Pass the signal. Passage of fall-off processes through linear lancets What works with the separated material
Let's look at a linear inertial system with a known transfer function or impulse reaction. Let the input of such a system be a stationary phased process with given characteristics: intensity, correlation function or energy spectrum. The characteristics of the process at the system output are important:
The simplest way is to find out the energy spectrum of the process at the output of the system. It is possible that while the process is being implemented, there are deterministic functions at the input, and the Fourieux apparatus is installed before them. Let's go
the implementation of trivality T in the fall process at the input has been accelerated, and
This is the spectral thickness. The spectral width of the implementation at the output of the linear system is more modern
The energy spectrum of the process at the output (1.3) is significant
tobto. corresponds to the energy spectrum of the input process, multiplied by the square of the amplitude-frequency characteristics of the system, and is not dependent on the phase-frequency characteristics.
The correlation function of the process at the output of the linear system can be defined as a Fourier transformation of the energy spectrum:
Also, when a stationary phased process is injected into the Linear system, the output is also a stationary phased process with an energy spectrum and a correlation function, which are indicated by the expressions (2.3) and (2.4). The pressure on the process at the output of the system is more modern
As a first step, let us look at the passage of white noise with spectral strength through an ideal low-pass filter, for which
Corresponding to (2.3), the energy spectrum of the process at the output is equal to the spectral strength in a mixture of frequencies, and the correlation function is significant
The intensity of the phase process at the output of an ideal low-pass filter will be equal to
As another example, let’s look at the passage of white noise through an ideal black filter, the amplitude-frequency characteristic of which for positive frequencies (Fig. 1.6) is indicated by:
The correlation function is significant using the additional Fourth cosine transformation:
The graph of the correlation function is shown in Fig. 1.7
The examined applications show from this point of view that they confirm the findings in § 3.3 of the connections between the correlation functions of low-frequency and high-frequency processes with the same form of the energy spectrum. The pressure of the process at the output of an ideal smoky filter will be equal to
The law of the division of velocities of the fall process at the output of a linear inertial system is divided into the law of division at the input, and it is determined by even complex problems, behind the culprit of two adjacent drops, which I'm stuck here.
Since the episodic process flows into the cosmic linear system, the amount of transmission which is significantly less than the width of the spectrum, then at the output of the system there is a phenomenon normalization to the law of the law. This lies in the fact that the law of division at the output of the musculoskeletal system of the normal process is independent of the distribution of the wide-skinned episodic process at the input. Physically it can be explained this way.
The process at the output of the inertial system at any given moment of the hour is a superposition of the adjacent outputs of the system on the chaotic flow of the input process at different times of the hour. The greater the system throughput and the wider the spectrum of the input process, the greater the number of elementary outputs the output process is created. The law is consistent with the central boundary theorem of the theory of certainties to the subdivision of the process, which becomes the sum of a large number of elementary effects, a deviation from the normal.
With the guidance of the world comes another serious, but even more important, outburst. If the process at the input of a linear system has a normal (Gausian) distribution, then it is not normal at the output of the system. In this case, both the correlation function and the energy spectrum of the process change.
It is acceptable that there is a presence of vibration at the input of the linear stationary system, which is the implementation of the fall process. Since this implementation is assigned in advance, then each new task does not arise - the signal must be followed as before a deterministic function. Knowing the mathematical model of the system, for example the frequency transfer coefficient, you can find the output reaction.
However, the specificity lies in the fact that additional information about the input signal is not available - we can only obtain information about the average and universal characteristics of the fallout process.
Meta - explore connections between statistical characteristics of processes that can be found on the basis of a mathematical model of the system.
Let us introduce the exchange - we can only see the stationary input and fall processes. It is more mathematical to calculate the mitt values of the implementation of the hour (), so that the correlation function lies below the value of the absolute value between the points on the hour axis.
Let's look at the implementation of the input signal and imagine it in terms of the Fourie integral
de – spectral thickness.
The output signal of the system will be found based on its frequency transmission ratio
(1)
The assumption of stationarity to the process imposes a conclusion: the average value of the spectral power.
Vikonuyuchi statistically averaged in both parts of the virazu (1), may
(2)
In order to calculate the correlation function, it is necessary to know the value of the output signal at the time.
(3)
Because speech function, therefore formula (3) will not change if in its right side we go to complex-derived quantities
(4)
de; - spectrum of tension in the stationary and seizure process. (The filtering power of the delta function is discussed).
(6)
The intensity spectrum of the output signal of the connections with a similar spectrum to the signal at the input of the connections
In applied problems, it is often necessary to use one-sided spectra i, which are valued more than positive frequencies,
Therefore, the dispersion of the output signal
(9)
It is often possible to observe the influx on linear frequency-selective lancets of bursty broadband signals created, for example, by a chaotic sequence of short pulses. And here, the effective width of the spectrum of the input fallout process significantly outweighs the width of the signal transmission of the system, the real fallout process can be replaced by an equivalent white noise with a one-sided tension spectrum, the point between the passivity of the lancet .
Todi formula (9) say goodbye
In engineering applications, a linear frequency-selective lance, which under the influx of a wide-dark signal, can be easily characterized by noise transmittance. The value is calculated as the transmission coefficient of an ideal black filter with a speech transmission coefficient equal to the maximum module of the transmission coefficient of a real lancet. When the ideal and real systems are awakened by white noise and a spectrum of intensity, the dispersion of noise signals at the outputs of both Lantzugs will inevitably be avoided
(11)
Otje
(12)
For example, for an integrated RC-lanc
; 
Otje 
For what it's worth.
Since the input phased process is normal (the Gaussian nature of the laws of division), then the phased process at the output yields power independently of the dynamic powers of the linear system.
