Carrier vibration. Amplitude-modulated oscillations

The carrier oscillation (CV) recovery system of demodulators of bandpass digital modulation signals is designed to generate a reference harmonic oscillation, the phase of which coincides with the phase of the carrier on the basis of which the demodulated signal is formed.

Already in the 30s of the last century, it became clear that FM-2 signals have the highest noise immunity. To use these signals in transmission systems, it was necessary to solve the problem of restoring the carrier (reference) oscillation in the demodulator, which is necessary for the operation of a synchronous detector. In those years it was proposed carrier oscillation recovery circuit with frequency multiplication by 2 (Fig. 13.1).

In the case of FM-2. Odds a i specified by the signal constellation (Fig. 11.1). Channel symbols:

“Weakly” filtered pulses have been used for many decades A(t), which were close in shape to the P-pulse over an interval of duration T

(13.2)

After multiplying the frequency by 2, as a signal s 1 (t), and the signal s 0 (t) give . A notch filter has a middle passband frequency of 2 f 0 . It is designed to reduce interference. A frequency divider by 2 can produce one of two possible reference oscillations:

Case 1:

Case 2:

Both oscillations are possible, since the result depends on the initial conditions in the divider circuit. The reference vibration is said to have phase uncertainty about 180°.

In case 1, an algorithm for optimal signal demodulation is implemented
FM-2. In case 2, the output of the multiplier, and then the matched filter and sampler, will have voltages opposite to those in case 1. The decision circuit will make inverse decisions: instead of 1, it produces 0 and vice versa. This phenomenon is called inverse (reverse) demodulator robot. It turned out that during the operation of the demodulator, random jump-like transitions from the oscillation u op1 ( t) to oscillation u op2 ( t) and vice versa.

In the FM-4 signal demodulator it is necessary to use a frequency multiplier by 4, a filter with an average passband frequency of 4 f 0 and a frequency divider by 4. After the frequency divider, one of the reference oscillations occurs, which differ in phase in increments of 90°. There is an uncertainty in the phase of the reference vibration of the order of 90°.

It is possible to eliminate the manifestation of uncertainty in the phase of the reference oscillation in the demodulator by using difference (relative) coding. Such transmission methods are called phase-difference (relative phase) modulation.

The VN system with exponentiation is discussed above. However, it works well when the pulse amplitude A(t) is close to a rectangular shape. Nowadays Nyquist pulses are used - pulses with a significantly smoothed shape A(t). With this pulse shape, the VN system with exponentiation does not work well.

The reference oscillation is necessary for the operation of a synchronous detector (Fig. 13.2). Let the FM-2 signal arrive at the detector input. The channel symbol is described

If the oscillation phase from the generator

differs from the carrier phase of the input signal by the amount Dj, then the signal at the output of the synchronous detector receives the multiplier cosDj:

Since the maximum cosine value is equal to unity and is achieved only in the case of Dj = 0, the presence of a phase difference leads to a decrease in the signal level at the detector output. If Dj = p/2, then there is no signal at the output of the detector at all: .

Nowadays the HV system is phase automatic frequency control system(PLL) (Fig. 13.3) with a special phase error detector, which is capable of operating in the absence of a carrier in the signal spectrum. Here the VCO is a voltage controlled oscillator. When a phase error voltage e appears, this voltage adjusts the frequency and phase of the oscillation produced by the VCO so as to reduce the magnitude of the phase error.

Let's consider the construction of a phase error detector in the case of an PM-2 signal. The detector circuit contains one more additional synchronous detector, the reference vibration of which is . Recall that the operation of a synchronous detector can be considered as calculating the projection s(t) on u op ( t). The two synchronous detectors feature reference oscillations that are 90° out of phase. Therefore, the voltages obtained from the outputs of synchronous detectors are quadrature components of the detected signal.

