The concept of a linear system and the analysis of linear systems. Application of differential equations. Application impulse response. Application of frequency characteristics. Nonlinear inertialess transformations of random processes.
Detection and estimation of signal parameters in the presence of noise.
6.1. Description of the signal and interference. Types of tasks to be solved. Testing statistical hypotheses (sample, sample space, likelihood function, simple and complex hypotheses, solutions, decision rules, error probabilities). Statistics, decision quality criterion, loss matrix, conditional risk, average risk.
6.2. Testing two alternative hypotheses:
Bayes test, minimax test, maximum posterior probability test, maximum likelihood test, Neyman-Pearson test, sequential Wald analysis, operating characteristic.
6.3. Processing of continuous signals. Likelihood functional. The likelihood ratio functional.
6.4. Application of the likelihood ratio functional to detect a completely known signal (algorithm, error probabilities) and a signal with a random phase (algorithm, error probabilities).
Estimation of signal parameters.
7.1. Point estimate, interval estimate. Properties of point estimates (consistency, unbiasedness, efficiency, sufficiency). Rao-Kramer's inequality. Estimation of the mathematical expectation and variance of the normal distribution.
7.2. Application of the likelihood functional to estimate signal parameters. Estimation of the temporal position of the signal. Statement of the problem, noise and signal functions and their characteristics, signal-to-noise ratio, determination of the signal function for a rectangular video signal, signal burst processing (signal-to-noise ratio).
7.3. Implementation of the algorithm for estimating the time position of the signal. Correlation receiver, matched filter (pulse and frequency characteristics, the signal at the output of the matched filter, the signal-to-noise ratio, the optimality of the matched filter, the ratio between the real and matched filters).
Information theory
Kotelnikov's theorem for random processes.
Signal quantization.
measure of information.
3.1. Measure of information according to Fisher, according to Hartley.
3.2. Shannon measure of information (definition, entropy and its properties, ensemble product entropy, continuous ensemble entropy Entropy of a signal with a limited domain of definition. Entropy of a signal with an unlimited domain of definition but with limited power), Amount of mutual information, partial amounts of mutual information. Epsilon entropy (ε-entropy). Compression ratio, redundancy ratio.
4. Coding the source of independent messages Key words: code tree, code prefix, uniform coding, Shannon coding, Huffman coding, Shannon's source coding theorem. Characteristics of the source and source encoder.
5. Communication channel. Classification. Information transfer rate and bandwidth.
5.1. Noiseless channel: capacity, Shannon's theorem for a noiseless channel.
5.2. Noisy channel: binary symmetric channel (capacity), Shannon's theorem on bandwidth for a noisy channel.
Guidelines for laboratory work
Laboratory work on the study of the transformation of the spectra of signals in nonlinear circuits is used in the process of studying the course “ Radio circuits and Signals” by students of specialty 201600 "Radioelectronic systems". The laboratory work “Passing signals through non-linear circuits” is based on discrete Fourier transform algorithms and is made in the form of an application for Windows 95...98/2000/Millennium/NT .
Il. 7, list lit. 4 titles
Approved by the educational and methodological commission of the instrument-making faculty for the specialty 201600 "Radioelectronic systems".
Reviewer N. G. Gaisov.
SUSU Publishing House, 2002
1. Introduction
Laboratory work is performed using a digital software model of a laboratory stand, made in the form Windows - applications. An enlarged block diagram of the model is shown in Fig.1.
Rice. 1.
The purpose of all elements of this scheme is obvious and does not require further explanation.
signal conditioning module, enters the non-linear conversion module, which calculates the implementation of the output signal. Spectral analysis module, calculates spectra of input and output signals. The calculated amplitude spectra of the signals are displayeddisplay modulein the respective windows.
Simultaneously, the corresponding windows display the realizations of the input and output signals
More detailed description stand models are given in the appendix to these guidelines.
- Purpose of the Lab
Familiarize yourself with the methods of representing the characteristics of nonlinear circuits;
To consolidate the theoretical provisions of the analysis of the passage of signals through nonlinear circuits;
Experimentally investigate the dependence of the characteristics of the spectrum, shape and main parameters of the signal at the output of a nonlinear circuit, on the shape and parameters of the input signal and the type and characteristics of the nonlinear circuit ( Special attention should be given to the study of the deformation of the signal spectrum by a nonlinear circuit);
Check the degree of agreement between the experimental data and the corresponding theoretical provisions.
3. Settlement task
Calculate and plot the spectra at the input and output of a nonlinear circuit for two three signals specified by the teacher and for two forms of the nonlinear characteristic, also specified by the teacher (the task is given at the preparatory lesson).
4. The order of work and guidelines
Before starting the lab, you must:
- Familiarize yourself with the description of the digital software model of the laboratory stand given in the appendix.
