A Signal in Which the Intensity Varies Continuously and Smoothly Over a Period of Time

Satellite-Based Location Systems

David Munoz , ... Rogerio Enriquez , in Position Location Techniques and Applications, 2009

7.3.2 Precision and Accuracy

The signal carrier gives the most precise PL; for example, given a wavelength of 20 cm for a 1.5-GHz carrier, we may have a precision of that magnitude. However, as a result of many sources of error, signal carrier precision is difficult and thus precision degrades. Although precision can be very high, as Equations 7.12 and 7.14 show, there are still many uncertainties, mainly because receivers lose cycles in detection.

From these equations, one of the terms used to refer to how precision is degraded is the dilution of precision (DOP), which is nothing more than the root–sum–squared measurement of the size of a confidence region where one wants to place a preselected level of probability. In a statistical sense, DOP is the square root of the traces of various submatrixes of the covariance matrix divided by the measured standard deviation, where the confidence regions are related to the eigenvalues of such matrix. In short, DOP relates the calculations with the geometry of the section of space where satellites are.

In practice, we want DOP to be the smallest possible figure, which means that the geometry of the satellites is as large as possible such that the matrixes will not be ill conditioned. However, with movement of satellites, geometry changes and therefore DOP may deteriorate For example, if we want our positioning precision to be 50 m, for a measured accuracy of 10 m, a DOP of 5 m is required. Afterward, if DOP is 1 m, then we have a positioning accuracy of 10 m. From a purely mathematical point of view, equations can be solved analytically. When using computers to solve equations, the numerical calculations may introduce errors and that could cause the matrixes involved to become ill conditioned.

In GNSS, when performing calculations for PL, there are three different ways of dealing witherror correction: one uses only raw code pseudoranges; the second uses the carrier phase information to smooth code measurements; finally, the most precise correction is obtained by using only the carrier phase. When error correction is carried out using carrier phase, we are talking of precise DGNSS. When error correction using raw code pseudoranges or part of carrier phase information, we are dealing with the common DGNSS. To improve the precision in the carrier phase, it is necessary to deal with phase carrier ambiguities. Thus, instead of talking about precision, it is better to talk about accuracy in the statistical sense; that is, the end PL is calculated as an expected value of several measurements.

Care must be taken when a commercial receiver claims very high accuracy because the accuracy depends on how they actually define the dispersion of data; sometimes such dispersion is expressed as being inside 4σ(!) of a region of confidence. Also, each reference axis has different dispersions, which is why using the smallest variance and not the global variance is more marketable.

Also, in trying to reduce sources of error, much effort was invested in trying to reduce uncertainty in knowing the signal ecology system [9]. Thus, ionospheric reduction, channel fading, and other effects are important to predict because they form the ecological surroundings of the communications signals.

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Fourier Analysis in Communications and Filtering

Luis F. Chaparro , Aydin Akan , in Signals and Systems Using MATLAB (Third Edition), 2019

Quadrature Amplitude Modulation (QAM)

QAM enables two AM-SC signals to be transmitted over the same frequency band-conserving bandwidth. The messages can be separated at the receiver. This is accomplished by using two orthogonal carriers, such as a cosine and a sine. See Fig. 7.4. The QAM modulated signal is

Figure 7.4. QAM transmitter and receiver: s(t) is the transmitted signal and r(t) the received signal.

(7.5) $s(t)=m_1(t)\cos ({\mathrm{\Omega }}_{c}t)+m_2(t)\sin ({\mathrm{\Omega }}_{c}t),$

where $m_1(t)$ and $m_2(t)$ are the messages. You can think of $s(t)$ as having a phasor that is the sum of two phasors perpendicular to each other (the cosine leading the sine by $\pi /2$ ), indeed

$s(t)=\mathcal{R}e[(m_1(t)e^{j0}+m_2(t)e^{-j\pi /2})e^{j{\mathrm{\Omega }}_{c}t}]$

has a phasor $m_1(t)e^{j0}+m_2(t)e^{-j\pi /2}$ , which is the sum of two perpendicular phasors. Since

$m_1(t)e^{j0}+m_2(t)e^{-j\pi /2}=m_1(t)-j{m}_2(t)$

we could interpret the QAM signal as the result of amplitude modulating the real and the imaginary parts of a complex message $m(t)=m_1(t)-j{m}_2(t)$ .