On the base of the Duhamel-mitteve formula, the meaning of vidguku
is the result of summing the forward values of the input signal, multiplied by the damaged impulse characteristic of the Lancsug.
Meta robots: To gain initial knowledge of the researched statistical characteristics of the waveform signals. To experimentally determine the laws of the distribution of the waveform signals at the outputs of linear and nonlinear radio-technical devices.
SHORT THEORETICAL VIEWS
1. Classification of radiotechnical Lancsugs
Radiotechnical lancets, which are used to reconvert signals, vary widely in their composition, structure and characteristics. In the process of development and analytical research, various mathematical models are used, which satisfy their adequacy and simplicity. In this case, any kind of radio engineering lancet can be described by formalized relationships, which means the transformation of the input signal x(t) into the output y(t), which symbolically can be applied to the view
y(t) = T,
De T is an operator that defines the rule that transforms the input signal.
As a mathematical model of a radio-technical lancet, the combination of the operator T and two factors X=(xi(t)) and Y=(yi(t)) of signals at the input and output of the lancet can serve as such, so that
(yI(t)) = T(xI(t)).
Based on the type of transformation of the input signals at the output, as seen by operator T, a classification of radiotechnical devices is carried out.
The radio-technical lancet is linear, since the operator T is such that the lancet satisfies the minds of additiveness and uniformity, so that fairness is fair
T = T : T = c T
i I
Des is a constant.
These minds express the essence of the principle of superposition, which has power over linear Lantzugs.
The functioning of linear lancets is described by linear differential equations with constant coefficients. It is characteristic that the linear transformation of a signal of any form is not accompanied by the appearance of harmonic warehouses with new frequencies in the spectrum of the output signal, so as not to enrich the spectrum of the signal.
Radiotechnical lancet Nonlinear, since the operator T does not ensure the achievement of minds of aditivity and homogeneity. The functioning of such cells is described by nonlinear differential equations.
Structurally, linear lances do not accommodate linear devices (pre-tensioners, filters, long lines, etc.). Nonlinear lancets can accommodate one or more nonlinear devices (generators, detectors, multipliers, intermediaries, etc.)
Depending on the nature of the time duration of the output signal, the input signal is divided into inertial and non-inertial radio-technical lancets.
Radio engineering lantzug, the value of the output signal y(t) at the moment t=t0 lies not only with the value of the input signal x(t) at this moment, and the value x(t) at the moment t0 is called Inertial Lanzug. Since the value of the output signal y(t) at the moment t=t0 is directly assigned to the values x(t) at the same moment t0, then such a loop is called Without inertia.
2. Reversal of episodic processes in linear lancets
The ongoing transformation of the phenomenal processes in linear radio-technical lancers can be seen in the current situation. At the input of a linear lance with a frequency characteristic K(jw), let us find a phased process x(t) from the given statistical powers. It is necessary to calculate the statistical characteristics of the fall process y(t) at the output of the lancet. It is important to analyze the characteristics of the phase processes x(t) and y(t) to consider two variants of the initial command:
1. The value of the energy spectrum and the correlation function of the fallout process at the output of the linear lancet.
2. The importance of the laws of the division of the dynamics of the fall process at the output of the linear stake.
The simplest is the first task. The solutions in the frequency domain are based on the fact that the energy spectrum of the input process at the output of the linear lance Wy(w) in the stationary mode is equal to the energy spectrum of the input process Wx(w), multiplied by the square of the module of the lance frequency characteristic south, tobto
Wy(W)= Wx(W) ∙│ K(Jw)│ A (1)
It is clear that the energy spectrum Wx(w) of the fall process x(t) with mathematical calculations mx=0 is related to its function of quartzation Bx(t) by Four'e transformations, so
Wx(W)= INX(T) E— JWTDT
INX(T)= Wx(W) To herWTDW.
Therefore, the covariance function Вy(t) of the fall process at the output of the linear lancet can be calculated as follows:
INY(T)= Wy(W) To herWTDW= Wx(W))│ K(Jw)│ A To herWTDW
Ry(T)=BY(T)+ Mya.
In this case, the dispersion Dy and the mathematical calculation of my output and fall process become more advanced
Dy = Ry (0) = Wx (w)) │ K (jw) │ adw
My= Mx∙ K(0) .
De mx - mathematical calculation of the fallout input process:
K(0) - coefficient of transmission of a linear lance along a stationary stream, then
K(0)= K(Jw)/ W=0
Formulas (1,2,3,4) are actually outside the solution to the task set in the frequency domain.
The method is based on another task, which allows one to directly determine the intensity of the process y(t) at the output of the linear inertial lance for a given intensity of the process x(t) at the input I can't seem to sleep. This holds true only for certain phased phases and for phased processes with a Gaussian (normal) law of subdivision, as well as Markovian phased processes.
In the process of a normal law, it was decided to say goodbye to this framework, so that when such a process is carried out linearly, the law of the division does not change. Since the normal process is determined by mathematical calculations and correlation functions, then to find the strength of the process it is sufficient to calculate its mathematical calculations and correlation functions Yu.
The law of the distribution of strengths of the signal at the output of the linear inertialess lancet is observed in the functional sense according to the law of the distribution of the input signal. You can change any of your parameters. Thus, since a linear inertial lance implements a functional transformation of the form y(t) = a x(t) + b, where a and b are constant coefficients, then the strength of the reactivity p(y) of the phased process to the output and the lancet is identified according to the familiar formula for the functional transformation of episodic processes
P(Y)= =
De p(x) is the density of the precipitation process x(t) at the entrance of the lancet.
