Passing signals through linear circuits. Passing signals through non-linear circuits Passing signals through linear circuits

There is no general procedure for determining the distribution law of the response of a linear FU to an arbitrary random action. However, correlation analysis is possible, i.e., the calculation of the reaction correlation function according to the given correlation function of the effect, which is convenient to carry out by the spectral method according to the scheme shown in Fig. 5.5.

To calculate the energy spectrum G Y(f) the reaction of a linear FU with a transfer function H(jω) we use its definition (4.1)

Correlation function B Y(t) is defined by the Fourier transform of the energy spectrum G Y(f)

Let us return to the definition of the distribution law of the reaction of a linear FU in individual special cases:

1. Linear transformation of a normal PN also generates a normal process. Only the parameters of its distribution can change.

2. The sum of normal SP (adder reaction) is also a normal process.

3. When a SP with an arbitrary distribution passes through a narrow-band filter (i.e., when the bandwidth of the filter is D F significantly smaller than the width of the energy spectrum of the action D f X) there is a phenomenon of normalization of the distribution of the reaction Y(t). It consists in the fact that the distribution law of the reaction approaches normal. The stronger the inequality D F<< Df X(fig. 5.6).

This can be explained as follows. As a result of the passage of the LB through a narrow-band filter, there is a significant decrease in the width of its energy spectrum (with D f X to D F) and, accordingly, an increase in the correlation time (c t X until t Y). As a result, between uncorrelated samples of the filter response Y(k t Y) is located approximately D f X / D F uncorrelated exposure counts X(l t X), each of which contributes to the formation of a single response count with a weight determined by the type of the filter impulse response.

Thus, in uncorrelated sections Y(k t Y) there is a summation of a large number of also uncorrelated random variables X(l t X) with limited mathematical expectations and variances, which, in accordance with the central limit theorem (A.M. Lyapunov), ensures that the distribution of their sum approaches the normal one with an increase in the number of terms.

5.3. Narrowband random processes

Joint venture X(t) with a relatively narrow energy spectrum (D f X << f c) as well as narrow-band deterministic signals, it is convenient to represent in quasi-harmonic form (see Section 2.5)

where is the envelope A(t), phase Y ( t) and the initial phase j ( t) are random processes, and ω c is a frequency chosen arbitrarily (usually as the average frequency of its spectrum).

To define the envelope A(t) and phase Y ( t) it is advisable to use the analytical joint venture

The main moment functions of the analytical joint venture:

1. Mathematical expectation

2. Dispersion

3. Correlation function

An analytical SP is called stationary if

Let us consider the problem typical in communication technology of passing a normal LB through a bandpass filter (BPF), amplitude (AM) and phase (PD) detectors (Fig.5.7). The signal at the output of the PF becomes narrowband, which means that its envelope A(t) and the initial phase j ( t) will be slowly varying functions of time in comparison with, where is the average frequency of the PF passband. By definition, the signal at the output of the blood pressure will be proportional to the envelope of the input signal A(t), and at the output of the PD - its initial phase j ( t). Thus, to solve this problem, it is sufficient to calculate the distribution of the envelope A(t) and phase Y ( t) (the distribution of the initial phase differs from the distribution Y ( t) only by mathematical expectation).

End of work -

This topic belongs to the section:

The theory of electrical communication. Lecture notes - part 2

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All topics in this section:

Spectral analysis of random processes
Spectral analysis of deterministic signals x (t) presupposes the use of the direct Fourier transform

Properties of energy spectra of random processes
1., which immediately follows from its definition (4.1). From this fact and relation

random process research
To consolidate the knowledge gained during the study of section 4 on the basis of a virtual laboratory, you can conduct experimental studies of random processes using:

signal converters
In the general case, the solution to the problem of passing a given LB through a concretion

through inertialess chains
An inertialess chain (an inertialess functional unit - IFC) is fully described by the functional dependence y = f (x), which connects the instantaneous values ​​of the

Functional transformation of two random processes
Problem statement: Two random processes X1 (t) and X2 (t) are given with a known joint probability density of their values ​​in coincidence

passing random processes through various FU
To consolidate the knowledge gained during the study of this section, it is recommended to perform work No. 20 “Passage of random processes through various

The ideal observer criterion
(Kotelnikov criterion) This criterion requires the provision of a minimum average probability of erroneous reception. For binary system

Maximum likelihood criterion
Assuming that all transmitted messages are equally probable,

Minimum average risk criterion
(Bayesian criterion) To take into account the different consequences of errors in the transmission of various messages, it is necessary to generalize the Kotelnikov criterion, minimizing the sum of conditional probabilities

Neumann-Pearson criterion
The Neumann-Pearson criterion is used in binary systems in situations where it is impossible to determine the prior probabilities of individual messages, and the consequences of various kinds of errors are not

on matched filters
Keeping the formulation of the demodulator synthesis problem from the previous section and relying on algorithms (6.13) and (6.14), we will try to replace the correlator (active filter) calculating the scalar

Matched filter properties
1. The impulse response of the SF is a "mirror image" of the signal with which it is matched, relative to the time instant 0.5t0 (accurate to a constant coefficient

Phase-frequency response of SF
differs in sign from the phase spectrum of the signal with which it is matched (b

Rectangular video pulses
The signal in the form of a rectangular video pulse s (t) (Fig. 6.8, a) and the impulse response gSF (t) of the filter matched to it (Fig. 6.8, b) are described by the expressions

Rectangular radio pulses
The signal in the form of a rectangular radio pulse s (t) is described by the expression

