Modulation - what is the difference between the types of modulation AM, FM (FM) and SSB: simple about the complex. Types of signal modulation

Understanding Modulation

Modulation— This is the process of converting one or more information parameters of a carrier signal in accordance with the instantaneous values ​​of the information signal.

As a result of modulation, signals are transferred to higher frequencies.

Using modulation allows you to:

• coordinate the signal parameters with the line parameters;
• increase the noise immunity of signals;
• increase signal transmission range;
• organize multi-channel transmission systems (MSP with CRC).

Modulation is carried out in devices modulators. The conventional graphic designation of the modulator looks like:

Figure 1 - Graphic designation of the modulator

When modulating, the following signals are supplied to the modulator input:

u(t) — modulating, this signal is informational and low-frequency (its frequency is designated W or F);

S(t)— modulated (carrier), this signal is non-informational and high-frequency (its frequency is designated w 0 or f 0);

Sм(t) — modulated signal, this signal is informational and high-frequency.

The following can be used as a carrier signal:

• harmonic oscillation, in which modulation is called analog or continuous;
• a periodic sequence of pulses, with modulation called pulse;
• direct current, and modulation is called noise-like.

Since the information parameters of the carrier oscillation change during the modulation process, the name of the type of modulation depends on the changed parameter of this oscillation.

1. Types of analog modulation:

• amplitude modulation (AM), the amplitude of the carrier vibration changes;
• frequency modulation (FM), there is a change in the frequency of the carrier vibration;
• phase modulation (PM), the phase of the carrier oscillation changes.

2. Types of pulse modulation:

• pulse amplitude modulation (PAM), the amplitude of the carrier signal pulses changes;
• pulse frequency modulation (PFM), the pulse repetition rate of the carrier signal changes;
• Pulse phase modulation (PPM), the phase of the carrier signal pulses changes;
• Pulse width modulation (PWM), the duration of the carrier signal pulses changes.

Amplitude modulation

Amplitude modulation- the process of changing the amplitude of the carrier signal in accordance with the instantaneous values ​​of the modulating signal.

amplitude modulated(AM) signal with a harmonic modulating signal. When exposed to a modulating signal

u(t)= Um u sin? t (1)

to carrier vibration

S(t)= Um sin(? 0 t+ ? ) (2)

the amplitude of the carrier signal changes according to the law:

Uam(t)=Um+and amUm u sin? t(3)

where a am is the proportionality coefficient of amplitude modulation.

Substituting (3) into the mathematical model (2) we obtain:

Sam(t)=(Um+and amUm u sin? t)sin(? 0 t+? ). (4)

Let's take Um out of brackets:

Sam(t)=Um(1+and amUm u/Um sin? t)sin(? 0 t+? ) (5)

The relation a am Um u / Um = m am is called amplitude modulation ratio. This coefficient should not exceed unity, since in this case distortions of the modulated signal envelope appear, called overmodulation. Taking into account m am, the mathematical model of the AM signal with a harmonic modulating signal will have the form:

Sam(t)=Um(1+mamsin ? t)sin(? 0 t+ ? ). (6)

If the modulating signal u(t) is non-harmonic, then the mathematical model of the AM signal in this case will have the form:

Sam(t)=(Um+and amu(t))sin(? 0 t+ ? ) . (7)

Let's consider the spectrum of the AM signal for a harmonic modulating signal. To do this, let's open the brackets of the mathematical model of the modulated signal, i.e., imagine it as a sum of harmonic components.

Sam(t)=Um(1+mamsin? t)sin (? 0 t+ ? ) = Um sin (? 0 t+ ? ) +

+mamUm/2 sin( (? 0 — ? )t+j) — mamUm/2 sin((? 0 + ? )t+j). (8)

As can be seen from the expression, there are three components in the spectrum of the AM signal: the carrier signal component and two components at the combination frequencies. Moreover, the component at frequency ? 0 —? called lower side component, and at frequency ? 0 + ? — upper side component. The spectral and time diagrams of the modulating, carrier and amplitude-modulated signals look like (Figure 2).

Figure 2 - Time and spectral diagrams of modulating (a), carrier (b) and amplitude-modulated (c) signals

D ? am=(? 0 + ? ) — (? 0 — ? )=2 ? (9)

If the modulating signal is random, then in this case in the spectrum the components of the modulating signal are symbolically designated by triangles (Figure 3).

Components in the frequency range ( ? 0 — ? max) ? ( ? 0 — ? min) form lower side band (LSB), and the components in the frequency range ( ? 0 + ? min) ? ( ? 0 + ? max) form upper side band (UPS)

Figure 3 - Time and spectral diagrams of signals with a random modulating signal

The spectrum width for a given signal will be determined

D? am=(? 0 + ? max) — (? 0 — ? min)=2 ? max (10)

Figure 4 shows time and spectral diagrams of AM signals at various m am indices. As can be seen when m am =0 there is no modulation, the signal is an unmodulated carrier, and accordingly the spectrum of this signal has only the carrier signal component (Figure 4,

Figure 4 - Time and spectral diagrams of AM signals at different mam: a) at mam=0, b) at mam=0.5, c) at mam=1, d) at mam>1

a), with the modulation index m am = 1, deep modulation occurs; in the spectrum of the AM signal, the amplitudes of the side components are equal to half the amplitude of the carrier signal component (Figure 4c), this option is optimal, since the energy falls to a greater extent on the information components. In practice, it is difficult to achieve a coefficient equal to unity, so they achieve a ratio of 0 1, overmodulation occurs, which, as noted above, leads to distortion of the AM signal envelope; in the spectrum of such a signal, the amplitudes of the side components exceed half the amplitude of the carrier signal component (Figure 4d).

The main advantages of amplitude modulation are:

• narrow spectrum width of the AM signal;
• ease of obtaining modulated signals.

The disadvantages of this modulation are:

• low noise immunity (because when interference affects the signal, its shape is distorted - the envelope, which contains the transmitted message);
• inefficient use of transmitter power (since the largest part of the modulated signal energy is contained in the carrier signal component up to 64%, and information sidebands account for 18% each).

Amplitude modulation has found wide application:

• in television broadcasting systems (for transmitting television signals);
• in sound broadcasting and radio communication systems on long and medium waves;
• in a three-program wire broadcasting system.

