The starting point for determining the vibration spectra for harmonic angular modulation is expression (1.27). To simplify the expressions, we accept and rewrite (1.27) in the form
Expression (1.28) represents the sum of two quadrature frequency oscillations, each of which is modulated in amplitude by frequency. Angular modulation is usually divided into narrowband and broadband. Broadband is most widespread in communications technology. Let's start by determining the spectrum of narrowband angular modulation. Assuming we have
Thus, the spectrum of narrowband angle modulation signals is similar to the spectrum of the simplest AM oscillation shown in Fig. 1.2. It contains components of the carrier frequency and two side frequencies. The parameter that determines the amplitudes of the side frequencies is the modulation index. The width of the spectrum of narrowband angular modulation is the same as with AM: it is equal to twice the modulation frequency.
Despite the identity of the spectra, the vibration in question differs from the AM vibration, which is a consequence of the difference in the signs (i.e., in the phase shift by 180°) of the components of the lower side frequency in expressions (1.30) and (1.10). This means that it is possible to convert AM oscillations into narrowband FM oscillations by rotating the phase of one of the side frequencies by 180°. To illustrate what has been said in Fig. 1.8a a vector diagram of AM oscillations is constructed. Changing the phase of the lower side frequency by 180°, we obtain the vector diagram in Fig. 1.86, in which the end of the vector of the resulting oscillation moves with a low frequency along a horizontal line, which corresponds to a change in phase. In this case, the amplitude also changes somewhat. However, the change in amplitude is negligible. According to Fig. 1.8 b. Replacing the tangents with their arguments at small values, we obtain the phase change corresponding to the FM oscillation.
For broadband angular modulation, both expressions (1.29) and (1.30) are invalid. We have to determine the vibration spectrum directly from (1.28). The expressions are periodic functions of frequency and therefore they can be expanded into Fourier series. The first of these functions is even, the second is odd. In Bessel theory
functions it is proved that the Fourier series for these functions have the form
where the Bessel function of the first kind of order on the argument In Fig. 1.9 shows the graphs of the Bessel functions. Substituting (1.31) into (1.28), we get
Thus, the spectrum of FM and FM oscillations modulated by a harmonic signal turns out to be discrete, relatively symmetrical and containing an infinite number of side frequencies of the form with amplitudes For, it is plotted in Fig. 1.10. The relationships between Bessel functions of different orders, and, consequently, between the amplitudes of various side components are determined by the modulation index. At some values, individual components may disappear (if the same applies to the carrier amplitude goes to zero at
The presence of an infinitely large number of side components of the spectrum means that theoretically the spectrum of PM and FM oscillations is infinitely wide. However, the Bessel function, starting from some, quickly decreases with growth, as can be seen in Fig. 1.9 and 1.10. This makes it possible to limit the useful (practical) spectrum of such signals to a certain number of side frequencies. When limiting the spectrum, it is necessary to take into account the influence of two contradictory factors: in a narrower frequency band, the influence of interference is weakened, but at the same time signal distortion increases due to the lack of omitted components. In practice, a compromise solution is chosen.
If we limit ourselves in the spectrum to side components, the amplitudes of which do not exceed the maximum amplitude of the spectral component (see Fig. 1.10), then for each we can calculate the corresponding spectrum width. It will turn out to be slightly larger than from Fig. 1.10 it follows that at For 4 the spectrum width At large modulation indices (of the order of tens
and hundreds) the practical spectrum width calculated in this way is close to twice the frequency deviation
Concluding our consideration of the issue of the width of the spectrum of harmonic angular modulation signals, we emphasize its difference from the frequency interval within which the instantaneous frequency of the signal changes:
1) theoretical spectrum width
2) its practical value at turns out to be slightly higher and is considered only approximately equal to it (1.33).
Let us consider the influence of the parameters of the modulating signal on the spectra of PM and FM oscillations, using the approximate expression (1.33) to determine the spectrum width. When the amplitude X of the modulating signal changes, the spectra of PM and FM oscillations change equally. As X increases, the modulation index increases proportionally, and the spectra expand due to an increase in the number of spectral components.
A change in the frequency of the modulating oscillation has different effects on the change in the spectra of PM and FM oscillations. With PM, the change does not affect the value of the modulation index, and therefore the number of spectral components (Fig. 1.11a, b).
When FM decreases, the modulation index increases, which leads to an increase in the number of spectral components (Fig. 1.11 c, d). As a result, the width of the spectrum of FM oscillations is almost independent of frequency, and with FM it changes proportionally
Instead of amplitude modulation, as in AM, DSBSC and SSB, information can be transmitted by modulating the frequency or phase of the carrier signal:
FM and PM are closely related and are sometimes referred to together as “angle modulation.” FM is well known as the type of modulation used in the 88-108 MHz microwave broadcast band (VHF band), while AM is used in the MHz broadcast band. Anyone who has a tuned FM receiver has probably noticed the "calming" of background noise in FM reception. This property (increasing ratio or increasing channel) is what makes broadband FM preferable to AM for high-quality transmissions.