Complex binary signals
Consider signals in the form of n-sequences of rectangular pulses

Optimal coherent reception in non-white noise
Consider the problem of synthesizing a matched filter that provides the maximum SNR at its output for the case when an additive mixture of the known signal s (

optimal coherent reception
To consolidate the knowledge gained during the study of Sections 6.1-6.3, it is advisable to perform laboratory work No. 15 "Study of coherent demodulators" (Fig. 6.19, 6.20) and No. 22 "Coherent f

noise immunity of the main types of digital modulation
To compare the noise immunity of the main types of digital modulation AM, FM (when using orthogonal signals) and PM, it is sufficient for each of them to determine the equivalent ene

incoherent reception in a binary communication system
To determine the average error probability of optimal incoherent reception in a binary system with equal probabilities of transmitted messages P (b0) = P (b

research incoherent reception
To consolidate the knowledge gained during the study of Sections 6.6 and 6.7, it is advisable to perform laboratory work No. 16 "Study of incoherent demodulators" (Fig. 6.40, 6.41) and

Objective:

study of the processes of passing harmonic signals and rectangular signals through linear circuits, such as differentiating and integrating circuits, serial and parallel oscillatory circuits, transformer;

study of transient processes in linear circuits;

obtaining the skill of working with measuring instruments;

learn to perform calculations of RCL-circuits using the symbolic method;

processing and analysis of the obtained experimental data.

Tasks:

measure the amplitude-frequency characteristics of seven linear circuits;

measure the phase-frequency characteristics of the above listed linear circuits;

get and investigate the transient characteristics of seven linear circuits;

1 Linear circuits

In electronics, electrical circuits are a collection of connected circuit elements such as resistors, capacitors, inductors, diodes, transistors, operational amplifiers, current sources, voltage sources, and others.

Circuit elements are connected using wires or printed buses. Electrical circuits made up of idealized elements are classified according to a number of characteristics:

By energy characteristics:

active (containing power supplies);

passive circuits (do not contain current and (or) voltage sources);

By topological features:

planar (flat);

nonplanar;

branched;

unbranched;

simple (one-, two-circuit);

complex (multi-circuit, multi-node);

By the number of external conclusions:

two-pole networks;

quadripoles;

multipoles;

From the frequency of the measuring field:

circuits with lumped parameters (in circuits with lumped parameters, only a resistor has resistance, only a capacitor has a capacity, only an inductor has an inductance);

circuits with distributed parameters (in circuits with distributed parameters, even connecting wires have capacitance, conductivity and inductance, which are distributed along their length; this approach to circuits in the microwave region is most typical);

From the type of elements:

linear chains if they consist of linear idealized elements;

nonlinear circuits, if the circuit includes at least one nonlinear element;

In this paper, passive circuits are considered, consisting of three circuit elements. The elements
- called idealized circuit elements. The current flowing through such elements is a linear function of the applied voltage:

for resistor
:
;

for capacitor :
;

for coil inductor :

Therefore, chains consisting of
elements are called linear.

Strictly speaking, in practice, not all
the elements are linear, but in many cases the deviations from linearity are small and the real element can be taken as an idealized linear one. An active resistance can be considered as a linear element only if the current flowing through it is so small that the heat released does not lead to a noticeable change in the value of its resistance. Similar considerations can be made for inductors and capacitors. If the parameters
circuits remain unchanged during the time when the studied electrical process is taking place, then we speak of a circuit with constant parameters.

Since the processes in linear circuits are described by linear equations, the principle of superposition is applicable to them. This means that the result of an action in a linear chain of a signal of a complex shape can be found as the sum of the results of actions of signals of simpler ones, into which the original, complex signal is decomposed.

Two methods are used to analyze linear circuits: the frequency response method and the transient response method.

Objective: Acquire primary skills in the study of statistical characteristics of random signals. Experimentally determine the distribution laws of random signals at the output of linear and nonlinear radio circuits.

BRIEF THEORETICAL INFORMATION

1. Classification of radio circuits

Radio engineering circuits used to convert signals are very diverse in their composition, structure and characteristics. In the process of their development and analytical research, various mathematical models are used that meet the requirements of adequacy and simplicity. In the general case, any radio engineering circuit can be described by a formalized relation defining the transformation of the input signal x (t) into the output y (t), which can be symbolically represented as

y (t) = T,

Where T is an operator indicating the rule according to which the input signal is converted.

Thus, a set of operator T and two sets X = (xi (t)) and Y = (yi (t)) of signals at the input and output of the circuit can serve as a mathematical model of a radio engineering circuit so that

(yI(t)) = T (xI(t)).

By the type of conversion of input signals into outputs, that is, by the type of operator T, radio circuits are classified.

A radio circuit is linear if the operator T is such that the circuit satisfies the conditions of additivity and homogeneity, that is, the equalities are true

T = T: T = c T

i I

Where c is a constant.

These conditions express the essence of the superposition principle inherent only to linear chains.

The operation of linear circuits is described by linear differential equations with constant coefficients. It is characteristic that the linear transformation of a signal of any shape is not accompanied by the appearance of harmonic components with new frequencies in the spectrum of the output signal, that is, it does not lead to an enrichment of the signal spectrum.

The radio circuit is Non-linear if the operator T does not ensure the fulfillment of the additivity and homogeneity conditions. The functioning of such circuits is described by nonlinear differential equations.

Structurally linear circuits contain only linear devices (amplifiers, filters, long lines, etc.). Non-linear circuits contain one or more non-linear devices (generators, detectors, multipliers, limiters, etc.)