Balanced and single sideband modulation

As noted above, one of the disadvantages of amplitude modulation is the presence of a carrier signal component in the spectrum of the modulated signal. To eliminate this drawback, balanced modulation is used. At balanced modulation a modulated signal is formed without a component of the carrier signal. This is mainly done by using special modulators: balanced or ring. The timing diagram and spectrum of the balanced modulated (BM) signal is presented in Figure 5.

Figure 5 - Time and spectral diagrams of modulating (a), carrier (b) and balanced-modulated (c) signals

Another feature of the modulated signal is the presence in the spectrum of two side bands carrying the same information. Suppression of one of the bands allows you to reduce the spectrum of the modulated signal and, accordingly, increase the number of channels in the communication line. Modulation in which a modulated signal with one sideband (upper or lower) is formed is called single lane. The formation of a single-sideband modulated (SB) signal is carried out from the BM signal using special methods, which are discussed below. The spectra of the OM signal are presented in Figure 6.

Figure 6 - Spectral diagrams of single-sideband modulated signals: a) with an upper sideband (UPS), b) with a lower sideband (LSB)

Frequency modulation

Frequency modulation- the process of changing the frequency of the carrier signal in accordance with the instantaneous values ​​of the modulating signal.

Consider the mathematical model frequency modulated(FM) signal with a harmonic modulating signal. When exposed to a modulating signal

u(t) = Um u sin? t

to carrier vibration

S(t) = Um sin(? 0 t+ ? )

the frequency of the carrier signal changes according to the law:

wworld championship(t) =? 0 + and the world championshipUm u sin? t(9)

where a fm is the proportionality coefficient of frequency modulation.

Since the value of sin ? t can change in the range from -1 to 1, then the largest deviation of the FM signal frequency from the carrier signal frequency is

? ? m = a chmUm u (10)

The quantity Dw m is called frequency deviation. Hence, frequency deviation shows the greatest deviation of the frequency of the modulated signal from the frequency of the carrier signal.

Meaning ? hm (t) cannot be directly substituted into S(t), since the argument of the sine ? t+j is the instantaneous phase of the signal?(t) which is related to the frequency by

? = d? (t)/ dt (11)

What follows from this is what to determine? hm(t) must be integrated ? hm (t)

And in expression (12)? is the initial phase of the carrier signal.

Attitude

Mchm = ?? m/ ? (13)

called frequency modulation index.

Taking into account (12) and (13), the mathematical model of the FM signal with a harmonic modulating signal will have the form:

Sworld championship(t)=Um sin(? 0 t — Mchmcos? t+? ) (14)

Timing diagrams explaining the process of forming a frequency-modulated signal are shown in Figure 7. The first diagrams a) and b) show the carrier and modulating signals, respectively, and Figure c) shows a diagram showing the law of change in the frequency of the FM signal. Diagram d) shows a frequency-modulated signal corresponding to a given modulating signal, as can be seen from the diagram, any change in the amplitude of the modulating signal causes a proportional change in the frequency of the carrier signal.

Figure 7 - FM signal generation

To construct the spectrum of an FM signal, it is necessary to decompose its mathematical model into harmonic components. As a result of the expansion we get

Sworld championship(t)= Um J 0 (Mworld championship) sin(? 0 t+? ) —

— Um J 1 (Mworld championship) (cos[(? 0 — ? )t+j]+cos[(? 0 + ? )t+ ? ]} —

— Um J 2 (Mworld championship) (sin[(? 0 — 2 ? )t+j]+ sin[(? 0 +2 ? )t+ ? ]}+

+ Um J 3 (Mworld championship) (cos[(? 0 — 3 ? )t+j]+cos[(? 0 +3 ? )t+? ]} —

— Um J 4 (Mworld championship) (sin[(? 0 — 4 ? )t+j]+ sin[(? 0 +4 ? )t+? ]} — … (15)

where J k (Mchm) are proportionality coefficients.

J k (Mchm) are determined by Bessel functions and depend on the frequency modulation index. Figure 8 shows a graph containing eight Bessel functions. To determine the amplitudes of the components of the FM signal spectrum, it is necessary to determine the value of the Bessel functions for a given index. And how

Figure 8 - Bessel functions

It can be seen from the figure that different functions begin at different values ​​of the MFM, and therefore, the number of components in the spectrum will be determined by the MFM (as the index increases, the number of spectrum components also increases). For example, it is necessary to determine the coefficients J k (Mchm) for Mchm=2. The graph shows that for a given index, it is possible to determine the coefficients for five functions (J 0, J 1, J 2, J 3, J 4). Their value for a given index will be equal to: J 0 = 0.21; J 1 =0.58; J 2 =0.36; J 3 =0.12; J 4 =0.02. All other functions begin after the value Mhm = 2 and are equal, accordingly, to zero. For the example given, the number of components in the spectrum of the FM signal will be equal to 9: one component of the carrier signal (Um J 0) and four components in each sideband (Um J 1; Um J 2; Um J 3; Um J 4).

Another important feature of the FM signal spectrum is that it is possible to achieve the absence of a carrier signal component or make its amplitude significantly smaller than the amplitudes of the information components without additional technical complications of the modulator. To do this, it is necessary to select a modulation index Mchm at which J 0 (Mhm) will be equal to zero (at the intersection of the function J 0 with the Mhm axis), for example Mhm = 2.4.

Since an increase in components leads to an increase in the spectrum width of the FM signal, this means that the width of the spectrum depends on the FM signal (Figure 9). As can be seen from the figure, at MFM? 0.5, the width of the spectrum of the FM signal corresponds to the width of the spectrum of the AM signal, and in this case the frequency modulation is narrowband, as the MFM increases, the spectrum width increases, and the modulation in this case is broadband. For an FM signal, the spectrum width is determined

D? world championship=2(1+Mhm) ? (16)

The advantages of frequency modulation are:

• high noise immunity;
• more efficient use of transmitter power;
• comparative simplicity of obtaining modulated signals.

The main disadvantage of this modulation is the large width of the spectrum of the modulated signal.

Frequency modulation is used:

• in television broadcasting systems (for transmitting audio signals);
• satellite television and radio broadcasting systems;
• high-quality stereo broadcasting systems (FM range);
• radio relay lines (RRL);
• cellular telephone communications.

Figure 9 - Spectra of the FM signal with a harmonic modulating signal and with various FM indices: a) with FM = 0.5, b) with FM = 1, c) with FM = 5

Phase modulation

Phase modulation- the process of changing the phase of the carrier signal in accordance with the instantaneous values ​​of the modulating signal.

