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Main characteristics of signals. Characteristics introduction Forms of analytical description of signals

Radio signals are called electromagnetic waves or electrical high-frequency oscillations that contain the transmitted message. To form a signal, the parameters of high-frequency oscillations are changed (modulated) using control signals, which are voltages that change according to a given law. Harmonic high-frequency oscillations are usually used as modulated ones:

where w 0 \u003d 2π f 0 – high carrier frequency;

U 0 is the amplitude of high-frequency oscillations.

The simplest and most commonly used control signals include harmonic oscillation

where Ω is low frequency, much smaller than w 0 ; ψ is the initial phase; U m - amplitude, as well as rectangular pulse signals, which are characterized by the fact that the voltage value U ex ( t)=U during the time intervals τ and, called the duration of the pulses, and is equal to zero during the interval between pulses (Fig. 1.13). Value T and is called the pulse repetition period; F and =1/ T and is the frequency of their repetition. Pulse Period Ratio T and to the duration τ and is called the duty cycle Q impulse process: Q=T and /τ and.

Fig.1.13. Rectangular pulse train

Depending on which parameter of the high-frequency oscillation is changed (modulated) with the help of a control signal, amplitude, frequency and phase modulation is distinguished.

With amplitude modulation (AM) of high-frequency oscillations by a low-frequency sinusoidal voltage with a frequency of Ω mod, a signal is formed, the amplitude of which changes with time (Fig. 1.14):

Parameter m=U m / U 0 is called the amplitude modulation factor. Its values are in the range from one to zero: 1≥m≥0. Modulation factor expressed as a percentage (i.e. m×100%) is called the amplitude modulation depth.

Rice. 1.14. Amplitude modulated radio signal

With phase modulation (PM) of a high-frequency oscillation by a sinusoidal voltage, the signal amplitude remains constant, and its phase receives an additional increment Δy under the influence of the modulating voltage: Δy= k FM U m sinW mod t, where k FM - coefficient of proportionality. A high-frequency signal with phase modulation according to a sinusoidal law has the form

With frequency modulation (FM), the control signal changes the frequency of high-frequency oscillations. If the modulating voltage changes according to a sinusoidal law, then the instantaneous value of the frequency of the modulated oscillations w \u003d w 0 + k World Cup U m sinW mod t, where k FM - coefficient of proportionality. Biggest Change frequency w with respect to its average value w 0 equal to Δw М = k World Cup U m, is called the frequency deviation. The frequency modulated signal can be written as follows:

The value equal to the ratio of the frequency deviation to the modulation frequency (Δw m / W mod = m FM) is called the frequency modulation ratio.

Figure 1.14 shows high-frequency signals for AM, PM and FM. In all three cases, the same modulating voltage is used. U mod, changing according to the symmetrical sawtooth law U mod( t)= k Maud t, where k mod >0 on time interval 0 t 1 and k Maud<0 на отрезке t 1 t 2 (Fig. 1.15, a).

With AM, the signal frequency remains constant (w 0), and the amplitude changes according to the law of the modulating voltage U AM ( t) = U 0 k Maud t(Fig. 1.15, b).

The frequency modulated signal (Fig. 1.15, c) is characterized by a constant amplitude and a smooth change in frequency: w( t) = w0 + k World Cup t. In the time span from t=0 to t 1 the oscillation frequency increases from the value w 0 to the value w 0 + k World Cup t 1 , and on the segment from t 1 to t 2 the frequency decreases again to the value w 0 .

The phase-modulated signal (Fig. 1.15, d) has a constant amplitude and frequency hopping. Let's explain this analytically. With FM under the influence of modulating voltage

Fig.1.15. Comparative view of modulated oscillations with AM, FM and FM:
a - modulating voltage; b – amplitude modulated signal;
c – frequency-modulated signal; d - phase modulated signal

signal phase receives an additional increment Δy= k FM t, therefore, a high-frequency signal with phase modulation according to the sawtooth law has the form

Thus, on the segment 0 t 1 the frequency is w 1 >w 0 , and on the segment t 1 t 2 it is equal to w 2

When transmitting a sequence of pulses, for example, a binary digital code (Fig. 1.16, a), AM, FM and FM can also be used. This type of modulation is called keying or telegraphy (AT, CT and FT).