- Plan the laboratory research program in accordance with the purpose of the laboratory work.
- Select and agree with the teacher the types of signals and types of nonlinear circuits that will allow you to most fully and clearly explain the influence of the characteristics of a nonlinear circuit on the characteristics of the signal spectrum.
When performing laboratory work, it is necessary to obtain a family of graphs that characterize the dependence of the characteristics of the spectra on the shape and parameters of the signals and the shape and parameters of the nonlinear circuit.
When performing work, pay attention to possible deviations of the calculated and experimental data.
Due to the fact that the physical meaning of the factors that determine the relationship between the spectral and temporal characteristics of signals is quite difficult to find out without having graphic illustrations, it is recommended to save for inclusion in the report the most characteristic oscillograms of signals and their spectra shown in the main application window (in graphical form , or in the form text file).
5. Requirements for the content of the report
The lab report should contain the following materials:
- Materials of the experimental study indicating the conditions of the experiment, including the indication of the temporal structure of the signal and its parameters.
- The results of the calculation task. Graphic images calculated and experimental dependencies for the same conditions, it is necessary to buildon common coordinate axes and on the same scale.
- Analysis of the results of the experiment with the justification of the reasons for the identified deviations of the experimental results from the calculated data.
- List of literature used in preparation for laboratory work and in the performance of the calculation task.
6.Control questions
1. Describe the main methods for approximating the characteristics of nonlinear elements.
2. What is cutoff angle? How to determine the cutoff angle for a cutoff amplifier?
3. Give comparative characteristic applicability conditions for two types of Berg coefficients ().
4. Find the spectral composition of the output signal if its characteristic has the form of a complete polynomial of the third degree, and the input is: a) a harmonic signal with a frequency; b) biharmonic signal of the form.
5. What members of the polynomial approximating the characteristic of a nonlinear circuit are involved in determining the amplitudes of the third and sixth harmonics of the output signal if a harmonic signal is applied to the input?
6. When not line element can be considered as a linear element with variable parameters?
7. Explain the operation of a resonant cutoff amplifier in the large oscillation mode. Draw its equivalent circuit.
8. Draw a diagram of a resonant frequency multiplier on n and explain the requirements for the parameters of the non-linear circuit element.
9. From what considerations is the optimal cutoff angle chosen?in the resonant frequency multiplier circuit.
10. Draw the equivalent circuit of the amplitude limiter and explain its principle of operation. What is a constraint characteristic?
- Gonorovsky I. R. Radio engineering circuits and signals: A textbook for universities 4th ed., Revised. And extra. M.: Radio and communication, 1986. 512 p.: ill.
- Baskakov S. I. Radio circuits and signals: Tutorial for universities on special "Radio engineering" - 2nd ed., revised. and additional M.: Higher School, 1988. 208 p.: ill.
- Radio engineering circuits and signals. Examples and tasks: Textbook for universities / Ed. I. S. Gonorovsky M.: Radio and communication, 1989. 248 p.: ill.
- Baskakov, S. I. Radio engineering circuits and signals. Guide to problem solving: Proc. allowance for radio engineering. specialist. universities. 2nd ed., revised. And extra. M.: Higher. school, 2002. 214 p.: ill.
Application
DESCRIPTION OF THE LABORATORY STAND MODEL
1.P. General provisions.
For researchcharacteristics of the spectral analysis of periodic signalsYour attention is invited to a software digital model equipped with user-friendly interface control of signal parameters and visual control of spectrum deformation when changing signal parameters.
Structural scheme model is shown in Figure 1.P. The purpose of all elements of this scheme is obvious and does not require further explanation.
Rice. 1.P.
The signal generated by the controlledsignal conditioning module, enters the modulenonlinear transformation, which calculates the implementation of the output signal. Spectral Analysis Module, calculates the spectra of the input and output signals. Calculated amplitude spectra and signal realizations are displayeddisplay modulein the corresponding windows in the form of oscilloscope images.
The conditions of the experiment, determined by the shape and parameters of the signal, as well as the characteristics of the nonlinear converter, are set in the main working window of the application.
The parameters and shape of the signal and the non-linear converter are set using the corresponding data input and editing elements located on the field of the main working window.
2.P. The main working window of the application
The signals and their spectra at the input and output of the non-linear circuit, as well as the characteristic of the non-linear circuit are displayeddisplay modulein the main working window, in the field for visual control, in the form of an oscilloscope image. An approximate view of the main working window is shown in Figure 2.P.
Rice. 2.P.
Arrays of signal counts and amplitude spectrum values are formed and updated with any changes in the signal parameters and can be saved in the form of text files for use in reports on laboratory work. To save the experiment data, use the menu of the main window "Save/Window Image", "Save/Signal Values", "Save/Signal Spectra" or "Save/All Data" (see Fig. 3.P.)