In Fig. Figure 13.4 shows the signal constellation of the demodulated FM-2 signal and the calculated quadrature components at the sampling time, provided that the channel symbol with amplitude is demodulated A: I– common-mode component, Q– quadrature component. In Fig. 13.4, A reference phase error Dj = 0; in this case, synchronous detectors calculate I = A, Q= 0. In Fig. 13.4, b reference phase error Dj > 0; in this case, synchronous detectors calculate I = A×cosDj, Q < 0. На рис. 13.4, V reference phase error Dj< 0; при этом синхронные детекторы вычисляют I = A×cosDj, Q > 0.

We see that the sign of the value Q corresponds to a phase error: namely, if Q < 0, то Dj >0 and it is necessary to reduce the frequency and phase of the VCO, if Q> 0, then Dj< 0 и необходимо увеличивать частоту и фазу ГУН. Таким образом, значение Q can be taken as phase error e. But the situation is with a sign Q the opposite when demodulating a channel symbol with amplitude – A.

The operation of modulating a continuous harmonic oscillation with a constant amplitude, called a carrier oscillation or simply “carrier”, is carried out in order to transfer the spectrum of the signal to be transmitted into the radio frequency region provided for transmission.

When modulating such a high-frequency oscillation, one (or several) of its parameters change according to the law of the modulating signal. The amplitude, phase and frequency of harmonic oscillation can be modulated. In accordance with this use:

Amplitude modulation (AM)

where is the modulating function (modulating signal);

And - respectively, the amplitude, frequency and initial phase of the carrier vibration;

Frequency modulation (FM)

where is frequency deviation;

Phase modulation (PM)

where is the phase deviation.

In digital communication systems, the modulating function takes only discrete values, the number of which is determined by the selected modulation position. This discrete modulation is often called keying.

When =2, the modulating function can take only two values ​​- plus or minus one, and the corresponding types of modulation are usually designated as AM-2, FM-2 and FM-2, where the number indicates the position of the modulation.

When AM takes the values ​​plus one and zero (AM-2). In this case, at =1, oscillations with a frequency are emitted, and at =0 there is no radiation. This transmission mode in a radio channel is called a passive pause mode.

It should also be noted that when manipulating a harmonic oscillation, a quasi-ternary code sequence (for example, when using PRF codes) takes on three possible values ​​- plus one, zero and minus one, and in this case = 3, although the modulation speed and information transfer rate V numerically coincide (the previously given relationship does not hold).

Binary types of modulation have different noise immunity under the same reception conditions. In coherent consistent reception, the bit error probability (the probability of a binary symbol being received erroneously) is given by

where q is the signal/noise ratio in terms of power at the input of the signal discriminator, r is the cross-correlation coefficient of the distinguished binary signals.

At FM-2. In this case, binary signals, which are segments of a cosine wave with opposite values ​​of the initial phase, are opposite signals having a cross-correlation coefficient r (opposite signals).

During FM-2, it is chosen so that the binary radio signals - segments of cosine waves with different frequencies - are orthogonal. Orthogonal signals have.

Figure 22 shows the amplitude spectra of radio signals corresponding to the transmission of binary symbols “1” and “0”.

Figure 22

Frequency deviation. In this case, at the reference points on the frequency axis (i), the amplitude spectrum of one of these signals is maximum, and the other is equal to zero. The difference frequency in this case numerically coincides with the manipulation speed.

Figure 23

Figure 23 shows the values ​​of the cross-correlation coefficient of frequency-shift keyed signals depending on

From this expression it follows that when, and.

At AM-2 r0.5 and

In the given expressions, the Crump function.

When there are high requirements for noise immunity, when it is convenient to calculate the error probability using the approximate formula of the Crump function obtained from its asymptotic representation: . The calculation error is no worse than 10%, if.

Thus, FM-2 turns out to be the most noise-resistant, FM-2 occupies an intermediate position between FM-2 and AM-2.

AM-2 amplitude keying is used very rarely in modern digital radio communications.

The minimum radio frequency bandwidth required to transmit a binary sequence from AM-2 is estimated by the previously given relationship

(specific information transmission speed) in this case

Phase shift keying (PM-2, FM-4 and FM-8) is currently widely used in terrestrial and satellite radio links.

