Amplitude Modulation - an overview (original) (raw)
Amplitude modulation consists of translating the varying voltage signal into variations in the amplitude of a carrier sine wave at a frequency of several kilohertz.
From: Measurement and Instrumentation, 2012
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Biomimetic bidirectional hand neuroprostheses for restoring somatosensory and motor functions
Francesco Iberite, ... Silvestro Micera, in Somatosensory Feedback for Neuroprosthetics, 2021
10.6.2 Amplitude modulation
Amplitude modulation is, currently, one of the most precise strategies to deliver tactile information to the prosthesis user (Valle, Petrini, et al., 2018), which was assessed with psychophysical methods for comparison with healthy subjects. The amplitude of neural stimulation controls the section of the nerve influenced by the electrical field produced by the electrode, in this way modulating the number of fibers activated, known as the recruitment. Recruitment can indeed be used to encode sensation magnitude, but, as it is expected from the underlying physiological phenomena, it has the side effect of also influencing the projective field of the elicited sensation referred to the skin. This can lead to a loss of spatial precision in the sensory encoding. As stated before, pulse width concurs with amplitude in defining the charge delivered at the interface by each pulse, and so they can be considered somewhat equivalent. Neither of these approaches, while simple and precise, evoke natural sensation due to synchronous firing (Valle, Mazzoni, et al., 2018), and may induce paresthesia or unpleasant sensations. In order to limit the risk of such side effects, amplitude linear modulation is always defined in an interval strictly limited by the minimal stimulation amplitude perceived and the maximal stimulation amplitude, not causing any discomfort.
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Broadband Interface Concepts
Louis E. FrenzelJr, in Handbook of Serial Communications Interfaces, 2016
Amplitude Modulation
In amplitude modulation, it is the voltage level of the signal to be transmitted that changes the amplitude of the carrier in proportion, see Figure 63.1. With no modulation, the AM carrier is transmitted by itself. When the modulating information signal (a sine wave) is applied, the carrier amplitude rises and falls in accordance. The carrier frequency remains constant during amplitude modulation.
Figure 63.1. Analog amplitude modulation.
Analog amplitude modulation is widely used in radio. AM broadcast stations are, of course, amplitude modulated. So are citizen’s band radios and aircraft radios. A special form of amplitude modulation, known as quadrature amplitude modulation (QAM), is also widely used in modems to transmit digital data over cable or wireless, more on that discussed later.
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Light Polarization and Signal Processing in Chiroptical Instrumentation
Kenneth W. Busch, Marianna A. Busch, in Chiral Analysis (Second Edition), 2018
3.9.1 Amplitude modulation
As discussed previously, amplitude modulation occurs when the amplitude of a high-frequency signal (the carrier) is varied or modulated by a lower frequency signal containing the information to be transmitted. Consider a high-frequency carrier signal given by
(3.23)Carrier=Ac cos ωct
If this signal is multiplied by a low-frequency signal given by
(3.24)Signal=As cos ωst
where Ac>As and ωc≫ωs, the result will be
(3.25){(As cos ωst)Ac}¯cos ωct
Notice now that the amplitude of the high-frequency carrier (indicated by the bar over the terms) is now a function of the low-frequency signal (i.e. the amplitude of the carrier has been modulated by the instantaneous value of As cos ωst). Fig. 3.28 shows the effect of the process graphically.
Figure 3.28. Modulated waveform generated by multiplying the signal waveform by the carrier waveform.
Notice that the modulated waveform given by Eq. (3.25) is a pure AC signal with no DC level. The information that was previously present in the low-frequency signal (Eq. (3.24)) is now encoded in the amplitude of the high-frequency carrier. As a consequence, the information has been shifted from a low-frequency domain (where 1/f noise is dominant) to a high-frequency domain that is less subject to 1/f noise fluctuations in the transmission channel.
Fig. 3.29 shows the process graphically with a computer-generated plot of a modulated waveform obtained by multiplying a low-frequency cosine wave by a higher-frequency cosine wave (the carrier).