Rice. 3.P.
The values of the signal and the spectral range of the signal, saved in text files, can be used in other works of the laboratory complex at the course "Radio circuits and signals".
The data format of a text file of signal values is as follows:
Character string (free-form title containing experiment number)
Character string (possibly table header: Count Level)
Data line: Unsigned integer ( Integer ) Real with a sign(float)
Data line: Unsigned integer ( Integer ) Real with sign ( float)
The data format of the text file of signal spectra is as follows:
Character string (“The spectrum of the input signal in experiment No. ” Unsigned integer ( Integer))
Character string
Data line: Unsigned integer ( Integer ) Real with a sign(Float ) Real with a sign(float)
… (Total 135 lines)
Data line: Unsigned integer ( Integer ) Real with sign ( float ) Real with a sign(float)
Character string (“Output signal spectrum” Unsigned integer( Integer))
Character string (possibly table header: Count Amplitude Phase)
Data line: Unsigned integer ( Integer ) Real with a sign(Float ) Real with a sign(float)
… (Total 135 lines)
Data line: Unsigned integer ( Integer ) Real with sign ( float ) Real with a sign(float)
The type of input signal is set in the main working window of the application using the menu"Input Signal" and its amplitude using editing window equipped with buttons like " up / down ”.All changes are immediately reflected in the display of waveforms of signals and spectra.
Reading the numerical values of the signal samples or the values of any of the amplitude spectra can be carried out by approximately aligning the position of the "mouse" cursor with necessary element oscillograms and pressing the left mouse button (see Fig. 4.P.).
Rice. 4.P.
The type of characteristic of a nonlinear circuit is selected using the menu of the main window "Characteristics of N.E.". The clipping level or the cutoff level in the non-linear circuit characteristic is controlled by sliders (See Fig. 5.P.).
Rice. 5.P
At the bottom of the main working window of the application (see Fig. 2.P.) there is a window for editing the number of the experiment. The number of the experiment is necessary for the correct recognition of the stored data, and when changing the type of signal, it is increased automatically. However, when changing only the parameters of the input signal and the nonlinear circuit, it is necessary to correct it manually if the experimental data are stored for the same signal shape, but with different values of its parameters.
3.P. Arbitrary waveform input window.
To study the nonlinear transformation of an arbitrary waveform signal, a special window "Setting the waveform" is used, which is called from the menu of the main working window"Signal type / Arbitrary".
The view of the "Arbitrary signal" window is shown in Figure 6.
Rice. 6.
In this window, you can edit the waveform using the corresponding buttons, or load data from a file with the . txt , containing readings in text form. Such a file can be called from a special library or prepared when performing a calculation task in preparation for laboratory work.
To load signal readings from a text file, you must call the download dialog by pressing the "Load" button.
To load signal readings from a text file, you must call the download dialog by pressing the "Load" button. The text file data format is described above.
To use the loaded or edited signal readings, press the "Accept" key, and to cancel the data, press the "Cancel" key.
The transmission of signals over real communication channels is always accompanied by changes (transformations) of these signals, as a result of which the received signals differ from those transmitted. These differences are primarily due to linear and nonlinear transformations of the input signals, as well as the presence of additive noise in the channel, which most often exists independently of the transmitted signals. From the point of view of information transmission over a channel, it is important to divide signal transformations into reversible and irreversible ones. As will be shown (see § 4.2), reversible transformations do not involve loss of information. With irreversible transformations, information loss is inevitable. For reversible signal transformations, the term "distortion" is often used, and irreversible transformations are called noise (additive and non-additive).
An example of the simplest deterministic reversible transformation of the input signal X(t), which does not change its shape, is
Y(t) = kX(t-τ). (3.1)
In this case, the output signal of the channel Y(t) differs from the input signal only by the known scale k, which is easily compensated by the corresponding amplification or attenuation of the signal and a constant time delay τ. It is usually small. As such, it is only with space-scale communications or with a very large number of reactive elements of the communication line that the delay can be noticeable * .
* (Here we are talking about the delay in the communication line itself, and not about the delays in the demodulator and decoder, which can be significant and sometimes limit the possibility of improving noise immunity.)
If the input signal X (t) in (3.1) is narrowband, it is convenient to represent it in the quasi-harmonic form (2.68): X(t) = A(t)cos× X [ω 0 t+Φ(t)], where A(t ) and Φ(t) are slowly varying functions. Therefore, with a sufficiently small delay t, in the first approximation, we can consider A (t-τ) ≈ A(t) and Φ(t-τ)≈Φ(t), and write the output signal in (3.1) as follows:
Y (t) = kA(t-τ) cos[ω 0 (t-τ) + Φ(t-τ) ≈ kA (t) cos[ω 0 t+Φ(t)-θ K], (3.2)
where θ K =ω 0 τ - phase shift in the channel. Thus, with a narrowband signal, a small delay is reduced to some phase shift.