By the nature of the time dependence of the output signal from the input signal, inertial and inertial radio circuits are distinguished.

An electronic circuit, the value of the output signal of which y (t) At the moment t = t0 depends not only on the value of the input signal x (t) at this moment in time, but also on the values ​​of x (t) at the moments of time preceding the moment t0 is called Inertial chain. If the value of the output signal y (t) and the moment t = t0 are completely determined by the value of x (t) at the same moment of time t0, then such a chain is called Inertialess.

2. Transformation of random processes in linear circuits

The problem of transforming random processes in linear radio engineering circuits in the general case is considered in the following setting. Let a random process x (t) with given statistical properties arrive at the input of a linear circuit with a frequency characteristic K (jw). It is required to determine the statistical characteristics of the random process y (t) at the output of the circuit. Depending on the analyzed characteristics of random processes x (t) and y (t), two versions of the general problem are considered:

1. Determination of the energy spectrum and correlation function of a random process at the output of a linear circuit.

2. Determination of the probability distribution laws of a random process at the output of a linear chain.

The simplest is the first task. Its solution in the frequency domain is based on the fact that the energy spectrum of the random process at the output of the linear circuit Wy (w) in the stationary mode is equal to the energy spectrum of the input process Wx (w) multiplied by the square of the modulus of the frequency response of the circuit, that is

Wy(W)= Wx(W) ∙│ K(Jw)│ A (1)

It is known that the energy spectrum Wx (w) of a random process x (t) with mathematical expectation mx = 0 is related to its covariance function Bx (t) by Fourier transforms, that is

Wx(W)= VX(T) E— JWTDT

VX(T)= Wx(W) EjWTDW.

Therefore, the covariance function Вy (t) of a random process at the output of a linear chain can be defined as follows:

VY(T)= Wy(W) EjWTDW= Wx(W))│ K(Jw)│ A EjWTDW

Ry(T) = BY(T)+ Mya.

In this case, the variance Dy and the mathematical expectation my of the output random process are equal

Dy = Ry (0) = Wx (w)) │K (jw) │adw

My= Mx∙ K(0) .

Where mx is the mathematical expectation of the input random process:

K (0) is the transfer coefficient of a linear circuit for direct current, that is

K(0)= K(Jw)/ W=0

Formulas (1,2,3,4) are essentially a complete solution to the problem in the frequency domain.

There is no general method for solving the second problem, which would make it possible to directly find the probability density of the process y (t) at the output of a linear inertial circuit from a given probability density of the process x (t) at the input. The problem is solved only for some special cases and for random processes with a Gaussian (normal) distribution, as well as Markov random processes.

With regard to the process of a normal distribution law, the solution is simplified on the basis that the distribution law does not change with a linear transformation of such a process. Since the normal process is completely determined by the mathematical expectation and the correlation function, then to find the probability density of the process, it is sufficient to calculate its mathematical expectation and the correlation function.

The distribution law of the signal probabilities at the output of the linear inertialess chain coincides in the functional sense with the distribution law of the input signal. Only some of its parameters are changed. So, if a linear inertialess chain implements a functional transformation of the form y (t) = a x (t) + b, where a and b are constant coefficients, then the probability density p (y) of a random process at the output of the chain is determined by the well-known formula of the functional transformation random processes

P(Y)= =

Where p (x) is the probability density of the random process x (t) at the input of the circuit.

In some cases, an approximate solution to the problem of determining the probabilistic characteristics of a random process at the output of inertial circuits allows the use of the effect of normalization of a random process by inertial systems. If a non-Gaussian process x (t1) with a correlation interval tk acts on an inertial linear circuit with a time constant t »tk (in this case, the width of the energy spectrum of the random process x (t) is greater than the bandwidth of the circuit), then the process y (t) at the output of such a circuit approaches Gaussian as the t / tk ratio increases. This result is called the random process normalization effect. The narrower the bandwidth of the circuit, the stronger the effect of normalization is.

3. Transformation of random processes in nonlinear circuits

Nonlinear inertial transformations are considered in the course of the analysis of nonlinear circuits, the inertia of which under given influences cannot be neglected. The behavior of such circuits is described by nonlinear differential equations, the general methods of solving which do not exist. Therefore, problems associated with the study of nonlinear inertial transformations of random processes are almost always solved approximately, using various artificial methods.

One such technique is to represent a nonlinear inertial circuit as a combination of linear inertial and nonlinear inertialless circuits. The problem of studying the influence of random processes on a linear chain was considered above. It was shown that in this case it is quite simple to determine the spectral density (or correlation function) of the output signal, but it is difficult - the distribution law. In nonlinear inertialess chains, the main difficulty lies in finding the correlation function. At the same time, there are no general methods for analyzing the effect of random signals on nonlinear circuits. They are limited to solving some particular problems of practical interest.

3.1. Statistical characteristics of a random process at the output of nonlinear circuits

Consider the transformation of a random process with a one-dimensional probability density by a nonlinear inertialess chain with the characteristic

Y= f (x).

Obviously, any implementation of a random process x (t) is transformed into a corresponding implementation of a new random process y (t), that is

y (t) =F[ X(T)] .

A. Determination of the distribution law of a random process y (t)

Let the probability density p (x) of a random process x (t) be known. It is necessary to determine the probability density p (y) of the random process y (t). Let's consider three typical cases.