More about FM: if the frequency deviation is large compared to the modulating frequency (the highest frequencies are retained), you have “wideband FM”, as in the VHF broadcast band. Modulation index, equal to the ratio of frequency deviation to the modulating frequency, in this case is greater than one. Wideband FM is preferable because, under the right reception conditions, it increases by 6 dB for every doubling of FM deviation. True, this increases the channel bandwidth, since the signal during wideband FM occupies approximately , where is the maximum deviation of the carrier frequency. FM broadcasting in the band 88-108 MHz uses a maximum deviation/dev, i.e. each station occupies a band of about . This explains why wideband FM is not used, for example, in the medium wave AM band (MHz): in this case, only six stations in a given broadcasting zone could operate in the entire range.
Rice. 13.44. Broadband FM spectrum.
World Cup spectrum.
The spectrum of a carrier wave frequency-modulated by a sine wave is similar to that shown in Fig. 13.44. Numerous side frequencies are separated from the carrier frequency at distances that are multiples of the modulating frequency, and their amplitudes are determined by Bessel functions. The number of significant sidebands, roughly speaking, corresponds to the modulation index. For narrowband FM (modulation index) there is only one sideband on each side of the carrier frequency. Outwardly this is similar to the AM spectrum, but if you take into account the phase of the sidebands, it turns out that these waves have a constant amplitude and a variable frequency, rather than a constant frequency and Variable amplitude (AM) In wideband FM, the amplitude of the carrier can be very small, resulting in high FM efficiency, which means that most of the transmitted energy is contained in the side frequencies that carry information.Generation and detection.
FM is easily obtained by changing the parameters of the elements of the configurable generator circuit; A varicap (diode used as a voltage-controlled capacitance (sec.) is ideal here. Other methods involve integrating the modulating signal followed by phase modulation. In each case, it is better to modulate at small deviations, and then apply frequency multiplication to increase the index modulation This is based on the fact that the rate of frequency deviation does not change as the frequency is multiplied, while the value of the deviation itself is multiplied along with the carrier frequency.For detection, a conventional superheterodyne receiver with two features is used. The first is the presence of a limiter in the final IF amplification stage, at this stage the amplitude is constant (saturation). The second is that the detector following the limiter (called a discriminator) must convert frequency deviations into amplitude. Here are some common detection methods.
1. “The detector is just a parallel LC circuit tuned to one side with respect to the intermediate frequency; as a result, it produces an increasing sensitivity curve as a function of frequency throughout the entire IF band; in this case, FM is converted to AM, and a conventional detector then converts AM to audio frequencies. Improved tilt detectors use a balanced pair of -circuits tuned symmetrically about the central IF.
2. The Foster-Seely detector, or its variant "ratio detector", consists of a single resonant circuit connected to a fiendishly clever diode device to produce a linear amplitude-frequency output across the entire IF bandwidth. Such discriminators are superior to simple tilt detectors (Fig. 13.45).
3. Phase-locked loop (PLL). This device changes the frequency of the internal voltage-controlled oscillator to match the frequency of the output signal; it was described in Sect. 9.31. If there is an IF signal at its input, then the voltage in the PLL circuit that controls the generator linearly depends on the frequency of the input signal, i.e. it can be used as an audio frequency output.
4. An averaging circuit in which the IF signal is converted into a sequence of identical pulses having the frequency of the input signal.
Rice. 13.45. FM discriminators. A-fractional detector; B-balanced quadrature detector.
As a result of averaging this sequence of pulses, a signal is generated at the output that is proportional to the IF, i.e., the audio signal added to some constant component.
5. A “balanced quadrature detector” is a combination of a phase detector (see Sections 9.27 and 9.31) and a phase-shifting circuit. The IF signal is passed through a circuit in which the phase shift varies linearly with frequency in the IF passband (-circuits perform this function perfectly). The phase-shifted and primary signals are fed to a phase detector, the output of which changes the signal in proportion to the relative phase shift. This output is the desired sound signal (Fig. 13.45).
It is often pointed out that FM, if the channel has a sufficient ratio , provides reception with significantly less noise compared to AM, where interference decreases little with increasing signal power. Recall that this becomes noticeable if FM signals are amplitude limited before detection. In this case, the system becomes relatively insensitive to interfering signals and noise, which appear as amplitude changes superimposed on the transmitted signal.