To simplify the computation of the spectrum of $s(t)$ , let us consider the message $m(t)=m_1(t)-j{m}_2(t)$ , i.e., a complex message, with spectrum $M(\mathrm{\Omega })=M_1(\mathrm{\Omega })-j{M}_2(\mathrm{\Omega })$ so that

$s(t)=\mathcal{R}e[m(t)e^{j{\mathrm{\Omega }}_{c}t}]=0.5[m(t)e^{j{\mathrm{\Omega }}_{c}t}+m^{⁎}(t)e^{-j{\mathrm{\Omega }}_{c}t}]$

where ⁎ stands for complex conjugate. The spectrum of $s(t)$ is then given by

$\begin{array}{ccc}\hfill S(\mathrm{\Omega })& \hfill =\hfill & 0.5[M(\mathrm{\Omega }-{\mathrm{\Omega }}_c)+M^{⁎}(\mathrm{\Omega }+{\mathrm{\Omega }}_c)]\hfill \\ \hfill & \hfill =\hfill & 0.5[M_1(\mathrm{\Omega }-{\mathrm{\Omega }}_c)-j{M}_2(\mathrm{\Omega }-{\mathrm{\Omega }}_c)+M_1(\mathrm{\Omega }+{\mathrm{\Omega }}_c)+j{M}_2(\mathrm{\Omega }+{\mathrm{\Omega }}_c)]\hfill \end{array}$

where the superposition of the spectra of the two messages is clearly seen. At the receiver, if we multiply the received signal—for simplicity assume it to be $s(t)$ —by $\cos ({\mathrm{\Omega }}_{c}t)$ we get

$\begin{array}{ccc}\hfill r_1(t)& \hfill =\hfill & s(t)\cos ({\mathrm{\Omega }}_{c}t)=0.25[m(t)e^{j{\mathrm{\Omega }}_{c}t}+m^{⁎}(t)e^{-j{\mathrm{\Omega }}_{c}t}][e^{j{\mathrm{\Omega }}_{c}t}+e^{-j{\mathrm{\Omega }}_{c}t}]\hfill \\ \hfill & \hfill =\hfill & 0.25[m(t)+m^{⁎}(t)]+0.25[m(t)e^{j2{\mathrm{\Omega }}_{c}t}+m^{⁎}(t)e^{-j2{\mathrm{\Omega }}_{c}t}],\hfill \end{array}$

which, when passed through a low-pass filter, with the appropriate bandwidth, gives

$\begin{array}{ccc}\hfill 0.25[m(t)+m^{⁎}(t)]& \hfill =\hfill & 0.25[m_1(t)-j{m}_2(t)+m_1(t)+j{m}_2(t)]\hfill \\ \hfill & \hfill =\hfill & 0.5m_1(t).\hfill \end{array}$

Likewise, to get the second message we multiply $s(t)$ by $\sin ({\mathrm{\Omega }}_{c}t)$ and pass the resulting signal through a low-pass filter.

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Image reconstruction for hard field tomography

U. Hampel , in Industrial Tomography, 2015

13.1.1 Assumption 1: the signal carrier propagates straight within the object

Straight propagation of the signal carrier particularly holds for ionizing electromagnetic and particle radiation. Therefore computed tomography (CT) with X-rays, gamma rays, and neutrons, but also with electrons, falls into this category. Moreover it holds for visible and infrared light tomography in gases, liquids, or semitransparent solids with purely absorbing constituents. Also low-frequency electromagnetic waves, such as microwaves or radar waves and sound waves, do in principle propagate straight. But they produce interference patterns when they are deflected, scattered and refracted at impedance discontinuities. Therefore, tomography methods based on wave–material interaction require different reconstruction schemes, such as diffraction tomography approaches. The interested reader may find further information on such approaches in other textbooks, such as Kak and Slaney (1988).

With the assumed straight propagation of the signal carrier we can understand the measurement process in transmission tomography as attenuation of rays. For emission tomography we would rather speak of thin measurement channels through which emitted radiation propagates towards external detectors.