Consider the mathematical model phase modulated(PM) signal with a harmonic modulating signal. When exposed to a modulating signal

u(t) = Um u sin? t

to carrier vibration

S(t) = Um sin(? 0 t+ ? )

the instantaneous phase of the carrier signal changes according to the law:

? fm(t) =? 0 t+? + a fmUm u sin? t(17)

where a fm is the proportionality coefficient of frequency modulation.

Substituting ? fm(t) in S(t) we obtain a mathematical model of the fm signal with a harmonic modulating signal:

Sfm(t) = Um sin(? 0 t+a fmUm u sin? t+? ) (18)

The product a fm Um u =Dj m is called phase modulation index or phase deviation.

Since a change in phase causes a change in frequency, using (11) we determine the law of change in the frequency of the FM signal:

? fm(t)= d ? fm(t)/ dt= w 0 +a fmUm u? cos ? t (19)

Product a fm Um u ? =?? m is the deviation of the phase modulation frequency. Comparing the frequency deviation with frequency and phase modulations, we can conclude that with both FM and FM, the frequency deviation depends on the proportionality coefficient and the amplitude of the modulating signal, but with FM, the frequency deviation also depends on the frequency of the modulating signal.

Timing diagrams explaining the process of forming an FM signal are shown in Figure 10.

When the mathematical model of an FM signal is decomposed into harmonic components, the same series will be obtained as with frequency modulation (15), with the only difference being that the coefficients J k will depend on the phase modulation index? ? m(Jk(? ? m)). These coefficients will be determined in the same way as in the case of FM, i.e., using the Bessel functions, with the only difference being that along the abscissa axis it is necessary to replace FM with? ? m. Since the spectrum of an FM signal is constructed similarly to the spectrum of an FM signal, it is characterized by the same conclusions as for an FM signal (clause 1.4).

Figure 10 - Formation of an FM signal

The spectrum width of the FM signal is determined by the expression:

? ? fm=2(1+ ? jm) ? (20).

The advantages of phase modulation are:

• high noise immunity;
• more efficient use of transmitter power.
• The disadvantages of phase modulation are:

• large spectrum width;
• comparative difficulty of obtaining modulated signals and their detection

Discrete binary modulation (harmonic carrier manipulation)

Discrete binary modulation (keying)- a special case of analog modulation, in which a harmonic carrier is used as a carrier signal, and a discrete, binary signal is used as a modulating signal.

There are four types of manipulation:

• amplitude manipulation (AMn or AMT);
• Frequency Shift Keying (FSK or TBI);
• phase shift keying (PSK or FMT);
• relative phase shift keying (RPMn or RPM).

Time and spectral diagrams of modulated signals for various types of manipulation are presented in Figure 11.

At amplitude keying, as well as with any other modulating signal, the envelope S AMn (t) repeats the shape of the modulating signal (Figure 11, c).

At frequency shift keying Are there two frequencies? 1 and? 2. When there is a pulse in the modulating signal (message), is a higher frequency used? 2, in the absence of a pulse (active pause), a lower frequency w 1 corresponding to an unmodulated carrier is used (Figure 11, d)). The spectrum of the frequency-keyed signal S FSK (t) has two bands near the frequencies? 1 and? 2.

At phase shift keying the phase of the carrier signal changes by 180° at the moment the amplitude of the modulating signal changes. If a series of several pulses follows, then the phase of the carrier signal does not change during this interval (Figure 11, e).

Figure 11 - Time and spectral diagrams of modulated signals of various types of discrete binary modulation

At relative phase shift keying the phase of the carrier signal changes by 180° only at the moment the pulse is applied, i.e., during the transition from an active pause to a send (0?1) or from a send to a send (1?1). When the amplitude of the modulating signal decreases, the phase of the carrier signal does not change (Figure 11, e). The spectra of signals for PSK and OFPS have the same appearance (Figure 9, f).

Comparing the spectra of all modulated signals, it can be noted that the spectrum of the FSK signal has the greatest width, the smallest - AMn, PSK, OPSK, but in the spectra of PSK and OPSK signals there is no component of the carrier signal.

Due to greater noise immunity, frequency, phase and relative-phase manipulations are most widespread. Various types of them are used in telegraphy, data transmission, and mobile radio communication systems (telephone, trunking, paging).

Pulse modulation

Pulse modulation is a modulation in which a periodic sequence of pulses is used as a carrier signal, and an analog or discrete signal can be used as a modulating signal.

Since a periodic sequence is characterized by four information parameters (amplitude, frequency, phase and pulse duration), there are four main types of pulse modulation:

• pulse amplitude modulation (AIM); the amplitude of the carrier signal pulses changes;
• pulse frequency modulation (PFM), the pulse repetition rate of the carrier signal changes;
• pulse phase modulation (FIM), the phase of the carrier signal pulses changes;
• pulse width modulation (PWM), the duration of the carrier signal pulses changes.

Timing diagrams of pulse-modulated signals are presented in Figure 12.

During AIM, the amplitude of the carrier signal S(t) changes in accordance with the instantaneous values ​​of the modulating signal u(t), i.e., the pulse envelope repeats the shape of the modulating signal (Figure 12, c).

With PWM, the pulse duration S(t) changes in accordance with the instantaneous values ​​of u(t) (Figure 12, d).

Figure 12 - Timing diagrams of signals during pulse modulation

During PFM, the period, and therefore the frequency, of the carrier signal S(t) changes in accordance with the instantaneous values ​​of u(t) (Figure 12, e).

With PPM, the carrier signal pulses are shifted relative to their clock (time) position in the unmodulated carrier (clock moments are indicated on the diagrams by points T, 2T, 3T, etc.). The PIM signal is presented in Figure 12, f.

Since in pulse modulation the message carrier is a periodic sequence of pulses, the spectrum of pulse-modulated signals is discrete and contains many spectral components. This spectrum is a spectrum of a periodic sequence of pulses in which near each harmonic component of the carrier signal there are components of the modulating signal (Figure 13). The structure of the sidebands near each component of the carrier signal depends on the type of modulation.