Figure 3.29. Computer-generated plot obtained by multiplying a high-frequency cosine wave by a low-frequency cosine wave.
It is clear from the figure that the information that was once present in the lower-frequency signal is now encoded in the variation of the amplitude of the carrier wave with time (i.e. the envelope of the carrier signal).
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Frequency Analysis: The Fourier Transform
Luis Chaparro, in Signals and Systems Using MATLAB (Second Edition), 2015
5.6.1.1 Why Amplitude Modulation?
The use of amplitude modulation to change the frequency content of a message from its baseband frequencies to higher frequencies makes the transmission of the message over the airwaves possible. Let us explore why it is necessary to use AM to transmit a music or a speech signal. Typically, music signals are audible up to frequencies of about 22 kHz, while speech signals typically have frequencies from about 100 Hz to about 5 kHz. Thus music and speech signals are relatively low-frequency signals. When radiating a signal with an antenna, the length of the antenna is about a quarter of the wavelength
λ=3×108fmeters
where f is the frequency in Hz (or 1/sec) of the signal being radiated and 3 × 108 meters per second is the speed of light. Thus if we assume that frequencies up to f = 30 kHz are present in the signal (this would include music and speech in the signal) the wavelength is 10 kilometers and the size of the antenna is 2.5 kilometers—a mile-and-a-half long antenna! Thus, for music or a speech signal to be transmitted with a reasonable-size antenna requires to increase the frequencies present in the signal. Amplitude modulation provides an efficient way to shift an acoustic or speech signal to a desirable frequency.
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Analog Communications
Revised by Michael B. Pursley, in Reference Data for Engineers (Ninth Edition), 2002
AMPLITUDE MODULATION
In amplitude modulation, the frequency components of the modulating signal are translated to occupy a different position in the spectrum. It is essentially a multiplication process in which the time functions that describe the modulating signal and carrier are multiplied together. The following amplitude-modulation systems are discussed.
(A)
Double-sideband suppressed carrier (DSB-SC), also called DSB
(B)
Conventional amplitude modulation (AM)
(C)
Vestigial sideband
(D)
Single sideband (SSB)
Double Sideband (DSB)
In DSB modulation, the message signal g(t), whose Fourier transform is G(jω), is considered to have zero dc component. The product
e(t)=Acg(t)cosωct
represents a double-sideband suppressed-carrier signal, and Ac = amplitude of unmodulated carrier. The radio-frequency envelope follows the waveform of the modulating signal g(t) as shown in Fig. 2. The spectral components of the DSB signal e(t) are given by its Fourier transform
Fig. 2. Double-sideband waveforms.
(From P. F. Panter, Modulation, Noise, and Spectral Analysis, _Fig. 5_–3, © 1965, McGraw-Hill Book Co.)Copyright © 1965
E(jω)=12G[j(ω−ωc)]+12G[j(ω+ωc)]
as shown in Fig. 3. Note that the upper and lower sidebands are translated symmetrically ±ωc about the origin.
Fig. 3. Baseband signal and double-sideband spectra.
(From P. F. Panter, Modulation, Noise, and Spectral Analysis, Fig. 5_–_2, © 1965, McGraw-Hill Book Co.)Copyright © 1965
Conventional Amplitude Modulation (AM)
In amplitude modulation, a dc term is added to the modulating signal g(t). The resulting waveform, shown in Fig. 4, is given by
Fig. 4. Amplitude modulation. The modulating signal is at top and the modulated carrier at bottom.
(From P. F. Panter, Modulation, Noise, and Spectral Analysis, Fig. 5_–_4, © 1965, McGraw-Hill Book Co.)Copyright © 1965
e(t)=[A0+as(t)]cosωct=A0[1+mas(t)]cosωct
where,
a = maximum amplitude of modulating function, g(t) = as(t), |s(t)| ≤ 1,
ma = a/_A_0 = modulation index or degree of modulation, 0 ≤ ma ≤ 1,
_A_0 = amplitude of unmodulated carrier, |mas (t)| ≤ 1, to ensure an undistorted envelope.
