Waves That Are Continuous With No Breaks Represent a N Signal Digital

Basic applied mathematics

BARRY J ELLIOTT , in Cable Engineering for Local Area Networks, 2000

2.5 Sine waves and phase

A sine wave or sinusoidal wave is the most natural representation of how many things in nature change state. A sine wave shows how the amplitude of a variable changes with time. The variable could be audible sound for example. A single pure note is a sine wave, although it would sound a very plain and flat note indeed with none of the harmonics we normally hear in nature. A straightforward oscillating or alternating current or voltage within a wire can also be represented by a sine wave. The number of times the sine wave goes through a complete cycle in the space of 1 second is called the frequency. Indeed the unit used to be cycles per second, but now the unit of measurement is hertz (Hz). A frequency of 1000Hz, or 1 kHz, means that the sine wave goes through 1000 complete cycles in 1 s. If we are considering audible sound waves then the human ear has a frequency range of about 20Hz-20kHz. The electrical mains frequency in Europe is 50Hz and 60Hz in America. Figure 2.1 shows a sine wave.

Fig. 2.1. A sine wave.

The sine of any angle can vary from −1 to +1. For example the sine of 0° is 0 and the sine of 90° is 1. The sine of 270° is −1 and when we get to 360° we are back to zero again. A cosine is 90° out of phase with a sine wave as we can see in Fig. 2.2.

Fig. 2.2. A sine and cosine wave.

The cosine of 0° is 1 and the cosine of 90° is 0. So we say that a cosine is 90° out of phase with a sine wave. Any number of sine waves can exist at any one time and have any manner of angular phase differences to each other. Whenever a phase angle is mentioned it is always relative to something else. Digital 'ones' and 'zeroes' can be encoded as two signals of identical amplitude and frequency but with different phases to each other or some other reference marker. This would be called phase modulation.

When we apply an alternating voltage across a resistor, a current flows through the resistor. If we looked at the voltage and current waveforms on an oscilloscope we would see two sine waves that superimpose each other, when differences of amplitude are taken into account. The two signals are in-phase with each other. If we add a capacitor in series with the resistor we would see the current and voltage signals diverge so they were out of phase with each other. When an electrical current flows in a circuit we are observing the effect of the flow of the fundamental particles called electrons flowing through the wire from a negative to a positive terminal. We can imagine a capacitor is like a big bucket for electrons. When the voltage is applied to the circuit, the electrons flood into the bucket. But as the rising voltage reaches its peak, the bucket is nearly full, and the flow of electrons, or current tails off. The flow of current therefore seems to lead the voltage and is out of phase with the voltage. An inductive load works the other way round. The rising voltage is needed to draft the electrons into the inductor where they fight against the magnetic field they have created. The current therefore lags the voltage. A circuit with a capacitive and/or inductive load is called reactive. The actual phase of the current relative to the voltage will depend on the values of resistance, capacitance and inductance in the circuit and may be represented as a complex number.

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Mass Spectrometry | Ion Traps☆

D.T. Snyder , ... R.G. Cooks , in Encyclopedia of Analytical Science (Third Edition), 2019

Digital Ion Traps

Digital ion traps operate by substituting sinusoidal waves commonly used to trap ions with square waves which have high and low states. The key operating parameter in digital ion traps is the duty cycle which is the ratio of the time spent in the high state over the period of the square wave. Duty cycle alterations unfortunately cause changes within the stability diagram for these traps but also enable digital asymmetric waveform isolation (DAWI) and boundary activation capabilities without the need for supplementary waveforms that are typical for sinusoidal traps. This simplifies the electronics required to perform mass analysis and MS/MS, but it also increases the instrument's power consumption compared to sinusoidal traps. The ability to perform mass analysis by altering the duty cycle could be advantageous in that it is typically easier to precisely scan a frequency than it is to scan the voltage of a waveform. Other principal advantages of altering the duty cycle for mass analysis include increased mass range, greater frequency resolution compared to the voltage resolution of sinusoidal traps, and the ability to alter the trapping waveform nearly instantaneously, which increases the versatility of the analyzer. Even so, digital traps have yet to gain a foothold commercially despite their promising capabilities.