The disadvantage of PM is the need for coherent demodulation. In this case, the formation of a reference oscillation from the received signal, as was shown earlier, entails the appearance of the effect of reverse operation of the demodulator.

The use of relative phase modulation makes it possible to eliminate this effect, however, at the cost of complicating the signal generation and processing equipment.

Relative phase modulation (RPM), also called phase difference or differential phase modulation, allows demodulation in two ways. The first of them, using relative decoding, was mentioned and discussed earlier. The second is differential-coherent (autocorrelation) detection of a PPM radio signal, in which the previous radio pulse, delayed exactly by the duration of the binary element (), is used as a reference oscillation. In this case, the operations of detection and relative decoding are combined. However, the problem remains in ensuring an accurate delay of the preceding radio pulse.

The spectrum width of the OFM radio signal depends on the speed of manipulation.

Frequency efficiency factor

Frequency shift keying (FM-2, FM-3, FM-4 and FM-8) is widely used in modern digital radio communication systems.

The frequency band required to transmit an FM radio signal depends on the maximum frequency deviation and the position of the modulation

Frequency efficiency factor

An FM radio communication channel using a non-coherent reception method (non-coherent demodulation) has these characteristics.

Of great interest is the use of minimum shift frequency shift keying (MSMS), which is a special case of continuous phase keying.

With this type of modulation, the phase of the manipulated radio signal, changing continuously, does not have jumps at the boundaries of the radio pulses. With FMMS, two frequencies are used to transmit “1” and “-1”, as with conventional FM-2, however, their difference is selected so that the mutual correlation coefficient is equal to the first zero of the function (see Figure 23). This correlation coefficient value corresponds to the argument

and therefore .

With such a difference frequency, the phase of the manipulated radio signal changes by exactly . In this case, if “1” is transmitted, then the frequency of the radio signal

so that at the moment the radio pulse ends, its phase is shifted by 2. When transmitting “-1”, the frequency of the radio pulse

As a result, the phase of the pulse at the moment of its end acquires a shift of minus 2. Thus, FMMS is very similar to OFM-2 in which the phase of the manipulated signal also changes by 2 during each interval. The difference is that with FMMS the phase does not change abruptly, but continuously.

FMMS demodulation uses coherent detection. This complicates the construction of the demodulator.

Frequency band required for transmitting the FMMS signal

Frequency efficiency factor

Amplitude modulation- a type of modulation in which the variable parameter of the carrier signal is its amplitude.

Amplitude modulation (AM) is a modulation in which undamped oscillations change in amplitude in accordance with the lower frequency oscillations modulating it.

With amplitude modulation (AM), the amplitude of the high-frequency oscillation (carrier) changes according to the law of the modulating (primary) signal.

With AM, the spectrum of the modulating signal is transferred to the carrier frequency region, forming the upper and lower side components of the spectrum. Since this transformation produces new frequencies, the modulation procedure is a nonlinear transformation. But since with AM the spectrum of the modulating signal does not change, but is only transferred to the high frequency region, AM is considered a linear type of modulation.

The goal of any modulation is undistorted signal transmission over a given communication line with less interference.

The principles of spectrum conversion in AM are widely used in technology,

for example, in the development of circuits for broadcasting and television receivers, multi-channel telephony systems with frequency division multiplexing of communication lines and, in particular, form the basis of a spectrum analyzer device.

Carrier frequency, the frequency of harmonic oscillations that are modulated by signals for the purpose of transmitting information. Low-frequency vibrations are sometimes called carrier vibrations. The oscillations with low frequencies themselves do not contain information, they only “carry” it. The spectrum of modulated oscillations contains, in addition to low frequencies, side frequencies that contain transmitted information.

If we take a signal having a sinusoid formula as the primary signal, then the amplitude-modulated signal will have the form shown in the figure.

On the qualitative side, amplitude modulation (AM) can be defined as a change in the amplitude of the carrier in proportion to the amplitude of the modulating signal.

