Is a Carrier Wave a Continuous Tone

Communication Systems, Civilian

Simon Haykin , in Encyclopedia of Physical Science and Technology (Third Edition), 2003

II.A.1 Double-Sideband Suppressed-Carrier Modulation

The carrier wave c(t) is completely independent of the information-carrying signal or baseband signal m(t), which means that the transmission of the carrier wave represents a waste of power. This points to a shortcoming of amplitude modulation, namely, that only a fraction of the total transmitted power is affected by m(t). To overcome this shortcoming, we may suppress the carrier component from the modulated wave, resulting in double-sideband suppressed-carrier (DSBSC) modulation. Thus, by suppressing the carrier, we obtain a modulated wave that is proportional to the product of the carrier wave and the baseband signal.

To describe a DSBSC-modulated wave as a function of time, we write

(4) $\text{s}\left(\text{t}\right)=\text{c}\left(\text{t}\right)\text{m}\left(\text{t}\right)={\text{A}}_{\text{c}}\text{cos}\left(2\pi f_{\text{c}}t\right)m\left(t\right)$

This modulated wave undergoes a phase reversal whenever the baseband signal m(t) crosses zero. Accordingly, unlike amplitude modulation, the envelope of a DSBSC-modulated wave is different from the baseband signal. For obvious reasons, a device that performs the operation described in Eq.(4) is called a product modulator.

The transmission bandwidth required by DSBSC modulation is the same as that for amplitude modulation, namely, 2W. Figure 2d illustrates the DSBSC-modulated wave for single-tone modulation and the corresponding amplitude spectrum.

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Data Transmission Media

John S. Sobolewski , in Encyclopedia of Physical Science and Technology (Third Edition), 2003

VI.D Pulse Modulation

All the modulation techniques described previously use a continuous sinusoidal carrier wave that is modulated by analog or digital signals. Because of the continuous carrier wave, these are sometimes referred to as continuous-wave (CW) modulation techniques. In pulse modulation, the carrier is not a continuous wave but a periodic pulse train whose amplitude, duration, or position is varied in accordance with the message. Pulse amplitude (PAM), pulse duration (PDM), and pulse position (PPM) modulation are illustrated in Fig. 7. Note that PPM consists of equal-width pulses derived from the trailing edge of PDM pulses. PPM has an advantage over PDM since the latter can require significant transmitter power for transmitting pulses of long duration.

FIGURE 7. PAM, PDM, and PPM using a sinusoidal modulating wave and a pulse carrier.

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Active Geophysical Monitoring

Rudolf Unger , in Handbook of Geophysical Exploration: Seismic Exploration, 2010

5 Conclusions

Pseudo-random sequences modulated onto a low-frequency carrier wave form ideal force-source signals in a 4D active, continuous seismic transmission system to monitor time-lapse changes in Earth parameters. Complex-envelope seismograms obtained by coherent stacking and quadrature time domain matched-filter processing enable high-resolution signal detection and timing. For large snr's, the timing error is inversely proportional to the square root of signal transmission duration. For a 2.5 Hz carrier wave frequency, at 36 dB MF seismogram snr, the timing error is less than 0.5 ms. In addition to application in our proposed transmission system, the method seems extremely suitable for incorporation in high-precision short baseline experiments using a piezo-electric source ( Silver et al., 2007; Niu et al., 2008), potentially resulting in nano-second timing precision, and in EM monitoring. The method may be applicable also with the Russian large mechanical vibrators (Alekseev et al., 2005) operating with eccentrics, and producing linear sweep (chirp) force signals rather than our prs signals since they cannot accommodate the prs phase flips. It seems unlikely that existing hydraulic vibrators are capable of long-term coherent, low-frequency prs transmission. For our transmission system concept, therefore, we conceived new technology vibrators, a magnetic levitation P-wave vibrator and a linear synchronous motor S-wave vibrator (Unger, 2002, 2004).