The methods of analyzing primary signals discussed above make it possible to determine their spectral and energy characteristics. Primary signals are the main carriers of information. At the same time, their spectral characteristics do not correspond to the frequency characteristics of transmission channels of radio engineering information systems. As a rule, the energy of primary signals is concentrated in the low frequency region. For example, when transmitting speech or music, the energy of the primary signal is concentrated approximately in the frequency range from 20 Hz to 15 kHz. At the same time, the UHF range, which is widely used for transmitting information and music programs, occupies frequencies from 300 to 3000 megahertz. The problem arises of transferring the spectra of primary signals to the appropriate radio frequency ranges for transmitting them over radio channels. This problem is solved through the modulation operation.
Modulation is the procedure for converting low-frequency primary signals into radio frequency signals.
The modulation procedure involves a primary signal and some auxiliary oscillation, called carrier vibration or simply a carrier. In general, the modulation procedure can be represented as follows
where is the rule for converting (operator) the primary signal into a modulated oscillation.
This rule indicates which parameter (or several parameters) of the carrier oscillation changes according to the law of change. Since it controls the change in parameters, then, as noted in the first section, the signal is control (modulating), and is modulated by signals. Obviously, it corresponds to the operator of the generalized block diagram of RTIS.
Expression (4.1) allows us to classify the types of modulation, which is presented in Fig. 4.1.
Rice. 4.1
As classification features, we will choose the type (shape) of the control signal, the shape of the carrier vibration and the type of the controlled parameter of the carrier vibration.
In the first section, the classification of primary signals was carried out. In radio engineering information systems, continuous and digital signals are most widely used as primary (control) signals. In accordance with this, by the type of control signal we can distinguish continuous And discrete modulation.
Harmonic oscillations and pulse sequences are used as carrier oscillations in practical radio engineering. In accordance with the shape of the carrier vibrations, they are distinguished harmonic carrier modulation And pulse modulation.
And finally, according to the type of controlled parameter of the carrier oscillation in the case of a harmonic carrier, they distinguish amplitude, frequency And phase modulation. Obviously, in this case, the amplitude, frequency or initial phase of the harmonic oscillation act as the controlled parameter, respectively. If a pulse sequence is used as a carrier oscillation, then an analogue of frequency modulation is pulse width modulation, where the controlled parameter is the pulse duration, and the analogue of phase modulation is time pulse modulation, where the controlled parameter is the position of the pulse on the time axis.
In modern radio systems, harmonic oscillation is most widely used as a carrier oscillation. Considering this circumstance in the future, the main attention will be paid to signals with continuous and discrete modulation of a harmonic carrier.
4.2. Continuous Amplitude Modulation Signals
Let’s begin our consideration of modulated signals with signals in which the variable parameter is amplitude carrier vibration. The modulated signal in this case is amplitude modulated or amplitude modulated signal (AM signal).
As noted above, the main attention will be paid to signals whose carrier oscillation is a harmonic oscillation of the form
where is the amplitude of the carrier vibration,
– frequency of the carrier vibration.
As modulating signals, we will first consider continuous signals. Then the modulated signals will be signals with continuous amplitude modulation. Such a signal is described by the expression
where is the envelope of the AM signal,
– amplitude modulation coefficient.
From expression (4.2) it follows that the AM signal is the product of the envelope and the harmonic function. The amplitude modulation coefficient characterizes modulation depth and in the general case is described by the expression
. (4.3)
Obviously, when the signal is simply a carrier wave.
For a more detailed analysis of the characteristics of AM signals, let’s consider the simplest AM signal, in which a harmonic oscillation acts as a modulating signal
, (4.4)
where , are the amplitude and frequency of the modulating (control) signal, respectively, and . In this case, the signal is described by the expression
, (4.5)
and is called a single-tone amplitude modulation signal.
In Fig. 4.2 shows the modulating signal, the carrier frequency oscillation and the signal.
For such a signal, the amplitude modulation depth coefficient is equal to
Using the well-known trigonometric relation
after simple transformations we get
Expression (4.6) establishes the spectral composition of a single-tone AM signal. The first term represents the unmodulated oscillation (carrier oscillation). The second and third terms correspond to new harmonic components resulting from modulation of the amplitude of the carrier vibration; the frequencies of these vibrations 
And 
are called the lower and upper side frequencies, and the components themselves are called the lower and upper side components.
The amplitudes of these two oscillations are the same and amount to
, (4.7)
In Fig. Figure 4.3 shows the amplitude spectrum of a single-tone AM signal. From this figure it follows that the amplitudes of the lateral components are located symmetrically relative to the amplitude and initial phase of the carrier vibration. Obviously, the spectrum width of a single-tone AM signal is equal to twice the frequency of the control signal
In the general case, when the control signal is characterized by an arbitrary spectrum concentrated in the frequency band from to , the spectral character of the AM signal is not fundamentally different from a single-tone signal.