Vestigial Sideband
Vestigial-sideband modulation is derived from a DSB signal by passing the output of the product modulator through a filter whose transfer function is Hv(jω), as shown in Fig. 5. The transfer function Hv(jω) of the filter treats the two sidebands of the DSB signal in such a manner as to attenuate one sideband differently from the other. The process of vestigial-sideband modulation by the use of the filter network Hv(jω) may be replaced by the equialent vestigial system shown in Fig. 6, where the transfer functions Hi(jω) and Hq(jω) are given by
Fig. 5. Vestigial-sideband transmission system.
(From P. F. Panter, Modulation, Noise, and Spectral Analysis, Fig. 5_–_7, © 1965, McGraw-Hill Book Co.)Copyright © 1965
Fig. 6. Equivalent vestigial-sideband transmission system.
(From P. F. Panter, Modulation, Noise, and Spectral Analysis, Fig. 5_–_8, © 1965, McGraw-Hill Book Co.)Copyright © 1965
Hi(jω)=12{Hv[j(ω−ωc)]+Hv[j(ω+ωc)]}Hq(jω)=(1/2j){Hv[j(ω−ωc)]−Hv[j(ω+ωc)]}
Single Sideband (SSB)
Single-sideband transmission may be produced in the same manner as vestigial sideband by using a high-pass filter Hs(jω) which completely eliminates all signals on one side of the carrier frequency. The transfer function Hs(jω) of the ideal high-pass filter is defined by
Hs(jω)=[12+12sgn(ω−ωc)]+[12−12sgn(ω−ωc)]
where sgnω is the signum function. The output spectrum Es(jω) is given by
Es(jω)=Hs(jω)E(jω)=12G[j(ω−ωc)][12+12sgn(ω−ωc)]+12G[j(ω+ωc)][12−12sgn(ω+ωc)]
The SSB signal can also be regarded as the resultant of quadrature modulation of a carrier by a pair of signals in phase quadrature. The modulated wave
es(t)=s(t)cosωct−σ(t)sinωct
represents an upper-sideband signal with no spectral components below the carrier angular frequency ωc, where s(t) is an arbitrary message function and σ(t) its harmonic conjugate.
This equation can be written in the form
es(t)=[s2(t)+σ2(t)]1/2cos{ωct+tan−1[σ(t)/s(t)]}=α(t)cos[ωct+ϕ(t)]
regarding the single-sideband signal as a hybrid amplitude-modulated and phase-modulated wave. The envelope α(t) and phase ϕ(t) are related by the analytic signal
ψ(t)=s(t)+jσ(t)=α(t)exp[jϕ(t)]
where σ(t) = s(t), the Hilbert transform of s(t). The amplitude and phase of the complex signal ψ(t) are identical to the envelope and phase of the single-sideband wave. The Fourier transform of the analytic signal ψ(t) is
ψ(jω)=S(jω)+jS(jω)=S(jω)+S(jω)=2S(jω),ω>0=S(jω)−S(jω)=0,ω<0
Thus, a study of single sideband can be made through the analytic signal without reference to the arbitrary carrier frequency ωc.
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Analogue modulation
ProfessorJ E Flood OBE DSc FInstP CEng FIEE, in Telecommunications Engineer's Reference Book, 1993
18.2 Amplitude modulation
18.2.1 Simple amplitude modulation
The simplest form of modulation is amplitude modulation. The modulator causes the envelope of the carrier wave to follow the waveform of the modulating signal and the demodulator recovers it from this envelope.
If a carrier, given by Equation 18.2, is modulated to a depth m by a sinusoidal modulating signal given by Equation 18.3, the resulting AM signal is as in Equation 18.4.