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High-precision GNSS for agricultural operations

Manuel Perez-Ruiz , ... Shrini K. Upadhyaya , in GPS and GNSS Technology in Geosciences, 2021

2 GPS signal and structure

The signals received from the GPS satellite segment consist of two carrier waves (L ₁  =   1575.42   MHz or 19   cm and L₂  =   1227.60   MHz or 24.4   cm) together with two or more digital codes (coarse acquisition code or C/A on L₁ and P-code on both L₁ and L₂) and a navigation message. While civilians have limited access restricted only to the C/A-code on L₁, the P-code is a military code encrypted with an unknown W-code resulting in a Y-code (i.e., P (Y)) and is not available to civilians. This is called "antispoofing." P-code provides highly accurate positioning (Precise Positioning Service), as ionospheric distortion can be completely removed by using L₁ and L₂ wave signals. In contrast, the use of the coarse acquisition code does not provide accurate estimates of position (Standard Positioning Service). The more modern satellites (belonging to the class II-RM) transmit two additional codes (L₂ CM—civilian moderate—and L₂ CL—civilian long), which are intended to minimize errors due to atmospheric effects of the ionosphere. The navigation message includes useful data and information about the almanac, ephemeris, clock correction, satellite health, atmospheric correction, etc., which are added to both the C/A code and the P-code. These codes also identify the referring satellite with a unique number (PRN) and include timing information. The codes are then added on to L₁ (both C/A- and P-codes) and L₂ (P-code only). Fig. 15.2 is a schematic diagram of the GPS signal. More recent satellites such as IIF carry an additional carrier wave, L₅ specifically is meant for improving aviational and safety services, and the most recent ones (III and IIF) carry yet another code, L₁C, which is designed to enhance interoperability between various GNSS.

Figure 15.2. Composition of the signals from GPS satellites.

Source: Peter H. Dana, The GPS Overview, The Geographer's Craft Project.

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PROCESS ANALYSIS | Acoustic Emission

R.M. Belchamber , in Encyclopedia of Analytical Science (Second Edition), 2005

RMS Conversion

The signal from a resonant transducer resembles an AM radio signal. It has a carrier wave, at the resonance frequency of the transducer, which is amplitude modulated by the process. The information about the process is in the modulation envelope. An RMS-to-DC (root mean square-to-direct current) converter is used to demodulate the signal. The output of this device is the amplitude of the envelope.

The envelope is digitally resampled at a frequency appropriate for the process. For instance, 50   Hz is a typical digital sampling rate for a fluid bed reactor.

In some simple applications it is sufficient to record the RMS signal. An example of this would be detecting fine material in a gas stream downstream of a cyclone.

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Space Gravimetry Using GRACE Satellite Mission: Basic Concepts

Guillaume Ramillien , ... Lucía Seoane , in Microwave Remote Sensing of Land Surface, 2016

6.2.2 The instruments on board

There are five instruments on board each satellite component of the GRACE mission:

–

the KBR system provides distance measurements between the two satellites, accurate to 10 μm, using the phases of carrier waves in the K-band ( f  =   26   GHz) and the Ka-band (f  =   32   GHz);

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the Ultra-Stable Oscillator (USO) generates electromagnetic waves in the K-band for the KBR system at the desired frequency;

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SuperSTAR accelerometers (ACC) with three axes accurately measures the non-conservative forces applied to each satellite;

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the Stellar Camera ASSEMBLY (SCA) determines the orientation of the satellite relative to the position of fixed stars;

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Black-Jack GPS receivers and Instrument Processing Unit – the GPS provides the three components of the position and speed of each of the satellites.

The assembly diagram of these components in each satellite component of the GRACE mission is shown in Figure 6.2.

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Radar

William Emery , Adriano Camps , in Introduction to Satellite Remote Sensing, 2017

5.2.6 Radar Waves at an Interface

The surface of the Earth can be interpreted as a resistive dielectric, whose dielectric constant depends on factors such as the material, the moisture, and the radar carrier frequency. In the case of a simple carrier wave, the refractive index of the soil can be expressed as

(5.35) $n_2=\sqrt{{\epsilon }^{\prime }-j·\frac{{\sigma }_{\text{cond}}}{2\mathrm{\pi }·f·{\epsilon }_0}}\text{,}$

where ${\epsilon }^{\prime }$ is the relative permittivity of the soil, σ _cond its conductivity, ⁵ f is the frequency, and ε ₀ is the permittivity of a vacuum. The soil will behave as a good conductor when the relative permittivity is negligible with respect to the value of the second term inside the square root, i.e.,

(5.36) $\frac{{\sigma }_{\text{cond}}}{2\mathrm{\pi }·f·{\epsilon }_0}\gg \epsilon \prime \text{,}$

a condition which is naturally more easily fulfilled at lower frequencies.

Let us first consider a simple example: an incident radar wave impinging on a flat surface separating the air from a dielectric soil. A surface will be flat whenever the length of the surface is much larger than the wavelength, i.e., L  ≫ λ. As predicted by geometrical optics, the wave will be reflected in the specular direction (i.e., θ _r   = θ _i ), as depicted in Fig. 5.17.