Figure 13 - Spectrum of a pulse-modulated signal

Another important feature of the spectrum of pulse-modulated signals is that the width of the spectrum of the modulated signal, except for PWM, does not depend on the modulating signal. It is completely determined by the pulse duration of the carrier signal. Since with PWM the pulse duration changes and depends on the modulating signal, then with this type of modulation the width of the spectrum also depends on the modulating signal.

The pulse repetition rate of the carrier signal can be determined by the theorem of V. A. Kotelnikov as f 0 = 2Fmax. In this case, Fmax is the upper frequency of the spectrum of the modulating signal.

Transmission of pulse-modulated signals over high-frequency communication lines is impossible, since the spectrum of these signals contains low-frequency components. Therefore, for transfer they carry out re-modulation. This is a modulation in which a pulse-modulated signal is used as a modulating signal, and a harmonic oscillation is used as a carrier signal. With repeated modulation, the spectrum of the pulse-modulated signal is transferred to the carrier frequency region. For re-modulation, any type of analog modulation can be used: AM, CS, FM. The resulting modulation is denoted by two abbreviations: the first indicates the type of pulse modulation and the second indicates the type of analog modulation, for example AIM-AM (Figure 14, a) or PWM-PM (Figure 14, b), etc.

Figure 14 - Timing diagrams of signals during pulse re-modulation

Types of analog modulation

where A 0 ,ω 0 = 2πf 0 , are the amplitude, angular frequency and initial phase of the carrier, respectively; k = A m /A 0 - proportionality coefficient between the modulating signal and variations in the amplitude of the AM oscillation or modulation coefficient; A t Ω= 2πF φ- amplitude, angular frequency and initial phase of the modulating oscillation; t- time.

In Fig. Figure 5.2 shows a graph of AM oscillation versus time, which shows that the envelope has the form of a harmonic modulating oscillation.

Expression (5.1) can be transformed into the form (for simplicity, the initial phases are omitted)

This form of recording shows that in the spectrum of a modulated oscillation, in addition to the carrier, there are two side components with an amplitude proportional to the modulation coefficient and with frequencies above and below the carrier at the modulation frequency Ω = 2πF (Fig. 5.3). The spectrum width of such an AM signal

If the low-frequency modulating oscillation is complex, then the spectrum of the modulated oscillation will contain, in addition to the carrier, two sidebands - upper and lower. They represent the spectrum of the modulating signal transferred to the carrier frequency region without change and with inversion, respectively. To determine the full width of the AM oscillation spectrum in this case, substitute the maximum frequency of the modulating oscillation spectrum into (5.3).

The vector diagram of the modulated signal is very clear (Fig. 5.4). The carrier harmonic oscillation is represented by the vector

Rice. 5.2 AM oscillation graph Fig. 5.3 AM oscillation spectrum

rotating counterclockwise at constant speed ω 0 radians per second. The lateral components, in turn, are represented by vectors /2 and /2, symmetrical with respect to the first vector and fixed at its end. They

rotate counterclockwise and clockwise with angular modulation speed Ω, moving along with the carrier vector. The resulting vector of the modulated oscillation changes its length depending on the position of the two symmetrical vectors; its rotation frequency remains constant.

The power of AM oscillation depends on the modulation depth. The carrier frequency power is constant and proportional. The power of each side component is proportional to the square of its amplitude, that is, the value.

At the deepest modulation (k=1), the power of the AM oscillation (equal to the sum of the powers of all three components) is only one and a half times greater than the power of the unmodulated oscillation. In practice, the average value of the amplitude modulation coefficient does not exceed 0.5 in order to reduce the likelihood of overmodulation at peak values ​​of the modulating function.

In order to increase the efficiency and use of the transmitter and save the frequency band occupied by the modulated signal, not the entire spectrum, but one sideband of the AM wave can be transmitted. In this case, the carrier and the other side are suppressed. This modulation is called single sideband AM (SSBAM). It should be noted that in a strict sense this will already be an oscillation with complex amplitude-phase modulation.

The following types of amplitude modulation are distinguished:

Two-way AM (Double Sideband - DSB);

Double Sideband Suppressed Carrier (DSBSC);

Single Sideband AM;

Single Sideband Suppressed Carrier (SSBSC) AM with Lower and Upper Sideband (LSB) and Upper Sideband (USB) options;

AM with one of the sidebands partially suppressed (Vestigal Sideband - VSB);

AM with two independent sidebands (Independent Single Sideband - ISSB).

Another way to increase the efficiency of AM is to use dynamic AM (DAM), in which the carrier power is adjusted depending on the amplitude of the modulating oscillation.

Amplitude modulation and its variations have found application mainly in radio and television broadcasting. In the DV and MW bands, two-band AM is used, in the HF and VHF bands, single-band AM is used. In the VHF range, TV systems use AM with one sideband partially suppressed to transmit the image signal (brightness component), and to transmit color difference signals in the PAL_ and NTSC systems, a type of balanced modulation, the so-called quadrature AM, is used. The principle of AM OBP is used to form groups of channels in multi-channel communication systems with frequency division multiplexing. In addition, this type of modulation is used in mobile communication systems and for communication with aircraft (118...136 MHz).

Frequency Modulation (FM) is a special case of angular modulation. In FM, the variable parameter is the carrier frequency, i.e. at each moment of time its deviation from its nominal value is proportional to the level of the modulating signal. In the case of a harmonic modulating oscillation, the instantaneous frequency

where is the amplitude of the deviation of the carrier frequency from the nominal value or frequency deviation.

The total instantaneous phase is related to its instantaneous frequency through the integral

Magnitude

called the frequency modulation index. For a complex modulating signal, the maximum frequency of its spectrum is substituted into (5.6). The analytical expression for the FM signal U(t) is written as follows:

Rice. 5.5 FM oscillation graph Fig. 5.6 FM signal spectrum

The FM signal graph is shown in Fig. 5.5.

The spectrum of FM oscillations with single-tone modulation can be obtained by representing oscillation (5.7) in the form of an infinite trigonometric series

where is the special Bessel function of order n of the argument x. For a fixed argument, the Bessel function decreases in absolute value with increasing order and at t > p has a small value. Therefore, in practice, we are limited to considering a finite number of spectrum components.

The spectrum of FM oscillations when modulated by a harmonic signal is shown in Fig. 5.6.

There are broadband T() and narrowband T() frequency modulation. In the first case, as a rule, components with numbers are taken into account n. This corresponds to the width of the FM spectrum during harmonic modulation, in which 99% of the signal energy is concentrated.

