Known waveform

If the detector noise originates from stationary, Gaussian random processes and the instrument is calibrated with zero-offset, this matched Alter output is a zero-mean Gaussian random variable with variance equal to the power signal-to-noise ratio of the known signal  

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While the BSR was originally considered as a means of obtaining the areal extent of hydrate, a classic article by Lee et al. (1993) provided a method to determine the amount of hydrate, assuming that the porosity of the sediment is known. Waveform inversion techniques proposed by Minshull and coworkers... 

We have only been able to generate the square waveform because we knew the appropriate values for au and k for every harmonic. The converse problem, of finding unknown and (j)fc for a known waveform function x t) is termed Fourier analysis and will be described in section 10.1.4. 

Many seismic events have very similar waveforms if they originate from the same or nearby sites. This could also be the case for volcanic events even though they might not be from exactly the same area. A known waveform shape can then be used to detect unknown arrivals by cross correlating the known wavelet with the signal of unknown events. An example of similar events is seen in Fig. 13. Despite the large distances, the signal shapes are nearly identical for the two events. 

Listing 7.4. Some examples of Fourier Analysis with known waveforms. 

We first supposed that the field radiated into the piece by the transducer is known, thanks to the Champ-Sons model. Then, the main approximation used consists in making far field assumptions in the beam defect interaction area. In the case of a focused transducer we assume that the incident wavefronts on the defect are plane. This is equivalent to say that the defect is located in or near the transducer focal area and that a defect located outside this zone does not cause a significant echo. In the case of planar contact transducer, the incident wavefronts on the defect are assumed to be spherical The incident field on the defect is therefore approximated by the product of a spatial function gfp,0,z)describing the amplitude distribution in the beam and a time-delayed waveform < ) ft) representing the plane or spherical propagation in the beam. The incident field on the defect can therefore be approximated for ... 

This crude approximation gives a rather simple result but is probably not accurate enough for most purposes. However, it turns out that the quantities on the right side of equation 10 can be obtained from a more careful analysis of the interferometer waveform. And once these quantities are known, n2 can then be found directly from them without recourse to equations 11 or 12. 

Let us now suppose that the waveform of figure 16.3 is applied to study the reversible oxidation of a species R to R in a given solvent. The reaction occurs at the working electrode (anode), and /i°(R/R ) is the standard potential of the R/R- couple. Because the standard potential of the reference electrode in our cell is known accurately relative to the standard potential of the SHE (E° = 0 by definition), we can write the cell reaction and the Nernst equation as... 

As an alternative to a stepwise variation with time, a continuously changing potential may be imposed. Though other possibilities have been used [42, 43], a linearly changing potential—time waveform, known as a potential ramp [Fig. 17(a)], is the most common. The technique has many names, including linear sweep voltammetry [44]. If the direction of the ramp is reversed [Fig. 17(b)], the technique is often termed cyclic voltammetry (see Chap. 3), though this name is more appropriately applied after sufficient ramp reversals [Fig. 17(c)] have caused the experiment to become periodic. 

The constant current may be reversed in direction at, or before, the transition time, as shown in Fig. 19(a), or repeatedly reversed, thus becoming periodic [Fig. 19(b)]. Less frequently employed is a current waveform which varies as a known function of time [66—69], such as the linear current ramp in Fig 19(c). 

Figure 27.16B shows the method known as pulsed coulometric detection (PCD). In this case, the current is integrated over a longer period and the time period is an integral number of 16.7-ms segments with typical total integration times of greater than 200 ms. The use of this type of waveform eliminates the most common electrical interference (60 Hz sinusoidal) encountered in pulsed electrochemical detection, and thereby increases the detection limits for most compounds. 

If the time warping function fw() is known then it is possible to regenerate the undistorted waveform s(t) as... 

The Fourier expression is an infinite series. In this equation, V, represents the constant or the DC component of the waveform. Vu V2, V3,..., Vn are the peak values of the successive terms of the expression. The terms are known as the harmonics of the periodic waveform. The fundamental (or first harmonic) frequency has a... 

Under this definition, the value of IHDX is always 100%. This method of quantifying the harmonics is known as harmonic distortion based on the fundamental. This is the convention used by the Institute of Electrical and Electronic Engineers (IEEE) in the U.S. The European International Electrotechnical Commission (IEC) quantifies harmonics based on the total RMS value of the waveform. Using the same example shown above, the RMS value of the waveform is ... 

To determine whether alternative ANN architectures can lead to improved resolution and successful agent detection, Radial Basis Function (RBF) networks [106] were considered for the same problem. RBFs are networks with one hidden layer associated with a specific, analytically known function. Each hidden layer node corresponds to a numerical evaluation of the chosen function at a set of parameters Gaussian waveforms are often the functions of choice in RBFs. The outputs of the nodes are multiplied by weights, summed, and added to a linear combination of the inputs, yielding the network outputs. The unknown parameters (multiplicative weights, means and spreads for the Gaussians, and coefficients for the linear combination of the inputs) are determined by training the RBF network to produce desired outputs for specific inputs. 

Linear sweep voltammetry (LSV), also known as linear sweep chronoamperometry, is a potential sweep method where the applied potential (E) is ramped in a linear fashion while measming cnrrent (i). LSV is the simplest technique that uses this waveform. The potential range that is scanned begins at an initial or start potential and ends at a final potential. It is best to start the scan at rest potential, the potential of zero current. For a reversible couple, the peak potential can be calcnlated nsing equation (6). ... 

To improve chemical selectivity, a triangular input waveform can be used that separates compounds into resolvable peaks. This form—cyclic voltammetry—can be carried out with high temporal resolution using high scan rates to allow the waveform to be completed in a short time. In fast-scan cyclic voltammetry (also known as fast cyclic voltammetry), waveforms last around 10 ms, and measurements are typically made every 10-200 ms. 

Some points are noteworthy. According to Fourier, formally, the series must be summed over all integral frequencies from —oo to +00 to be mathematically exact. In practice of course, this is never possible. As the number of terms increases, however, as higher frequency terms are included, the approximation to the exact resultant wave function becomes more nearly correct. As shown in Figure 4.9, it often doesn t require all that many terms before a quite acceptable result is obtained. The difference between the exact waveform and the one we obtain from summing a limited series of Fourier terms is known as series termination error. As illustrated by the two-dimensional case in Figure 4.11, the phases of the component waves in the synthesis play a crucial role in determining the form of the resultant wave. 

Finally, one could ask just how accurate the amplitudes and phases of the individual wave components have to be before the synthesis accurately resembles the true waveform. Well, infinity is a lot of terms. With that many terms, every one doesn t have to be right on. In fact an individual term only has to be sort of close, but if enough of these rather poorly determined components are available, the final result still looks pretty good. Fortunately X-ray crystallography provides a lot of these terms, more than enough. This is known in the church as God s blessing on X-ray crystallographers. ... 

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