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Digital Signal Processing

Lars Wanhammar , in DSP Integrated Circuits, 1999

3.3 SIGNALS

A signal conveys information by using a signal carrier. Typically, information is modulated onto a physical quantity, e.g., a speech signal can be represented by a voltage variation. Voltages or currents are commonly used as signal carriers in electronic signal processing systems. In practice, several types of modulations are used to encode the information and to exploit different types of signal carriers and circuit devices. The simplest type of modulation is to let the signal carrier (voltage) vary according to the continuous time signal, e.g., a speech signal. Such a signal, which varies continuously both over a range of signal values and in time, is called an analog signal. An analog signal is denoted

$t\to y:y=f\left(t\right),y\in C,t\in C$

The signal may be a complex function in the complex domain, but usually both y and t are real quantities. Note that we usually refer to the independent variable(s) as "time" even if in certain applications the variable may represent, for example, the spatial points (pixels) in an image. It is not a trivial matter to extend the theory for one-dimensional systems to two- or multidimensional systems, since several basic properties of one-dimensional systems do not have a direct correspondence in higher-dimensional systems.

In many cases, the signal does not vary continuously over time. Instead, the signal is represented by a sequence of values. Often, these signals are obtained from measurements (sampling) of an analog quantity at equidistant time instances. A sampled signal with continuously varying signal values is called a discrete-time signal:

$n\to y:y=f\left(nT\right),y\in C,n\in Z,T>0$

where T is an associated positive constant. If the sequence is obtained by sampling an analog signal, then T is called the sample period.

If signal values are restricted to a countable set of values, the corresponding signal is referred to as a digital signal or sequence:

$n\to y:y=f\left(nT\right),y\in Z,n\in Z,\mathrm{I}>0$

Unfortunately, it is common practice not to distinguish between discrete-time and digital signals. Digital signals are, in principle, a subset of discrete-time signals. Digital signals are usually obtained by measurements of some physical quantity using an A/D converter with finite resolution.

We will frequently make use of the following special sequences:

(3.1) $\mathrm{I}\mathrm{m}\mathrm{pulse}\mathrm{sequence}:\delta \left(nT\right)=\left\{\begin{array}{l}1\mathrm{if}n=0\\ 0\mathrm{otherwise}\end{array}\right.$

(3.2) $\mathrm{Unit}\mathrm{step}\mathrm{sequence}:u\left(nT\right)=\left\{\begin{array}{l}1\mathrm{if}n\ge 0\\ 0\mathrm{if}n<0\end{array}\right.$

The impulse sequence is sometimes called the unit sample sequence. Whenever convenient, we choose to drop the T and simply write x(n) instead of x(nT).

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Cellular radio systems

Malcolm Appleby MA , Fred Harrison BSc, CEng, MIEE , in Telecommunications Engineer's Reference Book, 1993

47.3.2 Co-channel interference

Generally, a mobile will receive a wanted carrier signal (C) from the base station serving the cell in which it is located, and in addition, interfering signals (I) from other cells. The carrier to interference ratio C/I is related to the re-use ratio D/R. Cellular radio systems are designed to tolerate a certain amount of interference, but beyond this, speech quality will be severely degraded. The TACS cellular system, for example, will work with a C/I down to around 17dB. This lower limit on C/I effectively sets the minimum D/R ratio that can be used.

The two key factors in ensuring that good quality transmission can occur between a mobile and base station are that the wanted signal strength is sufficiently large, that is, above the receiver threshold sensitivity, and that the interference level is low enough to give an adequate C/I ratio. Both of these factors depend on the radio propagation between the mobile and base stations.

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Fiber Optics in Sensor Instrumentation

B.T. Meggitt , in Instrumentation Reference Book (Fourth Edition), 2010

17.2.2 Modulation Parameters

Fiber optic sensors operate by the modulation of an optical carrier signal by some optical mechanism present in the sensing region that is itself responsive to the external parametric measurand field. Subsequent signal processing of the modulated carrier signal then relates these changes to variations in the measurand field of interest. There are a limited number of such possible optical properties that can be modulated in an optical sensor system. These can be identified as follows:

•

Intensity modulation.

•

Wavelength/frequency modulation.

•

Temporal modulation.

•

Phase modulation.

•

Polarization modulation.