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Optical microscopy for forensic samples

Chaudhery Mustansar Hussain , ... Maithri Tharmavaram , in Handbook of Analytical Techniques for Forensic Samples, 2021

3.5 Polarizing microscope

Typically, light is made up of the sinusoidal waves of the electric and magnetic components propagating at right angles to each other; such a light is commonly referred to as nonpolarized light. However, when light is polarized, there is only one direction or plane of vibration. Light that is completely polarized propagates in a sinusoidal uniform helical motion and can be visualized as an ellipse at the end; this is known as elliptically polarized light. Apart from this, linearly and plane polarized light also exist. The polarizing microscope uses the linearly polarized light to visualize various samples and is used to image samples with higher contrast, resolution, and magnification.

Initially, the source used in the polarizing microscope to generate polarized light was a crystal called a tourmaline. Later, these were switched to calcite. Currently, a Nicol prism is the most commonly used polarized light source. From the source, a condenser is kept that is responsible for carefully diverting the polarized light onto the sample, which is set on a rotating stage. The light obtained from the specimen is further passed through the objective lens with low numerical apertures. It also passes through a compensator and analyzer that are highly useful in determining the birefringence. Birefringence is a phenomenon that typically occurs in anisotropic materials and whose refractive index changes based on the polarization and the direction of light propagation. Birefringence is also responsible for double refraction, due to which the polarized light may take two different paths, thus resulting in a distorted image. The image analysis is also done in which background subtraction and any errors are removed (Fig. 5) (Wayne, 2019c).

Fig. 5. Schematic of a polarized light microscope (Wayne, 2019c).

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Signal processing for tool condition monitoring: from wavelet analysis to sparse decomposition

Zhu Kunpeng , ... Hong Geok Soon , in Mechatronics and Manufacturing Engineering, 2012

4.2.3 Linear system, sampling theorem and convolution

The Fourier transform is powerful and dominates signal processing because the sinusoidal waves e ^{− jωt} are the eigenvectors of time invariant linear system. A system or transform maps an input signal x(t) into an output signal y(t),

(4.8) $y\left(t\right)=T\left[x\left(t\right)\right]$

where T denotes the system or transform, a function from input signals to output signals. Systems come in a wide variety of types.

One important class is known as linear systems. A linear system obeys rules of linearity, which are homogeneity and additivity. Specifically, a linear system T that maps a vector space X into a vector space Y is said to be linear if for every scalar α, β we have,

(4.9) $T\left(\mathit{\alpha x}+\mathit{\beta y}\right)=\mathit{\alpha T}\left(x\right)+\mathit{\beta T}\left(y\right)$

Please note that the linear system here refers to the signal processing system, not to a claim that our machining system is linear. Linear signal processing brings a lot of good properties and is favourable with signal analysis. Fourier analysis, wavelet analysis and sparse decomposition are all linear. The non-linear time frequency approaches such as Wigner-Wille distribution (Cohen, 1989) are not discussed in this chapter.

In practice, the continuous signals are sampled at discrete time to process. It can be completely represented by a set of equally spaced samples, if the samples occur at more than twice the frequency of the highest frequency component of the signal. This sampling frequency is called the Nyquist frequency (Shannon, 1949). If we sample the signal x(t) with sampling frequency f_s (sampling interval T_s  =  l/f_s ), then we generate the sequence {…, x(− nT_s ), …, x(− T_s ), x(0), x(T_s ),…,x(nT_s ),…},

(4.10) $y\left(t\right)={\displaystyle \sum _{n=-\infty }^{\infty }x\left(n{T}_s\right)\cdot \delta \left(t-n{T}_s\right)}=x\left(t\right)\cdot {\displaystyle \sum _{k=-\infty }^{\infty }\delta \left(t-n{T}_s\right)}$

where δ(t – nT_s ) =  1 at t – nT_s and equals 0 elsewhere. The maximum seeable frequency in the X(ω) by DFT is f_s /2 according to this theory.

The Nyquist frequency has dominated the sampling principles in signal processing, but it has been discovered lately (Candès et al., 2006) that this constraint is not necessary when the signal is sparse in some domain (either time, frequency, or another). This finding leads to the newly developed theory called sparse decomposition (or compressive sensing/sampling in information theory), which is to be discussed in 'Sparse coding' below.