(18.2)vc=Vccosωct
(18.3)vm=Vmcosωmt
(18.4)v=(1+mcosωmt)Vccosωct =Vc[cosωct+12mcos(ωc+ωm)t +12mcos(ωc−ωm)t]
If the modulating signal contains several components, f1, f2, …, etc., then the modulated signal contains fc–f1, fc–f2, …, etc., and fc+f1, fc+f2, …, etc. in addition to fc. If the modulating signal consists of a band of frequencies, as shown in Figure 18.1(a), the modulated signal consists of two sidebands, each occupying the same bandwidth as the baseband signal, as shown in Figure 18.1(b). In the upper sideband, the highest frequency corresponds to the highest frequency in the baseband; this is therefore known as an erect sideband. In the lower sideband, the highest frequency corresponds to the lowest frequency in the baseband; this is known as an inverted sideband.
Figure 18.1. Frequency spectra for amplitude modulation: (a) baseband signal; (b) simple amplitude modulation (AM); (c) double sideband suppressed carrier (DSBSC) modulation; (d) Single sideband suppressed carrier (SSBSC) modulation; (e) vestigial sideband (VSB) modulation
Simple amplitude modulation makes inefficient use of the transmitted power, as information is transmitted only in the sidebands but the majority of the power is contained in the carrier. If a carrier as in Equation 18.2 is modulated to a depth m by the sinusoidal baseband signal of Equation 18.3, the output power is given by Equation 18.5.
(18.5)v2¯=Vc2(12+18m2+18m2) =12Vc2(1+12m2)
The maximum sideband power is obtained with 100% depth of modulation. The power in the sidebands is then a third of the total transmitted power. For smaller modulation depths, it is even less.
One method of producing AM is to add the baseband signal to the carrier and apply them to a non-linear amplifier, as shown in Figure 18.2. If the input/output characteristic of the non-linear circuit is given by Equation 18.6 and its input voltage by Equation 18.7, then the output voltage is as in Equation 18.8.
Figure 18.2. Low level amplitude modulator
(18.6)vo=ao+a1vi+a2vi2+……
(18.7)vi=vmcosωmt+Vccosωct
(18.8)vo=ao+a2(Vm2+Vc2) +a1(Vmcosωmt+Vccosωct) +a2(12Vm2cos2ωmt+12Vc2cos2ωct) +VcVm[cos(ωc−ωm)t +cos(ωc+ωm)t])+……..
The bandpass filter is required to remove all components except those which comprise the AM wave. These are shown in bold in the above equation.
Another form of modulator is shown in Figure 18.3. This uses a gain controlled amplifier whose input signal is the carrier and whose control voltage is the modulating signal. For example, the carrier may be applied to the base of a transistor and the modulating signal superimposed on the collector supply voltage.
Figure 18.3. High level amplitude modulator
If the gain of the amplifier is given by Equation 18.9 and the carrier and modulating waveforms by Equations 18.2 and 18.3 then the output voltage is given by Equation 18.10, which is the required AM wave.
(18.9)A=Ao(1+kvm)
(18.10)vo=AoVccosωct+kAoVmVccosωmtcosωct =AoVc(cosωct+12kVm[cos(ωo-ωm)t+cos(ωc+ωm)t])
The first method is used for low level modulators before the power amplifier of a transmitter. The second method is used for high level modulators in the final stage of amplification.
An AM wave can be demodulated by the simple diode circuit shown in Figure 18.4. The rectified output voltage across the load resistor follows the envelope of the modulated input signal as in Figure 18.5. The time constant CR must be large compared with the period of the carrier to prevent the output voltage decaying substantially between the peaks of the carrier. However, time constant CR in Figure 18.4 must be sufficiently small for the output voltage to decay as rapidly as the envelope changes when the baseband signal has its maximum frequency, Fm. If fc >> Fm, this is easily arranged.
Figure 18.4. Envelope demodulator circuit
Figure 18.5. Envelope demodulator waveforms: (a) input; (b) output
Other demodulators, which give a better performance at poor input signal/noise ratios, are the coherent demodulator (described later) and the phase locked loop AM demodulator (Gardner, 1979;Gosling, 1986).