All these optical modulation parameters are well known in optical metrology. The purpose in optical fiber sensors is to adopt and extend such methods for use with the optical fiber medium.

The first method in the list, intensity modulation, is perhaps the simplest technique to consider for optical fiber use (see, e.g., Senior and Murtaza, 1989). However, since there are many processes in an optical fiber network, such as coupler loss and bend loss, that can also modulate the intensity of the transmitted radiation, it cannot be used directly without providing an additional processing technique that can unambiguously identify the intensity changes induced solely by the measure and interaction of interest. The most common method for providing this facility in the intensity-modulated sensor is to use a two-wavelength referencing technique (e.g., for gas sensing, see Bianco and Baldini, 1993). Here, two different wavelength sources are used (see Figure 17.3) whereby one, the signal channel, has its wavelength variably absorbed by the sensor interaction and the second channel acts as a reference wavelength that is unaffected by the sensing interaction and against which the sensing signal can be normalized for system losses. It is, however, necessary to ensure that the optical fiber link to and from the sensor head has the same spectral response to both beam wavelengths. This requires the two beams to have closely spaced wavelengths; these are usually produced from the spectral division of a single source element.

FIGURE 17.3. Typical configuration for a two-wavelength intensity modulated optical fiber sensor system.

In principle, the other listed optical modulation techniques are immune to intensity modulation changes in the fiber link, provided that some form of temporal modulation is applied to the radiation and that the link does not introduce addition modulation effects.

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Modulation

Rodger Ziemer , in Encyclopedia of Physical Science and Technology (Third Edition), 2002

II.A.1 Double-Sideband and Amplitude Modulation

In linear analog modulation, the amplitude of the carrier signal depends linearly on the analog message signal. Three examples of linear modulation are DSB, AM, and PAM. PAM is discussed later with the other possible analog pulse modulation methods. In the case of AM and DSB, the carrier is sinusoidal as expressed by ( 7) and the modulated signal is written as

(9) ${\text{x}}_{\text{c},\text{DSB}}\left(\text{t}\right)={\text{A}}_{\text{c}}\text{m}\left(\text{t}\right)\text{cos}\left(2\pi {\text{f}}_{\text{c}}\text{t}\right)$

for DSB, and

(10) ${\text{x}}_{\text{c},\text{AM}}\left(\text{t}\right)={\text{A}}_{\text{c}}\left[1+\text{a}\text{m}\left(\text{t}\right)\right]\text{cos}\left(2\pi {\text{f}}_{\text{c}}\text{t}\right)$

for AM, where m(t) is the message signal, f _c is the carrier frequency, and a is a parameter called the modulation index. A modulator for either DSB or AM is implemented as a multiplier.

If the Fourier transforms of Eqs. (9) and (10) are taken, the resulting spectra are

(11) ${\text{X}}_{\text{c},\text{DSB}}(\text{f})=\frac{1}{2}{\text{A}}_{\text{c}}[\text{M}(\text{f}-{\text{f}}_{\text{c}})+\text{M}(\text{f}+{\text{f}}_{\text{c}})]$

for DSB, and

(12) $\begin{array}{ll}{\text{X}}_{\text{c},\text{AM}}(\text{f})& =\frac{1}{2}{\text{A}}_{\text{c}}[\delta (\text{f}-{\text{f}}_{\text{c}})+\delta (\text{f}+{\text{f}}_{\text{c}})]\\ & +\frac{1}{2}\text{a}{\text{A}}_{\text{c}}[\text{M}(\text{f}-{\text{f}}_{\text{c}})+\text{M}(\text{f}+{\text{f}}_{\text{c}})]\end{array}$

for AM. The main difference between DSB and AM is the presence of a carrier component in AM, which manifests itself as the term A _c cos(2πf _c t) in the time domain and as the pair of delta functions located at frequencies ±f _c in the frequency domain. Waveforms and spectra for DSB and AM modulation where the message signal is sinusoidal are compared in Fig. 1. Note that in both cases, the spectrum corresponding to the message signal possesses a mirror-image symmetry about the carrier (the carrier is absent in the case of DSB). These paired message spectral components are known as the upper and lower sidebands and carry duplicate information.