Figure 5.17. Reflection and refraction of an incident wave onto a flat surface, L  ≫ λ. The angle of reflection θ _r is equal to the incident angle θ _i . The angle of refraction is subject to Snell's law. The amplitude and phase balance of the reflection and refraction effects are described by the Fresnel coefficients.

When the material is a good conductor, all incident energy is reflected. In the opposite case, some of the energy of the incident wave penetrates and is refracted into the dielectric, with the geometry of refraction following Snell's law.

The amount of energy reflected and refracted can be computed with help of the Fresnel's coefficients, derived from the boundary conditions of the electrical and magnetic fields at the interface. The Fresnel coefficients vary depending on the polarization of the incident wave. The reflection coefficients for vertically (V) and horizontally (H) polarized waves take the following forms (Hecht, 1997)

(5.37) $\begin{array}{l}{\rho }_{\text{Vr}}=\frac{E_{\text{Vr}}}{E_{\text{Vi}}}=\frac{n_2·\cos {\theta }_i-n_1·\cos {\theta }_t}{n_1·\cos {\theta }_i+n_2·\cos {\theta }_t}\text{,}\\ {\rho }_{\text{Hr}}=\frac{E_{\text{Hr}}}{E_{\text{Hi}}}=\frac{n_1·\cos {\theta }_i-n_2·\cos {\theta }_t}{n_1·\cos {\theta }_i+n_2·\cos {\theta }_t}\text{.}\end{array}$

When $\left|n_2\right|>\left|n_1\right|$ , $\left|{\rho }_{\text{Hr}}\right|>\left|{\rho }_{\text{Vr}}\right|$ , this is a characteristic that will be relevant in the analysis of the scattering by rough surfaces. Note that ρ _Hr and ρ _Vr can take negative values depending on the sign of the numerator, which corresponds to a phase shift of 180°. This is the particular case of reflection from a good conductor, with ρ _Vr  =   1, and ρ _Hr  =   −1. The refraction coefficients for the V and H components are

(5.38) $\begin{array}{l}{\rho }_{\text{Vt}}=\frac{E_{\text{Vt}}}{E_{\text{Vi}}}=\frac{2·n_1·\cos {\theta }_i}{n_1·\cos {\theta }_t+n_2·\cos {\theta }_i}\text{,}\\ {\rho }_{\text{Ht}}=\frac{E_{\text{Ht}}}{E_{\text{Hi}}}=\frac{2·n_1·\cos {\theta }_i}{n_1·\cos {\theta }_i+n_2·\cos {\theta }_t}\text{.}\end{array}$

Eqs. (5.37) and (5.38) encompass all the necessary information needed to understand the scattering processes occurring in typical Earth observation radars. Reflection will be the relevant phenomenon in the analysis of simplified targets, such as urban areas and land and ocean surfaces. Refraction will play a role in the penetration of the radar waves and the imaging of volumetric structures such as forests, ice, or dry soil.

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DYNAMICAL METEOROLOGY | Solitary Waves

J.P. Boyd , in Encyclopedia of Atmospheric Sciences (Second Edition), 2015

A Bestiary of Solitary Waves and Coherent Vortices

Figure 5 shows the diversity of solitary waves and coherent structures. The six species illustrated are only a set of interesting creatures from a much larger zoo.

Figure 5. A selection of soliton species. All are snapshots at a single time except for the breather, which shows the oscillation at intervals of one-quarter of the temporal period.

Bell solitons, such as those that solve the KdV equation, have been described above.

An 'envelope solitary wave' is the product of a sinusoidal 'carrier wave' with a slowly varying amplitude factor called the 'envelope,' which is dashed in the figure. The envelope solves the nonlinear Schrodinger equation and propagates at approximately the group velocity of infinitesimal waves of the wavelength of the 'carrier wave.'

'Breathers' are solitary waves whose amplitude oscillates in time. The breather may be either stationary or propagating, but the period and amplitude of the 'breathing' oscillations never changes. The sine–Gordon, self-induced transparency, and ${\varphi }^4$ field theory equations also have breathers.

'Kinks,' also known in some contexts as 'traveling shocks,' occur in both inviscid models (such as the sine–Gordon equation) and viscous equations, such as Burgers' equation and the Kuramoto–Sivashinsky equation. Viscous shocks seem a paradox since mechanical energy is being damped and yet the shocks, like solitary waves, are independent of time except for a steady propagation. The plateaus, extending indefinitely away from the shock, act as limitless reservoirs of energy to sustain the shock. Real kinks do not extend indefinitely, but are consistent local approximations to coherent structures of finite width.