18.2.2 Suppressed carrier modulation
It is possible, by using a balanced modulator (Tucker, 1953) to eliminate the carrier and generate only the sidebands, as shown in Figure 18.1(c). This is known as double sideband suppressed carrier modulation (DSBSC). The modulator acts as a switch, which multiplies the baseband signal by a quasi square wave carrier. Its output thus contains upper and lower sidebands about the fundamental and harmonics of the carrier frequency. As shown in Figure 18.6(a), a bandpass filter is used to remove all components except the wanted sidebands. Thus, for a sinusoidal baseband input signal, the output signal is given by Equation 18.11.
Figure 18.6. Applications of balanced modulator: (a) product modulator; (b) coherent demodulator
(18.11)vo=vmcosωmtcosωct =12Vm(cos(ωc−ωm)t+cos(ωc+ωm)t)
A balanced modulator used in this way is often called a product modulator.
To demodulate a DSBSC signal, it is necessary to use a coherent demodulator, consisting of a balanced modulator supplied with a locally generated carrier as shown in Figure 18.6(b) instead of the envelope demodulator used with simple AM. If the incoming DSBSC signal is given by Equation 18.12 and the coherent demodulator multiplies this with a local carrier given by Equation 18.13, its output voltage is as in Equation 18.14.
(18.12)vi=12mVc(cos(ωc+ωm)t+cos(ωc−ωm)t)
(18.13)vc=cos(ωct+θ)
(18.14)v=14mVc(cos[(2ωc+ωm)t+θ] +cos[(2ωc−ωm)t+θ] +cos[θ+ωmt)+cos(θ−ωmt))
The components at frequencies (2ω_c_ ± ω_m_) are removed by a low pass filter and the baseband output signal is given by Equation 18.15.
(18.15)vo=14mVc(cos(θ+ωmt)+cos(θ−ωmt)) =12mVccosθcosωmt
Thus vo represents the original baseband signal, provided that the phase θ of the local carrier is stable.
A further economy in power, and a halving in bandwidth, can be obtained by producing a single sideband suppressed carrier (SSBSC) signal, as shown in Figure 18.1(d). If the upper sideband is used, the effect of the modulator is simply to produce a frequency translation of the baseband signal to a position in the frequency spectrum determined by the carrier frequency. If the lower sideband is used, the band is inverted as well as translated.
The SSBSC signal requires the minimum possible bandwidth for transmission. Consequently, the method is used whenever its complexity is justified by the saving in bandwidth (Pappenfus, 1964). An important example is the use of SSBSC for multichannel carrier telephone systems (Kingdom, 1991). An error in the frequency of the local carrier of the demodulator results in a corresponding shift in the frequencies of the components in the baseband output signal. For speech transmission, frequency shifts of the order of ±10Hz are not noticeable, but the errors that can be tolerated for telegraph and data transmission are less. The CCITT specifies that the frequency shift should be less than ±2Hz.
A SSBSC signal can be generated by using a balanced modulator and a bandpass filter, as for DSBSC. However, the filter in Figure 18.6(a) is designed to pass only one of the sidebands, instead of both.
An alternative method of generating a SSBSC signal is the quadrature method shown in Figure 18.7. This uses two product modulators. The baseband signal, vm, and the carrier, vc, are applied to one directly, and to the other after a phase shift of 90°. If the carrier and modulating signals are as in Equations 18.2 and 18.3, the outputs of the two modulators are given by Equations 18.16 and 18.17 and vo by Equation 18.18.
Figure 18.7. Quadrature method of SSBSC generation
(18.16)v1=Vmsinωmtsinωct
(18.17)v2=Vmcosωmtcosωct
(18.18)vo=v1+v2=Vmcos(ωc−ωm)t
Alternatively, subtracting v1 from v2 gives Equation 18.19.
(18.19)vo=Vmcos(ωc+ωm)t
It is straightforward to provide a 90° phase shift at the single carrier frequency, but not over the band of frequencies of a modulating signal. Instead, the baseband signal is applied to the modulators through a pair of networks whose phase shifts differ by approximately 90° over the required band (Coates, 1975). The quadrature method is not widely used, since small phase errors result in unwanted frequency components in the output signal.