Convolution is another important concept in signal processing. For a linear system, let h(t) be the impulse response (the output of the system to the input of delta function) of the system, then for any input x(t), the output y(t) is obtained by convolution,

(4.11) $y\left(t\right)=\mathit{Lx}\left(t\right)=h\left(t\right)*x\left(t\right)={\displaystyle {\int }_{-\infty }^{\infty }h\left(\tau \right)x\left(t-\tau \right)\mathit{d\tau }}$

then the system is called linear.

The convolution has a clear physical meaning that: at any time, the system's output y(t) is obtained by the input x(t) and the convolution of the impulse response of the system h(t). And if y(t – t ₀)   = Lx(t – t ₀), any input delay will result in an output delay, the system is called linear time invariant system (LTI). Convolution has a special importance in the understanding of wavelet. As a matter of fact, Mallat and Hwang (1992) defined the wavelet transform by convolution.

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SCATTERING | Scattering from Surfaces and Thin Films

A. Duparré , in Encyclopedia of Modern Optics, 2005

Scattering from Rough Surfaces

A randomly rough surface can be considered as a Fourier series of sinusoidal waves with different amplitudes, periods, and phases. Following the grating equation, a single grating with spacing D causes scatter into the angle Θ according to

[1] $\text{sin}\text{Θ}=\lambda /D$

where λ is the wavelength of light. D represents one spatial wavelength in the Fourier series. Accordingly, f = 1/D represents one single spatial frequency. A randomly rough surface contains many different spatial frequencies. This is quantitatively expressed by the power spectral density (PSD), giving the relative strength of each roughness component of a surface microstructure as a function of spatial frequency:

[2] $\text{PSD}\left(\text{f}\right)=\underset{A\to \infty }{\text{lim}}\frac{1}{A}{\left|{\displaystyle {\int }_A\zeta \left(\mathbf{r}\right)\text{exp}\left(-2\pi \text{i}\mathbf{f}\cdot \mathbf{r}\right)\text{d}\mathbf{r}}\right|}^2$

where ζ(r) represents the height of the surface roughness profile, r is the position vector, and f is the spatial frequency vector in the x–y plane. A is the area of the measured region A. We confine our discussion to isotropic surfaces, which represent the majority of cases in optical surface and thin film scattering studies. Thus, a PSD(f) independent of the surface direction Φ of vector f is obtained by averaging the two-dimensional function PSD(f) over all surface directions after transformation into polar coordinates:

[3] $\text{PSD}\left(f\right)=\frac{1}{2\pi }{\displaystyle {\int }_0^{2\pi }\text{PSD}\left(f,\text{Φ}\right)\text{d}\text{Φ}}$

Well-established vector scattering theories developed, for instance, by Bousquet et al. or Elson provide the link between the PSD and the scattering intensity per solid angle of a surface:

[4] $\frac{\text{d}P}{P_0\text{d}\text{Ω}}=F\left(\lambda ,n,\text{Θ},\text{Φ}\right)\text{PSD}\left(f\right)$

This theory is valid for surfaces whose rms roughness is small compared to the wavelength. dP/(P ₀  dΩ) denotes the differential power scattered into the direction (Θ, Φ) per unit solid angle dΩ=sin   Θ   dΘ   dΦ divided by the incident power P ₀. Θ and Φ are the polar and azimuthal angles of scattering, respectively. The optical factor F contains all information on the corresponding perfect surface (without the roughness properties), i.e., the refractive index n, wavelength, and the conditions of illumination and observation. Both backscattering and forward scattering can be expressed by eqn [4], according to the illumination and observation conditions chosen. Without loss of generality, all formulas have been written here for normal incidence. The formalism, however, allows consideration of all possible cases, including oblique angles of incidence and arbitrary polarization properties. dP/(P ₀  dΩ) is called angle resolved scattering (ARS) which is related to the well-established term BRDF/BTDF (bidirectional reflectance/transmittance distribution function) by multiplication with cos   Θ:

[5] $\frac{\text{d}P}{P_0\text{d}\text{Ω}}=\text{ARS}=\text{BRDF}\left(\text{or}\text{BTDF}\right)\cdot \text{cos}\text{Θ}$

Total scattering (TS), which is defined as the power P scattered into the backward or forward hemisphere divided by the incident power P ₀ (see also section on scattering measurement below), is obtained by integrating eqn [4] over the forward or backward hemisphere:

[6] $\text{TS}=2\pi {\displaystyle {\int }_0^{\frac{\pi }{2}}\left(\frac{\text{d}P}{P_0\text{d}\text{Ω}}\right)}\text{sin}\text{Θ}\text{d}\text{Θ}$

If the correlation length of surface roughness is much larger than the wavelength, scalar scattering theories like the one from Carniglia can also be employed. Moreover, in this case, the well-known simple approximate formula for total backscattering can be derived from both vector and scalar theories:

[7] ${\text{TS}}_{\text{back}}=R_0{\left(\frac{4\pi \sigma }{\lambda }\right)}^2$

where σ is the rms roughness and R ₀ the specular reflectance.

It must be emphasized that this formula is only valid if the above-mentioned condition is met and as long as only single surfaces without coatings are considered.

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Coherent Raman scattering processes

Hervé Rigneault , in Stimulated Raman Scattering Microscopy, 2022

1.2.1 Harmonic oscillator

The harmonic oscillator is an ideal physical object whose temporal oscillation is a sinusoidal wave with constant amplitude and with a frequency that is solely dependent on the system parameters. It is found in many fields of physics and it is a good approximation of physical systems that are close to a stable position. In the mechanical framework, the simplest harmonic oscillator is a mass m attached to a spring with a stiffness k (Fig. 1.1). Being set vertical, the mass feels the gravitational attraction g and its center of mass is described by

Fig. 1.1. A mass-spring system. x the relative displacement.

(1.1) $\mathit{mg}-k{x}_0=0.$

It is possible to study the mass displacement x from its equilibrium position x ₀. Using the fundamental principles of dynamics (energy conservation), it can be shown [23] that the mass displacement x follows

(1.2) $\frac{{\mathrm{d}}^{2}x}{\mathrm{d}{t}^2}+{{\omega }_0}^{2}x=0$

where ω ₀  = k/m is the system resonant frequency.

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SOLITARY WAVES

J.P. Boyd , in Encyclopedia of Atmospheric Sciences, 2003

Dispersion, Frontogenesis and the Bell Soliton

The left curve in Figure 1 is a schematic of the simplest solitary wave. It is called a "bell soliton" because its shape resembles a church bell.

Figure 1. A bell-shaped crest (left) will dissolve into little ripples under pure wave dispersion; it will steepen and eventually break if advective steepening is unopposed by dispersion. In a solitary wave, dispersion and steepening exactly balance so that a bell-shaped curve propagates steadily without change of shape.

Waves are said to be 'dispersive' if the propagation speed c of a sinusoidal wave varies with the wavelength λ. It is possible to superimpose many sine waves of different wavelengths to make a bell shape, which is centered where all the crests are in phase. However, the bell shape rapidly disperses into an ever-widening patch of ever-shrinking ripples as illustrated in the upper right of Figure 1. In a marathon race, the runners are elbow-to-elbow at the start, but disperse into an ever-widening pack with the fastest runners in front and the slowest runners falling farther and farther behind at the rear. Wave dispersion is the same as track-and-field dispersion: When waves travel at different speeds, the disturbance must spread over time unless some other mechanism intervenes.

One such mechanism is advective steepening. If the fluid velocity is proportional to height, then an initial bell shape will evolve a leading-edge front (Figure 1). As the fast-moving tip overtakes the lower, slower fluid, the trailing (left) edge is stretched while the leading edge steepens ('frontogenesis').

In a solitary wave, dispersion and nonlinear steepening exactly balance so as to create a wave which propagates without change of shape.