A harmonic oscillation of high frequency w is modulated in amplitude by a harmonic oscillation of low frequency W (t = 1/W is its period), t is time, A is the amplitude of the high-frequency oscillation, T is its period.

Amplitude modulation by a sinusoidal signal, w - carrier frequency, W - frequency of modulating oscillations, Amax and Amin - maximum and minimum amplitude values.

For a large amplitude modulating signal, the corresponding amplitude of the modulated carrier must be large and for small amplitude values, this modulation scheme can be implemented by multiplying two signals.

Amplitude modulation depth- maximum relative deviation of the amplitude from the average

The spectral density of the modulated signal represents two spectra of the modulating function, constructed relative to the frequencies w = w 0 and w = -w 0 (shifted to the carrier frequencies).

Example. Single tone modulation spectrum

A radio signal consists of a carrier wave and two sine waves called sidebands.

In conventional amplitude modulation, information is contained in each of the two sidebands

Carrier signal- a signal, one or more parameters of which are subject to change during the modulation process. The degree of parameter change is determined by the instantaneous value of the information (modulating) signal.

Any stationary signal can be used as a carrier signal. Most often, a high-frequency (relative to the information signal) harmonic oscillation is used as a carrier signal, which is due to the simplicity of demodulation and a narrow spectrum. However, in some cases it is advisable to use other types of carrier signal, for example, square wave.

The carrier signal is often called simply carrier(from carrier frequency), or carrier (oscillation). All these terms mean practically the same thing. In English terminology, a carrier signal is denoted by the word carrier.

The ratio U /U 0 is called the modulation coefficient mAM. It is often expressed as a percentage. If U 0 >=Umax, then the coefficient mAM will vary from 0 to 1.

Amplitude modulation coefficient(AM coefficient, legacy modulation depth) - the main characteristic of amplitude modulation 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, expressed as a percentage

AM oscillations are the result of the addition of three high-frequency oscillations; oscillations with frequency f 0 and amplitude U 0 and two oscillations with frequencies f 0 + F and f 0 - F and amplitude 0.5 mAM*U 0 .

In amplitude modulation (AM) systems, the modulating wave changes the amplitude of a high-frequency carrier wave. Analysis of the output frequencies shows the presence of not only the input frequencies f 0 and F, but also their sum and difference: f n + F and f n - F. If the modulating wave is complex, such as a speech signal, which consists of many frequencies, then the sums and differences of various frequencies will occupy two bands, one below and the other above the carrier frequency. The frequencies f n + F and f n - F are called the upper and lower side frequencies, respectively.

Top side stripe is a copy of the original conversational signal, only shifted to the Fc frequency. The lower band is an inverted copy of the original signal, i.e. the high frequencies in the original are the low frequencies in the lower side.

Lower side strip this is a mirror image of the upper side with respect to the carrier frequency Fc.

An AM system that transmits both sidebaud and carrier is known as a double sidebaud (DSB) system. The carrier carries no useful information and can be removed, but with or without the carrier, the DSB signal has twice the bandwidth of the original signal. To narrow the band, it is possible to displace not only the carrier, but also one of the side ones, since they carry the same information. This type of operation is known as single sideband suppressed carrier modulation (SSB-SC - Single SideBand Suppressed Carrier).

Amplitude modulation of a complex signal

Any transmitting radio station operating in amplitude modulation mode emits not just one frequency, but a whole set (spectrum) of frequencies. In the simplest case (with a sinusoidal signal), this spectrum contains only three components - a carrier and two side ones. If the modulating signal is not sinusoidal, but more complex, then instead of two side frequencies in the modulated oscillation there will be two side bands, the frequency composition of which is determined by the frequency composition of the modulating signal.