In real communication channels, even when additive noise can be neglected, signal transformations are complex and usually lead to a difference in the shape of the output signal from the input.
The study of transformations of random processes during their passage through dynamic systems (both with regular and randomly changing parameters) is associated with the solution of problems of two types:
determination of the correlation function (energy spectrum) of the response Y(t) at the output of the dynamic system, given by its characteristics according to the given correlation function (or energy spectrum) of the input action X(t);
determination of the multidimensional distribution of the response Y(t) at the output of a given dynamic system by the multidimensional distribution of the input action X (t).
The second of these tasks is more general. From its solution, obviously, the solution of the first problem can also be obtained. However, in what follows, we will mainly confine ourselves to a brief consideration of the first problem and only indicate possible ways solution of the second, more difficult problem.
Passage of random signals through deterministic linear circuits. As is known, a linear circuit with constant parameters is characterized by its impulse response g(t) or its Fourier transform transfer function k(iω). If, for example, a centered process X(t) arrives at the input of the circuit, then the process Y (t) at the output is determined by the Duhamel integral *
In a physically realizable circuit at t
* (Here and in what follows, the integration of random processes is understood in the root-mean-square sense [see. f-lu (2.8)].)
Let's find the correlation function of the centered output process Y (t):
where θ 1 = t 1 -τ 1 θ 2 = t 2 -τ 2; B X (θ 1 -θ 2) is the correlation function of the input signal.
Let the input process be stationary. Then B X (θ 1 -θ 2) = B(θ), where θ=θ 2 -θ 1 . We also introduce the notation t 2 -t 1 =τ, t 1 -θ 1 = τ 1 . Then t 2 -θ 2 = τ+τ 1 -θ and
where the "temporal correlation function" (TFC) from a non-random impulse response is used
In this case, β = τ - θ.
It can be seen from (3.4) that for a stationary input process and the output process turns out to be stationary, since B Y (t 1 ,t+τ) does not depend on t 1 . Therefore, one can write
The resulting equality is an analog of the Duhamel integral for correlation functions. Thus, the FC of the output process is an integral convolution of the FC of the input process and the CFC of the impulse response of the circuit.
Note that the PFC of the impulse response is related by the Fourier transform to the square of the modulus of the transfer function |k(iω)| 2 or frequency response (AFC) of the circuit. Really,
It is known from the theory of the Fourier transform that the Fourier transform of the convolution of two functions is equal to the product of the Fourier transforms of these functions. Applying this to (3.5), we obtain a simple relation between spectral densities stationary processes at the input and output of a linear circuit with a constant transfer function k (iω):
G Y (J) = G X (f)|k(i2πf)| 2 (3.7)
From (3.5) and (3.7) it follows that the PC and the spectrum of the process at the output of the circuit are completely determined by the PC or the spectrum of the process at the input and the frequency response of the circuit, i.e., they do not depend either on the probability distribution of the input process or on the phase-frequency characteristic of the circuit .
Consider an example of the passage of random processes through deterministic linear systems - the passage of white noise with an energy spectrum N 0 through a serial oscillatory circuit with parameters R, L, C. If output voltage removed from the tank, then the complex gain of the loop
resonant frequency,
In the region of small detunings |k(ω)| 2 = ω 2 0 /(4[β 2 + (ω-ω 0) 2 ]), β = R/(2L), and according to (3.7) the energy spectrum at the output
G Y (ω) = N 0 ω 2 0 /(4[β 2 + (ω - ω 0) 2 ]).
Output correlation function
When applying the signal X(t) to a deterministic linear circuit with variable parameters, the output signal is Y(t). as you know, can be expressed by the convolution integral:
where g(t, τ) is a function of two variables that determines the response of the system at time t to a δ-impulse applied to the input at time t-τ.
represents the transfer function of a linear circuit with variable parameters, which, of course, is a function not only of frequency, but also of time.
Since in a physically realizable circuit the response cannot occur before the impact, then g(t, τ)=0 for τ
The problem of finding the probability distribution of the response of a linear system under an arbitrary random action turns out to be very difficult in the general case, even if we confine ourselves to finding a one-dimensional distribution. Note, however, that if a Gaussian process is fed to the input of a linear deterministic system, then the output process turns out to be Gaussian, which follows from the well-known properties of the normal distribution, which remains normal under any linear transformations. If the process at the input is not Gaussian, then when passing through the linear system, its probability distribution sometimes changes quite significantly.