In Fig. Figure 4.4 shows the spectra of the control signal and the signal with amplitude modulation. Unlike a single-tone AM signal, the spectrum of an arbitrary AM signal includes lower and upper sidebands. In this case, the upper sideband is a copy of the spectrum of the control signal, shifted along the frequency axis by
value, and the lower side stripe is a mirror image of the upper one. Obviously, the spectrum width of an arbitrary AM signal
those. equal to twice the upper limit frequency of the control signal.
Let's return to the single-tone amplitude modulation signal and find its energy characteristics. The average power of the AM signal over the period of the control signal is determined by the formula:
. (4.9)
Since , a , let's put 
, Where . Substituting expression (4.6) into (4.9), after simple but rather cumbersome transformations, taking into account the fact that and using trigonometric relations
Here, the first term characterizes the average power of the carrier vibration, and the second – the total average power of the lateral components, i.e.
Since the total average power of the side components is divided equally between the lower and upper, which follows from (4.7), it follows
Thus, more than half the power is spent on transmitting the carrier wave in an AM signal (taking into account that) than on transmitting side components. Since the information is contained precisely in the side components, the transmission of the carrier vibration component is impractical from an energy point of view. The search for more efficient methods of using the principle of amplitude modulation leads to balanced and single-sideband amplitude modulation signals.
4.3. Balanced and SSBAM signals
Balanced amplitude modulation (BAM) signals are characterized by the absence of a carrier vibration component in the spectrum. Let us proceed immediately to the consideration of single-tone balanced modulation signals, when a harmonic signal of the form (4.4) acts as a control oscillation. Elimination from (4.6) of the carrier vibration component
leads to results
Let's calculate the average power of the balanced modulation signal. Substituting (4.12) into (4.9) after transformations gives the expression
.
It is obvious that the energy gain when using balanced modulation signals compared to classical amplitude modulation will be equal to
When this gain is .
In Fig. Figure 4.5 shows one of the options for the block diagram of a balanced amplitude modulation signal generator. The shaper contains:
- Inv1, Inv2 – signal inverters (devices that change the polarity of voltages to the opposite);
- AM1, AM2 – amplitude modulators;
- SM – adder.
The carrier frequency oscillation is supplied to the inputs of modulators AM1 and AM2 directly. As for the control signal, it is supplied directly to the second input AM1, and to the second input AM2 through the inverter Inv1. As a result, oscillations of the form are formed at the outputs of the modulators
The inputs of the adder receive oscillations and 
. The resulting signal at the output of the adder will be
In the case of single-tone amplitude modulation, expression (4.13) takes the form
Using the formula for the product of cosines, after transformations we get
which coincides with (4.12) up to a constant factor. Obviously, the width of the spectrum of BAM signals is equal to the width of the spectrum of AM signals.
Balanced amplitude modulation eliminates the transmission of carrier vibration, which leads to energy gain. However, both sidebands (sidebands in the case of single-tone AM) carry the same information. This suggests the advisability of generating and transmitting signals with one of the sidebands suppressed. In this case, we come to single-sideband amplitude modulation (SAM).
If we exclude one of the side components from the spectrum of the BAM signal (say, the upper side component), then in the case of a harmonic control signal we obtain
Since the average power of the BAM signal is divided equally between the side components, it is obvious that the average power of the OAM signal will be
The energy gain compared to amplitude modulation will be
and when it will be equal to .
The formation of a single-sideband AM signal can be carried out on the basis of balanced modulation signal conditioners. The block diagram of a single-sideband AM signal shaper is shown in Fig. 4.6.
The single-sideband amplitude modulation signal conditioner includes:
The following signals are received at the inputs of BAM1:
Then at its output, in accordance with (4.15), a signal is generated
The inputs of BAM2 receive signals
And 
.
An oscillation is removed from the output of BAM2, described in accordance with (4.14) with the replacement of cosines by sines
Taking into account the known trigonometric relation
the BAM2 output signal is converted to the form
Adding signals (4.17) and (4.18) in the adder SM gives
which coincides with (4.16) up to a constant factor. As for the spectral characteristics, the width of the spectrum of OAM signals is half that of the AM or BAM signals.
Thus, with the same values, single-sideband AM provides a significant energy gain compared to classical AM and balanced modulation. At the same time, the implementation of balanced amplitude and single-sideband amplitude modulation signals is associated with some difficulties regarding the need to restore the carrier wave when processing signals on the receiving side. This problem is solved by synchronization devices of the transmitting and receiving sides, which in general leads to more complex equipment.
