Exponentially damped sinusoid

Although most implementations of VARPRO have concentrated on using exponentially damped sinusoids for the basis functions (i.e. Lorentzian lineshapes), there is no reason why more realistic models (e.g. Gaussian or Voigt32) should not be used. 

These radiation-induced resistance oscillations can be empirically described as exponentially damped sinusoidal waves ini , i.e., -A exp (-A/B) sin(27T Bf/B), in... 

In addition to GFT and BPR, a host of other methods can be applied to radially-sampled data like BPR, these methods reconstmet the fully-dimensional spectmm. Zhou and colleagues employed radial FT [28] to process data collected along concentric rings in fi/f2 [29]. MLM methods that fit a model (typically consisting of a sum of exponentially-damped sinusoids) can also be used to analyze radially sampled data, as can regularization methods that do not model the signal (e.g., li-norm, MaxEnt). 

A sine wave whose amplitude decays exponentially with time. The analog signal induced in the receiver coil of the NMR instrument will consist of one or more exponentially damped sinusoids. 

Kasemo s laboratory at Chalmers University of Technology and Goteborg University was the source of the QCM-D technique [10], now embodied in instrumentation offered by Q-Sense AB. They define the dissipation, A as the inverse of the quality factor Q of the quartz crystal resonator [47] (see equation (4)). In the Q-Sense instrumentation, the driving RF power to the oscillator, causing it to respond at resonant frequency f, is switched off and the exponentially damped sinusoidal wave decays with a time constant r, where D -. .More than 100... 

The change in D can be obtained by measuring the impedance spectroscopy [9] or by fitting the oscillation decay [14], In the former method, a broader resonance peak is indicative of a larger dissipation factor. In the latter case, the measurement of D is based on the fact that the amplitude of oscillation (A) or the output voltage over the crystal decays as an exponentially damped sinusoid when the driving power of the piezoelectric oscillator is switched off at t = 0 (Fig. 1.2) [16] ... 

Theoretically, it is not relevant, whether the quantification is performed in the frequency or in the time domain, since Fourier transform is a linear operator. The Fourier transform of a sinusoidal, exponentially damped signal in time... 

This may not be too clear. The full sinusoidal source can be exponentially damped, have a DC offset, and have a time delay as well as a phase delay. In the above equation, the phase ( ) is specified in degrees and is converted into radians by the constant 2jt/360. We note that for Td > t > 0, V, is constant. The sinusoid does not start until t = tphase delays, the above equation reduces to the exponentially damped sine wave ... 

As a general rule, the larger the losses, the larger are the discrepancies between these two solutions to the free vibration problem. However, it is possible to have greater discrepancies even for a low loss material due to changes in the damped sinusoidal term. For a more complex linear viscoelastic material consisting of a finite number of elementary viscoelastic elements, the solution would include a sum of decreasing exponential terms and damped sinusoidal waves. 

Since for P0 > 0, these roots have negative real parts, this singular point is a stable focus, and the steady state values given by Equation 40 are approached either by a damped sinusoid or an exponential (63). Note that for P0 — 0, the classical case, the roots are purely imaginary, and the oscillation persists indefinitely. 

This represents a damped sinusoidal wave, as shown in Fig. 8.3. For stronger damping, such that y>coQ, x(t) decreases exponentially with no oscillation. 

For the underdamped ease the eomplementary solution eonsist of damped sinusoidal terms and for the overdamped ease it eonsists of two real exponential terms. In either ease two equations ean be formulated for the boundary eonditions as ... 

The complementary solution consists of oscillating sinusoidal terms multiplied by an exponential. Thus the solution is oscillatory or underdamped for ( < 1. Note that as long as the damping coefficient is positive (C > 0), the exponential term will decay to zero as time goes to infinity. Therefore the amplitude of the oscillations will decrease to zero. This is sketched in Fig. 6.6. 

Equation (17.118) suggests that the viscoelastic response of the material is the sum of a sinusoidal wave delayed with respect to the input wave and a damped exponential. This last component corresponds to the simple three-element standard model governed by a single relaxation time. For a distribution of relaxation times, the response obtained would include a sum of such exponentials. 

It should be stressed that the form of the response is not altered if = 0, that is, if the material that surrounds the vibrating slab is viscoelastic. The obtained solution consists of a decreasing exponential and a damped out-of-phase sinusoidal wave. 

Predictably, most systems that produce sound are more complex than the ideal mass/spring/damper system. And of course, most sounds are more complex than a simple damped exponential sinusoid. Mathematical expressions of the physical forces (thus the accelerations) can be written for nearly any system, but solving such equations is often difftcult or impossible. Some systems have simple enough properties and geometries to allow an exact solution to be written out for their vibrational behavior. A string under tension is one such system, and it is evaluated in great detail in Chapter 12 and Appendix A. For... 

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