A coherent demodulator is required for demodulating a SSBSC signal. Since only one sideband is present, the demodulated output signal, from Equation 18.15, is as in Equation 18.20.
(18.20)vo=14mVccos(ωmt±θ)
If a SSBSC signal is corrupted by noise, the coherent demodulator acts on each component of the noise spectrum in the same way as it does on each component of the signal. Consequently, the signal to noise ratio at the output of the demodulator is the same as the input signal to noise ratio.
For a DSBSC signal, the output of the demodulator produced by a signal component at fc + fm is in phase with that produced by the component at fc – fm. Thus, the output signal voltage is twice the input signal voltage (c.f. Equation 18.15 with θ = 0) and the output signal power is four times that for SSBSC. For noise, the outputs produced by components at fc – fm and fc + fm have a random phase difference. Thus, the output noise power is twice the input noise power. Consequently, the output signal to noise ratio is twice the input signal to noise ratio, an apparent improvement of 3dB over SSBSC. However, the bandwidth required for DSBSC transmission is twice that for SSBSC. This doubles the received noise power, so no overall improvement is obtained.
For simple AM, a coherent demodulator acts in the same way as for DSBSC. So does the envelope demodulator of Figure 18.4 at good signal to noise ratios. The diode acts as a switch operated by the incoming carrier and this multiplies the input signal by cos ω_c_ t, as does a coherent demodulator.
At poor input signal to noise ratios, this is no longer true. The diode is switched by peaks of the noise voltage and the output signal to noise ratio is worse than for a coherent demodulator (Brown and Glazier, 1974; Coates, 1975). The signal to noise ratio performance of simple AM is, of course, worse than that of DSBSC because the input power includes that of the carrier as well as the sidebands. For 100% depth of modulation, the sideband power is only a third of the total power. Thus, for equal transmitter powers and receiver noise power densities, simple AM gives an output signal to noise ratio 4.8dB worse than for DSBSC or SSBSC transmission.
By using SSBSC, it is possible to transmit two channels through the bandwidth needed by simple AM for a single channel; one uses the upper sideband of the carrier and the other uses the lower sideband. This is known as independent sideband modulation and is used in h.f. radio communication (Hills, 1973). However, it is also possible to do this using DSBSC. The transmitter uses two modulators whose carriers are in quadrature. The receiver uses two coherent demodulators whose local carriers are in quadrature. Equation 18.15 shows that each demodulator produces a full output from the signal whose carrier is in phase (since cos 0=1) and zero output from the signal whose carrier is in quadrature (since cos π/2=0). This is called quadrature amplitude modulation (QAM). In practice, the method is not used for analogue baseband signals since small errors in the phases of the local carriers cause a fraction of the signal of each channel to appear as crosstalk in the output from the other. However, the method is used for transmitting digital signals.
18.2.3 Vestigial sideband modulation
If the sideband signal extends down to very low frequencies, as in television, it is almost impossible to suppress the whole of the unwanted sideband without affecting low frequency components in the wanted sideband. Use is then made of vestigial sideband (VSB) transmission instead of SSBSC. A conventional AM signal (as shown in Figure 18.1(b)) is first generated and this is then applied to a filter having a transition between its pass and stop band that is skew symmetric about the carrier frequency. This results in an output signal having the spectrum shown in Figure 18.1(e).
If a coherent demodulator is used, the original baseband signal can be recovered without distortion. It is also possible to use a simple envelope demodulator for VSB, but some non linear distortion then results (Black, 1953). VSB transmission does, of course, require a greater channel bandwidth than SSB. However, for a wideband signal such as television, the bandwidth saving compared with DSB is considerable.
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Fourier Transform Properties
Steven W. Smith, in Digital Signal Processing: A Practical Guide for Engineers and Scientists, 2003
Multiplying Signals (Amplitude Modulation)
An important Fourier transform property is that convolution in one domain corresponds to multiplication in the other domain. One side of this was discussed in the last chapter: time domain signals can be convolved by multiplying their frequency spectra. Amplitude modulation is an example of the reverse situation: multiplication in the time domain corresponds to convolution in the frequency domain. In addition, amplitude modulation provides an excellent example of how the elusive negative frequencies enter into everyday science and engineering problems.
