Signal passed. Passage of signals through non-linear lanterns Passage of signals through non-linear lanterns

Understanding the linear system and analysis line systems... Gagging Differential Equivalents... Stagnation of impulse indicators. Zasosuvannya frequency indicators. Nonlinear noninertial reimagination of vypadkovyh processes.

Finding and evaluating the signal parameters in the presence of noise.

6.1. Describe the signal that pereshkodi. Tipi rozv'yazuvanih zavdan. Revision of statistical hypotheses (vibrating, vibrating space, the function of credibility, hypotheses of simplicity and folding, solution, rules for the establishment of grants). Statistics, criterion of quality, value of solution, matrix of input, clearing risik, middle risik.

6.2. Revision of two-alternative hypotheses:

Bayus criterion, minimum criterion, criterion for maximum a posteriori imovity, maximum likelihood criterion, Neumann-Pirson criterion, Wald's last analysis, working characteristic.

6.3. Processing without interruptions. Functional likelihood. Likelihood assessment function.

6.4. Stasis in the function of determining the likelihood of a given signal (algorithm, number of pardons) and a signal with a drop-in phase (algorithm, number of pardons).

Estimation of signal parameters.

7.1. Tochkova estimation, interval estimation. The power of point assessments (power, independence, efficiency, sufficiency). Rao-Kramer's unfaithfulness. Assessment mathematical ochіkuvannya the dispersion of the normal growth rate

7.2. Zastosovuє functionality of likelihood for the estimation of parameters in the signal. Assessment of the team hour camp to the signal. Problem statement, noise and signal function and characteristics, signal / noise ratio, signal function assignment for a direct video signal, processing of a signal burst (signal / noise signal).

7.3. Realization of the algorithm for the estimation of the time station to the signal. Korelyats_yniy priymach, uzgodzheniy filter (impulse ta frequency characteristics signal to the inputs of a narrow filter, signal / noise output, optimal efficiency of a narrow filter, performance versus real and narrow filters).

Theory of information

Kotelnikov's theorem for complete processes.

Quantum signal.

world of information.

3.1. Access information by Fisher, by Hartley.

3.2. Informácie measure of Shannon (viznachennya, entropіya that її vlastivostі, entropіya creates ensemble entropіya bezperervnogo ensemble Entropіya signal of obmezhenoyu region viznachennya. Entropіya signal of neobmezhenoyu region viznachennya, ale obmezhenoyu potuzhnіstyu s) Quantity vzaєmnoї Informácie, okremі kіlkostі vzaєmnoї Informácie. Epsilon entropia (ε-entropia). Coefficacy of stiffness, performance of overworld.

4. Koduvannya dzherela nezalezhnykh povidomlen: code tree, code preference, equal code, code for Shannon's method, code for Huffman's method, Shannon's theorem about code for dzherel. The characteristics of a dzherel that of a dzherel coder.

5. Channel connection... Classification. Information transmission and bandwidth.

5.1. Noiseless channel: building capacity, Shannon's theorem for a noiseless channel.

5.2. Channel with noise: double symmetric channel (capacity), Shannon's theorem about throughput capacity for the channel due to noise.

The hodograph will be unnecessarily binding for the design of the stiffness. For a complete analysis of the frequency response and frequency response. Also, the third alternative to the Nyquist criterion is: if the frequency response is greater than one at frequencies beyond the phase response of the road 0 or de n € z, then the system s ringing not stiyka, innakche stiyka (Malunok 3.10).

Small. 3.9 Frequency response and phase response of a closed system with a ringing sound

4 Passage of video signals through line stationary lanterns

The main characteristics of the lipid process are mittєvih meaning signal, the correlation function and the spectral intensity of the pressure. The knowledge of the efficiency of the lanceyug is of the significance of the signal at the outputs of the line lancer for the appearance of the gap of the lancetness at the inputs of the lanceug and of the characteristics of the lanceuga, which is even more efficient. However, if the input signal is Gaussian, the outbound signal will be Gaussian. This means that the task of solving the problem is to say goodbye and to be brought up to the value of the parameters of the output signal (mathematical calculation and variance).

Zavdannya znakhozhennya correlation function and the spectral density of the pressure of the visual signal is much simpler.