We note a general property inherent in linear systems. If the bandwidth F C occupied by the input signal X(t) is much wider than the bandwidth of a given linear system, then the distribution of the output process tends to approach normal. This can be roughly explained on the basis of (3.8). The narrow bandwidth means that the duration of the impulse response g(t, τ) as a function of τ is large compared to the input process correlation interval X(t). Therefore, the cross section of the output process Y(t) at any moment t is determined by the integral (3.8), whose integrand with a sufficiently large weight includes many uncorrelated cross sections of the process X(t). The probability distribution of such an integral, according to the central limit theorem, should be close to normal, the closer, the greater the ratio of the width of the spectrum of the input signal to the bandwidth of the circuit. In the extreme case, if the input of the circuit is affected by white noise, whose spectrum width is infinite, and the circuit has a limited bandwidth, then the output process will be strictly Gaussian.
Passage of narrow-band random signals through linear strip circuits. As noted in § 2.4, it is convenient to represent relatively narrow-band processes (i.e., those in which the width of the spectrum is much narrower than the average frequency) in the quasi-harmonic form (2.68). If the average frequency ω 0 is given, then such a narrow-band signal is completely determined by its complex envelope A(t) (2.70) or its real and imaginary parts (quadrature components) A C (t) and A S (t), which are low-frequency processes, i.e. That is, their spectra occupy the region of frequencies lower than the spectrum of the signal itself. Such a representation in many cases, at the stages of synthesis and analysis of signal transmission systems (messages), is very useful. So, to represent (2.72) on the interval T by a Kotelnikov series, 2T(f 0 + F) samples are required, but to represent on the same interval T two independent low-frequency real functions A C (t) and A S (t) (or one complex function A(t)), 4FT samples are sufficient, i.e., approximately f 0 /2F times less.
We also note that, if necessary, to simulate narrow-band signals and a communication system with such signals on computer or if it is necessary to implement various transformations of such signals on the basis of a modern microelectronic base, difficulties arise, most often practically insurmountable, due to the limited speed of these machines or the corresponding microcircuits. Naturally, it is much easier in these cases to operate with low-frequency signal equivalents, which are the components of the envelope.
The expression for the low-frequency equivalent Ȧ x (t) of a narrowband signal (2.72), determined from (2.70, a):
A X (t) \u003d X (t) exp [-iω 0 t]
has according to (2.32) the Fourier spectrum
S Ȧ X (iω) = Sx.
Figure 3.1 illustrates the spectral relationships for a real narrow-band signal X * (t) (Fig. 3.1, a), an analytical signal X (t) (Fig. 3.1.6) and its low-frequency equivalent А̇ X (t) (Fig. 3.1, c ).
* (It is useful to recall that the spectrum S X (iω) of the real signal X(t) is symmetrical about the origin, S * X (-iω) = S X (iω) (i.e., the amplitude spectrum is an even function of frequency, and the phase spectrum is odd, or the real part S X (iω) is an even function of frequency, and the imaginary part is odd).)
The main part of real continuous communication channels is linear and narrow-band, so the signals at their output can be considered as a response to the narrow-band signal Х(t) of a band-pass filter with a transfer function k(iωt), the modulus of which has the character of Fig. 3.1a. The advantages of representing signals using a low-frequency equivalent (complex envelope) arise from the fact that band-pass filtering of a narrow-band signal can be interpreted as filtering complex low-frequency signals with complex low-pass filters.
Consider the passage of a narrow-band signal X(t) through a narrow-band channel (band-pass filter) with constant parameters and transfer function k(iω) (Fig. 3.2, a).
Narrowband input (2.72)
Given the preceding footnote, it is easy to show that the spectrum of the conjugate complex envelope A * X (t) = A C (t) - iA S (t) is equal to S * Ȧ X (-iω), where (iω) is the Fourier spectrum of A X ( t). Since the multiplication of the time function by e ±itω 0 corresponds to a shift of the spectrum along the frequency axis by ±ω 0 , then for the Fourier spectrum of the function X(t) defined by (3.10), we can write
Similarly, assuming that the average frequency of the input signal ω 0 coincides with the center pass frequency of the filter, we can represent the transfer function of the band pass filter (Fourier transform of the impulse response of the filter g(t) *
where Γ is the Fourier spectrum of the complex (analytical) signal ġ(t) = g(t) +ig̃(t) = γ̇(t)e itω 0 formed from g(t). The value Γ(iω) is the spectral characteristic of the complex envelope γ̇(t) of the impulse response of the filter g(t), i.e., the low-frequency equivalent of a narrow-band channel.
* (Note that the functions Γ and Γ*[-i(ω+ω 0)], being modulo symmetrical with respect to the y-axis for the band-pass filter, do not overlap, since the first lies almost entirely in the region of positive frequencies, and the second is negative. A similar statement is true for the functions S Ā and S* Ȧ [-i(ω+ω 0)] narrowband signal.)