FIGURE 1. Waveforms and spectra for (a) DSB modulation and (b) AM modulation (a=0.8); sinusoidal modulating signal frequency of 1   Hz and carrier frequency of 10   Hz.

The demodulation of either DSB or AM can be accomplished by a coherent demodulator shown schematically in Fig. 2a. The implementation of such a demodulator presupposes the availability of a coherent carrier reference at the receiver. A circuit for achieving this is the squarer-frequency-divider circuit of Fig. 2b; the Costas loop circuit of Fig. 2c establishes a coherent reference and provides the demodulated signal at its output.

FIGURE 2. (a) Coherent demodulator for DSB and AM; (b) squarer-frequency-divider to obtain reference for demodulating DSB; (c) Costas loop to demodulate DSB; (d) envelope detector.

The use of coherent demodulation is necessary for the successful demodulation of DSB because of the 180° phase reversals of the carrier that occur each time m(t) changes sign. Because of the presence of a carrier component in AM, these phase reversals do not occur if a  >   1, thus allowing the use an envelope detector as illustrated in Fig. 2d. The diode passes the positive halves of the modulated carrier sine wave, and these pulses are then smoothed into a replica of the modulating signal by the low-pass RC filter that follows the rectifer. The simplicity of this scheme is one reason for its use in commercial AM broadcast radio.

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Conventional Multilevel Inverter

Pradyumn Chaturvedi , in Modeling and Control of Power Electronics Converter System for Power Quality Improvements, 2018

3.3.2.1 Based on carrier signals

Various SPWM techniques can be derived based on the placement of carrier signals, their magnitude and frequency, and the amplitude of overlapping signal with each other.

1.

Phase disposition SPWM (PD SPWM)

In this SPWM technique, all the carrier signals are in phase and level shifted. Fig. 3.11 shows the principle of pulse generation for five-level PD SPWM. The carrier signals are C ₁, C ₂, C ₃, and C ₄ while three-phase reference or modulating signals are $V_{\mathit{r}}$ , $V_{\mathit{y}}$ , and $V_{\mathit{b}}$ . Comparison of these four carrier signals with the corresponding modulating signal generates the control signal, which has to be given to the corresponding switches of that phase-leg devices.

Figure 3.11. PD SPWM technique for five-level inverter.

2.

Phase opposition disposition SPWM (POD SPWM)

The carrier signals above the reference/zero line are in the same phase and the carrier signals below the zero line are in the same phase, but the carriers below and above the zero line are out of phase by 180 degrees as shown in Fig. 3.12.

Figure 3.12. POD SPWM technique for five-level inverter.

3.

Alternate phase opposition disposition SPWM (APOD SPWM)

It is similar to PD SPWM technique; however, the carriers are phase displaced from one another by 180 degrees alternatively as shown in Fig. 3.13.

Figure 3.13. APOD SPWM technique for five-level inverter.

4.

Phase shift SPWM (PS PWM)

All the carriers are phase shifted by appropriate angle as shown in Fig. 3.14. The performance of the technique depends on phase-shift angle between carriers.

Figure 3.14. PS SPWM technique for five-level inverter.

5.

Variable frequency carrier bands SPWM (VFCB SPWM)

In this technique, the frequency of all the carriers is not the same. Some carriers have different frequency than others, as shown in Fig. 3.15. The techniques mentioned above in (1)–(4) can be modified to achieve VFCB SPWM.

Figure 3.15. VFCB SPWM for five-level inverter.

6.

Carrier overlapping PWM (CO PWM)

All the carriers are overlapping each other by some definite magnitude as shown in Fig. 3.16. The amount of overlapping magnitude will decide the output performance of inverter.

Figure 3.16. CO PWM for five-level inverter.

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Navigation Receiver

Rajat Acharya , in Understanding Satellite Navigation, 2014

5.2.4.3.1 Very Early – Very Late correlator

We have seen that the ideal autocorrelation characteristic of a BOC signal is a regular bipolar triangular variation with amplitude enveloped within the autocorrelation function of a normal spreading code, as shown in Figure 5.15. The profile has a main peak at the middle for exact superimposition of the signals with diminishing adjacent side peaks at regular delay intervals which corresponds to the shifted matching of the subcarriers.