For small FM indices (from 1 to 2.5), you should use

formula

Outside this band, the amplitude of the components is 100 times less than the amplitude of the unmodulated carrier

At T FM oscillation (5.7) is approximately described as

those. we can assume that the spectrum of such a frequency-modulated signal contains only the carrier and two side components separated from it by the modulation frequency. However, unlike amplitude modulation, the second side component has a phase shift of π radians.

The vector diagram in this case has the form shown in Fig. 5.7. Unlike AM vibrations, the sum of the vectors of lateral vibrations is perpendicular to the vector of the carrier vibration, which leads to acceleration and deceleration of the rotation of the resulting vector. The length of this vector, representing the amplitude of the modulated oscillation, changes slightly due to the approximations made. In the general case, a larger number of vectors will be added, and the end of the resulting vector, when swinging, will move along a circular arc, i.e. the length of the resulting vector will not change.

Since the spectrum of the FM signal is wider than with AM, the noise immunity of such modulation is higher. FM is used because of its broadband, mainly in the meter and shorter wavelength range. Narrowband FM (Narrow Frequency Modulation - NFM) is used in mobile communication systems, wideband (Wide Frequency Modulation - WFM) in radio and television broadcasting. In stereo broadcasting, the baseband signal contains a subcarrier with additional modulation depending on the broadcast standard. In addition, FM was widely used in radio relay and satellite communication systems; the carrier modulation was carried out with a broadband group signal, but at present, such signals are practically replaced by digital ones.

In radar, FM is used as intrapulse in the variants of linear FM, symmetrical, zigzag, etc.

Phase modulation (PM) is also a special case of angular modulation. The frequency-modulated oscillation discussed above is at the same time phase modulated. However, with phase modulation, the change in phase, and not frequency, must coincide with the law of change of the modulating oscillation. In the case of a sinusoidal modulating oscillation, the analytical representation of the FM oscillation has the form

where is the amplitude of the phase deviation (deviation).

When angular modulation is carried out with a harmonic signal, it is possible to distinguish frequency modulation from phase modulation only by comparing changes in the instantaneous phase of the modulated oscillation with the law of change in the modulating voltage.

A comparison of (5.7) and (5.12) shows that the frequency modulation index is affected by the amplitude of the phase deviation, measured in radians. However, with frequency modules, the modulation index is inversely proportional to the modulating frequency, and with phase modules, the phase deviation is fixed and does not depend on the modulation frequency.

The spectrum of a phase-modulated harmonic oscillation signal will be the same as a frequency-modulated one if the modulation indices are the same. When the spectrum of the FM signal will contain a carrier and two side components, spaced from the carrier by the modulation frequency. The only difference from the AM signal spectrum is that the side components are phase shifted by 90°.

At large modulation indices, the spectrum width of the FM signal should be calculated using formulas for FM signals. The width of the spectrum in both cases is determined by the frequency deviation. But it should be noted that with an increase in the modulation frequency of an FM signal, the spectrum width will remain the same with a smaller number of spectral components, and with PM, the spectrum width will increase with a constant number of these components.

The FM vector diagram is no different from the FM vector diagram. You just need to keep in mind that PM is determined by the angular deviation of the resulting vector from the position of the carrier frequency vector, and FM by the speed of this deviation, i.e. derivative of the phase with respect to time. Phase modulation is used mainly in radio navigation systems.

To carry out effective signal transmission in any medium, it is necessary to transfer the spectrum of these signals from the low-frequency region to the region of sufficiently high frequencies. This procedure is called modulation in radio engineering.

The essence of modulation is as follows. A certain oscillation (most often harmonic) is formed, called a carrier oscillation or simply a carrier, and any of the parameters of this oscillation changes over time in proportion to the original signal. The original signal is called modulating, and the resulting oscillation with time-varying parameters is called a modulated signal. The reverse process - separating the modulating signal from the modulated oscillation - is called demodulation.

Classification of modulation types:

1) by type of information signal (modulating signal);

Continuous modulation (analog signal);

Discrete modulation (discrete signal);

2) by type of carrier (or carrier frequency)

Harmonic (sinusoidal signal);

Pulse (rectangular periodic pulse).

3) by the type of carrier frequency parameters that undergo changes under the influence of the information signal.

Amplitude modulation;

Frequency modulation;

Phase modulation;

Width modulation;

Pulse width modulation (Figure 1.1).

Figure 1.1 – Types of modulation

General harmonic signal:

S (t) = A cos(ω 0 t+ φ 0).

This signal has three parameters: amplitude A, frequency ω 0 and initial phase φ 0. Each of them can be associated with a modulating signal, thus obtaining three main types of modulation: amplitude, frequency and phase. Frequency modulation and phase modulation are very closely related, since they both affect the argument of the cos function. Therefore, these two types of modulation have a common name - angular

modulation.

Currently, an increasing part of the information transmitted through various communication channels exists in digital form. This means that it is not a continuous (analog) modulating signal that is to be transmitted, but a sequence of integers P 0 , P 1, P 2 , ..., which can take values ​​from some fixed finite set. These numbers, called symbols, come from a source of information with a period T, and the frequency corresponding to this period is called the symbol rate: f T = 1/T.

A variant often used in practice is binary sequence of characters when each of the numbers n i can take one of two values ​​- 0 or 1.

The sequence of transmitted symbols is obviously a discrete signal. Since the symbols take values ​​from a finite set, this signal is actually quantized, that is, it can be called digital signal.

A typical approach to transmitting a discrete sequence of symbols is as follows. Each of the possible symbol values ​​is associated with a certain set of carrier vibration parameters. These parameters are maintained constant during the interval T, that is, until the arrival of the next symbol. This actually means converting a sequence of numbers { n k } to step signal S n (t) using piecewise constant interpolation:

s n (t)=f(n k ), kT

Here f is some transformation function. Received signal S n (t) is then used as a modulating signal in the usual way.

This method of modulation, when the parameters of the carrier oscillation change abruptly, is called manipulation. Depending on which parameters are changed, they distinguish between amplitude (AM), phase (PM), and frequency (FM). In addition, when transmitting digital

information, a carrier wave of different shape can be used

from harmonic. Thus, when using a sequence of rectangular pulses as a carrier oscillation, pulse amplitude (APM), pulse width (PWM) and pulse time (PMT) modulation are possible. PAM - pulse amplitude modulation is that the amplitude of the pulse carrier changes according to the law of changes in the instantaneous values ​​of the primary signal.