Now let's find the Fourier spectrum of the signal at the channel output y(t). On the one hand, since this signal is narrow-band with the average frequency of the spectrum ω 0 , we can write similarly to (3.11)
where S Ȧ y is the Fourier spectrum of the complex (analytical) signal ẏ(t) = y(t) + iȳ(t) = Ȧ y e itω 0 , while S Ȧ y (iω) is the spectrum of the complex envelope Ay(t) of the output signal . On the other hand, for a linear system with constant parameters spectral characteristics signals at the input and output are related by the relation
S y (i ω) - Sx (iω)k(iω). (3.14)
Substituting relations (3.11) and (3.12) into (3.14) and taking into account the footnote on page 78, we obtain
From (3.13) and (3.15)
As a consequence, the complex envelope of the signal at the output of the narrowband channel A y (t) is obtained as a convolution of the complex envelope of the input signal A x (t) and the complex envelope of the impulse response of the filter γ̇(t)
If the filter is non-distorting, i.e., Γ(iω) = γe -it 0 ω or ġ(t) = γδ(t-t 0), then, using the filtering property of the b-function, from (3.17) we obtain
We write the complex envelopes in terms of the in-phase and quadrature components:
Ȧ X (t) = A X,C (t) + iA X,S (t);
γ̇(t) = γ C (t) + iγ S (t);
Ȧ y (t) = A Y,C (t) + iA Y,S (t), (3.18)
Then from (3.17)
In a particular region, relation (3.19) takes the form:
So, bandpass filtering with transfer function k(iω) of narrowband
process x(t) is equivalent to the low-pass filtering with the transfer function Γ(iω) of the complex low-frequency process Ȧ x (t) (see Fig. 3.2).
Processes A X, C and A X, S can be obtained from x(t) in the device, functional diagram which is shown in Fig. 3.3a. Indeed, multiplying x(t) by 2cos ω 0 t we get
[ A X,C (t) cos ω 0 t + A X,S (t) sin ω 0 t] 2 cos ω 0 t = A X,C (t) + A X,C (t) cos 2 ω 0 t + A X, S (t) sin 2ω 0 t, (3.21)
and the LPF will pass only the first low-frequency, the other two terms are high-frequency and will be delayed by the filter. Similarly, the quadrature component A X,S (t) is distinguished in the second branch.
Now consider how you can implement complex low-pass filtering (3.19) or (3.20) using three real low-pass filters (for such a filter, the response to a real signal is real or the transfer function satisfies the footnote condition on page 77), operating with quadrature components. This is done according to (3.19) or (3.20) by two-channel filtering of the real low-frequency in-phase and quadrature components (Fig. 3.3.6).
Passage of random signals through non-linear circuits. We confine ourselves to considering only inertial non-linear systems with regular parameters, in which the input and output are connected by some non-linear dependence, called the characteristic of the system:
y(t) = φ, (3.22)
Relation (3.22) can quite accurately characterize the operation of a number of links of real communication channels, for example, those included in demodulators, limiters, modulators, etc. The transformation x(t) → y(t), as a rule, is unambiguous, which is not always possible to say about inverse transformation y(t)→x(t) (for example, quadratic circuit with characteristic y = kx 2). Due to the inapplicability of superposition to nonlinear systems, consideration of a complex effect (for example, the sum of a deterministic and random terms) cannot be reduced to considering the passage of each of the components separately.
With nonlinear transformations, a transformation (change) of the spectrum of the input action occurs. So, if the input of a nonlinear system is affected by a mixture of a regular signal and additive noise X(t) = u(t) + N(t) in a narrow frequency band F c , grouped around the average frequency f 0 , then in the general case the output will contain components of combination frequencies of three types, grouped around frequencies nf 0 (n = 0, 1,...), products of beats of input signal components among themselves (s×s), products of beats of input noise components (w×w); signal and noise beat products (s×w). It is usually not possible to separate them at the output of the system.
If the characteristic y \u003d φ (x) of the nonlinear system and the two-dimensional distribution function of the input action w (x 1, x 2, t 1 , t 2) are known, then the main statistical characteristics of the output process, in principle, can always be determined. So, the mathematical expectation of the response
and its correlation function
The inverse Fourier transform can also be used to find the energy spectrum using (3.24).
Using the rules for finding distribution laws for functions from random variables(random processes), it is possible, in principle, to find the distribution of the output process of any order, if the distribution of the input process is known. However, the determination of the probabilistic characteristics of the response of nonlinear systems (circuits) even to stationary input actions turns out to be very cumbersome and difficult, despite the fact that a number of special techniques have been developed to solve this problem. In many cases, especially for narrowband signals, these calculations are greatly simplified when using a quasi-harmonic representation of the process.