Therefore, each transmitting station occupies a certain frequency slot on the air. To avoid interference, the carrier frequencies of different stations must be separated from each other by a distance greater than the sum of the sidebands. The width of the sideband depends on the nature of the transmitted signal: for radio broadcasting - 10 kHz, for television - 6 MHz. Based on these values, the interval between the carrier frequencies of different stations is selected. To obtain an amplitude-modulated oscillation, the oscillation of the carrier frequency and the modulating signal are fed to a special device - a modulator.

Demodulation of an AM signal is achieved by mixing the modulated signal with a carrier of the same frequency as the modulator.

The original signal is then obtained as a separate frequency (or frequency band) and can be filtered from other signals. The demodulation carrier is generated locally and may not coincide in any way with the carrier frequency at the modulator. The slight difference between the two frequencies causes frequency mismatch, which is inherent in telephone circuits.

Due to amplitude modulation of a complex signal, the data transmission speed increases.

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 variable 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 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 FM signal formation 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 signal spectra for PSK and OFPS have the same appearance (Figure 9, e).

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

Signals coming from a message source (microphone, transmitting television camera, telemetry system sensor), as a rule, cannot be directly transmitted over a radio channel. It's not just that these signals are not large enough in amplitude. Much more significant is their relative low frequency. To carry out effective signal transmission in any medium, it is necessary to move 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.

4.1. Amplitude Modulated Signals

Before studying this simplest type of modulated signals, let's briefly consider some issues related to the principles of modulation of any kind.

The concept of carrier vibration. The idea of ​​a method that allows you to transfer the signal spectrum to the high frequency region is as follows. First of all, an auxiliary high-frequency signal called a carrier wave is generated in the transmitter. Its mathematical model is such that there is a certain set of parameters that determine the shape of this oscillation. Let be a low-frequency message to be transmitted over a radio channel. If at least one of these parameters changes over time in proportion to the transmitted message, then the carrier oscillation acquires a new property - it carries: information that was originally contained in the signal

The physical process of controlling the parameters of a carrier vibration is modulation.

In radio engineering, modulation systems using a simple harmonic oscillation as a carrier wave have become widespread.

having three free parameters

By changing one or another parameter over time, you can obtain different types of modulation.

The principle of amplitude modulation.

If the amplitude of the signal turns out to be variable and the other two parameters are unchanged, then there is amplitude modulation of the carrier oscillation. The form of recording an amplitude-modulated, or AM, signal is as follows:

The oscillogram of the AM signal has a characteristic appearance (see Fig. 4.1). Noteworthy is the symmetry of the graph relative to the time axis. In accordance with formula (4.2), the AM signal is the product of the envelope and harmonic filling. In most practically interesting cases, the envelope changes over time much more slowly than the high-frequency filling.

Rice. 4.1. AM signals at different modulation depths: a - shallow modulation: b - deep modulation; c - overmodulation

In amplitude modulation, the relationship between the envelope and the modulating useful signal is usually defined as follows:

Here is a constant coefficient equal to the amplitude of the carrier vibration in the absence of modulation; M - amplitude modulation coefficient.

The value M characterizes the depth of amplitude modulation. The meaning of this term is illustrated by the oscillograms of AM signals shown in Fig. 4.1, a-c.

At a small modulation depth, the relative change in the envelope is small, i.e. at all times, regardless of the signal shape

If, at times when the signal reaches extreme values, there are approximate equalities

then they talk about deep amplitude modulation. Sometimes an additional relative modulation coefficient is introduced upwards

and relative modulation factor down

AM signals with a shallow modulation depth in radio channels are impractical due to incomplete use of transmitter power.

At the same time, 100% upward modulation doubles the amplitude of oscillations at the peak values ​​of the modulating message. A further increase in this amplitude, as a rule, leads to unwanted distortion due to overloading the output stages of the transmitter.

No less dangerous is too deep downward amplitude modulation. In Fig. 4.1, c shows the so-called overmodulation. Here the shape of the envelope ceases to follow the shape of the modulating signal.

Single tone amplitude modulation.

The simplest AM signal can be obtained when the modulating low-frequency signal is a harmonic oscillation with a frequency of . Such a signal

called a single tone AM signal.