In certain situations, the task of identifying the most important characteristics of the fallout process at the output of the inertial lancets allows the vicarious effect of normalization of the fallout process by inertial systems. If the non-Gausian process x(t1) with a correlation interval tk flows into the inertial linear lancet at a stationary hour t»tk (at which the width of the energy spectrum of the fall process x(t) is greater than the transmission of the lancet), then process y(t) at the output of such The lantzug is approaching the Gaussian world, the world has an increased rate of t/tk. This result is called the effect of normalization of the relapse process. The effect of normalization is manifested in the strongest way, even though the transmission of the lancet is lower.
3. Reversal of episodic processes in nonlinear lancets
Non-linear inertial transformations are seen during the analysis of non-linear lances, the inertia of which cannot be missed for given inflows. The behavior of such lantsugs is described by nonlinear differential relationships, and there are no secret methods for solving them. Therefore, the problem associated with the investigation of nonlinear inertial transformations of episodic processes may soon appear approximately, distorted by various piecemeal techniques.
One of these techniques is based on a combination of linear inertial and nonlinear inertial lances. The background investigation into the influx of fall processes onto the linear lancet was seen more clearly. It was shown that in this case it is possible to simply calculate the spectral density (or correlation function) of the output signal, or simply the law of division. In nonlinear inertialess lances, the main foldability lies in the well-known correlation function. There are no conventional methods for analyzing the influx of spiking signals on nonlinear lancets. Exchange the most important private tasks to create practical interest.
3.1. Statistical characteristics of the fall process at the output of nonlinear Lantzugs
Let's take a look at the transformation of the fall process with the one-dimensional strength and strength of the nonlinear inertia-free lance with the characteristic
Y= f(x).
Obviously, if any implementation of the fall process x(t) is transformed into a new implementation of the new fall process y(t), then
y(t)=F[ X(T)] .
A. According to the law of subdivision of the phased process y(t)
Let the thickness of the homovirality of the p(x) in the fall process of x(t) be known. It is necessary to calculate the density of the homovirality p(y) of the fall process y(t). Let's look at three characteristic episodes.
1. The function y= f(x) of a nonlinear lance means an unambiguous relationship between x(t) and y(t). It is important that the return function x = j(y), which also means a unique relationship between y(t) and x(t). In this case, the probability of finding the implementation of the fall process x(t) in the interval (x0, x0+dx) is the same as the probability of finding the implementation of the fall process y(t)=f in the interval (y 0, y0+dу) with y0= f(x0) y0+dy= f(x0+dx), then
P(X) Dx= P(Y) Dy
Otje,
P(Y)= .
The similarity is taken by absolute value since the strength of the homovirality p(y) > 0, so the similarity can be negative.
2. The return function x = j(y) is ambiguous, so one value is confirmed by the value of x. For example, the values y1 = y0 indicate the values x = x1, x2,…, xn.
From the fact that y0≤ y(t)≤ y0+dy, one of n mutually unreasonable possibilities emerges
X1 ≤ X(T)≤ X1 + Dx, or X2 ≤ X(T)≤ X2 + Dx, or... Xn≤ X(T)≤ Xn+ Dx.
The stagnation rule for adding surrogacy is abolished
P(Y)= + +…+ .
/ X= X1 / X= X2 / X= Xn
3, Characteristic of a nonlinear element = f(x) there are one or more horizontal sections (sections, where y = const.). Todi Viraz
P(Y)=
It should be added that it is possible to restart y(t) at the interval, where y = const.
The easiest way to look at this is at the butt.
Let the function y = f(x) be represented in Fig. 1 by the formula
Rice. 1 Injection of the fall-out process into a double-sided boundary.
At 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 strength of the virality
P1(y) = P1∙δ(y).
Similar to sizing for the box x(t)> b, can be removed
Pa = P = P = P (x) dx,
pa(Y) = Pa∙ δ (Y— C).
/ Y= C
For the equation a≤x≤b the following formula is valid:
Pa(Y) =
/0≤ Y≤ C
The intensity of the output process is determined by the expression
P(Y)= P1 ∙ δ (Y)+ Pa∙ δ (Y— C)+ .
Please note that in order to remove the residual viscosity, it is necessary to use the functions p(x) and dy/dx, and change the functions of x to the functions of y, using the reverse function x = j(y). Thus, the assigned thickness to the subdivision of the fall process at the output of the nonlinear inertialess lancet is determined analytically for simple characteristics y = f(x).
B. Value of the energy spectrum and correlation function of the fall process y(t)
It is impossible to directly determine the energy spectrum of the fall process at the output of the nonlinear lancet. The main method is to assign a correlation function to the signal at the output of the Lantzug with further stagnation of direct Fourier transformation of the spectrum value.
If the input of a nonlinear inertia-free lancet is a stationary fall process x(t), the correlation function of the fall process y(t) at the output can be represented in the form
Ry(T)= By(T)- My2 ,
De By(t) - accessible function;
my - mathematical calculation of the fall process y(t). The covariance function of the fall process is a statistically averaged additional value of the fall process y(t) at time t and t+t, then
By(T)= M[ Y(T)∙ Y(T+ T)].
To implement the fall process y(t) solid y(t)∙y(t+t) is a number. For the process as a whole of its implementation, this solid creates a drop-in value, the division of which is characterized by a two-dimensional strength of the molecular weight p2 (y1, y2, t), where y1 = y (t), ya = y (t + t). It is important to note that in the remaining formula the variable t does not appear, because the stationary process is the result of which t may lie.
When specifying the function p2 (y1, y2, t), the operation of averaging over a multiplier follows the formula
By(T)=У1∙у2∙р2 (у1, у2,T) Dy1 Dy2 = F(X1 )∙ F(X2 )∙ P(X1 , X2 , T) Dx1 Dx2 .