The revolutionary transformation of Fur on the basis of the spectral strength of the exertion is based on the theory of Winer - Khinchin:

- Correlated function of the signal

Zvorotnі redevelopment of Fur'є from the performance of transmissions for the effort:

- Correlation function of impulse characteristics to the signal

Oscillations of additional spectra in two signals according to the spectrum of the short-circuited signals, it is possible to record:

That is, the correlation function of the signal at the inputs of the linear lantern is connected to the signal at the inputs of the lantern and the function of the correlative impulse characteristics of the lantern.

When analyzing other systems as a transient, there is often a loud noise, so there is a low spectral intensity of the exertion after the end of the entire frequency range:

that correlation function

Otzhe, the correlation function of the output signal to the autocorrelation function and the impulse characteristics of the function.

5 Passing signals through non-linear lanterns

Linear stationary cola does not change the spectral warehouse of the signal. The main radio-technical re-implementation, connected from a different spectral warehouse to a signal, can be used for additional help nonlinear lantsyugiv, abo lіnіynyh lantsyugіv іf change parameters.

Pre-development of nonlinear lantsyugiv є foldable zavdannya, like a polyagaє at the latest nonlinear differential rivnya. Analysis of non-lantsyugiv goodbye, when non-linear elementє bezinertsіynim, that is, the reaction to the change of input into the flow is mittєvo. Strictly seemingly, non-inertial elements (BNE) are dumb, even if the hour changes to the input signal, it means that the hour of the process in the non-linear element is changed, the element can be considered non-inertial. In radiotechnics and non-linear elements, the most common thing is that they are attached to the conductors (diode, transistor). For the description of such attachments, the VACs are vicious, such as tying straps between themselves, applied to the straps that flow through the attachments.

Injection. 6 pending transmission new signals through line lantsyugi z with the same parameters... The connection between the input and output signals in such lancers began behind an additional transmission function (spectral method) or for additional impulse characteristics (integral overlay method).

Similar performance can be used for linear lances due to the changing parameters. Obviously, in some lantsyugah the nature of fallowness and input and output signals in the process of transmission changes. In addition, apparently, the transfer function of the lanceyug is to lie down only every hour; The impulse characteristic can also be stored from two changes: from the interval between the moment of stopping a single impulse and the moment of guarding the output signal t (as for the lancer with constant parameters) and, in addition, on the basis of that. In addition, for a lancer with small parameters, an impulse characteristic should be recorded in a zagalny form

Even if there is a high signal s (t) at the inputs of a chotiripole with an impulse characteristic (Fig. 10.2), then, based on the principle of superposition, the output signal for an analogue with virase (6.11) can be attributed to an additional virase

(10.12)

Trying now to back up the transfer function for the lancet with the new parameters. For the whole function of the viglyadi Integral Fur'є:

(10.13)

de - spectral power signal s (t).

Todi viraz (10.13) cross at the offensive:

Small. 10.2. Parametric chotiripolus

Having designated the internal integral through the rewritable last viraz as follows:

(10.14)

З (10.14) vyplyaє, where the function, which should be started by viraz

The initial procedure for determining the law of the distribution of the reaction of the linear FU on a fairly large volume of fuel injection is mute. However, mozhliviy correlation analysis, That is, the development of the correlation function of the reaction for the given correlation function in the injection, which is carried out manually by the spectral method for the circuit, shown in Fig. 5.5.

For calculating the energy spectrum G Y(f) the reaction of the linear FU with the transfer function H(jω) the speed of the value to the values ​​(4.1)

Function of correlation B Y(t) is meaningful to the transformations of Fur'є energetic spectrum G Y(f)

Turning to the value of the law of the rise in the reaction of the linear FU in the same order of magnitude as:

1. Linear transformation of a normal SP is also a normal process. You can change the parameters of your system.

2. Sum of normal SP (sumator reaction) є normal process.

3. When passing through a joint venture with a sufficient rosette through a vuzkosmugovy filter (tobto with a width of swamp, passing a filter D F at least a smaller width of the energy spectrum in the flow D f X) to prevent the phenomenon of normalization of the reaction Y(t). It’s a good thing if the law of the reaction is getting closer to normal. The step of the closeness is the bigger, the stronger the insecurity D F<< Df X(Fig. 5.6).

The price can be explained in such a way. As a result of the passage of the joint venture through the high-density filter, the sutta changes in the width of the energy spectrum (z D f X to D F) і, apparently, an increase in the hour of the correlation (c t X until t Y). As a result of non-referenced indications of the filter reaction Y(k t Y) swash around D f X / D F non-localized views in the water X(l t X), leather additions in the form of a single view of the reaction from the car, which is indicated by the type of pulse characteristics of the filter.