1. The function y = f (x) of a nonlinear circuit determines a one-to-one correspondence between x (t) and y (t). We assume that there is an inverse function x = j (y), which also determines a one-to-one correspondence between y (t) and x (t). In this case, the probability of finding the implementation of the random process x (t) in the interval (x0, x0 + dx) is equal to the probability of finding the implementation of the random process y (t) = f in the interval (y0, y0 + dу) with y0 = f (x0) and y0 + dy = f (x0 + dx), that is

P(X) Dx= P(Y) Dy

Hence,

P(Y)= .

The derivative is taken in absolute value because the probability density p (y)> 0, while the derivative can be negative.

2. The inverse function x = j (y) is ambiguous, that is, one value of y corresponds to several values ​​of x. For example, let the value у1 = y0 correspond to the values ​​х = x1, x2,…, xn.

Then the fact that у0≤ y (t) ≤ у0 + dy implies one of n mutually incompatible possibilities

X1 ≤ X(T)≤ X1 + Dx, or X2 ≤ X(T)≤ X2 + Dx, or … Xn≤ X(T)≤ Xn+ Dx.

Applying the rule of addition of probabilities, we obtain

P(Y)= + +…+ .

/ X= X1 / X= X2 / X= Xn

3, The characteristic of a nonlinear element y = f (x) has one or more horizontal sections (sections where y = const.). Then the expression

P(Y)=

It should be supplemented with a term that takes into account the probability of staying y (t) on the interval where y = const.

The easiest way to consider this case is not an example.

Let the function y = f (x) have the form shown in Fig. 1 and the formula

Rice. 1 Influence of a random process on a two-way limiter.

For x (t)<а выходной сигнал y(t)=0, Это значит, что вероятность принятия случайным процессом y(t) нулевого значения равна

P1 = P = P = P (x) dx,

And the probability density

P1 (y) = P1 ∙ δ (y).

Arguing similarly for the case x (t)> b, we obtain

Pa = P = P = P (x) dx,

pa(Y) = Pa∙ δ (Y— C).

/ Y= C

For the case a≤ x≤ b, the following formula is valid

Pa(Y) =

/0≤ Y≤ C

In general, the probability density of the output process is determined by the expression

P(Y)= P1 ∙ δ (Y)+ Pa∙ δ (Y— C)+ .

Note that to obtain the final expression, it is necessary to transform the functional dependences p (x) and dy / dx, which are functions of x, into functions of y, using the inverse function x = j (y). Thus, the problem of determining the distribution density of a random process at the output of a nonlinear inertialess chain is solved analytically for fairly simple characteristics y = f (x).

B. Determination of the energy spectrum and correlation function of a random process y (t)

It is not possible to directly determine the energy spectrum of a random process at the output of a nonlinear circuit. There is only one method - determination of the correlation function of the signal at the output of the circuit with the subsequent application of the direct Fourier transform to determine the spectrum.

If a stationary random process x (t) arrives at the input of a nonlinear inertialess chain, then the correlation function of the random process y (t) at the output can be represented in the form

Ry(T)= By(T)- My2 ,

Where By (t) is the covariance function;

my is the mathematical expectation of a random process y (t). The covariance function of a random process is a statistically averaged product of the values ​​of a random process y (t) at times t and t + t, that is,

By(T)= M[ Y(T)∙ Y(T+ T)].

For realizations of a random process y (t), the product y (t) ∙ y (t + t) is a number. For a process as a set of realizations, this product forms a random variable, the distribution of which is characterized by a two-dimensional probability density p2 (y1, y2, t), where y1 = y (t), ya = y (t + t). Note that the variable t does not appear in the last formula, since the process is stationary - the result depends on t but depends.

For a given function p2 (y1, y2, t), the operation of averaging over a set is carried out according to the Formula

By(T) = У1 ∙ у2 ∙ р2 (у1, у2,T) Dy1 Dy2 = F(X1 )∙ F(X2 )∙ P(X1 , X2 , T) Dx1 Dx2 .

The mathematical expectation my is determined by the following expression:

My= Y∙ P(Y) Dy.

Taking into account that p (y) dy = p (x) dx, we obtain

My= F(X)∙ P(X) Dx.

The energy spectrum of the output signal in accordance with the Wiener - Khinchin theorem is found as the direct Fourier transform of the covariance function, that is

Wy(W)= By(T) E— JWTDT

Practical application of this method is difficult, since the double integral for By (t) cannot always be calculated. We have to use various simplifying methods related to the specifics of the problem being solved.

3.2. Impact of narrowband noise on an amplitude detector

In statistical radio engineering, broadband and narrowband random processes are distinguished.

Let ∆ fe be the width of the energy spectrum of a random process, determined by the formula (Fig. 2.)

Rice. 2. Width of the energy spectrum of a random process

Narrowband a random process is a process for which ∆fe «f0, where f0 is the frequency corresponding to the maximum of the energy spectrum. A random process, the width of the energy spectrum of which does not satisfy this condition, is Broadband.

It is customary to represent a narrow-band random process as a high-frequency oscillation with slowly varying (in comparison with the oscillation at frequency f0) amplitude and phase, that is

X (t) = A (t) ∙ cos,

Where A (t) = √x2 (t) + z2 (t),

J (t) = arctan,

z (t) is the Hilbert conjugate function with the original function x (t), then

z (t) = -DT

All parameters of this oscillation (amplitude, frequency and phase) are random functions of time.

An amplitude detector, which is an integral part of the receiving path, is a combination of a non-linear inertialess element (for example, a diode) and an inertial linear circuit (low-pass filter). The voltage at the output of the detector reproduces the envelope of the amplitudes of the high-frequency oscillation at the input.