Audio signals are great for short-distance communication; when you speak, someone across the room hears you. On the other hand, radio frequencies are very good at propagating long distances. For instance, if a 100-volt, 1-MHz sine wave is fed into an antenna, the resulting radio wave can be detected in the next room, the next country, and even on the next planet. Modulation is the process of merging two signals to form a third signal with desirable characteristics of both. This always involves nonlinear processes such as multiplication; you can't just add the two signals together. In radio communication, modulation results in radio signals that can propagate long distances and carry along audio or other information.
Radio communication is an extremely well-developed discipline, and many modulation schemes have been developed. One of the simplest is called amplitude modulation. Figure 10-14 shows an example of how amplitude modulation appears in both the time and frequency domains. Continuous signals will be used in this example, since modulation is usually carried out in analog electronics. However, the whole procedure could be carried out in discrete form if needed (the shape of the future!).
FIGURE 10-14. Amplitude modulation. In the time domain, amplitude modulation is achieved by multiplying the audio signal, (a), by the carrier signal, (c), to produce the modulated signal, (e). Since multiplication in the time domain corresponds to convolution in the frequency domain, the spectrum of the modulated signal is the spectrum of the audio signal shifted to the frequency of the carrier.
Figure (a) shows an audio signal with a DC bias such that the signal always has a positive value. Figure (b) shows that its frequency spectrum is composed of frequencies from 300 Hz to 3 kHz, the range needed for voice communication, plus a spike for the DC component. All other frequencies have been removed by analog filtering. Figures (c) and (d) show the carrier wave, a pure sinusoid of much higher frequency than the audio signal. In the time domain, amplitude modulation consists of multiplying the audio signal by the carrier wave. As shown in (e), this results in an oscillatory waveform that has an instantaneous amplitude proportional to the original audio signal. In the jargon of the field, the envelope of the carrier wave is equal to the modulating signal. This signal can be routed to an antenna, converted into a radio wave, and then detected by a receiving antenna. This results in a signal identical to (e) being generated in the radio receiver's electronics. A detector or demodulator circuit is then used to convert the waveform in (e) back into the waveform in (a).
Since the time domain signals are multiplied, the corresponding frequency spectra are convolved. That is, (f) is found by convolving (b) and (d). Since the spectrum of the carrier is a shifted delta function, the spectrum of the modulated signal is equal to the audio spectrum shifted to the frequency of the carrier. This results in a modulated spectrum composed of three components: a carrier wave, an upper sideband, and a lower sideband.
These correspond to the three parts of the original audio signal: the DC component, the positive frequencies between 0.3 and 3 kHz, and the negative frequencies between −0.3 and −3 kHz, respectively. Even though the negative frequencies in the original audio signal are somewhat elusive and abstract, the resulting frequencies in the lower sideband are as real as you could want them to be. The ghosts have taken human form!
Communication engineers live and die by this type of frequency domain analysis. For example, consider the frequency spectrum for television transmission. A standard TV signal has a frequency spectrum from DC to 6 MHz. By using these frequency shifting techniques, 82 of these 6-MHz wide channels are stacked end-to-end. For instance, channel 3 is from 60 to 66 MHz, channel 4 is from 66 to 72 MHz, channel 83 is from 884 to 890 MHz, etc. The television receiver moves the desired channel back to the DC to 6 MHz band for display on the screen. This scheme is called frequency domain multiplexing.