Vortices, whether monopoles or modons, are not always identified as solitary waves. If the diameter of the vortex is sufficiently small compared to the radius of the earth, then wave effects may be only a small correction to vortex dynamics. However, vortices often exhibit the same robustness and nonlinear self-preservation as KdV solitons.

Monopole vortices have vorticity which is everywhere of the same sign except perhaps for an annular ring surrounding the core. Modons are pairs of contra-rotating vortices as described earlier. These have a strong nonlinear translation, indicated by the hollow arrow in the figure, which is augmented or opposed by westward Rossby wave propagation.

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SOLITARY WAVES

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

A Bestiary of Solitary Waves and Coherent Vortices

Figure 5 shows the diversity of solitary waves and coherent structures. The six species illustrated are only a set of interesting creatures from a much larger zoo.

Figure 5. A selection of soliton species.

Bell solitons, such as those that solve the KdV equation, have been described above.

An 'envelope solitary wave' is the product of a sinusoidal 'carrier wave' with a slowly varying amplitude factor called the 'envelope', which is dashed in the figure. The envelope solves the nonlinear Schrödinger (NLS) equation.

'Breathers' are solitary waves whose amplitude oscillates in time. The breather may be either stationary or propagating, but the period and amplitude of the 'breathing' oscillations never changes. The sine-Gordon, self-induced transparency (SIT), and ϕ⁴ field theory equations have breathers.

'Kinks', also known in some contexts as 'travelling shocks', occur in both inviscid models (such as the sine-Gordon equation) and viscous equations, such as Burgers' equation and the Kuramoto–Sivashinsky (KS) equation. Viscous shocks seem a paradox since mechanical energy is being damped and yet the shocks, like solitary waves, are independent of time except for a steady propagation. The plateaus, extending indefinitely away from the shock, act as limitless reservoirs of energy to sustain the shock. Real kinks do not extend indefinitely, but are consistent local approximations to coherent structures of finite width.

Vortices, whether monopoles or modons, are not always identified as solitary waves. If the diameter of the vortex is sufficiently small compared to the radius of the Earth, then wave effects may be only a small correction to vortex dynamics. However, vortices often exhibit the same robustness and nonlinear self-preservation as KdV solitons.

Monopole vortices have vorticity which is everywhere of the same sign except perhaps for an annular ring surrounding the core. Modons are pairs of contra-rotating vortices as described earlier. These have a strong nonlinear translation, indicated by the hollow arrow in the figure, which is augmented or opposed by westward Rossby wave propagation.

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DYNAMICAL METEOROLOGY | Baroclinic Instability

R. Grotjahn , in Encyclopedia of Atmospheric Sciences (Second Edition), 2015

Other Issues

Baroclinic instability has links with barotropic instability. First, each instability draws energy from mean flow shear. Second, barotropic instability has a similar stability criterion (AV gradient changing sign in the domain). Third, there can be interference between the two instabilities. The most unstable baroclinic eigenmode has optimal structure for a flow having the vertical shear alone, but when horizontal shear is added to that flow a different structure is needed otherwise the eddy will be sheared apart. The subsequent structure is unlikely to be as optimal for baroclinic energy conversion. Hence, the baroclinic conversion will usually be reduced, though the barotropic growth mechanism may compensate. Figure 7(c) illustrates such a calculation; in this case adding a purely barotropic flow reduced the growth rate even though the barotropic growth mechanism was activated.

Figure 7. Baroclinic energy conversion (A _z →A _e) for four models. (a) Lowest order, square wave solution for an Eady-type model but including compressibility, increasing vertical shear in U, β = 1. (b) Solution when a surface frontal zone, centered at y = 0, is added to the lowest order mean flowU ₀ and leading ageostrophic advective effects are included (using geostrophic coordinates). The frontal zone adds wind field: 0.2(2z−z ²) U ₁ whereU ₁ =b ₁(1−tanh²(ay))−b ₂−3b ₃ y ² to U ₀. The geostrophic coordinate transform causes the asymmetry. (c) Modification to the conversion shown in (a) when barotropically unstable horizontal shear U ₁ is added to U ₀. If the total wind is U =U ₀ +μU ₁, then the total conversion is (a) +μ(c). The barotropic shear reduces the growth rate. (d) Modification due to all leading order ageostrophic corrections. If those corrections are order μ, then the total conversion is (a) +μ(d). Ageostrophic conversions reduce the conversion and introduce asymmetry.

Adapted with permission from Grotjahn, R., 1979. Journal of Atmospheric Sciences 36, 2049–2074.