Let the input of the amplitude detector receive a narrow-band random signal (for example, from the output of the IF amplifier, which has a narrow passband relative to the intermediate frequency), which has the properties of an ergodic random process with a normal distribution law. It is obvious that the signal at the output of the detector will represent the envelope of the input random signal, which is also a random function of time. It is proved that this envelope, that is, the envelope of a narrow-band random process, is characterized by a probability density called the Rayleigh distribution and having the form:

Where A - envelope values;

Sx2 is the variance of the random signal at the detector input.

The Rayleigh distribution graph is shown in Fig. 3.

Fig. 3. Rayleigh distribution law

The function p (A) has a maximum value equal to

When A = sx. This means that the A = sx value is the most likely envelope value.

The mathematical expectation of the envelope of a random process

MA= = =

Thus, the envelope of a narrow-band random process with a normal distribution law is a random function of time, the distribution density of which is described by Rayleigh's law.

3.3. The law of distribution of the envelope of the sum of a harmonic signal and a narrow-band random noise

The problem of determining the distribution law of the envelope of the sum of a harmonic signal and a narrow-band random noise arises when analyzing the process of linear detection in radar and communication systems operating in conditions when the intrinsic or external noise is comparable in level with the useful signal.

Let the sum of a harmonic signal a (t) = E ∙ cos (wt) and a narrow-band noise х (t) = A (t) ∙ cos with a normal distribution law arrive at the input of the receiver. The total fluctuation in this case can be written

N(T) = S(T)+ X(T) = Е ∙ сS(Wt)+ A(T)∙ Cos[ Wt+ J(T)]=

= [E +A(T)∙ Cos(J(T))] ∙ сS(Wt)- A(T)∙ Sin(J(T))∙ Sin(Wt)= U(T)∙ Cos[ Wt+ J(T)],

Where U (t) and j (t) are the envelope and phase of the total signal, determined by the expressions

U(T)= ;

J(T)= Arctg

When the total oscillation u (t) acts on the amplitude detector, an envelope is formed at the output of the latter. The probability density p (U) of this envelope is determined by the formula

P(U)= (5)

Where sxa is the variance of the noise x (t);

I0 is the zero-order Bessel function (modified).

The probability density determined by this formula is called the generalized Rayleigh law, or Rice's law. The p (U) function plots for several values ​​of the signal-to-noise ratio E / sx are shown in Fig. 4.

In the absence of a useful signal, that is, at E / sx = 0, expression (5) takes the form

P(U)=

That is, the envelope of the resulting signal is distributed in this case according to the Rayleigh law.

Fig. 4. Graphs of the generalized Rayleigh distribution law

If the amplitude of the useful signal exceeds the rms noise level, that is, E / sx »1, then for U≃E one can use the asymptotic representation of the Bessel function with a large argument, that is

≃≃.

Substituting this expression in (5), we have

P(U)= ,

That is, the envelope of the resulting signal is described by the normal distribution law with variance sx2 and mathematical expectation E. In practice, it is believed that already at E / sx = 3, the envelope of the resulting signal is normalized.

4. Experimental determination of the laws of distribution of random processes

One of the methods for the experimental determination of the distribution function of a random process x (t) is a method based on the use of an auxiliary random function z (t) of the form

Where x is the value of the function x (t), for which z (t) is calculated.

As follows from the semantic content of the function z (t), its statistical parameters are determined by the parameters of the random process x (t), since changes in the values ​​of z (t) occur at the moments when the random process x (t) crosses the level x. Therefore, if x (t) is an ergodic random process with a distribution function F (x), then the function z (t) will also describe an ergodic random process with the same distribution function.

Figure 5 shows realizations of random processes x (t) and z (t), which illustrate the obviousness of the relation

P[ Z(T)=1]= P[ X(T)< X]= F(X);

P[ Z(T)=0]= P[ X(T)≥ X]= 1- F(X).

Fig. 5 Realizations of random processes x (t), z (t), z1 (t)

The mathematical expectation (statistical mean) of the function z (t), which has two discrete values, is determined in accordance with the formula (see Table 1)

M[ Z(T)]=1∙ P[ Z(T)=1]+0 ∙ P[ Z(T)=0]= F(X).

On the other hand, for an ergodic random process

In this way,

Analyzing this expression, we can conclude that a device for measuring the distribution function of an ergodic random process x (t) should contain a level discriminator to obtain a random process described by the function z (t) in accordance with expression (6), and an integrating device , made, for example, in the form of a low-pass filter.

The method of experimental determination of the distribution density of a random process x (t) is essentially similar to that considered above. In this case, an auxiliary random function z1 (t) of the form

The mathematical expectation of the function z1 (t), which has two discrete values ​​(Fig. 5), is

M[ Z1 (T)]=1∙ P[ Z1 (T)=1]+0 ∙ P[ Z1 (T)=0]= P[ X< X(T)< X+∆ X].

Taking into account the ergodicity of the random process described by the function z1 (t), we can write

In this way,

It is known that

P(X≤ X(T)< X+∆ X) ≃ P(X)∙∆ X.

Hence,

Thus, the device for measuring the distribution density of an ergodic random process x (t) has the same structure and composition as the device for measuring the distribution function.

The measurement accuracy of F (x) and p (x) depends on the duration of the observation interval and the quality of the integration operation. It is quite obvious that in real conditions we obtain Evaluations distribution laws, since the averaging (integration) time is finite. Returning to expression (6) and Fig. 5.Note that

Z(T) Dt= ∆ T1 ,

Where ∆ t1 is the 1st time interval of the stay of the function x (t) below the level x, that is, the time interval when the function z (t) = l.