4.4. Continuous angle modulated signals
4.4.1. Generalized representation of angle modulated signals
In the previous section, the modulation procedure was considered, when the information parameter changed in accordance with the law of the control (modulating) signal was the amplitude of the carrier oscillation. However, in addition to the amplitude, the carrier oscillation is also characterized by frequency and initial phase
where is the total phase of the carrier oscillation, which determines the current value of the phase angle.
Changing either or in accordance with the control signal corresponds to angular modulation. Thus, the concept of angular modulation includes both frequency(World Cup) and phase(FM) modulation.
Let us consider generalized analytical relations for signals with angular modulation. At frequency modulation in accordance with the control signal, the instantaneous frequency of the carrier oscillation changes in the range from the lower to the boundary frequencies
The largest value of frequency deviation from is called deviation frequencies
.
If the boundary frequencies are located symmetrically relative to , then the frequency deviation
. (4.22)
It is precisely this case of frequency modulation that will be considered further.
The law of change in the total phase is defined as the integral of the instantaneous frequency. Then, taking into account (4.21) and (4.22), we can write
Substituting (4.23) into (4.20), we obtain a generalized analytical expression for a signal with frequency modulation
Term 
represents the total phase component due to the presence of frequency modulation. It is easy to verify that full phase frequency modulated signal changes according to the law of integral from .
At phase modulation, in accordance with the modulating signal, the initial phase of the carrier oscillation changes within the range from the lower to the upper phase limit values
The greatest deviation of the phase shift from is called phase deviation. If and are located symmetrically relative to , then 
. In this case, the total phase of the phase-modulated signal is
Then, substituting (4.26) into (4.20), we obtain a generalized analytical expression for a signal with phase modulation
Let's consider how the instantaneous frequency of the signal changes during phase modulation. It is known that the instantaneous frequency and current half-
phase are related by the relation
.
Substituting formula (4.26) into this expression and performing the differentiation operation, we obtain
Where 
– frequency component due to the presence of phase modulation of the carrier oscillation (4.20).
Thus, a change in the initial phase of the carrier oscillation leads to a change in the instantaneous frequency values according to the law of derivative of time.
The practical implementation of angle modulation signal generation devices can be carried out by one of two methods: direct or indirect. With the direct method, in accordance with the law of change in the control signal, the parameters of the oscillatory circuit of the carrier oscillation generator change. The output signal is modulated in frequency. To receive a phase modulation signal, a differentiating circuit is turned on at the input of the frequency modulator.
Phase modulation signals in the direct method are formed by changing the parameters of the oscillating circuit of the amplifier connected to the output of the carrier oscillator. To convert phase modulation signals into a frequency modulation signal, the control oscillation is applied to the input of the phase modulator through an integrating circuit.
Indirect methods do not involve the direct influence of the control signal on the parameters of the oscillatory circuit. One of the indirect methods is based on converting amplitude-modulated signals into phase modulation signals, and those, in turn, into frequency modulation signals. The issues of generating frequency and phase modulation signals will be discussed in more detail below.
4.4.2. Frequency Modulated Signals
We will begin our analysis of the characteristics of signals with angle modulation by considering single-tone frequency modulation. The control signal in this case is a unit amplitude oscillation (this form can always be reduced to)
,
(4.29)
and the modulated parameter of the carrier oscillation is the instantaneous frequency. Then, substituting (4.29) into (4.24), we get:
Having performed the integration operation, we arrive at the following expression for a single-tone frequency modulation signal
Attitude
called index frequency modulation and has a physical meaning of the part of the frequency deviation per unit frequency of the modulating signal. For example, if the deviation of the carrier frequency MHz is 
, and the frequency of the control signal is kHz, then the frequency modulation index will be . In expression (4.30) the initial phase is not taken into account as having no fundamental significance.
The signal timing diagram for single-tone FM is shown in Fig. 4.7
Let’s begin our consideration of the spectral characteristics of an FM signal with a special case small frequency modulation index. Using the ratio
let us represent (4.30) in the form
Since , then we can use approximate representations
and expression (4.31) takes the form
Using the well-known trigonometric relation
and assuming and , we get:
This expression resembles expression (4.6) for a single-tone AM signal. The difference is that if in a single-tone AM signal the initial phases of the side components are the same, then in a single-tone FM signal with small frequency modulation indices they differ by angle, i.e. are in antiphase.
The spectral diagram of such a signal is shown in Fig. 4.8
The values of the initial phase of the lateral components are indicated in parentheses. Obviously, the spectrum width of the FM signal at small frequency modulation indices is equal to
.
Signals with low frequency modulation are used quite rarely in practical radio engineering.