Mathematical calculation of my is indicated by the next expression:
My= Y∙ P(Y) Dy.
In reality, p(y)dy = p(x)dx can be removed
My= F(X)∙ P(X) Dx.
The energy spectrum of the output signal is consistent with the Wiener-Hinchin theorem and is found to be a direct transformation of the Fourier covariance function, so
Wy(W)= By(T) E— JWTDT
In practice, this method is difficult to implement, since the sub-integral for By(t) can be repeated again. It is necessary to use different methods to make things simpler, depending on the specifics of the task at hand.
3.2. Injection of cosmic noise into an amplitude detector
Statistical radio engineering is divided into broad-spectrum and university-specific processes.
Nehai ∆ fе – the width of the energy spectrum of the fall process, calculated by the formula (Fig. 2)
Rice. 2. Width of the energy spectrum of the fall process
Vuzkosmugovim A burst process is a process in which ∆fe «f0 de f0 is the frequency that corresponds to the maximum of the energy spectrum. A fallow process, the width of the energy spectrum does not satisfy the mind, it Widespread.
The high-frequency vibration process is usually represented by high-frequency vibrations with a completely continuous (equivalent to vibrations at frequency f0) amplitude and phase, so
X(t)= A(t)∙cos,
De A(t) = √x2(t) + z2(t) ,
J(t) = arctan,
z(t) is a function related to Hilbert with the output function x(t), then
z(t)= -DT
All parameters of this vibration (amplitude, frequency and phase) have variable functions.
An amplitude detector, which is a storage part of the primary tract, is connected to a nonlinear inertia-free element (for example, a diode) and an inertial linear lanyard (low-pass filter). The voltage at the detector output creates the amplitude that cancels out the high-frequency vibration at the input.
Don’t let a high-frequency waveform signal come to the input of the amplitude detector (for example, at the output of the amplifier, which carries a high-frequency signal at the intermediate frequency of the signal transmission), which is subject to the power of an annual waveform process with the normal distribution law. Obviously, the signal at the output of the detector will be the output signal of the input signal, which is also the function of the clock. It has been proven that this is the same as the high cosmos-smooth phased process, which is characterized by the intensity of its intensity, which is called the Rayleigh subdivision and has the appearance:
De A - meaning of the original;
Sx2-dispersion of the drop signal at the detector input.
The Rayleigh subdivision graph is presented in Fig. 3.
Fig.3. Graph of Rayleigh's law
The function p(A) has a maximum value that is higher than
When A = sx. This means that the values of A = sx and are the most significant values of the other.
Mathematical explanation of the episodic process
M.A.= = =
In this way, describing the variegated process with the normal law of the division, the division function of the hour, the strength of the division is described by Rayleigh's law.
3.3. The law of the division of the sum that burns out, the harmonious signal and the high-grade noise
The distribution of the law of the envelope sum of the harmonic signal and the high-voltage phased noise arises when analyzing the process of linear detection in radar and communication systems that operate in the minds, if there is power or outside And the noise can be equalized with the red signal.
Don’t let the receiver input the amount of harmonic signal a(t)=E∙cos(wt) and high-grade noise x(t)=A(t)∙cos with the normal law of distribution. Sumarne Kolivannya can be recorded at any time
N(T) = S(T)+ X(T)= E∙sS(Wt)+ A(T)∙ Cos[ Wt+ J(T)]=
=[E+A(T)∙ Cos(J(T))]∙ зіS(Wt)- A(T)∙ Sin(J(T))∙ Sin(Wt)= U(T)∙ Cos[ Wt+ J(T)],
Where U(t) and j(t) are the initial phase of the total signal, which is indicated by
U(T)= ;
J(T)= Arctg
When the total vibration u(t) is injected into the amplitude detector, the output of the remaining detector is formed again. The strength of the hardness p(U) is calculated according to the formula
P(U)= (5)
De sxa – noise dispersion x(t);
I0-bessel function of zero order (modified).
The strength of the intensity, as indicated by this formula, is called the formalized Rayleigh law or Rice’s law. Graphs of the function p(U) for several values of the signal-to-noise ratio E/sx are shown in Fig. 4.
Due to the absence of a cortical signal, then at E/sx=0, the expression (5) appears
P(U)=
So, the source of the resulting signal is divided in accordance with Rayleigh’s law.
Fig.4. Graphs of the law established by Rospodil Rayleigh
Since the amplitude of the core signal outweighs the mean square noise level, then E/sx»1, then at U≃E it is possible to speed up the asymptotic application of the Bessel function with a great argument, then
≃≃.
Having substituted this expression for (5), we can
P(U)= ,
Thus, the value of the resulting signal is described by the normal law of distribution with dispersion sx2 and mathematical calculations E. It is important to note that even at E/sx=3 the value of the resulting signal is normalized.
4. Experimental determination of the laws of the division of episodic processes
One of the methods for experimentally determining the function of the division of the phase process x(t) is the method based on the vicoristic of the additional phase function z(t) in the form
Where x is the value of the function x(t), which is covered by z(t).
As it follows from the substitution function z(t), its statistical parameters are determined by the parameters of the phase process x(t), so changes in the value of z(t) are determined at the moment of transition by the phase process x(t) of level x. Also, since x(t) is a periodic cyclical process with the function F(x), then the function z(t) will also describe a cyclical process with the same function.
Figure 5 shows the implementation of the phase processes x(t) and z(t), which illustrate the obviousness of the relationship
P[ Z(T)=1]= P[ X(T)< X]= F(X);
P[ Z(T)=0]= P[ X(T)≥ X]= 1- F(X).