In such a rank, among non-native pererez Y(k t Y) there is a sum of great quantities of also non-native large quantities X(l t X) with the encircled mathematical calculations and variances, which are similar to the central boundary theorem (A.M. Lyapunov), will prevent the rise of the sum to the normal for the increase in the number of additional ones.

5.3. Vuzkosmugovі vіpadkovі processes

Joint venture X(t) with a remarkably narrow energy spectrum (D f X << f c) as і vuzkosmugovі determined signals manually represent in quasіharmonic formі (div. razdіl 2.5)

de fiercely A(t), phase Y ( t) that pochatkova phase j ( t) є vypadkovy processes, and ω s - frequency, how to vibrate pretty much (start as the middle frequency of the spectrum).

For designation of fire A(t) that phase Y ( t) dozіlno speedy analytical joint venture

The main moment functions of the analytical joint venture:

1. Mathematical ochіkuvannya

2. Dispersion

3. Function of correlation

An analytical joint venture is called stationary,

An easy-to-understand technical connection is made to the problem of passing a normal SP through a swarthy filter (PF), amplitude (AT) and phase (PD) detectors (Fig. 5.7). The signal at the output of the PF becomes vuzkosmugovim, but it means that it is fired A(t) that pochatkova phase j ( t) will be functions for an hour, which will change by and large; As a result, the signal at the AT output will be proportional to the envelope of the input signal A(t), but in the FD output - the first cob phase j ( t). In such a rank, for the revival of the zavdannya it is enough to count the A(t) that phase Y ( t) (growth of the cob phase grows towards the growth of Y ( t) only for mathematical ochikuvannya).

Kinets of robots -

Qia theme should be distributed:

Theory of electrical connection. Lecture notes - 2nd part

Meanings for students, how to teach the discipline "Theory of electrical communication". Material for the course of the TEC course.

Do you need additional material on this topic, for you didn’t know those who were joking, it is recommended to hurry with a joke on our base of robots:

We will robitimo with the material:

As soon as the material has appeared cinnamon for you, you can save it to your side in social framing:

To all those tsiogo razdilu:

Spectral analysis of video processes
Spectral analysis of deterministic signals in x (t) transmission

The power of the energy spectrum in the process
1., without a forehand value (4.1). On the basis of the fact and the relationship

up to the last minute of the process
To consolidate the knowledge of the virtual laboratory, it is possible to carry out experimental pre-development of the vicious processes:

converting signals
At the top of the list is the passage of the given joint venture through the concretes

through bezinertsіynі lantsyugi
An inert lanceyug (an inertial functional university - BFU) is described more often by a functional fallowness y = f (x),

Functional transformation of two types of processes
Statement of the problem: Two types of processes X1 (t) and X2 (t) are assigned with a type of social efficiency and value for a zbig

passing of vypadkovyh processes through the development of FU
For the consolidation of the knowledge that is taken away during the vaccination, it is recommended to the Viconati within the framework of the virtual laboratory to the robot No. 20

The criterion of the ideal sposter
(Kotelnikov's criterion) Tsei criterion for ensuring that the minimum of average quality is guaranteed. For two systems

Maximum likelihood criterion
Vazhayuchi, what is all about, what is transmitted, rivnoymove,

Criterion of the minimum average rhizik
(Bajєsovskiy criterion) For urahuvannya іnіmіzuyuchi the sum of clever possibilities

Neumann-Pearson criterion
The criterion of Neumann-Pirson stagnates in two systems in situations, if it is unfortunate that there are a lot of apiorny and motivational reasons, and the inheritance of pardons of a kind of neo

on uzgodzhenikh filters
Saving the formulation of the problem of the synthesis of the demodulator from the front section and spiraling on the algorithms (6.13) and (6.14), we will try to replace the correlator (active filter) by calculating the scalar

The power of the uzgodzhenikh filters
1. The impulse characteristic of the UV "mirror image" signal, with a clear signal, at the moment of an hour 0.5t0 (in accuracy up to the constant

Phase-frequency characteristic of SF
be recognized by the sign of the phase spectrum of the signal, with the sign of the narrowing (b