In real radio systems, the frequency modulation index significantly exceeds one.
For example, in modern analog mobile communication systems that use frequency modulation signals for the transmission of voice messages at the upper frequency of the speech signal kHz and frequency deviation 
kHz, the index, as is easy to see, reaches a value of ~3-4. In meter-wave radio broadcasting systems, the frequency modulation index can exceed a value equal to 10. Therefore, we will consider the spectral characteristics of FM signals at arbitrary values of .
Let's return to expression (4.32). The following types of decomposition are known
where is the Bessel function of the first kind of the th order.
Substituting these expressions into (4.32), after simple but rather cumbersome transformations using the relations of products of cosines and sines already mentioned repeatedly above, we obtain
|
(4.36)Where 
.
The resulting expression represents the decomposition of a single-tone FM signal into harmonic components, i.e. amplitude spectrum. The first term of this expression is the spectral component of the carrier frequency oscillation with amplitude 
. The first sum of expression (4.35) characterizes the side components with amplitudes and frequencies, i.e. the lower sideband, and the second sum is the side components with amplitudes and frequencies, i.e. upper side band of the spectrum.
The spectral diagram of the FM signal at arbitrary is shown in Fig. 4.9.
Let us analyze the nature of the amplitude spectrum of the FM signal. First of all, we note that the spectrum is symmetrical with respect to the carrier frequency and is theoretically infinite.
The components of the lateral sidebands are located at a distance Ω from each other, and their amplitudes 
depend on the frequency modulation index. And finally, the spectral components of the lower and upper side frequencies with even indices have the same initial phases, while the spectral components with odd indices differ by an angle .
Table 4.1 shows the values of the Bessel function for various i And . Let's pay attention to the component of the carrier vibration. The amplitude of this component is equal to 
. From Table 4.1 it follows that when amplitude , i.e. there is no spectral component of the carrier wave in the spectrum of the FM signal. But this does not mean the absence of a carrier oscillation in the FM signal (4.30). Simply, the energy of the carrier vibration is redistributed between the components of the sidebands.
Table 4.1
As already emphasized above, the spectrum of the FM signal is theoretically infinite. In practice, the bandwidth of radio devices is always limited. Let us estimate the practical width of the spectrum at which the reproduction of an FM signal can be considered undistorted.
The average power of the FM signal is determined as the sum of the average powers of the spectral components
The calculations showed that about 99% of the FM signal energy is concentrated in frequency components with numbers . This means that frequency components with numbers 
can be neglected. Then the practical width of the spectrum for single-tone FM, taking into account its symmetry with respect to
and for large values
Those. equal to twice the frequency deviation.
Thus, the width of the spectrum of the FM signal is approximately times greater than the width of the spectrum of the AM signal. At the same time, it is used to transmit information all the energy signal. This is the advantage of frequency modulation signals over amplitude modulation signals.
4.5. Discrete modulated signals
The continuous modulation signals discussed above are mainly used in radio broadcasting, radiotelephony, television and others. At the same time, the transition to digital technologies in radio engineering, including in the listed areas, has led to the widespread use of signals with discrete modulation or manipulation. Since historically discrete modulation signals were first used to transmit telegraph messages, such signals are also called amplitude (AT), frequency (FT), and phase (PT) telegraphy signals. Below, when describing the corresponding signals, this abbreviation will be used, which will distinguish them from signals with continuous modulation.
4.5.1. Discrete Amplitude Modulation Signals
Discrete amplitude modulation signals are characterized by the fact that the amplitude of the carrier wave changes in accordance with the control signal, which is a sequence of pulses, usually rectangular in shape. When studying the characteristics of signals with continuous modulation, a harmonic signal was considered as a control signal. By analogy with this, for signals with discrete modulation, we use a periodic sequence of rectangular pulses as a control signal
Obviously, as follows from (4.39), the pulse duration is , and the duty cycle is .
In Fig. Figure 4.10 shows diagrams of a control signal, a carrier oscillation, and an amplitude-keyed signal. Here and further we will assume that the amplitude of the control signal pulses is equal to , and the initial phase of the carrier oscillation is equal to . Then the signal with discrete amplitude modulation can be written as follows
Previously, the expansion of a sequence of rectangular pulses into a Fourier series (2.13) was obtained. For the case under consideration, expression (2.13) takes the form
Substituting (4.41) into (4.40) and using the formula for the product of cosines, we obtain:
In Fig. Figure 4.11 shows the amplitude spectrum of a signal amplitude modulated by a sequence of rectangular pulses. The spectrum contains a carrier frequency component with amplitude and two side bands, each of which consists of an infinite number of harmonic components located at frequencies whose amplitudes vary according to the law 
. The sidebands, as with continuous AM, are located in a mirror image with respect to the spectral component of the carrier frequency. The zeros of the amplitude spectrum of the AT signal correspond to the zeros of the amplitude spectrum of the signal, but are shifted to the left and right by an amount.