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Two-dimensional nanomaterials as enhanced surface plasmon resonance sensing platforms: Design perspectives and illustrative applications
Yufeng Yuan, ... Junle Qu, in Biosensors and Bioelectronics, 2023
2.2.1 Amplitude modulation
Amplitude modulation, also called intensity modulation, is the simplest approach, which is performed to directly measure the intensity of p-polarized reflected light at a fixed incidence angle and excitation wavelength (Haider et al., 2018; Homola, 2008). According to Equation (4), the intensity is proportional to the square of amplitude (|rpmd|2). As the intensity of reflected light can reflect the resonance strength between the incident light and SPPs wave, it can serve as SPR signal output. The variation in obtained SPR signal is proportional to local RI variation approaching the metal-dielectric interface. To improve its performance, one needs to fix and optimize both incident angle and excitation wavelength to obtain the best conditions.
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Analog Communications
Revised by Michael B. Pursley, in Reference Data for Engineers (Ninth Edition), 2002
Conventional Amplitude Modulation (AM)
In amplitude modulation, a dc term is added to the modulating signal g(t). The resulting waveform, shown in Fig. 4, is given by
Fig. 4. Amplitude modulation. The modulating signal is at top and the modulated carrier at bottom.
(From P. F. Panter, Modulation, Noise, and Spectral Analysis, Fig. 5_–_4, © 1965, McGraw-Hill Book Co.)Copyright © 1965
e(t)=[A0+as(t)]cosωct=A0[1+mas(t)]cosωct
where,
a = maximum amplitude of modulating function, g(t) = as(t), |s(t)| ≤ 1,
ma = a/_A_0 = modulation index or degree of modulation, 0 ≤ ma ≤ 1,
_A_0 = amplitude of unmodulated carrier, |mas (t)| ≤ 1, to ensure an undistorted envelope.
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Measurement Signal Transmission
Alan S. Morris, Reza Langari, in Measurement and Instrumentation, 2012
10.2.3 Transmission Using an a.c. Carrier
Another solution to the problem of noise corruption in low-level d.c. voltage signals is to transfer the signal onto an a.c. carrier system before transmission and extract it from the carrier at the end of the transmission line. Both amplitude modulation (AM) and frequency modulation (FM) can be used for this.
Amplitude modulation consists of translating the varying voltage signal into variations in the amplitude of a carrier sine wave at a frequency of several kilohertz. An a.c. bridge circuit is used commonly for this as part of the system for transducing the outputs of sensors that have a varying resistance (R), capacitance (C), or inductance (L) form of output. Referring back to Equations (9.14) and (9.15) in Chapter 9, for a sinusoidal bridge excitation voltage of Vs=Vsin(ωt), the output can be represented by Vo=FVsin(ωt). Vo is a sinusoidal voltage at the same frequency as the bridge excitation frequency, and its amplitude, FV, represents the magnitude of the sensor input (R, C, or L) to the bridge. For example, in the case of Equation (9.15):
FV=(LuL1+Lu−R3R2+R3)V.
After shifting the d.c. signal onto a high-frequency a.c. carrier, a high-pass filter can be applied to the AM signal. This successfully rejects noise in the form of low-frequency drift voltages and mains interference. At the end of the transmission line, demodulation is carried out to extract the measurement signal from the carrier.
Frequency modulation achieves even better noise rejection than AM and involves translating variations in an analogue voltage signal into frequency variations in a high-frequency carrier signal. A suitable voltage-to-frequency conversion circuit is shown in Figure 10.3 in which the analogue voltage signal input is integrated and applied to the input of a comparator preset to a certain threshold voltage level. When this threshold level is reached, the comparator generates an output pulse that resets the integrator and is also applied to a monostable. This causes the frequency, f, of the output pulse train to be proportional to the amplitude of the input analogue voltage.
Figure 10.3. Voltage-to-frequency convertor.
At the end of the transmission line, the FM signal is usually converted back to an analogue voltage by a frequency-to-voltage converter. A suitable conversion circuit is shown in Figure 10.4 in which the input pulse train is applied to an integrator that charges up for a specified time. The charge on the integrator decays through a leakage resistor, and a balance voltage is established between the input charge on the integrator and the decaying charge at the output. This output balance voltage is proportional to the input pulse train at frequency f.
Figure 10.4. Frequency-to-voltage convertor.
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