PFM – pulse frequency modulation. According to the law of changes in the instantaneous values ​​of the primary signal, the repetition rate of the carrier pulses changes.

VIM is time-pulse modulation, in which the information parameter is the time interval between the synchronizing pulse and the information pulse.

PWM – pulse width modulation. The point is that, according to the law of changes in the instantaneous values ​​of the modulating signal, the duration of the carrier pulses changes.

PPM – pulse phase modulation, differs from VIM by the method of synchronization. The phase shift of the carrier pulse does not change relative to the synchronizing pulse, but relative to some conventional phase.

PCM – pulse-code modulation. It cannot be considered as a separate type of modulation, since the value of the modulating voltage is represented in the form of code words.

SIM – counting pulse modulation. It is a special case of PCM, in which the information parameter is the number of pulses in the code group.

At amplitude keying a single symbol is transmitted by HF padding, and a zero symbol by the absence of a signal. The amplitude-manipulated signal is described by the expression:

where the amplitude term can take M discrete values, and the phase term φ is an arbitrary constant. The AM signal shown in Figure 1.2 (c) can correspond to a radio transmission using two signals, the amplitudes of which are 0 and .

Amplitude manipulation is the simplest, but at the same time the least noise-resistant and is currently practically not used.

At frequency discrete modulation(FM, FSK–Frequency Shift Keying) values ​​0 and 1 of the information bit correspond to their own frequencies of the physical signal with its amplitude unchanged. The general analytical expression for a frequency-shift keyed signal is as follows:

Here the frequency ω i can take M discrete values, and the phase φ is an arbitrary constant. A schematic representation of the FM signal is shown in Figure 1.2 b, where you can observe a typical change in frequency at the moments of transitions between symbols.

Frequency modulation is very noise-resistant, since it is mainly the signal amplitude, not the frequency, that is distorted by interference. In this case, the reliability of demodulation, and therefore the noise immunity, is higher, the more signal periods fall into the baud interval. But increasing the baud interval, for obvious reasons, reduces the speed of information transmission. On the other hand, the signal spectrum width required for this type of modulation can be significantly narrower than the entire channel bandwidth. This leads to the area of ​​application of FM - low-speed, but highly reliable standards that allow communication on channels with large distortions of the amplitude-frequency response, or even with a truncated bandwidth.

At phase shift keying 1 and 0 differ in the phase of the high-frequency oscillation. The phase-keyed signal has the following form:

Here the phase component φ i (t) can accept M discrete values, usually defined as follows:

where E is the energy of the symbol;

T – symbol transmission time.

Figure 1.2a shows an example of binary (M=2) phase shift keying, where characteristic sharp phase changes are clearly visible during the transition between symbols.

In practice, phase shift keying is used when the number of possible values ​​of the initial phase is small - typically 2.4 or 8. In addition, it is difficult to measure when receiving a signal absolute initial phase value; much easier to determine relative phase shift between two adjacent symbols. Therefore, phase difference or relative phase shift keying is usually used.

At phase difference modulation(DOPSK, TOPSK, DPSK – Differential Phase Shift Keying) the parameter that changes depending on the value of the information element is the phase of the signal with constant amplitude and frequency. In this case, each information element is associated not with the absolute value of the phase, but with its change relative to the previous value.

According to CCITT recommendations, at a speed of 2400 bps, the data stream to be transmitted is divided into pairs of consecutive bits (dibits), which are encoded into a phase change with respect to the phase of the previous signal element. One signal element carries 2 bits of information. If the information element is a dibit, then depending on its value (00, 01, 10 or 11), the phase of the signal can change by 90, 180, 270 degrees or not change at all.

With triple relative phase modulation or eightfold

In phase difference modulation, the data stream to be transmitted is divided into triplets of consecutive bits (tribits), which are encoded into a change in phase with respect to the phase of the previous signal element. One signal element carries 3 bits of information.

Phase modulation is the most informative, however, increasing the number of coded bits above three (8 phase rotation positions) leads to a sharp decrease in noise immunity. Therefore, at high speeds, combined amplitude-phase modulation methods are used.

Amplitude-phase manipulation. Amplitude phase keying (APK) is a combination of ASK and PSK schemes. The ARC modulated signal is shown in Fig. 1.2 G and is expressed as

with indexing of amplitude and phase terms. In Fig.1. 2 G one can see characteristic simultaneous (at the moments of transition between symbols) changes in the phase and amplitude of the ARC-modulated signal. In the given example M=8, which corresponds to 8 signals (octal transmission). A possible set of eight signal vectors is plotted in phase-amplitude coordinates. Four of the vectors shown have one amplitude, four more have another. The vectors are oriented so that the angle between the two closest vectors is 45°.

Figure 1.2 – Types of digital modulations

If in the two-dimensional space of signals between M dialing signals at a straight angle, the scheme is called quadrature amplitude modulation (QAM).

Quadrature amplitude modulation

It should be noted that another type of linear modulation is quadrature amplitude modulation (QAM), the essence of which is the transmission of two different signals using AM or FM methods on the same carrier frequency. The spectra of these two signals completely overlap and their separation using filters is impossible. To maintain the possibility of signal separation at the receiving side, the oscillation carriers are supplied to the modulators with a phase shift of 90° (in quadrature).

Figure 1.3 shows the QAM signal generation diagram.

Figure 1.3 – Quadrature AM

The advantage of QAM compared to conventional AM or BM is twice the number of signals that can be independently transmitted in the same frequency band.

Angle (frequency and phase) modulation

Angle modulation is usually used when it is necessary to ensure high fidelity of reception of the transmitted message. This is explained by the fact that systems with angular modulation have increased resistance to noise and other types of interference compared to AM. It is known, for example, that FM systems have properties to suppress additive noise interference. This means that when FM is detected, the signal-to-noise ratio is significantly improved. However, this advantage is achieved at the cost of deterioration of other signal parameters, in particular at the cost of increasing the occupied frequency band. Frequency modulation is perhaps the most common example that illustrates methods for increasing the noise immunity of communication systems based on spreading the signal spectrum.

Figure 1.4 shows the timing diagram of the signal with single-tone angle modulation.