Due to the fact that the main part of the energy of the control signal is concentrated within the first lobe of the spectrum, the practical width of the spectrum in the case under consideration, based on Fig. 4.11 can be defined as
. (4.43)
This result is consistent with the spectrum calculations given in [L.4], where it is shown that most of the power is concentrated in the side components with frequencies and .
4.5.2. Discrete Frequency Modulation Signals
When analyzing signals with discrete angular modulation, it is convenient to use a periodic sequence of rectangular pulses of the “meander” type as a modulating signal. Then the control signal over the time interval takes the value 
, and on the time interval - the value . Again, as in the analysis of AT signals, we will assume .
As follows from subsection 4.3.1, a frequency modulated signal is described by expression (4.24). Then, taking into account the fact that on the interval there is a control signal , and on the interval the control signal , after performing the integration operation, we obtain the expression for the signal CT
Figure 4.12 shows the timing diagrams of the control signal, the carrier oscillation and the discrete frequency modulation signal.
On the other hand, the CT signal, as follows from Fig. 4.12, can be represented by the sum of two discrete amplitude modulation signals and , the frequencies of the carrier oscillations of which are respectively equal
|
,Another common type of modulation used in radio communications is frequency modulation (FM), in which the carrier frequency is changed in accordance with the modulating signal (Fig. 15.1).
Rice. 15.1. Frequency modulation.
Note that the carrier amplitude remains constant, but the frequency varies.
Frequency deviation
Frequency deviation is the degree to which the carrier frequency changes when the signal level changes by 1 V. Frequency deviation is measured in kilohertz per volt (kHz/V). Suppose, for example, that a carrier with a frequency of 1000 kHz is to be modulated with a square wave signal with an amplitude of 5 V (Fig. 15.2). Let us also assume that the frequency deviation is 10 kHz/V. Then, in the time interval from A to B, the carrier frequency will increase by 5 10 = 50 kHz (the product of the signal amplitude and the frequency deviation) and will become equal to 1000 kHz + 50 kHz = 1050 kHz. In the time interval from B to C, the carrier frequency will change by the same amount, namely 5 10 = 50 kHz, but this time in the negative direction with a decrease in the carrier frequency to 1000 - 50 = 950 kHz.
Rice. 15.2.
Maximum deviation
The change in carrier frequency when the signal level changes must be limited to a certain maximum value, exceeding which is unacceptable. This value is called maximum deviation. For example, BBC FM broadcasts use a frequency deviation of 15 kHz/V and a maximum deviation of 75 kHz. The maximum magnitude of the modulating signal is determined by the maximum permissible deviation.
Maximum deviation ±75
Maximum signal = -------------- = -- = ±5 V
Frequency deviation 15
or in other words, 5V to the positive or negative region.
Side frequencies and bandwidth
If the carrier is frequency modulated with a harmonic signal, an unlimited number of side frequencies are generated. The amplitudes of the side components gradually decrease as the frequency of these components moves away from the carrier frequency.
Thus, to accommodate all the side frequencies, the bandwidth of the FM system must be infinite. In practice, small amplitude side components of an FM signal can be rejected without introducing any noticeable distortion. For example, the BBC's FM broadcasts use a frequency band of 250 kHz.
ComparisonA.M.- and FM modulation systems
Amplitude Frequency
modulation modulation
1. Carrier amplitude Changes along Remains
With constant signal
2. Side frequencies Two for each Infinite
Frequencies in the spectrum number
Signal
3. Occupied bandwidth 9 kHz 250 kHz frequency band
4. Frequency range LW, MW. KB VHF
Benefits of Frequency Modulation
FM broadcasting has the following advantages over AM broadcasting of programs.
1. FM system provides better sound quality. This is due to the large frequency bandwidth of the FM signal, covering a much larger number of harmonics.
2. FM transmission achieves very low noise levels. Noise is unwanted signals that appear at the output, usually in the form of changes in carrier amplitude. In an FM system, these signals are easily eliminated by bidirectionally limiting the carrier amplitude. The information carried by the changing frequency is completely preserved.
This video talks about frequency modulation:
We continue the series of general education articles under the general title “Theory of Radio Waves.”
In previous articles we got acquainted with radio waves and antennas: Let's take a closer look at radio signal modulation.