Fig.5 Implementations of burst processes x(t), z(t), z1(t)
The mathematical calculation (statistical average) of the function z(t), which has two discrete values, is calculated according to the formula (div. Table 1)
M[ Z(T)]=1∙ P[ Z(T)=1]+0 ∙ P[ Z(T)=0]= F(X).
On the other hand, for a phased, annual process
In such a manner
Analyzing this data, it is possible to create a concept that a device for oscillating the function of a subdivision of an annual phasing process x(t) can place a peer discriminator in its warehouse to reject the cyclic process, which I describe There is a function z(t) corresponding to the expression (6) and the integrating device , for example, in the form of a low-pass filter.
The method of experimental determination of the thickness of the subdivision of the fall process x(t) is essentially similar to the above-mentioned method. Whose vikoryst has an additional fall function z1(t) in the form
The mathematical definition of the function z1(t), which has two discrete values (Fig. 5), is more advanced
M[ Z1 (T)]=1∙ P[ Z1 (T)=1]+0 ∙ P[ Z1 (T)=0]= P[ X< X(T)< X+∆ X].
The medical annuality of the episodic process, which is described by the function z1(t), can be written
In such a manner
Vidomo
P(X≤ X(T)< X+∆ X) ≃ P(X)∙∆ X.
Otje,
Thus, the device for varying the thickness of a subdivision of the annual fallout process x(t) has the same structure and storage as the device for varying the function of a subdivision.
The accuracy of the measurement F(x) and p(x) lies within the limits of the caution interval and the accuracy of the integration operation. It is completely obvious what is rejected in real minds Ratings According to the laws of the division, for about an hour the averaging (integration) is valid. Turn around until virazu (6) and fig. 5. respectfully
Z(T) Dt= ∆ T1 ,
Where ∆ t1 is the 1st hourly interval of the function x(t) lower than x, then the hourly interval if the function z(t)=l.
The validity of this formula is indicated by the geometric position of the integral integral (the area of the figure surrounded by the function z(t) and the segment (0,T) of the hour axis).
In this manner, you can write
Then the function of the division of the fall process x(t) is similar to the reference hour of the process implementation in the interval -< x(t) < х.
Similar to size, can be removed
De ∆ t1 - 1st hour interval of function x(t) at intervals (x, x+∆x).
In the practical implementation of the considered method of experimental determination of the laws of the division of the fall process, the fall signal x(t) is applied to the analysis between the changes in the fall values from xmin to xmax (Fig. 6). At these boundaries, the value of the process x(t) is mainly neutral (imovernis sense).
The values xmin and xmax are selected based on the required accuracy of the division laws. In which case the investigation is subject to truncated divisions so that
F(Xmin)+<<1.
The entire range (xmin, xmax) of the x(t) value is divided into N new intervals ∆x, so that
XMax— Xmin= N∙∆ X.
Rice. 6. Function of the division (a), intensity of gravity (b) and implementation (c) of the fall process x(t)
Intervals set the width of the differential corridors in which vibrating worlds. The assessment of credibility is calculated
Pi* ≃ P[ Xi-∆ X/2≤ X(T)< Xi-∆ X/2]
Re-implementation of x(t) at the boundaries of the differential corridor from the average values of x(t) at the boundaries equal to xi. The estimate of Pi* is determined as a result of the modification of the reference hour for the implementation of x(t) in the skin from the differential corridors, so that
Pi*=1/T Zi(t)dt= ,
I = 1, ..., N.
Vrahovoyuchi scho
Pi* ≃ P1 = P(X) Dx,
It is possible to estimate the thickness of the subsection of the skin from the differential corridors.
Pi* (X)= Pi*/∆ X.
Based on the results obtained, the values of pi*(x), xi, ∆x will be the steps of the curve p*(x), which is called a histogram of the thickness of the subsection (div. Fig. 7).
Fig.7. Histogram of thickness of subsection
The area under the skin fragment with histograms in the intervals ∆x is numerically larger than the area that the true curve occupies with the distribution p(x) at this interval.
The number N of differential corridors may be no more than 10...20. Further increase in their quantity does not lead to a more accurate law p(x), since with the increase in N the value of the interval ∆x changes, which reduces the need for an accurate measurement of ∆ti.
The results allow us to calculate estimates of the mathematical calculation and variance of the fall process x(t)
Mx* = Xi∙ Pi* ; Dx* = (Xi— Mx* )2∙ Pi* .
When calculated Mx* і Dx* These formulas ensure that the value of the implementation of the fall process x(t) is lost in the 1st differential corridor, to which the value is assigned (the middle of the differential corridor).
The considered method of identifying the laws of the subdivision of episodic processes forms the basis of the statistical analyzer in this laboratory robot.
DESCRIPTION OF THE LABORATORY INSTALLATION
The study of the laws of the distribution of surge signals is carried out using an additional laboratory setup, which includes a laboratory model, a statistical analyzer and an S1-72 oscillograph (Fig. 8).
Fig.8. Laboratory setup diagram
The laboratory model is used for the formation and transformation of waveform signals, ensuring their statistical analysis, generating a histogram of subdivision laws and graphical display of these laws on the indicator of a statistical analyzer. You should include the following functional units:
A. Signal generator block. Forms several different types of signals.
- Signal x1(t) = A∙sin - harmonic vibration with the fall-out cob phase, the law of which Rivnomirny in intervals 0
P(J)= 1/2
P, 0<
J<2
P.
The strength and intensity of the mittevs, the value of such a signal is older — Signal x2(t) — a saw-like periodic voltage with a constant amplitude and a peak-to-peak parameter q, the law of division
P(Q)= 1/
T0
; 0<
Q≤
T0
.