Fig.1.16. Comparative view of manipulated oscillations in AT, PT and FT

With amplitude telegraphy, a sequence of high-frequency radio pulses is formed, the amplitude of which is constant during the duration of the modulating pulses τ and, and is equal to zero the rest of the time (Fig. 1.16, b).

With frequency telegraphy, a high-frequency signal is formed with a constant amplitude and a frequency that takes two possible values (Fig. 1.16, c).

With phase telegraphy, a high-frequency signal is formed with a constant amplitude and frequency, the phase of which changes by 180 ° according to the law of the modulating signal (Fig. 1.16, d).

Lecture #5

T theme #2: Transmission of DISCRETE messages

Lecture topic: DIGITAL RADIO SIGNALS AND THEIR

Features Introduction

For data transmission systems, the requirement for the reliability of the transmitted information is most important. This requires logical control of the processes of transmission and reception of information. This becomes possible when digital signals are used to transmit information in a formalized form. Such signals make it possible to unify the element base and use correction codes that provide a significant increase in noise immunity.

2.1. Understanding Discrete Messaging

Currently, for the transmission of discrete messages (data), as a rule, the so-called digital communication channels are used.

Message carriers in digital communication channels are digital signals or radio signals if radio communication lines are used. The information parameters in such signals are amplitude, frequency and phase. Among the accompanying parameters, the phase of the harmonic oscillation occupies a special place. If the phase of the harmonic oscillation on the receiving side is precisely known and this is used when receiving, then such a communication channel is considered coherent. AT incoherent In the communication channel, the phase of the harmonic oscillation on the receiving side is not known and it is assumed that it is distributed according to a uniform law in the range from 0 to 2  .

The process of converting discrete messages into digital signals during transmission and digital signals into discrete messages during reception is illustrated in Fig. 2.1.

Fig.2.1. The process of converting discrete messages during their transmission

Here it is taken into account that the main operations for converting a discrete message into a digital radio signal and vice versa correspond to the generalized block diagram of the discrete message transmission system discussed in the last lecture (shown in Fig. 3). Consider the main types of digital radio signals.

2.2. Characteristics of digital radio signals

2.2.1. Amplitude-shift keyed radio signals (aMn)

Amplitude shift keying (AMn). Analytical expression of the AMn signal for any moment of time t looks like:

s AMn (t, )= A 0  (t) cos(  t ) , (2.1)

where A 0 ,   and  - amplitude, cyclic carrier frequency and initial phase of the AMn radio signal,  (t) – primary digital signal (discrete information parameter).

Another form of writing is often used:

s 1 (t) = 0 at  = 0,

s 2 (t) = A 0 cos(  t ) at  = 1, 0  t T ,(2.2)

which is used in the analysis of AMn signals in a time interval equal to one clock interval T. As s(t) = 0 at  = 0, then the AMn signal is often referred to as a signal with a passive pause. The implementation of the AMn radio signal is shown in Fig. 2.2.

Fig.2.2. Implementation of the AM radio signal

The spectral density of the AMn signal has both a continuous and a discrete component at the carrier frequency   . The continuous component is the spectral density of the transmitted digital signal  (t) transferred to the carrier frequency region. It should be noted that the discrete component of the spectral density takes place only at a constant initial phase of the signal  . In practice, as a rule, this condition is not met, since, as a result of various destabilizing factors, the initial phase of the signal randomly changes in time, i.e. is a random process  (t) and is uniformly distributed in the interval [-  ;  ]. The presence of such phase fluctuations leads to “blurring” of the discrete component. This feature is also characteristic of other types of manipulation. Figure 2.3 shows the spectral density of the AMn radio signal.

Fig.2.3. Spectral density of the AMn radio signal with a random, uniform

distributed in the interval [-  ;  ] initial phase 

The average power of the AM radio signal is equal to
. This power is equally distributed between the continuous and discrete components of the spectral density. Consequently, in the AMn radio signal, the share of the continuous component due to the transmission of useful information accounts for only half of the power emitted by the transmitter.