The validity of this formula is determined by the geometric meaning of a definite integral (the area of ​​the figure bounded by the function z (t) and the segment (0, T) of the time axis).

Thus, one can write

That is, the distribution function of a random process x (t) is equal to the relative time spent by the implementation of the process in the interval - ¥< x(t) < х.

Reasoning similarly, one can obtain

Where ∆ t1 is the 1st time interval of stay of the function x (t) within (x, x + ∆x).

In the practical implementation of the considered method of experimental determination of the distribution laws of a random process, a random signal x (t) is analyzed within the range of its instantaneous values ​​from xmin to xmax (Fig. 6). Within these limits, the main set (in the probabilistic sense) of the instantaneous values ​​of the process x (t) is concentrated.

The xmin and xmax values ​​are selected based on the required measurement accuracy of the distribution laws. In this case, the study will be subjected to truncated distributions so that

F(Xmin)+<<1.

The entire range (xmin, xmax) of x (t) values ​​is divided into N equal intervals ∆x, that is

XMax— Xmin= N∙∆ X.

Rice. 6. Distribution function (a), probability density (b) and implementation (c) of a random process x (t)

The intervals define the width of the differential corridors in which the measurements are taken. The probability estimate is determined

Pi* ≃ P[ Xi-∆ X/2≤ X(T)< Xi-∆ X/2]

Stays of realization x (t) within the differential corridor with the average value of x (t) within it, equal to xi. The estimate Рi * is determined as a result of measuring the relative residence time of the realization x (t) in each of the differential corridors, that is

Pi * = 1 / T Zi (t) dt =,

I = 1, ..., N.

Considering that

Pi* ≃ P1 = P(X) Dx,

Estimates of the distribution density in each of the differential corridors can be determined

Pi* (X)= Pi*/∆ X.

Using the results obtained, that is, the values ​​pi * (x), xi, ∆x, a stepped p * (x) curve is constructed, which is called the histogram of the distribution density (see Fig. 7).

Fig. 7. Distribution density histogram

The area under each fragment of the histogram within ∆x is numerically equal to the area occupied by the true distribution curve p (x) in this interval.

The number N of differential corridors should be within 10 ... 20. A further increase in their number does not lead to a more accurate p (x) law, since with an increase in N, the value of the interval ∆x decreases, which worsens the conditions for accurate measurement of ∆ti.

The results obtained make it possible to calculate the estimates of the mathematical expectation and variance of the random process x (t)

Mx* = Xi∙ Pi* ; Dx* = (Xi— Mx* )2∙ Pi* .

When calculating Mx* and Dx* according to these formulas, it is taken into account that if the value of the realization of the random process x (t) falls into the 1st differential corridor, then the value and is assigned to it (the middle of the differential corridor).

The considered method for determining the distribution laws of random processes is the basis for the work of the statistical analyzer used in this laboratory work.

DESCRIPTION OF THE LABORATORY UNIT

The study of the distribution laws of random signals is carried out using a laboratory setup, which includes a laboratory model, a statistical analyzer and an S1-72 oscilloscope (Fig. 8).

Fig. 8. Laboratory setup diagram

The laboratory model carries out the formation and transformation of random signals, providing their statistical analysis, the construction of histograms of distribution laws and a graphical display of these laws on the indicator of the statistical analyzer. It contains the following functional units:

A. Signal generator block. Generates four different random signals.

- Signal x1 (t) = A ∙ sin - harmonic oscillation with a random initial phase, the distribution law of which Uniform in the interval 0

P(J)= 1/2 P, 0< J<2 P.

The probability density of the instantaneous values ​​of such a signal is

- Signal x2 (t) - sawtooth periodic voltage with constant amplitude A and random shift parameter q, distribution law
whom Uniform in the interval where T0 is the signal period, that is, the probability density is

P(Q)= 1/ T0 ; 0< Q≤ T0 .

The probability density of the instantaneous values ​​of such a signal is determined by the expression

- Signal x3 (t) - a random signal with a normal distribution law (Gaussian law) of instantaneous values, that is

Pa(X)= ,

Where mx, sx are the mathematical expectation and variance of the random signal x3 (t).

- Signal x4 (t) - a random clipped signal, which is a sequence of rectangular pulses of constant amplitude A and random duration, arising at random times. Such a signal appears at the output of an ideal limiter when a random process with a normal distribution law acts at its input. The transformation characteristic has the form

Where x is the level of restriction.

Thus, the random process x4 (t) takes two values ​​(A and - A) with probabilities

P = P = F3 (x);

P = P = 1-F3 (x);

Where F3 (x) is the integral distribution law of the random process x3 (t).

Considering the above, the probability density of the clipped signal is

P4 (x) = F3 (x) ∙D(x + A) + ∙D(x - A).

Figure 9 shows the realizations of each of the random signals generated by the laboratory model iterator and their probability densities.

These signals, each of which is characterized by its characteristic distribution density, can be fed to the inputs of typical elements of radio engineering devices in order to transform and study the distribution laws of signals at their outputs.

B. Linear signal mixer. Forms the sum of two random signals xi (t) and x1 (t), supplied to its inputs, in accordance with the relation

Y(T)= R∙ Xi(T)+ (1- R)∙ X1 (T),

Where R is the coefficient set by the potentiometer knob within the range of 0 ... 1.