As an example, consider the passage through a quadratic detector of the sum of a harmonic signal s(t) = U 0 cos ω 0 t and stationary quasi-white narrow-band noise n(t) = Х cn (t) × X cos ω 0 t + X sn sin ω 0 t , where X cn (t), X sn (t) are uncorrelated quadrature Gaussian noise components, for which m X cn = m X sn = 0, B X cn (τ) = B X sn (τ) = B(τ ), and the energy spectrum is uniform and limited by the frequency band F n
There is no general procedure for determining the distribution law of the response of a linear FU to an arbitrary random action. However, it is possible correlation analysis, i.e., the calculation of the correlation function of the reaction according to the given correlation function of the impact, which is conveniently carried out by the spectral method according to the scheme shown in Fig. 5.5.
To calculate the energy spectrum G Y(f) reactions of a linear FU with a transfer function H(jω), we use its definition (4.1)
Correlation function B Y(t) we define by the Fourier transform of the energy spectrum G Y(f)
.
Let us return to the definition of the distribution law of the response of a linear FU in some particular cases:
1. Linear transformation normal SP also generates a normal process. Only the parameters of its distribution can change.
2. The sum of normal joint ventures (the reaction of the adder) is also a normal process.
3. When an SP with an arbitrary distribution passes through a narrow-band filter (i.e., with a filter bandwidth D F significantly smaller width of the impact energy spectrum D f X) there is a phenomenon of normalization of the distribution of the reaction Y(t). It lies in the fact that the distribution law of the reaction approaches normal. The degree of this approximation is the greater, 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 to t Y). As a result, between uncorrelated readings of the filter response Y(k t Y) is located approximately D f X / D F uncorrelated exposure readings X(l t X), each of which contributes to the formation of a single response sample with a weight determined by the type of 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 normal with an increase in the number of terms.
5.3. Narrow-band random processes
joint venture X(t) with a relatively narrow energy spectrum (D f X << fc) as well as narrow-band deterministic signals, it is convenient to represent it in a quasi-harmonic form (see Section 2.5)
where is the envelope A(t), phase Y( t) and 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 an envelope A(t) and phase Y( t) it is expedient to use the analytical joint venture
, (5.4)
The main moment functions of the analytical SP:
1. Mathematical expectation
2. Dispersion
3. Correlation function
,
,
.
An analytic SP is called stationary if
,
,
Let us consider a typical problem in communication technology of passing a normal SP through a band-pass filter (BP), amplitude (AD) 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 initial phase j( t) will be slowly varying functions of time compared to , where is the average frequency of the BPF passband. By definition, the signal at the output of the IM will be proportional to the envelope of the input signal A(t), and at the PD output, its initial phase j( t). Thus, to solve this problem, it suffices to calculate the distribution of the envelope A(t) and phase Y( t) (distribution of the initial phase 
differs from the distribution Y( t) only by mathematical expectation ).
Formulation of the problem
Given:
1) X(t) = A(t)cosY( t) is a narrow-band centered stationary normal SP (at the PF output),
2) 
.
Define:
1) w(A) is the one-dimensional probability density of the envelope,
2) w(Y) is the one-dimensional probability density of the phase.
To solve this problem, we outline three stages:
1. Transition to analytical SP and determination of the joint probability density .
2. Calculation of the Joint Probability Density Based on the Calculated at the First Stage and the Connections A(t), Y( t) with (5.3) ÷ (5.6) .
3. Definition of one-dimensional probability densities w(A) And w(Y) by the computed joint probability density .
Solution
Stage 1. Let us find the one-dimensional probability density of the process . Based on the linearity of the Hilbert transform 
we conclude that is a normal SP. Further, considering that 
, we get 
, and consequently
Thus, we have
.
Let's prove uncorrelatedness 
at coinciding times, i.e., that .
.
After substituting , , , taking into account that at , we obtain
The uncorrelatedness of the cross sections of normal processes implies their independence, hence
.
Stage 2. Joint Probability Density Calculation
,
where according to (5.2), (5.5) and (5.6)
.
Therefore, taking into account (5.3), we have
. (5.7)
Stage 3. Definition of one-dimensional probability densities
Finally
, (5.8)
. (5.9)
Expression (5.8) is known as Rayleigh distribution, its graph is shown in Fig. 5.8. On fig. 5.9 shows a graph of the uniform distribution of the phase (5.9).
Expression (5.7) can be represented as a product of (5.8) and (5.9)
which implies the independence of the envelope A(t) and phase w(Y) normal SP.
Let us consider a more complex problem of passing an additive mixture of the above-considered normal SP with a harmonic signal through the AD and PD. The problem statement remains the same except for the original process Y(t) , which takes the form
Where X(t) is a centered normal SP.
Because the
.
Let's write down Y(t) in the quasi-harmonic form
and we will solve the problem of determining the probability densities w(A) And w(j) according to the above plan.
Preliminary write down X(t) in quasi-harmonic form and through its quadrature components
, (5.10)
(5.11)
To find it, we turn to the analytical joint venture
.