Let's find out whether such a signal can be represented as a sum of simple harmonic oscillations with different frequencies. Using the well-known trigonometric formula for the product of cosines, from expression (4.4) we immediately obtain

Formula (4.5) establishes the spectral composition of a single-tone AM signal. The following terminology is accepted: - carrier frequency, - upper side frequency, - lower side frequency.

When constructing a spectral diagram of a single-tone AM signal using formula (4.5), you should first of all pay attention to the equality of the amplitudes of the upper and lower lateral oscillations, as well as the symmetry of the location of these spectral components relative to the carrier oscillation.

Energy characteristics of the AM signal.

Let us consider the question of the relationship between the powers of the carrier and lateral vibrations. A single-tone AM signal source is equivalent to three harmonic oscillation sources connected in series:

Let us assume for definiteness that these are EMF sources connected in series and loaded by a single resistor. Then the instantaneous power of the AM signal will be numerically equal to the square of the total voltage:

To find the average signal power, the value must be averaged over a sufficiently large period of time T:

It is easy to verify that, when averaging, all mutual powers will give a zero result - therefore, the average power of the AM signal will be equal to the sum of the average powers of the carrier and lateral oscillations:

It follows that

Thus, even with 100% modulation (M = 1), the share of the power of both lateral oscillations is only 50% of the power of the modulated carrier oscillation. Since the message information is contained in the lateral oscillations, there is an inefficiency in the use of power when transmitting an AM signal.

Amplitude modulation with a complex modulating signal.

In practice, single-tone AM signals are rarely used. A much more realistic case is when the modulating low-frequency signal has a complex spectral composition. A mathematical model of such a signal can be, for example, a trigonometric sum

Here the frequencies , form an ordered increasing sequence, while the amplitudes and initial phases Φ, are arbitrary.

Substituting formula (4.9) into (4.3), we obtain

Let us introduce a set of partial (partial) modulation coefficients

and write the analytical expression for a complex-modulated (multi-tone) AM signal in a form that generalizes expression (4.4):

Spectral decomposition is carried out in the same way as for a single-tone AM signal:

In Fig. 4.2, and shows the spectral diagram of the modulating signal constructed in accordance with formula (4.9). Rice. 4.2b reproduces the spectral diagram of a multi-tone AM signal corresponding to this modulating oscillation.

Rice. 4.2. Spectral diagrams of a - modulating signal; b - AM signal with multi-tone modulation

So, in the spectrum of a complex modulated AM signal, in addition to the carrier vibration, there are groups of upper and lower lateral vibrations. The spectrum of the upper lateral oscillations is a large-scale copy of the spectrum of the modulating signal, shifted to the high frequency region by an amount. The spectrum of the lower lateral oscillations also repeats the spectral diagram of the signal and is mirrored relative to the carrier frequency

An important conclusion follows from the above: the width of the spectrum of the AM signal is equal to twice the highest frequency in the spectrum of the modulating low-frequency signal.

Example 4.1. Estimate the number of broadcast radio channels that can be placed in the frequency range from 0.5 to 1.5 MHz (approximate boundaries of the mid-wave broadcast range).

To reproduce broadcast signals satisfactorily, it is necessary to reproduce audio frequencies between 100 Hz and 12 kHz. Thus, the frequency band allocated to one AM channel is 24 kHz. To avoid crosstalk between channels, a guard interval of 1 kHz should be provided. Therefore, the permissible number of channels

Amplitude-manipulated signals.

An important class of multi-tone AM signals are the so-called keyed signals. In the simplest case, these are sequences of radio pulses separated from each other by pauses. Such signals are used in radiotelegraphy and in systems for transmitting discrete information over radio channels.

If s(t) is a function that at each moment of time takes the value either 0 or 1, then the amplitude-manipulated signal is represented in the form

Let, for example, the function display the periodic sequence of video pulses considered in example 2.1 (see Chapter 2). Assuming that the amplitude of these pulses based on (4.14) we have at

where q is the duty cycle of the sequence.