To form the AMn radio signal, a device is usually used that provides a change in the amplitude level of the radio signal according to the law of the transmitted primary digital signal  (t) (for example, an amplitude modulator).

Signal is the physical process that displays the message. In technical systems, electrical signals are most often used. Signals are usually functions of time.

1. Signal classification

Signals can be classified according to various criteria:

1. Continuous ( analog) - signals that are described by continuous functions of time, i.e. take a continuous set of values in the interval of definition. Discrete - are described by discrete time functions i.e. take a finite set of values on the interval of definition.

Deterministic - signals that are described by deterministic functions of time, i.e. whose values are determined at any time. Random - are described by random functions of time, i.e. whose values at any given time is a random variable. Random processes (SP) can be classified into stationary, non-stationary, ergodic and non-ergodic, as well as Gaussian, Markovian, etc.

3. Periodic - signals whose values are repeated at intervals equal to the period

x (t) = x (t + nT), where n= 1,2,...,¥; T- period.

4. Causal - signals that originate in time.

5. finite - signals of finite duration and equal to zero outside the definition interval.

6. coherent - signals that coincide at all points of definition.

7. Orthogonal - signals opposite to coherent ones.

2. Characteristics of signals

1. Signal duration ( transmission time) T s- the time interval during which the signal exists.

2. Spectrum width Fc- frequency range within which the main signal power is concentrated.

3. signal base is the product of the signal spectrum width and its duration.

4. Dynamic Range Dc- logarithm of the ratio of the maximum signal power - Pmax to the minimum Pmin(minimally distinguishable at the level of interference):

D c = log (P max / P min).

In expressions where logarithms with any base can be used, the base of the logarithm is not indicated.

As a rule, the base of the logarithm determines the unit of measurement (for example: decimal - [Bel], natural - [Neper]).

5. Signal volume is determined by the relation V c = T c F c D c .

6. Energy characteristics: instantaneous power - P(t); average power - P cf and energy - E. These characteristics are determined by the relations:

P(t) =x 2 (t); ; (1)

where T=tmax -tmin.

3. Mathematical models of random signals

Deterministic, i.e. a pre-known message does not contain information, because the recipient knows in advance what the transmitted signal will be. Therefore, the signals are statistical in nature.

Random (stochastic, probabilistic) process - a process that is described by random functions of time.

random process X(t) can be represented by an ensemble of non-random time functions x i (t), called realizations or samples (see Fig. 1).

Fig.1. Implementations of a random process X(t)

The complete statistical characteristic of a random process is n- dimensional distribution function: F n (x 1 , x 2 ,..., x n ; t 1 , t 2 ,..., t n), or probability density f n (x 1 , x 2 ,..., x n ; t 1 , t 2 ,..., t n).

The use of multidimensional laws is associated with certain difficulties,

therefore, they are often limited to using one-dimensional laws f 1 (x, t), characterizing the statistical characteristics of a random process at certain points in time, called sections of a random process or two-dimensional f 2 (x 1, x 2; t 1, t 2), characterizing not only the statistical characteristics of individual sections, but also their statistical relationship.

Distribution laws are exhaustive characteristics of a random process, but random processes can be quite fully characterized with the help of the so-called numerical characteristics (initial, central and mixed moments). In this case, the following characteristics are most often used: mathematical expectation (initial moment of the first order)

; (2)

mean square (initial moment of the second order)

; (3)

dispersion (second-order central moment)

; (4)

correlation function, which is equal to the correlation moment of the corresponding sections of the random process

. (5)

In this case, the following relation is true:

(6)

Stationary processes - processes in which numerical characteristics do not depend on time.

Ergodic processes - a process in which the results of averaging and over the set coincide.

Gaussian processes - processes with a normal distribution law:

(7)

This law plays an extremely important role in the theory of signal transmission, since most interference is normal.

According to the central limit theorem, most random processes are Gaussian.

M Arkov process - a random process in which the probability of each subsequent value is determined only by one previous value.