FIGURE 5.15. Autocorrelation of a BOC signal.

The idea in VEVL is that it employs additional correlators-that is, very early (VE) and very late (VL) gates-implemented away from the prompt gate in addition to the nominal Early and Late gates. These correlators monitor the amplitude of adjacent peaks in a correlation function. Because of the symmetric nature of the envelope, adjacent peaks at an equal distance from the center peak at two opposite sides have the same relative peak height. Since the subcarrier period is known, the distances in terms of delays from the center that the adjacent peaks will occur can be predetermined. The VE and VL correlators are placed equally at these delay offsets on two sides of the prompt.

As the side peak heights fall off symmetrically on both sides of the main peak, the VE and VL correlators must get equal values of peak when the prompt is aligned with the incoming code, at the center of the envelope. If the prompt version has coincided with a peak based on the E-L gates, but still a comparison indicates a higher amplitude on either VE or VL, it means the prompt has got latched to a side peak and hence the codes need to be shifted for balancing. However, because the peaks occur at intervals equal to the period of the subcarrier, T_s, the receiver must make the appropriate jump of either +T_s or −T_s, in the direction of the correct peak until the autocorrelation values at VE and VL balance.

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Radio communication techniques

D.I. Crecraft , S. Gergely , in Analog Electronics: Circuits, Systems and Signal Processing, 2002

Demodulation of the DSBSC signal: synchronous detection

Clearly, an envelope detector cannot be used to demodulate the DSBSC signal, because its envelope is a distorted version of the modulating signal's waveform. Instead, the technique of synchronous detection is used. Figure 9.11 shows the block diagram. In essence, the idea is quite simple; in the receiver, the carrier is reintroduced and multiplied by the DSBSC signal. By now you should be getting used to the fact that multiplication of signals gives rise to sum and difference frequencies. Here, the output of the multiplier contains sum and difference frequencies of the DSBSC signal and the carrier. With a sinusoidal modulating signal, the DSBSC signal is, from Eqn (9.6)

Fig. 9.11. Synchronous detection of DSBSC signals.

${\nu }_{\text{in}}=\frac{V_{\text{m}}}{\text{2}}\left\{cos\left({\omega }_{\text{c}}+{\omega }_{\text{m}}\right)t+cos\left({\omega }_{\text{c}}-{\omega }_{\text{m}}\right)t\right\}$

The re-introduced carrier is

${\nu }_{\text{c}}=V_{\text{c}}cos{\omega }_{\text{c}}t$

The product is:

${\nu }_{\text{o}}=\frac{V_{\text{m}}{V}_{\text{c}}}{2}cos{\omega }_{\text{c}}t\left\{cos\left({\omega }_{\text{c}}+{\omega }_{\text{m}}\right)t+cos\left({\omega }_{\text{c}}-{\omega }_{\text{m}}\right)t\right\}$

From the first product, using the result of Eqn (9.6) for the product of two sinusoids, and noting that cos(−ω _m t)=cos ω_m t:

(9.8) $cos{\omega }_{\text{c}}tcos\left({\omega }_{\text{c}}+{\omega }_{\text{m}}\right)t=\frac{1}{2}\left\{cos\left(2{\omega }_{\text{c}}+{\omega }_{\text{m}}\right)t+cos{\omega }_{\text{m}}t\right\}$

From the second product

(9.9) $cos{\omega }_{\text{c}}tcos\left({\omega }_{\text{c}}-{\omega }_{\text{m}}\right)t=\frac{1}{2}\left\{cos\left(2{\omega }_{\text{c}}-{\omega }_{\text{m}}\right)t+cos{\omega }_{\text{m}}t\right\}$

The total output is the sum of these

(9.10) ${\nu }_{\text{o}}=\frac{V_{\text{m}}{V}_{\text{c}}}{2}\left\{\frac{1}{2}cos\left(2{\omega }_{\text{c}}+{\omega }_{\text{m}}\right)t+\frac{1}{2}cos\left(2{\omega }_{\text{c}}-{\omega }_{\text{m}}\right)t+cos{\omega }_{\text{m}}t\right\}$

The first and second terms are side frequencies lying each side of the second harmonic of the carrier, and are easily removed by a low-pass filter. The third term is the required baseband signal.

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