The strength of the intensity of the mittevs, the value of such a signal is indicated by the expression — Signal x3(t) is a fall signal from the normal law of division (Gauss’ law) of Mitt’s value, so that
Pa(X)=
, De mx, sx – mathematical calculation and dispersion of the fall signal x3(t). — Signal x4(t) is a spiking clipping signal, which is a sequence of direct-current pulses of constant amplitude A and spiking trivality, which occurs at spiking moments. Such a signal appears at the output of an ideal intermediary, if at its input there is a surge process with a normal distribution law. Characteristics of the transformation may appear De x – rhubarb exchange. Thus, the episodic process x4(t) takes two values (A and A) with properties
P=P=F3(x); P=P=1-F3(x); Where F3(x) is the integral law of the subdivision of the phase process x3(t). Doctors say that the intensity of the clipped signal is ancient
P4(x)= F3(x)∙D(x+ A)+ ∙D(x - A). Figure 9 shows the implementation of cutaneous episodic signals, which are formed by the iterator of the laboratory layout, and their strength. These signals, which are characterized by a powerful distribution, can be fed to the inputs of typical elements of radio engineering devices to transform and monitor the laws of the distribution of signals at their outputs. B. Linear signal mixer. Forms the sum of two waveform signals xi(t) and x1(t), which are fed to its inputs, connected until completion
Y(T)=
R∙
Xi(T)+ (1-
R)∙
X1
(T),
De R is a coefficient that is set with the potentiometer knob in the range 0...1. Vikorist is used to investigate the laws of division of the sum of two burst signals. Art. Sockets for connecting various terminals - functional converters. The laboratory installation kit includes 4 functional converters (Fig. 10). Rice. 9. Implementations of batch processes x1(t), x2(t), x3(t), x4(t) and their intensity Pidsilyuvach - intermediary (limited) with the characteristic of re-creation Where U1, U2 - lower and upper level of the interconnection; k is the coefficient that is similar to tg where the characteristics of the transformation are being improved. There is a non-linear, inertia-free transformation of input signals. Vuzkosmogovy filter (F1) about the resonant frequency f0=20 kHz. Vikorist is used to formulate university-related episodic processes according to the law of division, close to normal. Typical AM-Kolivan receiving path (high-pass filter F1 – linear detector D – low-pass filter F2). This is the formation of the final high-cosmetic phase signal with linear detection. The functional transformative windows are considered constructively as small interchangeable blocks. As another functional vikorist, the “ideal” booster is being created - an intermediary (electronic key), which is included in the warehouse of the signal generator block of the layout. VIN ensures the formation of a clipped signal, being a nonlinear, inertia-free converter of the input and output signal. Rice. 10. Functional transformations R. Uzgodzhuvalny pіdsiluvach. Ensures that the value of the signal and the amplitude range of the statistical analyzer is consistent with the range. This is achieved by using the “Powering” and “Displacement” potentiometers when installing the remitter P1 (Fig. 8) at the “Calibration” position. A convenient booster is also used as a functional converter (in addition to several of the above), ensuring linear, inertia-free conversion consistent with the formula
Y(T)=
A∙
X(T)=
B,
Dea is the strength coefficient, which is set using the “Power” knob; b - constant storage signal, which is set with the “Displacement” knob. The pointers on the diagram in Fig. 8 do not show the analyzer block near the layout warehouse in the robot. The laboratory installation transfers the stagnation of a digital statistical analyzer, connected to a nearby device. D. A digital statistical analyzer is used to measure and formulate laws based on the meaning of signals that are fed to its input. The analyzer operates in this way. The analyzer is turned on in the vibrating mode using the “Start” button. The clock time is 20 s. At the same time, the value of the input signal is taken from the input signal (at random moments), the total number of N of which is equal to 1 million. called differential corridors, or grouping of sample values at intervals) . The intervals are numbered from 0 to 31, their width is equal to 0.1 V, and the lower interval between the 0th interval is equal to 0, the upper interval between the 31st interval is equal to +3.2 V. Over time There are a number of items ni, what was consumed in the skin interval. The result is visible as histograms distributed on the monitor screen, where the horizontal is the entire scale grid and the entire signal value is no more than 0...+3.2 V, the vertical is the entire frequency range ni/N, i = 0.1...31. To read the results of calibration in a digital form, use a digital indicator, which displays the number of the selected interval and the corresponding frequency (rating of compatibility) ni/N. The selection of interval numbers for the digital indicator can be done by switching the interval. In this case, on the selection monitor screen, the interval is indicated by a marker. With the "Multiple" option you can select a manual scale for caution using the histograms of the vertical axis. When this value is selected, the range of input voltage of the analyzer (the range of analog-to-digital conversion) must be switched to a position of 0…+3.2 V. Before skin adjustments, it is necessary to press the “Reset” button Start" (when pressing the "Discount" button ") the memory device is reset, and the results of the previous modification are copied to the stack memory, from which they can be accessed using the "Storinka" switch. The halal procedure follows the law of the division of the reaction of the linear FU at a sufficiently high level of infusion. However, it is possible to carry out a correlation analysis, which can be carried out manually using the spectral method using the scheme shown in Fig. 5.5. To calculate the energy spectrum G Y(f) reaction of a linear FU with a transfer function H(jω) velocity of its values (4.1) Correlation function BY(t) is significant to the transformation of the Four energy spectrum G Y(f) Let's turn to the law of the subdivision of the reaction of the linear FU in the following phases: 1. Linear transformation of a normal joint venture gives rise to a normal process. You can change the parameters of your division. 