4. Forms of analytical description of signals

Signals can be represented in the time, operator or frequency domain, the relationship between which is determined using the Fourier and Laplace transforms (see Fig. 2).

Laplace transform:

L-1: (8)

Fourier transforms:

F-1: (9)

Fig.2 Signal representation areas

In this case, various forms of signal representation in the form of functions, vectors, matrices, geometric, etc. can be used.

When describing random processes in the time domain, the so-called correlation theory of random processes is used, and when describing in the frequency domain, the spectral theory of random processes is used.

Taking into account the parity of functions

and and in accordance with the Euler formulas: (10)

one can write expressions for the correlation function R x (t) and energy spectrum (spectral density) of the random process S x (w), which are related by the Fourier transform or the Wiener-Khinchin formulas

; (11) . (12)

5. Geometric representation of signals and their characteristics

Any n- numbers can be represented as a point (vector) in n-dimensional space, remote from the origin at a distance D,

where . ( 13)

Signal duration T s and spectrum width F with, in accordance with the Kotelnikov theorem is determined N counts, where N = 2F c T c.

This signal can be represented by a point in n-dimensional space, or by a vector connecting this point to the origin.

The length of this vector (norm) is:

; (14)

where x i =x (nDt)- signal value at time t = n.Dt.

Let's say: X is the message being sent, and Y- accepted. Moreover, they can be represented by vectors (Fig. 3).

X1 , Y1

0 a1 a2 x1 y1

Fig.3. Geometric representation of signals

Let's define connections between geometric and physical representation of signals. For the angle between vectors X and Y can be written

cosg =cos(a 1 -a 2) =cosa 1cosa 2 +sina 1sina 2 =

where w 0 \u003d 2π f 0 – high carrier frequency;

U 0 is the amplitude of high-frequency oscillations.

The simplest and most commonly used control signals include harmonic oscillation

where Ω is a low frequency, much less than w 0 ; ψ is the initial phase; U m - amplitude, as well as rectangular pulse signals, which are characterized by the fact that the voltage value U ex ( t)=U during the time intervals τ and, called the duration of the pulses, and is equal to zero during the interval between pulses (Fig. 1.13). Value T and is called the pulse repetition period; F and =1/ T and is the frequency of their repetition. Pulse Period Ratio T and to the duration τ and is called the duty cycle Q impulse process: Q=T and /τ and.

U ex ( t)

T and

τ and

Fig.1.13. Rectangular pulse train

Depending on which parameter of the high-frequency oscillation is changed (modulated) with the help of a control signal, amplitude, frequency and phase modulation is distinguished.

U AM ( t)

Rice. 1.14. Amplitude modulated radio signal

With frequency modulation (FM), the control signal changes the frequency of high-frequency oscillations. If the modulating voltage changes according to a sinusoidal law, then the instantaneous value of the frequency of the modulated oscillations w \u003d w 0 + k World Cup U m sinW mod t, where k FM - coefficient of proportionality. The greatest change in frequency w with respect to its average value w 0 equal to Δw М = k World Cup U m, is called the frequency deviation. The frequency modulated signal can be written as follows:

The value equal to the ratio of the frequency deviation to the modulation frequency (Δw m / W mod = m FM) is called the frequency modulation ratio.

With AM, the signal frequency remains constant (w 0), and the amplitude changes according to the law of the modulating voltage U AM ( t) = U 0 k Maud t(Fig. 1.15, b).

The phase-modulated signal (Fig. 1.15, d) has a constant amplitude and frequency hopping. Let's explain this analytically. With FM under the influence of modulating voltage

U AM ( t)

U World Cup ( t)

U mod( t)

t 1

t 2

w 0

U fM ( t)

w 1

w 2

in)

Fig.1.15. Comparative view of modulated oscillations with AM, FM and FM:
a - modulating voltage; b – amplitude modulated signal;
c – frequency-modulated signal; d - phase modulated signal

signal phase receives an additional increment Δy= k FM t, therefore, a high-frequency signal with phase modulation according to the sawtooth law has the form

Thus, on the segment 0 t 1 the frequency is w 1 >w 0 , and on the segment t 1 t 2 it is equal to w 2

When transmitting a sequence of pulses, for example, a binary digital code (Fig. 1.16, a), AM, FM and FM can also be used. This type of modulation is called keying or telegraphy (AT, CT and FT).