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X-Ray Crystallography of Macromolecules, Theory and Methods

Anthony C.T. North , in Encyclopedia of Spectroscopy and Spectrometry (Third Edition), 2017

Fourier Analysis

The mathematician Joseph Fourier showed that any periodic function could be represented by the sum of sinusoidal waves. The most familiar application is the harmonic analysis of the sound from a musical instrument, which can be built up by addition of the fundamental vibration and higher harmonics of it. The electron density of a crystal, with its regularly repeating pattern of unit cells, is a 3-D periodic function and it turns out that the amplitudes of the diffracted rays from a crystal are proportional to the amplitudes of the Fourier components of the 3-D electron density function. By measuring the amplitudes of all the X-ray reflections, it should therefore be possible to calculate their 'Fourier transform,' namely, the actual electron density distribution in the crystal. There is, however, a problem in doing this calculation: namely, the 'phase problem' referred to earlier, which means that, although the amplitudes of the components are known, they cannot be added up correctly without knowledge of their relative phases.

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X-Ray Crystallography of Macromolecules, Theory, and Methods☆

A.C.T. North , J.R. Helliwell , in Reference Module in Chemistry, Molecular Sciences and Chemical Engineering, 2013

Fourier Analysis

The mathematician Joseph Fourier showed that any periodic function could be represented by the sum of sinusoidal waves. The most familiar application is the harmonic analysis of the sound from a musical instrument, which can be built up by addition of the fundamental vibration and higher harmonics of it. The electron density of a crystal, with its regularly repeating pattern of unit cells, is a 3-D periodic function and it turns out that the amplitudes of the diffracted rays from a crystal are proportional to the amplitudes of the Fourier components of the 3-D electron density function. The possibility of using this Fourier analysis approach was realized by William Henry Bragg, the father of William Lawrence Bragg, around the 1920s. By measuring the amplitudes of all the X-ray reflections, it should therefore be possible to calculate their 'Fourier transform,' namely, the actual electron density distribution in the crystal. There is, however, a problem in doing this calculation: namely, the 'phase problem' referred to earlier, which means that, although the amplitudes of the components are known, they cannot be added up correctly without knowledge of their relative phases.

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Microbial Electrolysis Cell for Biohydrogen Production

René Cardeña , ... Germán Buitrón , in Biohydrogen (Second Edition), 2019

8.2.5 Electrochemical Impedance Spectroscopy

The method is based on the perturbation of a system either electrode–electrolyte or entire electrochemical cell by a sinusoidal wave input. The perturbation is applied around a potential E that can be set to a fixed value or relative to the equilibrium potential. Amplitude of the potential of 10   mV is commonly utilized for MEC systems. Electrochemical impedance spectroscopy (EIS) can be applied in potentiostatic or galvanostatic mode; however, the potentiostatic mode has been preferred for microbial electrochemical cells; nevertheless, there is no evidence to avoid the use of galvanostatic mode. This method measure the rate constant of fast electron transfer reactions at short times, is applied to processes in solutions, as well as in solids. The results of EIS provide information on reaction parameters, electrode surfaces porosity, coating, mass transport, interfacial capacitance measurements, and reaction mechanisms. Previous to the interpretation of EIS data, this data must be validated for:

•

Linearity. The potential is a linear response to current changes,

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Causality. The measured signal is caused only by the excitation of the potentiostat,

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Stability. Acceptable duplicates are obtained when measuring at steady-state conditions, and

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Finiteness. At the low-frequency, border is influenced by the capacitance of the system, the size of the electrode, and the presence of mass transfer limitations [84].

Although EIS technique has diverse applications, its use has been limited to the estimation of the distribution of resistances (ohmic and charge transfer), and rarely to estimate capacitances, all this in the electrode–electrolyte interphase or the entire MEC. The model circuits to adjust the EIS data also have been utilized; however, previous EIS data validation is rarely reported. A model circuit is a representation of the physical elements that conform the electrochemical system and it is utilized for modeling and predictive studies.

EIS data can be represented as Nyquist spectra, Bode-phase angle, and Bode-impedance modulus. In the Nyquist semicircle, the first intersection with the X-axis corresponds to the ohmic resistance, while the difference between the second and first intersections corresponds to the charge transfer resistances. In the Bode representations, the impedance modulus corresponds to the charge transfer resistance, and the changes of slope in the phase angle curve are linked with the elements that form of the circuit model.

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