2. The sum of normal SPs (reaction of the sumator) is also a normal process. 3. When the SP passes through a sufficiently dispersed filter (then with filter pass width D F The very small width of the energy spectrum of D f X) beware of normalization of the subdivision of the reaction Y(t). Look, the law of the subdivision of the reaction is approaching normal. The degree of proximity is greater than the greater inequality D F<< Df X(Fig. 5.6). It can be explained this way. As a result of the passage of the SP through a high-energy filter, there is a change in the width of its energy spectrum (from D f X to D F) and, obviously, an increase in the correlation time (c t X to t Y). As a result of the filter reaction between uncorrelated areas Y(k t Y) grows approximately D f X / D F non-correlated species in tide X(l t X), the skin of which gives inputs from the formation of a single reaction with the water, which is indicated by the type of impulse characteristic of the filter. In this manner, in non-correct cuts Y(k t Y) it is expected that there will be a large number of also uncorrelated variable quantities X(l t X) with interconnected mathematical calculations and dispersions, which is consistent with the central boundary theorem (A.M. Lyapunova) ensures that the distribution of their sums is close to normal and increases the number of additions. 5.3. Vuzkosmugovi fall processes JV X(t) with a remarkably narrow energy spectrum (D f X << f c) as well as musco-deterministic signals can be manually represented in a quasi-harmonic form (div. section 2.5) de oginayucha A(t), phase Y( t) that cob phase j( t) are episodic processes, and ω z is the frequency that is selected sufficiently (meaning the average frequency of the spectrum). For the purpose of this A(t) that phase Y( t) completely quickly become an analytical joint venture The main momentary functions of the analytical joint venture: 1. Mathematical calculation 2. Dispersion 3. Correlation function An analytical joint venture is called stationary because Let us look at the typical technical problem of passing a normal SP through a black filter (PF), amplitude (AT) and phase (PD) detectors (Fig. 5.7). The signal at the output of the PF becomes highly cosmic, which means that it A(t) that cob phase j( t) will be completely minimal functions of the hour equalized with , de - Average frequency of the transmission of the PF. Otherwise, the signal at the AT output will be proportional to the envelope of the input signal A(t), but at the exit of the PD – the th cob phase j( t). In this way, for the highest order, it is enough to calculate the division of the original A(t) that phase Y( t) (the division of the cob phase is subdivided into the division Y( t) Only mathematical insights). The end of the robot - This topic belongs in this section: Applications for students who study the discipline “Theory of Electrical Connection”. The material is based on the basic introductory program for the TEC course. If you need additional material on this topic, or you didn’t find what you were looking for, we recommend that you quickly search our database: If this material was interesting for you, you can save it to your social media page: Spectral analysis of episodic processes The power of energy spectra and episodic processes tracking of episodic processes reversing signals through inertia-free lancets Functional reversal of two phase processes passage of episodic processes through different FUs Criterion for an ideal sponsor Maximum likelihood criterion Criterion for minimum average risk Neyman-Person test on customized filters The power of convenient filters Phase-frequency characteristic of SF Direct-circuit video pulses Direct-path radio pulses Folding double signals Optimal coherent reception in low noise conditions optimal coherent reception the stability of the main types of digital modulation incoherent reception of a two-system coupling tracking of incoherent reception Let's look at a linear inertial system with a known transfer function or impulse reaction. Let the input of such a system be a stationary phased process with given characteristics: intensity, correlation function or energy spectrum. The most important parameters for the process at the system output are: , i . The simplest way is to find out the energy spectrum of the process at the output of the system. True, in addition to the implementation of the process at the input, it is deterministic. functions, and the Fourie device is assigned to them. Let it go - the implementation of trivalism has been truncated T due to the episodic process at the entrance, and This is the spectral thickness. The spectral width of the implementation at the output of the linear system is more modern The energy spectrum of the process at the output (3.3.3) will be determined by the virus tobto. corresponds to the energy spectrum of the input process, multiplied by the square of the amplitude-frequency characteristics of the system, and is not dependent on the phase-frequency characteristics. The correlation function of the process at the output of the linear system can be defined as a Fourier transformation of the energy spectrum: Also, when a phased stationary process is injected into a linear system, the output is also a stationary phased process with an energy spectrum and a correlation function, which are indicated by the expressions (3.4.3) and (3.4.4). The pressure on the process at the output of the system is more modern The strength of the power distribution and the numerical characteristics of the signal at the output of an inertia-free nonlinear lancet. Baskakov stor. 300 – 302 Passage of spiking signals through nonlinear inertia-free lancets.