Direct video impulses
The signal at the rectangular viewer to the video pulse s (t) (Fig. 6.8, a) and the pulse characteristic gSF (t) of the narrowed filter with it (Fig. 6.8, b) is described by the video

Direct radio impulses
The signal at the direct-current viewer to the s (t) pulse is described by the viraz

Folding two signals
Clearly visible signals at the n-endpoints of the impulses of the rectangular form

Optimal coherent reception with low noise
The task of synthesizing a narrow filter is understandable, so that the maximum output of the s / w to its input for the output is not guaranteed, if the input to the input is additive, the sum of the signal s (

optimal coherent reception
To consolidate the knowledge, take out the number of razdіlіv 6.1-6.3, subtotally viconati laboratory robots No. 15 "Pre-development of coherent demodulators" (Fig. 6.19, 6.20) and No. 22 "Uzgodzhena f

production of basic types of digital modulation
To adjust the performance of the main types of digital modulation AM, ES (with vicious orthogonal signals) and FM, it is sufficient for the skin of them to have an equivalent value.

incoherent reception in two communication systems
For the mean value of the average rate of change, the optimal incoherent acceptance in two systems is at the same rate of change, which is transmitted P (b0) = P (b

to the extent of incoherent acceptance
For consolidation of knowledge, from the distribution of 6.6 and 6.7, subtotal testing of laboratory robots No. 16 "Pre-development of incoherent demodulators" (Fig. 6.40, 6.41) and

Signal transmission in real channels depends on the supervised change (reimplementation) of the signals, in the result of received signals from the transmitted ones. The evidence for everything, the linear and non-linear reconfiguration of the input signals, as well as the obvious additive noise near the channel, which is most often not far from the signals that are transmitted. From the point of view of the transmission of information over the channel, it is important to rewrite the signal to the reverse and non-gate. As it will be shown (div. § 4.2), the werewolves are not reincarnated because of information. In case of non-turn-back revisions, the information is inevitable. For werewolves, the re-transformation of the signal is often vikoristoyuyut the term "self-creation", and the non-reversal of the re-transformation is called reversal codes (additive and non-additive).

The application of the simplest deterministic reverse transformation of the input signal X (t), as it does not change its form, serve

Y (t) = kX (t-τ). (3.1)

In this case, the output signal to the Y (t) channel is seen from the input signal on a scale of k, which is easy to compensate for when the signal is weakened or the signal is weakened and continuously received in the hour τ. Vona is most often small. In fact, for a call on a scale to space, and beyond the even great number of reactive elements of the line, the call can be visible *.

* (Here it is about capturing the connection itself, and not about capturing the demodulator and decoder, which can be significant and sometimes change the ability to adjust the performance.)

If the input signal X (t) in (3.1) is vuzkosmugovy, it is manually detected in the quasi-harmonic form (2.68): X (t) = A (t) cos × X [ω 0 t + Φ (t)], de A (t ) і Φ (t) - the function changes gradually. Therefore, when reaching a small notice, it is possible in the first approach to the value A (t-τ) ≈ A (t) і Φ (t-τ) ≈Φ (t), and the output signal (3.1) can be recorded with the next rank:

Y (t) = kA (t-τ) cos [ω 0 (t-τ) + Φ (t-τ) ≈ kА (t) cos [ω 0 t + Φ (t) -θ К], (3.2)

de θ К = 0 τ is the phase phase control at the channel. In such a rank, with a vuzkosmugovy signal, the coverage is low up to a dozen zsuvu phase.

In real channels, the sound, if it is possible to navigate through the additive noise, transforms the signals into a collapsible character and allows to produce up to the visibility of the form of the outgoing signal from the input.

The pre-development of the process of change when they pass through the dynamical systems (both with regular and with parameters, but also with varying degrees of variation) are linked to the latest versions of two types:

the assignment of a correlation function (energetic to the spectrum) to the input Y (t) on the inputs of the dynamical system, given by its characteristics of the given correlation function (or to the energy spectrum) of the input to the input X (t);

The value of the high-frequency output to the input Y (t) on the outputs of the given dynamic system and the high output input to the input X (t).

A friend from the meanings is a big house. From її development, obviously, the solution and the first task can be resolved. However, in the main, it is intertwined with a short look at the first plant, and it is impossible to see another, folded plant.