U AT ( t)

U Thu ( t)

τ and

w 0

U mod( t)

w 2

w 1

in)

U FT ( t)

w 0

Fig.1.16. Comparative view of manipulated oscillations in AT, PT and FT

With frequency telegraphy, a high-frequency signal is formed with a constant amplitude and a frequency that takes two possible values (Fig. 1.16, c).

With phase telegraphy, a high-frequency signal is formed with a constant amplitude and frequency, the phase of which changes by 180 ° according to the law of the modulating signal (Fig. 1.16, d).

1 Classification of types of modulation, the main characteristics of radio signals.

To implement radio communication, it is necessary to somehow change one of the parameters of the radio frequency wave, called the carrier, in accordance with the transmitted low-frequency signal. This is achieved by modulating the RF waveform.

It is known that the harmonic

characterized by three independent parameters: amplitude, frequency and phase.

Accordingly, there are three main types of modulation:

amplitude,

frequency,

Phase.

Amplitude modulation (AM) is such a type of influence on the carrier wave, as a result of which its amplitude changes according to the law of the transmitted (modulating) signal.

We assume that the modulating signal has the form of a harmonic oscillation with a frequency W

much lower frequency of the carrier wave w.

As a result of modulation, the amplitude of the voltage of the carrier oscillation should change in proportion to the voltage of the modulating signal uW (Fig. 1):

UAM = U + kUWcosWt = U + DUcosWt, (1)

where U is the voltage amplitude of the carrier radio frequency oscillation;

DU=kUW - amplitude increment.

The equation of amplitude-modulated oscillations, in this case, takes the form

UAM = UAM coswt = (U + DUcosWt) coswt = U (1+cosWt) coswt. (2)

According to the same law, the iAM current will also change during modulation.

The value characterizing the ratio of the magnitude of the change in the amplitude of oscillations DU to their amplitude in the absence of modulation U is called the modulation coefficient (depth)

It follows from this that the maximum oscillation amplitude Umax = U + DU = U (1+m) and the minimum amplitude Umin= U (1-m).

As it is easy to see from equation (2), in the simplest case, modulated oscillations are the sum of three oscillations

UAM = U(1+ mcosWt)coswt = Ucoswt U/2+ cos(w - W)t U/2+ cos(w + W)t . (four)

The first term is the oscillations of the transmitter in the absence of modulation (silence mode). The second - fluctuations of lateral frequencies.

If the modulation is carried out by a complex low-frequency signal with a spectrum from Fmin to Fmax, then the spectrum of the received AM signal has the form shown in Fig. The frequency band Δfc occupied by the AM signal does not depend on m and is equal to

Δfс = 2Fmax. (five)

The occurrence of oscillations of side frequencies during modulation leads to the need to expand the bandwidth of the transmitter circuits (and, accordingly, the receiver). She must be

where Q is the quality factor of the circuits,

Df - absolute detuning,

Dfk - loop bandwidth.

On fig. the spectral components corresponding to the lower modulating frequencies (Fmin) have smaller ordinates.

This is explained by the following circumstance. For most types of signals (for example, speech) entering the transmitter, the amplitudes of the high-frequency components of the spectrum are small compared to the components of low and medium frequencies. As for the noise at the input of the detector in the receiver, their spectral density is constant within the bandwidth

receiver. As a result, the modulation coefficient and the signal-to-noise ratio at the input of the receiver detector for high frequencies of the modulating signal turn out to be small. To increase the signal-to-noise ratio, the high-frequency components of the modulating signal during transmission are emphasized by amplifying the high-frequency components by a greater number of times compared to the components of low and medium frequencies, and are attenuated by the same amount when received before or after the detector. The attenuation of the high-frequency components before the detector almost always occurs in the high-frequency resonant circuits of the receiver. It should be noted that artificial emphasis on upper modulating frequencies is acceptable as long as it does not lead to overmodulation (m > 1).

05.02.2021

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