Figure 1.4 Angle modulation: a - modulating low-frequency signal; b - single-tone signal with angular modulation

The angular modulation (AM) signal with a harmonic carrier can be written as follows:

u UM (t)= U 0 cos[(t)]=U 0 cos[ω 0 t+φ(t)],

where (t)=ω 0 t+φ(t) – total phase of the signal;

φ(t) – phase, which carries information about the primary signal.

There are two types of PA: phase (PM) and frequency (FM). With PM, phase changes are directly proportional to the primary signal

Where φ 0 is the initial phase.

In FM, the instantaneous frequency of the signal is directly proportional to the primary signal

, where is the conversion coefficient of the control signal into a change in the frequency of the signal at the output of the frequency modulator.

The shapes of PM and FM signals do not differ from each other if the time derivative of the primary signal has the same form as the primary signal itself. This occurs with a sinusoidal primary signal, for example

b(t)=Usint .

The PA signal in this case can be written as follows:

u UM (t)=U 0 cos(ω 0 t+Msint),

where M is the modulation index.

The FM index is determined as

M FM ==K FM U  ( – phase deviation).

The World Cup index is

M FM ==K FM U  /,

where the frequency deviation is K FM U  . therefore, the World Cup index

M FM =/=f / F.

Let's find the signal spectrum for PA with one tone. Let us represent the signal with PA in one tone using the following expression:

(Re is the real part).

Because during the World Cup

M FM =/=f /F,

then we find that for large modulation indices

f mind 2f ,

i.e., the frequency bandwidth at FM is equal to twice the frequency deviation and does not depend on the modulation frequency F.

Figures 1.5 and 1.6 show schemes for obtaining angle modulation signals

where b(t) is the primary signal;

–carrier generator U0cosω0t ;

block -/2 rotates the phase by angle -/2;

Types of modulation

Modulation is the process of controlling one or more parameters of high-frequency oscillations in accordance with the law of the transmitted message. Modulation can also be thought of as the process of superimposing one vibration onto another. The transmitted signal is called modulating, the controlled high-frequency signal is called modulated. The frequency of the modulating signal must be one or more orders of magnitude lower than the modulated one.

Modulation methods can be classified according to three criteria depending on:

– from the controlled parameter of the high-frequency signal: amplitude (AM), frequency (FM) and phase (PM);

– number of modulation stages: one-, two-, three-stage;

– type of transmitted message – (analog, digital or pulse) - continuous, with an abrupt change in the controlled parameter (this modulation is called manipulation) and pulse.

Description of modulated signals is possible within the framework of time and spectral methods. For undistorted reception of a modulated signal, the bandwidth of all high-frequency paths of the radio transmitter and radio receiver must be equal to or greater than the width of the spectrum of the emitted signal. On the other hand, the spectrum of the modulated signal should not exceed the permissible emission band allocated to this channel (Fig. 19.1).

Rice. 19.1. Allowable emission band of the modulated signal spectrum

Emissions lying outside the allocated emission band are called out-of-band. Their level should not exceed a certain, strictly standardized value. Otherwise, this communication channel will interfere with other channels.

The spectrum width of the modulated high-frequency signal Df c p depends both on the spectrum of the transmitted message and on the type of modulation. The parameter characterizing the modulated signal, allowing comparison of different types of modulation, is the signal base:

B=TDf c p, (19.1)

where T is the duration of the signal element.

When transmitting analog messages, the upper frequency of its spectrum F is related to the parameter T, interpreted as the time of the reference interval, by the ratio T=l/(2F) and therefore (19.1) takes the form:

B=Df c p /(2F). (19.2)

When transmitting digital information in a binary code consisting of logical 1 and 0, with a speed V equal to the number of transmitted chips (bits) per second (bit/s = baud), the value T is interpreted as the duration of the chip T = 1/V, and That's why:

B=Df c p /V. (19.3)

When B=1, the high-frequency modulated signal is called narrowband, when B>3...4 - broadband. In accordance with this definition, depending on the type of signal used, the radio system as a whole is called narrow- or broadband.

With amplitude modulation, the signal is always narrowband; with frequency (depending on the frequency deviation parameter characterizing it) - narrow or broadband. The type of modulation and the value of parameter B have a significant impact on the noise immunity of the radio system and obtaining the required signal-to-noise ratio in the radio receiver.

An example of modulated signals of the same power, but with different spectrum widths is shown in Fig. 19.2.

Rice. 19.2. Example of modulated signals of the same power with different spectrum widths

Let's consider what caused the need to use two-stage, and in some cases even three-stage modulation. Suppose that at one frequency of carrier oscillations f there is no need to transmit messages from several sources. To be able to separate received messages in a radio receiver, proceed as follows. Each message first modulates its own individual carrier, in this case called a subcarrier (Fig. 19.3).

In addition to simple types of digital modulation, there are more complex types designed to maximize efficiency in some respects. Most modern telecommunication systems use efficient modulation.

The main two directions in which types of digital modulation are being improved are power efficiency and spectral efficiency.

Quadrature modulation. When describing digital modulation, signal vectors are often represented in terms of quadrature and in-phase components (“ Q" And " I" - rice. 2.10).

This is due to the fact that modulation and demodulation of signals in digital communications are most often carried out using quadrature modulators and demodulators, since their implementation is much simpler than directly controlling the phase and amplitude of the signal, especially when simultaneous AM and PM are required.

The simplest way to increase spectral efficiency is to increase the duration of the rectangular bit burst while maintaining the same transmission rate in the number of bits per unit time. Quadrature phase shift keying is based on this principle ( quadrature phase shift keying – QPSK).

In Fig. 2.11, A the original data stream is presented dk(t) = d 0 , d 1 , d 2,..., consisting of bipolar pulses, i.e. dk take the values ​​+1 or –1, representing a binary one and a binary zero.

This pulse stream is divided into an in-phase stream dI(t) = d 0 , d 2 , d 4 , … and quadrature d Q(t) = d 1 , d 3 , d 5, ..., as shown in Fig. 2.11, b. Stream speeds dI(t) And d Q(t) equal to half the flow rate dk(t). Convenient orthogonal implementation of the QPSK signal, S(t), can be obtained using amplitude modulation of in-phase and quadrature flows on sine and cosine functions of the carrier:

Using trigonometric identities, this equation can be represented as follows:

The QPSK modulator shown in Fig. 2.11, uses the sum of the sine and cosine terms.