Let us now look at the history of the fallout process through a nonlinear system. In the opposite case, the problem is even more complex, but it is significant to say goodbye if the nonlinear system is inertialess. In inertia-free nonlinear systems, the values of the output process at a given time are determined by the values of the input process at that very moment. For nonlinear, inertial-free transformations, the simplest tasks are assigned to the division function at the output, which is richly complex - the assigned correlation function or energy spectrum. As it was meant more, n - the peaceful function of the subdivision of the phased process essentially deals with the function of the division of n phased values, which are the values of the phased process at n different moments of the hour, the significance of the laws of the division is functional re-creation of temporary values - equal to the simple tasks. Let's take a look at the simplest butt of one-dimensional, low-grade size. Let it be - the strength of the intensity of the drop-in value, which lends itself to nonlinear transformation. The density of the dropwise value η is significant. It is acceptable that the function is such that the wrapping of its function is unambiguous. Since the drop in value ζ is found at small intervals we know the stars It is best to take the absolute value as the main value, since the intensity of the intensity may be negative. If the wrapped function is ambiguous, then. there is a small number of gels, then for the strength of the homogeneity with the vicoristic theories of folding properties can be removed It is significant that the importance of the numerical characteristics of nonlinear-reversible phase processes does not necessitate the importance of their strengths. In truth, the fallout for the cob moment is of the kth order Ale zgidno (3.4.13) It is easy to expand the lines (3.4.14) and (3.4.15) to fit any number of sizes. Let us bring here the residual result for the two-dimensional fallout. Since the variable values also indicate the strength of the properties, then for the variable values when the gate functions are unambiguous The strength of the properties is significant De value is called the Jacobian transformation of the relationships between elementary areas when moving from one coordinate system to another. If so, then jealousy is just de Meal №23 Discrete pulse sequence, its spectrum. Baskakov stor. 382-383 Sampling of periodic signals. Discrete transformation of Four'e (DFT). Update of the output DFT signal. Four's discrete transformation (ODPF). Baskakov stor. 388-392 Meal №24 The principle of digital signal processing based on discrete Fourier transformation. Baskakov stor. 400-405 Implementation of digital filtering algorithms (transversal digital filters, recursive digital filters, impulse response, output signal) Digital filters can be either recursive (RF) or non-recursive (NF). The advantages of non-recursive filters compared to recursive ones are reduced to the present: Non-recursive filters can produce exactly linear phase response; The intensity of loud noises NF, zazvichay, nabagato less, nizh in the Russian Federation; For NF it is simpler to calculate the coefficients. A few non-recursive filters, compared with recursive ones, are reduced to the present: Recursive filters allow signal processing to be carried out with greater accuracy, which allows them to more correctly implement the impulse response without removing the “tail”; The scheme of implementation of the Russian Federation is much simpler than that of the SF; Recursive filters allow you to implement algorithms that otherwise cannot be implemented using non-recursive filters. The impulse response of a recursive filter is unskewed, but non-recursive is finite. Baskakov st. 405-408, 409-411, 413 Meal №25 Understanding signal/noise ratio, filtration and optimal filter. Improved signal/noise- is a dimensionless quantity that relates the intensity of the core signal to the intensity of the noise. Filtration- this is the processing process signal frequency-selective devices using the method of changing the spectral composition of the signal Optimal linear filter We call it a frequency-selective system that controls the processing of signal and noise in the most precise manner. The output maximizes the signal-to-noise ratio. Baskakov stor. 423-424 The signal/noise ratio at the output of a customized filter. Baskakov st. 425, 431-432 Characteristics of the optimal (narrow) filter for signals of the visible form (AFC, PFC, IX). Signal to the output of the used filter.
what Rivnomirny in the interval, where T0 is the period of the signal, so the strength and intensity is equalThe theory of electronic coupling. Lecture notes – part 2
What we can do with the removed material:
Tweet
All topics in this section:
Spectral analysis of deterministic signals x(t) transmits the vicoristant of Four'e's direct transformation
1. , Which flows out of the middle from its meaning (4.1). About this fact and connection
To consolidate the knowledge gained from Section 4 in the virtual laboratory, you can carry out experimental investigations of the batch processes of vikoryst: · about
At the same time, it is necessary to carry out the passage of a given joint venture through a concrete
The inertia-free lancet (the inertia-free functional unit – BFU) is completely described by the functional location y = f(x), which relates to the mitigation values
Statement of the problem: Two types of fall processes X1(t) and X2(t) are given due to the known strength of their value at runoff
To consolidate the knowledge that is lost when completing this section, it is recommended that you sign up within the virtual laboratory for work No. 20 “Passage of episodic processes through various
(Kotelnikov’s criterion) This criterion emphasizes ensuring a minimum of average milk consumption. For twin system
Respectfully, all information that is being conveyed,
(Bayesian criterion) To identify different inheritances of transfer of different information, use the Kotelnikov criterion, minimizing the amount of mental capacity
The Neyman-Pearson criterion stagnates in two-dimensional systems in situations where it is impossible to determine the a priori certainty of certain information, and the inheritance of various types of information is not the same.
Keeping the formulation of the problem of synthesizing the demodulator from the front section and applying algorithms (6.13) and (6.14), we will try to replace the correlator (active filter), which calculates the scalar
1. Impulse characteristic of UV - “mirror reflection” signal, due to its benefits, up to the moment of hour 0.5t0 (accurate to the constant coefficient
is distinguished by the sign of the phase spectrum of the signal, for which it is necessary (b
The signal in the form of a direct-cut video pulse s(t) (Fig. 6.8, a) and the impulse characteristic gSF(t) of the filter associated with it (Fig. 6.8, b) are described in terms of
The signal in the form of a direct-cut radio pulse s(t) is described by the expression
Let's look at the signals as n-sequences of pulses of a rectangular shape
Let us consider the problem of synthesizing a narrow filter that will ensure maximum s/n ratio at its output for the output, if at its input there is an additive sum of the output signal s(
To consolidate the knowledge, withdrawing from sections 6.1-6.3, completely conclude laboratory work No. 15 “Advancement of coherent demodulators” (Fig. 6.19, 6.20) and No. 22 “Uzgodzhena f
To equalize the effectiveness of the main types of digital modulation AM, HF (with vicarious orthogonal signals) and FM, it is sufficient for the skin to determine the equivalent value
To determine the average speed of the optimal incoherent reception in a two-way system at equal speeds of information that are transmitted P(b0) = P(b
To consolidate the knowledge, from sections 6.6 and 6.7, complete laboratory work No. 16 “Investigation of incoherent demodulators” (Fig. 6.40, 6.41) and
(3.4.3)
(3.4.4)
(3.4.5)
, then, as a result, there is an unambiguous functional relationship between ζ and η and the variable value η is obligatory in the interval 
, yes, the credibility of these approaches may be the same, then. 
(3.4.13)
(3.4.14)
(3.4.15)
(3.4.16)
ta . Therefore, the remaining expression can be rewritten
(3.4.17)
(3.4.18)