Baroclinically unstable frontal cyclones prefer to develop in certain regions. The preference may arise from local conditions such as lower static stability or locally greater vertical shear. The illustrative model above assumes a wavetrain solution; when more localized development is considered, a variety of issues are raised.

For example, if one uses a single low as the initial condition, the solution typically evolves into a chain of waves as the modal constituents of the initial state disperse. Alternatively, a wave packet initial condition might be used consisting of a 'carrier wave' multiplied by an amplitude envelope. The packet evolution depends upon the mean flow properties and assumptions made in the model. However, for reasonable choices of parameters, one might find a packet that spreads while propagating downwind. The leading edge of the packet has mainly faster, wider, and deeper modes. The trailing edge has slower, shorter, and shallower waves. It is possible to construct a localized structure, which resists this dispersion by making a judicious combination of eigenmodes having similar phase speed, but different zonal wavenumber. One such example was used when discussing 'type B' cyclogenesis ( Figure 5). Figure 8 illustrates another example using neutral continuum modes. When this model is solved as an initial value problem the packet maintains a localized shape for a long time and almost no growth occurs since the normal modes were filtered out and there is very slow phase shifting of the constituent modes. However, when nonlinear advection is allowed, modes interact and soon amplitude is injected into all the eigenmodes including the growing normal modes, which grow rapidly in this example.

Figure 8. Initial value calculations for a linearly localized initial condition. (a) Zonal cross section showing contours of streamfunction initially. Values < −1.0 are shaded. (b) Horizontal pattern of streamfunction at tropopause level (z = 1.0) initially. Initial condition constructed from neutral modes having similar phase speed. Growing or decaying normal modes are excluded. (c) Time series of energy growth rate for three integrations. Linear model (dotted line) showing little growth since the nonmodal mechanism is weak and growing normal modes cannot develop. Also shown are nonlinear calculations for two amplitudes of the initial condition, where the solid line uses three times the initial amplitude of the dot-dashed line. Growing normal modes are activated by nonlinear interaction. Some evidence of nonlinear saturation is seen.

Studies of regional development spawned subcategories of baroclinic instability. 'Absolute' instability occurs when the wave packet expands faster than it propagates; the amplitude at a point keeps growing. 'Convective' (in the advection sense) instability occurs when the packet moves faster than it spreads so that growth then decay occurs as the packet moves past a point. 'Global' instability (like the eigensolutions shown here) has growth that is invariant to a Galilean transform. Such is not the case for 'locally' unstable modes. Normal modes for zonally varying basic states look like carrier waves modulated by a spatially fixed amplitude envelope; the envelope locally modifies the growth rate (sometimes called 'temporal' instability); enhancing the global growth locally where the carrier wave propagates from lower to higher amplitude of the envelope. 'Spatial' instability allows wave number to be complex while phase speed remains real.

Nonlinear calculations raise other issues related to baroclinic instability. One issue concerns equilibration. The growing wave modifies the mean flow while drawing energy from it. This places a limit upon the cyclone development. In PV theory, this may be where the distortion shown in Figure 4 becomes comparable to the cyclone width. Waves longer than the most unstable wave tend to reach larger amplitude than the linearly most unstable mode. One reason why is that they are deeper and so can potentially tap more APE in the mean flow. Another reason may be the larger scale in both horizontal dimensions provides a longer time for PV contour distortion. Another reason concerns the inversion of a PV anomaly: the streamfunction amplitude is larger for a broader PV anomaly.

'Life cycle' studies model cyclones from birth to peak amplitude to decay. These studies typically find baroclinic growth followed by barotropic decay. This cycle fits the observed facts that eddies have a net heat flux and a net momentum convergence. These studies also reveal a characteristic evolution of the eddy structure: upper level amplification compared to the linear eigenmodes. An explanation is that saturation is reached sooner at the critical level and at the surface while upper levels continue to grow. Another was given above regarding the Eliassen–Palm flux F. When averaged over the life cycle, the vertical distribution of the zonal mean eddy heat and momentum fluxes becomes more realistic.

Finally, the atmosphere has higher order processes than the QG system. The biggest impact of ageostrophy is to break symmetries in the solutions. Figure 7(d) shows the leading order ageostrophic effects for a linear model. Ageostrophy causes enhanced eddy development on the poleward side (mainly by negative baroclinic conversion on the equatorward side), builds mean flow meridional shear, and slows down the wave. Ageostrophy also causes contours to be more closely spaced around a low and more widely spaced around a high.

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