Pulse flow dI(t) is used for amplitude modulation (with amplitude +1 or –1) of a cosine wave. This is equivalent to shifting the phase of the cosine wave by 0 or π; hence the result is a BPSK signal. Similarly, the flow of impulses d Q(t) modulates a sine wave, which produces a BPSK signal orthogonal to the previous one. By summing these two orthogonal carrier components, a QPSK signal is obtained. Value θ( t) will correspond to one of four possible combinations dI(t) And d Q(t): θ( t) = 0, ±90, 180º; the resulting signal vectors are shown in the signal space in Fig. 2.12. Since cos(2π f 0 + π/4) and sin(2π f 0 + π/4) are orthogonal, the two BPSK signals can be detected separately.

Thus, QPSK is twice as economical as BPSK in terms of the use of frequency resources, since it has a spectrum of the same shape, but narrowed by half due to double stretching of the message. And this gain was achieved without deteriorating the noise immunity of reception (the Euclidean distance between adjacent vectors will remain the same, since at constant power the energy of the message will double due to doubling its duration).

However, the basic version of quadrature keying turns out to be not entirely favorable in terms of energy consumption. Since phase jumps of 180º are possible during transmission, the requirements for the linear range of the amplifier are excessive. To use the most favorable class C mode in terms of power consumption of the transmit amplifier, it is necessary to have a carrier with a constant envelope.

There are varieties of quadrature keying designed to reduce phase jumps. In the case of using quadrature shift keying ( OQPSK – Offset QPSK), threads dI(t) And d Q(t) are transmitted with a shift by T, as shown in Fig. 2.13.

Therefore, a simultaneous change in sign in both flows becomes impossible, which means that phase jumps by 180º are excluded, and the phase can only change by 90º.

Another approach to constant envelope modulation is called π/4-QPSK. Here, instead of shifting the parcels, a rotation by an angle π/4 of the alphabet of phase values ​​is introduced when moving from even to odd parcels. Thanks to this shift, when i = 2k φ i takes values ​​from the set 0, π, ±π/2, and when i = 2k+ 1 – from the set ±π/4, ±3π/4 (Fig. 2.14).

This type of modulation avoids a lot of demodulator complexity, although it is not as effective in mitigating dynamic range requirements as OQPSK.

QAM. Quadrature amplitude modulation ( QAM – Quadrature Amplitude Modulation) can be considered a logical continuation of QPSK, since the QAM signal also consists of two independent carriers (amplitude modulated). The transmission of QAM modulated signals can also be thought of as a combination of amplitude shift keying and phase shift keying (ASK and PSK). Due to the unequal length of signal vectors, optimization of their constellation is achieved, maximizing the minimum distance between signal vectors. Similar modulation formats with a wide variety of signal vectors and their constellation configurations (Fig. 2.15) are widely used in many telecommunications systems.

MSK. You can further enhance the QPSK format by eliminating discontinuous phase transitions. One of the schemes that implements modulation without a phase break is minimum shift keying ( minimum shift keying – MSK). It can be considered a special case of non-phase frequency shift keying (CPFSK) or a special case of QPSK with sinusoidal symbol weighting. In the first case, the MSK signal can be represented as follows:

Here f 0 carrier frequency, dk= ±1 represents bipolar data, and dk– phase constant for k th interval of binary data transmission. At dk= 1 transmitted frequency is f 0 + 1/4T, and when dk= –1 is f 0 – 1/4T. Therefore, the tone spacing in MSK is half that used in orthogonal FSK, hence the name - keying. minimal shift.

The type of modulation under consideration is essentially reduced to binary frequency shift keying. In this case, the frequency switches without phase jumps; the transmission of the next symbol begins with the phase that “ran up” during the transmission of the previous symbol. This principle can be illustrated by a tree of phase trajectories (Fig. 2.16, A). During each period of time, the phase increases or decreases linearly in accordance with the current frequency increment, and any of the possible phase trajectories turns out to be a continuous function (Fig. 2.16, b). This modulation ensures a constant envelope and, as a result, optimal mode of the transmitter power amplifier.

In quadrature representation, the signal can be written as follows:

Thus, the sending becomes an impulse with an envelope in the form of a half-wave cosine. Due to its smoothed shape, the spectrum is significantly narrowed compared to QPSK.

GMSK. When transmitting over a radio channel, a narrower signal spectrum band is often desirable than with MSK, where there are quite large side lobes extending beyond the 1/ T b. To achieve further narrowing of the spectrum, low-pass filtering is performed before modulation. If a filter with a Gaussian frequency response is used, then this modulation option is called GMSK ( Gaussian MSK). To characterize the passband of the low-pass filter, enter the following value:

Where f–3 dB – cutoff frequency at –3 dB level; R– bit rate. In Fig. 3.17, A the impulse characteristics of the Gaussian filter are given at BT= 0.3 and BT= 0.5. In Fig. 2.17, b You can see the bandwidth gain when using GMSK with these values ​​relative to MSK.

Rice. 2.17

However, as can be seen from Fig. 2.17, A, with increasing value BT The symbol length is stretched, which is fraught with increased intersymbol interference. That is, the gain in spectrum compactness is achieved by reducing the reliability of information transmission. In the GSM standard, the optimal value is taken BT = 0.3.

GMSK modulation can be seen as a further improvement on the principle of achieving phase continuity. In this case, not only discontinuities in the phase itself are eliminated, but also in its derivatives. In Fig. Figure 2.18 shows a phase tree for GMSK modulation, illustrating this principle.

As the above review shows, the digital modulation methods used are noticeably diverse. Therefore, when designing telecommunication systems, there are many ways to achieve optimal performance. In conclusion, we can give a brief comparison of some types of digital modulation with each other.

In Fig. Figure 2.19 shows a graph where the ordinate axis shows the specific spectral efficiency (bit/s/Hz), and the abscissa axis shows the energy efficiency (the ratio of the energy per message bit to the spectral noise density required to achieve an error probability of 10 –5).

Different types of modulation are marked on this graph by a dot characterizing the relationship between the spectral and energy efficiency of this type. The graph clearly shows the compromise nature of choosing the type of digital modulation with respect to these two parameters.

In table 2.1 provides examples of the use of some types of digital modulation in commercial telecommunication systems for various purposes.

Table 2.1

The choice of modulation type depends on the specifics of the application, system deployment, required transmission speed, and required transmission reliability.