Within the framework of this article, analog modulation of the following types will be considered:
- Amplitude modulation
- Amplitude modulation with one sideband
- Frequency modulation
- Linear frequency modulation
- Phase modulation
- Differential phase modulation
Amplitude modulation
With amplitude modulation, the envelope of the amplitudes of the carrier vibration changes according to a law that coincides with the law of the transmitted message. The frequency and phase of the carrier oscillation does not change.
One of the main parameters of AM is the modulation coefficient (M).
The modulation coefficient is the ratio of the difference between the maximum and minimum values of the amplitudes of the modulated signal to the sum of these values (%).
Simply put, this coefficient shows how much the amplitude of the carrier vibration at a given moment deviates from the average value.
When the modulation factor is greater than 1, an overmodulation effect occurs, resulting in signal distortion.
AM spectrum
This spectrum is characteristic of a modulating oscillation of a constant frequency.
On the graph, the X axis represents the frequency, and the Y axis represents the amplitude.
For AM, in addition to the amplitude of the fundamental frequency located in the center, amplitude values to the right and left of the carrier frequency are also presented. These are the so-called left and right side stripes. They are separated from the carrier frequency by a distance equal to the modulation frequency.
The distance from the left to the right side strip is called spectrum width.
In the normal case, with a modulation coefficient<=1, амплитуды боковых полос меньше или равны половине амплитуды несущей.
Useful information is contained only in the upper or lower side bands of the spectrum. The main spectral component, the carrier, does not carry useful information. The transmitter power during amplitude modulation is mostly spent on “heating the air”, due to the lack of information content of the most basic element of the spectrum.
Single sideband amplitude modulation
Due to the ineffectiveness of classical amplitude modulation, single sideband amplitude modulation was invented.
Its essence is to remove the carrier and one of the sidebands from the spectrum, while all the necessary information is transmitted over the remaining sideband.
But in its pure form, this type did not take root in household radio broadcasting, because In the receiver, the carrier must be synthesized with very high accuracy. Used in compaction equipment and amateur radio.
In radio broadcasting, AM with one sideband and a partially suppressed carrier is more often used:
With this modulation the quality/efficiency ratio is best achieved.
Frequency modulation
A type of analog modulation in which the carrier frequency changes according to the law of the modulating low-frequency signal. The amplitude remains constant.
a) - carrier frequency, b) modulating signal, c) modulation result
The largest frequency deviation from the average value is called deviation.
Ideally, the deviation should be directly proportional to the amplitude of the modulating oscillation.
The frequency modulation spectrum looks like this:
It consists of a carrier and sideband harmonics symmetrically lagging behind it to the right and left, at a frequency that is a multiple of the frequency of the modulating oscillation.
This spectrum represents a harmonic vibration. In the case of real modulation, the spectrum has more complex shapes.
There are broadband and narrowband FM modulation.
In broadband, the frequency spectrum significantly exceeds the frequency of the modulating signal. Used in FM radio broadcasting.
Radio stations mainly use narrowband FM modulation, which requires more precise tuning of the receiver and, accordingly, is more protected from interference.
Broadband and narrowband FM spectra are presented below
The spectrum of narrowband FM resembles amplitude modulation, but when you consider the phase of the sidebands, these waves appear to have constant amplitude and variable frequency, rather than constant frequency and variable amplitude (AM). With wideband FM, the carrier amplitude can be very small, which results in high FM efficiency; this means that most of the transmitted energy is contained in the side frequencies that carry information.
The main advantages of FM over AM are energy efficiency and noise immunity.
Linear frequency modulation is a type of FM.
Its essence lies in the fact that the frequency of the carrier signal changes according to a linear law.
The practical significance of linear frequency modulated (chirp) signals lies in the possibility of significant compression of the signal during reception with an increase in its amplitude above the noise level.
Chirps are used in radar.
Phase modulation
In reality, the term phase manipulation is more commonly used, because They mainly modulate discrete signals.The meaning of PM is that the phase of the carrier changes abruptly with the arrival of the next discrete signal, different from the previous one.
From the spectrum you can see the almost complete absence of a carrier, which indicates high energy efficiency.
The disadvantage of this modulation is that an error in one symbol can lead to incorrect reception of all subsequent ones.
Differential phase shift keying
In the case of this modulation, the phase does not change with each change in the value of the modulating pulse, but with a change in the difference. In this example, when each “1” arrives.
The advantage of this type of modulation is that if a random error occurs in one symbol, this does not entail a further chain of errors.
It is worth noting that there are also phase manipulations such as quadrature, which uses a phase change within 90 degrees and higher-order PM, but their consideration is beyond the scope of this article.
PS: I want to note once again that the purpose of the articles is not to replace a textbook, but to tell you “at a glance” about the basics of radio.
Only the main types of modulations are considered to create the reader’s idea of the topic.