It can be seen from its expression that are linear transformations of the centered normal SP X(t):
and therefore have a normal distribution with variances
.
Let us prove their uncorrelatedness (and hence independence) at coinciding times
.
It is taken into account here that B(t) and θ( t) – the envelope and phase of the normal SP are, as established above, independent.
Thus,
and taking into account (5.10) and (5.11) we obtain
. (5.12)
Since expression (5.12) cannot be represented as a product of one-dimensional functions , we can conclude that the processes are dependent.
To find the distribution of the envelope sum of a centered normal SP with a harmonic signal, we integrate (5.12) over all possible values of the random phase j( t)
.
Integral of the form
known in mathematics as the modified zero-order Bessel function. Taking it into account, we finally have
. (5.13)
Expression (5.13) is called generalized Rayleigh distribution or Rice distribution. Graphs of this expression are shown in fig. 5.10 for the following special cases:
1) U = 0 
is the usual Rayleigh distribution,
2) 
- case of absence Y(t) SP X(t),
3) 
is the generalized Rayleigh (Rice) distribution.
It can be seen from the graphs that the greater the signal-to-noise ratio, the more to the right the maximum of the probability density is shifted and the more symmetrical (closer to the normal distribution) the curve .
conclusions
1. If the instantaneous values of the centered SP X(t) have a normal distribution, then its envelope A(t) distributed according to the Rayleigh law
,
and phase Y( t) evenly
2. The distribution of the envelope of the additive mixture of a centered normal SP and a harmonic signal obeys the generalized Rayleigh distribution (it is also the Rice distribution)
.
Control questions
1. Formulate the problem of analyzing the passage of the SP through a given functional node.
2. How the probability density is calculated w(y) the reaction of the inertialess chain according to the known probability density w(x) impact?
3. How to calculate the mathematical expectation of the reaction of an inertialess chain to a random impact X(t)?
4. How to calculate the variance of the reaction of an inertialess chain to a random impact X(t)?
5. How to calculate the correlation function of the response of the inertialess chain to a random impact X(t)?
6. How the Joint Probability Density is Calculated w(at 1 , at 2; t) two joint ventures Y 1 (t) And Y 2 (t) connected by known functional dependencies 
And 
with two other joint ventures X 1 (t) And X 2 (t)?
7. How does the distribution of a normal SP change when it passes through a linear circuit?
8. How does the arbitrary distribution of the SP change when it passes through a narrow band filter?
9. What is the essence of the phenomenon of normalization of a broadband process when it passes through a narrowband filter? Give a mathematical justification for this phenomenon.
10. Describe the procedure for the correlation analysis of the passage of the SP through a linear circuit.
11. Define the envelope and phase of the SP.
12. Define the analytical joint venture, its mathematical expectation, variance and correlation function.
13. What conditions does a stationary analytical SP satisfy?
14. What is the distribution of the envelope of the centered normal SP?
15. What is the distribution of the phase of a centered normal SP?
16. What is the distribution of the envelope sum of a centered normal SP and a harmonic signal?
17. Write an analytical expression for Rayleigh's law. Which SP distribution does it characterize?
18. Write an analytical expression for the generalized Rayleigh's law (Rice's law). Which SP distribution does it characterize?
Consider a linear inertial system with a known transfer function or impulse response . Let the input of such a system be a stationary random process with given characteristics: probability density , correlation function or energy spectrum . 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 input process 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 implementation 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 amplitude-frequency characteristic of the system, and will not depend on the phase-frequency characteristic.
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 the Linear System, the output also turns out to be 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 output energy spectrum of the process 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 band-pass filter, whose amplitude-frequency response for positive frequencies (Fig. 1.6) is given by:
We define the correlation function using the cosine Fourier transform:
The graph of the correlation function is shown in fig. 1.7
The considered examples are indicative from the point of view that they confirm the relationship 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 process power at the output of an ideal band-pass filter will be equal to
The probability distribution law of a random process at the output of a linear inertial system differs from the distribution law at the input, and its determination is a very difficult task, with the exception of two special cases, which we will dwell on here.
If a random process acts on a narrow-band linear system, the bandwidth of which is much less than its spectral width, then the phenomenon occurs at the output of the system normalization distribution law. This phenomenon lies in the fact that the distribution law at the output of a narrow-band system tends to normal, regardless of what distribution the broad-band random process has at the input. Physically, this can be explained as follows.
The process at the output of the inertial system at some point in time is a superposition of individual responses of the system to the chaotic effects of the input process at different points in time. The narrower the bandwidth of the system and the wider the spectrum of the input process, the greater the number of elementary responses formed the output process. According to the central limit theorem of the theory of probability, the distribution law of the process, which is the sum of a large number of elementary responses, will tend to normal.
From the above reasoning follows the second particular, but very important case. If the process at the input of 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.
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