It is used to study the distribution laws of the sum of two random signals.

V. Sockets for connecting various two-port networks - functional converters. The set of the laboratory installation includes 4 functional transducers (Fig. 10).

Rice. 9. Realizations of random processes x1 (t), x2 (t), x3 (t), x4 (t) and their probability densities

Amplifier - Limiter (Limit) with Conversion Characteristic

Where U1, U2 are the lower and upper limitation levels, respectively;

k - coefficient equal to tg of the slope of the conversion characteristic.

Carries out non-linear inertialess conversion of input signals.

Narrow-band filter (F1) with resonant frequency f0 = 20 kHz. It is used to form narrow-band random processes with a distribution law close to normal.

Typical path of the AM oscillation receiver (narrow-band filter F1 - linear detector D - low-pass filter F2). Shapes the envelope of a narrow-band random signal with linear detection.

Structurally, the considered functional converters are made in the form of replaceable blocks of small size.

As another functional converter, an "ideal" amplifier - limiter (electronic key) is used, which is part of the block of signal generators of the breadboard. It provides the formation of a clipped signal, being a nonlinear inertialess converter of an input random signal.

Rice. 10. Functional converters

G. Matching amplifier. Provides agreement between the range of values ​​of the signal under investigation and the amplitude range of the statistical analyzer. Matching is carried out by potentiometers "Gain" and "Offset" when setting switch P1 (Fig. 8) to the position "Calibration."

The matching amplifier is also used as a functional converter (except for the four discussed above), providing a linear inertialess conversion in accordance with the formula

Y(T)= A∙ X(T)= B,

Where a is the gain set by the "Gain" knob;

b - constant component of the signal, set by the "Offset" knob.

The analyzer block shown in the diagram in Fig. 8 is not used as part of the model in this work. The laboratory setup provides for the use of a digital statistical analyzer, made in the form of a separate device.

D. The digital statistical analyzer is used to measure and form the distribution laws of the signal values ​​supplied to its input. The analyzer works as follows.

The analyzer is switched on to the measurement mode by pressing the "Start" button. The measurement time is 20 s. During this time, samples of the input signal values ​​are taken (at random times), the total number of N of which is 1 million. Samples are sampled by level so that each of them is in one of 32 intervals (called differential corridors, or sampled values). The intervals are numbered from 0 to 31, their width is 0.1 V, and the lower limit of the 0 interval is 0 V, the upper limit of the 31st interval is +3.2 V. During the measurement time, the number of samples is counted ni hitting each interval. The measurement result is displayed as a distribution histogram on the monitor screen, where the horizontal axis of the scale grid is the axis of signal values ​​within 0 ... + 3.2 V, the vertical axis is the axis of relative frequencies ni / N, i = 0.1 ... 31.

To read the measurement results in digital form, a digital indicator is used, which displays the number of the selected interval and the corresponding frequency (probability estimate) ni / N. Enumeration of the numbers of intervals for the digital indicator is carried out with the "Interval" switch. In this case, the selected interval is marked with a marker on the monitor screen.

Using the "Multiplier" switch, you can select a convenient for observation scale of the histogram along the vertical axis.

When performing this work, the switch for the input voltage range of the analyzer (the range of analog-to-digital conversion) must be set to the position 0 ... + 3.2 V. Before each measurement, it is necessary to alternately press the "Reset" and "Start" buttons (by pressing the "Reset" button the memory device is reset to zero, and the results of the previous measurement are written into the stack memory, from which they can be called using the "Page" switch).

Consider a linear inertial system with a known transfer function or impulse response. Let a stationary random process with given characteristics: probability density, correlation function, or energy spectrum arrive at the input of such a system. Let us determine the characteristics of the process at the output of the system: and

The simplest way is to find the energy spectrum of the process at the output of the system. Indeed, individual implementations of the process at the input are deterministic functions, and the Fourier apparatus is applicable to them. Let

a truncated implementation of the duration T of a random process at the input, and

Its spectral density. The spectral density of the realization at the output of the linear system will be equal to

The energy spectrum of the output process according to (1.3) will be determined by the expression

those. will be equal to the energy spectrum of the process at the input, multiplied by the square of the frequency response of the system, and will not depend on the phase response.

The correlation function of the process at the output of a linear system can be defined as the Fourier transform of the energy spectrum:

Consequently, when a random stationary process acts on a Linear system, the output is also a stationary random process with an energy spectrum and a correlation function defined by expressions (2.3) and (2.4). The power of the process at the output of the system will be equal to

As a first example, consider the passage of white noise with spectral density through an ideal low-pass filter, for which

According to (2.3), the energy spectrum of the process at the output will have a spectral density uniform in the frequency band, and the correlation function will be determined by the expression

The power of a random process at the output of an ideal low-pass filter will be equal to

As a second example, consider the passage of white noise through an ideal bandpass filter, the frequency response of which for positive frequencies (Fig. 1.6) is determined by the expression:

We define the correlation function using the Fourier cosine transform:

The graph of the correlation function is shown in Fig. 1.7

The examples considered are indicative from the point of view that they confirm the connection established in § 3.3 between the correlation functions of low-frequency and narrow-band high-frequency processes with the same shape of the energy spectrum. The power of the process at the output of an ideal bandpass filter will be equal to

The law of probability distribution of a random process at the output of a linear inertial system differs from the law of distribution at the input, and its determination is a very difficult task, with the exception of two special cases, which we will focus on here.