Vector diagram of an AM signal.

Sometimes it can be useful to graphically represent the AM signal as a sum of vectors rotating in the complex plane.

For simplicity, let's consider diotonal modulation. The instantaneous value of the carrier vibration is the projection of a time-neutral vector onto the angular reference axis, which rotates around the origin with angular velocity in the clockwise direction (Fig. 4.3).

The upper lateral oscillation is displayed on the diagram by a vector of length and its phase angle at is equal to the sum of the initial phases of the carrier and modulating signals [see. formula (4.5).

Rice. 4.3. Vector diagrams of a single-tone AM signal: a - at ; b - at

The same vector for the lower lateral oscillation differs only in the sign in the expression for its phase angle. So, on the complex plane it is necessary to construct the sum of three vectors

It is easy to see that this sum will be oriented along the vector ines. The instantaneous value of the AM signal at will be equal to the projection of the end of the resulting vector onto the horizontal axis (Fig. 4.3a).

Over time, in addition to the noted rotation of the angle reference axis, the following transformations of the drawing will be observed (Fig. 4.3,6): 1) the vector will rotate around the point of its application with angular velocity in a counterclockwise direction, since the phase of the upper lateral oscillation increases faster than the phase carrier signal; 2) the vector will also rotate with angular velocity, but in the opposite direction.

By constructing the total vector and projecting it onto the angle reference axis, you can find instantaneous values ​​at any time.

Balanced amplitude modulation.

As has been shown, a significant portion of the power of a conventional AM signal is concentrated in the carrier wave. To more efficiently use the transmitter power, AM signals can be generated with a suppressed carrier wave, implementing the so-called balance amplitude modulation. Based on formula (4.4), the representation of a single-tone AM signal with balanced modulation is as follows:

There is a multiplication of two signals - modulating and carrier. From a physical point of view, oscillations of the form (4.16) are beats of two harmonic signals with identical amplitudes and frequencies equal to the upper and lower side frequencies.

With multi-tone balanced modulation, the analytical expression of the signal takes the form

As with conventional amplitude modulation, two symmetrical groups of upper and lower lateral oscillations are observed here.

If we consider the beat oscillogram, it may seem unclear why there is no carrier frequency in the spectrum of this signal, although there is a presence of high-frequency filling that changes over time precisely at this frequency.

The fact is that when the beat envelope passes through zero, the phase of the high-frequency filling changes abruptly by 180°, since the function has different signs to the left and right of zero. If such a signal is applied to a high-quality oscillatory system (for example, an -circuit) tuned to a frequency, then the output effect will be very small, tending to zero as the quality factor increases. Oscillations in the system excited by one period of beating will be damped by the next period. This is exactly how it is customary to consider the question of the real meaning of the spectral decomposition of a signal from a physical point of view. We will return to this problem again in Chap. 9.

Single sideband amplitude modulation.

An even more interesting improvement on the principle of conventional amplitude modulation is to generate a signal with the upper or lower sideband frequencies suppressed.

Signals with one sideband (OBP or SSB signals - from the English single sideband) resemble ordinary AM signals in their external characteristics. For example, a single-tone OBP signal with the lower side frequency suppressed is written as

Carrying out trigonometric transformations, we get

The last two terms are the product of two functions, one of which changes slowly over time, and the other quickly. Taking into account that the “fast” factors are in relation to each other in time quadrature, we calculate the slowly changing envelope of the OBP signal:

Rice. 4.4. Envelopes of single-tone modulated signals with - OBP signal; 2 - regular AM signal

The graph of the OBP signal envelope calculated using formula (4.18) is shown in Fig. 4.4. Here, for comparison, the envelope of a conventional single-tone AM signal with the same modulation coefficient is constructed.

A comparison of the above curves shows that direct demodulation of the OBP signal along its envelope will be accompanied by significant distortion.

A further improvement of OBP systems is partial or complete suppression of carrier vibration. In this case, the transmitter power is used even more efficiently.