second harmonic and the DC component, respectively, the third-order function Gs(ft), ft), to) to the dominant term of the third harmonic, etc. This fact enables estimation of different FRFs from the harmonics of the output obtained for different input amplitudes, as shown by Lee [53]. 

In large plasmas the lower harmonics of the cyclotron frequency are reabsorbed before they escape, and the synchrotron losses are not as large as in Eq. (45). For harmonics lower than a critical harmonic m, the plasma radiates like a black body from its surface higher harmonics have lower amplitude but are not reabsorbed. An early estimate by Dawson for a p-B ... 

High-precision ND analysis allows the vibrations of the atom to be studied more accurately than with X-rays (see p. 224) and this, incidentally, requires small corrections ( 0.003 A) to be added to the bond lengths above. Apart from the rigid-body vibrations of the molecule as a whole, the analysis is accurate enough to allow assessment of the intramolecular torsional oscillation of the CHj and NH3+ groups. The former mode has a r.m.s. amplitude of 9.5°, the latter 5.8°. Using a harmonic approximation and assuming a threefold cosinusoidal barrier, the authors were able to estimate... 

Estimate the classical vibrational amplitude (the distance between the classical turning points) for the zero-point vibrations of H2 and of N2 using the experimental values of and the harmonic approximation. 

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