If a random process affects a narrow-band linear system, the bandwidth of which is much less than its spectrum width, then at the output of the system the phenomenon takes place normalization distribution law. This phenomenon consists in the fact that the distribution law at the output of a narrow-band system tends to normal, regardless of what distribution the broadband random process has at the input. Physically, this can be explained as follows.

The process at the output of an inertial system at a certain moment of time is a superposition of individual responses of the system to the chaotic influences of the input process at different moments of time. The narrower the system bandwidth and the wider the spectrum of the input process, the larger the number of elementary responses the output process is formed. According to the central limit theorem of probability theory, the distribution law of a process that is the sum of a large number of elementary responses will tend to normal.

A second special, but very important case follows from the above reasoning. If the process at the input to a linear system has a normal (Gaussian) distribution, then it remains normal at the output of the system. In this case, only the correlation function and the energy spectrum of the process change.

Electrical circuits are an integral part of electronic automation elements that perform a large number of different specific functions. The main difference between electrical circuits and electronic ones is that they are a set of passive linear elements, that is, those whose current-voltage characteristics obey Ohm's law, and they do not amplify input signals. Because of this, the electrical circuits of electronic devices are often called linear devices for converting and generating electrical signals.

Functionally linear devices for the formation and conversion of electrical signals can be divided into the following main groups:

Integration circuits used to integrate signals, and sometimes to expand (increase the duration) of pulses;

Differentiating (shortening) circuits used to differentiate signals, as well as to shorten pulses (receiving pulses of a given duration);

Resistor and resistor-capacitive dividers used to change the amplitude of electrical signals;

Pulse transformers used to change the polarity and amplitude of pulses, for galvanic isolation of pulse circuits, for forming positive feedback in generators and pulse shapers, for matching circuits by load, for receiving pulses from several output windings;

Electrical filters designed to extract frequency components located in a given region from a complex electrical signal and to suppress frequency components located in all other frequency regions.

Depending on the elements on which the linear devices are executed, they can be divided into RC-, RL- and RLC-circuits. In this case, linear devices can include a linear resistor R, a linear capacitor C, a linear inductor L, a pulse transformer without core saturation. The word "linear" emphasizes that we mean only those types of elements that have current-voltage characteristics of the linear type, or, in other words, the nominal value of the parameter (resistance, capacitance, etc.) for which it is constant and does not depend on the flowing current or applied voltage. For example, a conventional capacitor with mica dielectric spacers in a wide voltage range is considered linear, and the value of the pn-junction capacitance depends on the applied voltage, and it cannot be attributed to linear elements. In addition, there are always limitations on the amplitude or power of the signal, at which the element retains its linear properties. For example, the permissible voltage across a capacitor should not exceed the breakdown value. Other elements have similar restrictions, and they have to be taken into account when assigning an element to a particular class.

The most important property of linear devices is their ability to accumulate and release energy in capacitive and inductive elements and thereby convert input signals into a temporary change in output intervals. This property underlies the operation of generators, devices for suppressing impulse noise and "competition" in digital circuits that arise in the process of passing an electrical signal through circuits with different time delays.

Certain difficulties in using linear electric circuits in integral technology should be noted. This is due to the presence of a number of technological difficulties in the manufacture of resistors and capacitors, not to mention inductors, in an integral design.

Frequency independent voltage divider is designed to reduce the voltage of the signal source to the required value. DN is used to match the input stage with the voltage signal source, to set the operating point of the transistor in the amplifier, to form a reference (more often referred to as "reference") voltage. The diagram of the simplest voltage divider is shown in the figure just above.

When analyzing real electronic circuits, to avoid gross errors, it is always necessary to take into account the electrical characteristics of the signal source and load. The most important of them are:

The magnitude and polarity of the EMF of the signal source;

Internal resistance of the signal source (Rg);

Frequency response and phase frequency response of the signal source;

Load resistance (Rн);

The following figure shows the varieties of voltage dividers.

Figure (a) shows a voltage divider across a variable resistor. Used to adjust the EI sensitivity. In the same place, figure b shows a divider with several output voltages. Such a DN is used, for example, in a cascode amplifier. In some cases, when the resistance Rн is small, it is used as the lower arm of the divider. For example, when building an amplifier with an OE, the position of the operating point is set by the divider formed by Rb and the resistance of the base junction of the transistor rbe.

An important place in electronics is occupied by voltage dividers, in which the upper or lower shoulder is formed by variable resistance. If the divider is powered with a constant stable voltage, and, say, a resistance is put in the lower arm, the value of which is curled from temperature, pressure, humidity and other physical parameters, then a voltage proportional to temperature, pressure, humidity, etc. can be removed from the output of the voltage divider. ... A special place is occupied by dividers, in which one of the resistances depends on the frequency of the supply voltage. They form a large group of various filters for electrical signals.

Further improvement of the voltage divider led to the emergence of a measuring bridge, which consists of two dividers. In such a scheme, you can pick up the signal between the midpoint and the common wire, and between the two midpoints. In the second case, the output signal swing doubles with the same change in variable resistances. Amplifiers of electrical signals are also a voltage divider, in which the role of a variable resistance is played by a transistor controlled by an input voltage.

The simplest integrating chain is a voltage divider, in which the capacitor C plays the role of the lower arm of the divider

Differentiating linear circuits

The simplest differentiating chain is a voltage divider, in which the capacitor C plays the role of the upper arm of the divider

The integrating and differentiating links, when exposed to continuous random signals, behave as, respectively, low and high pass filters, elements R1 and C2 form a low-pass filter, and C1 and R2 form a high-pass filter