Passage of vipadkovy signals through the determination of the line cola. Yak vіdomo, lіnіyny lantsyug, from the post-life parameters, is characterized by its impulse reaction g (t) or to the reimagined Fur'є-transfer function k (iω). For example, at the input of the lantsyug, centering processes X (t), the process Y (t) at the outputs are initiated by the Duhamel integral *

In lantsyugu, it is physically realizable, at t

* (Here, it is necessary to integrate the processes of falling into the mean square value [div. f-lu (2.8)].)

We know the function of the correlation of the centered input process Y (t):

de θ 1 = t 1 - 1 θ 2 = t 2 - 2; B X (θ 1 -θ 2) - function of correlation of the input signal.

Let the input process be stationary. Todі B X (θ 1 - θ 2) = B (θ), de θ = θ 2 -θ 1. We also introduce the value t 2 -t 1 = τ, t 1 - θ 1 = τ 1. Todi t 2 -θ 2 = + 1 -θ i

de vikoristan "team clock function of correlation"

At times β = τ - θ.

From (3.4) it can be seen that during a stationary input process, the input process is stationary, so as B Y (t 1 t + τ) does not lie in t 1. This can be written down

Otrimana parity є analogue of Duhamel integral for correlation functions. In such a rank, the FC of the input process є the integral star of the FC of the input process of the VFK impulse reaction of the lantsyug.

Remarkably, the VFK of the impulse reaction is tied to the reconfigurations of the Fur'є with the square of the module of the transfer function | k (iω) | 2 abo the amplitude-frequency characteristic (AFC) of the lantsyug. Fair,

From the theory of the revision of Fur'є vidomo, which of the revision of Fur'є from the side of the two functions is to add to the revision of the Fur'є from these functions. Stagnant (3.5), it is easily recognized by the spectral parameters of stationary processes at the inputs and outputs of the line lance with the stationary transmission function k (iω):

G Y (J) = G X (f) | k (i2πf) | 2 (3.7)

From (3.5) і (3.7), the spectrum of the process at the input of the lantern is increased by the spectrum of the process at the input of the frequency response of the lancet, so that there is no reason for the difference in the output characteristics of the input process, but not the frequency of the phase.

A clear butt of passing through vypadkovy processes through the determination of the line system - passing through the white noise with the energy spectrum N 0 through the last colival contour with the parameters R, L, С.

Resonance frequency

In spheres of malikh rozladiv | k (ω) | 2 = ω 2 0 / (4 [β 2 + (ω-ω 0) 2]), β = R / (2L), і readily (3.7)

G Y (ω) = N 0 ω 2 0 / (4 [β 2 + (ω - ω 0) 2]).

Correlation function at the entrance

When the signal X (t) is applied to the deterministic line, the output signal Y (t) is colossal with the change parameters. as vidomo, it is possible to see it with the integral of the back:

de g (t, τ) is the function of two changes, which starts the reaction of the system at time t to a δ-pulse, feeds to the input at time t-τ.

representing the transfer function of the linear lancer with the change parameters, yak, naturally, є the function of the frequency, and the first hour.

Oskilki in lantsyugu, but it is physically realizable, if you can’t win it earlier, then g (t, τ) = 0 at τ

The head of the knowledge of the distribution of the parameters of the line system with a high level of infusion appears at the drop, even more foldable, to navigate, as it seems, to the knowledge of the one-dimensional distribution. Significantly, however, if a Gaussian process is applied to the input of a linear deterministic system, then the process at the inputs appears to be Gaussian, but it is driven by the powers of a normal generation, but it would be unnecessary to be normal. If the process at the entrance is not Gaussian, then with the passing of the linear system of the system, the change in the number of changes is made in some way.

Significantly, I will give power to the power of linear systems. When the smog of frequencies F is occupied by the input signal X (t), the transmission of this line system is wider than the smog, then the outgoing process tends to approach normal. The price can be roughly explained, s (3.8). Vuzka smuga transmission means that the triviality of the impulse reaction g (t, τ) as the function τ is large in proportion to the interval of correlation input to the process X (t). That overriding the outgoing process Y (t) at any moment t becomes an integral (3.8), in the integral function of how to complete the great wagon to enter a lot of unrelated ones intermittently in the process X (t). The distribution of the parameters of such an integral is close to the central boundary theorem, but also close to the normal value, the closest is the difference in the width of the spectrum of the input signal before the swamp of the passage of the lance. At the borderline, if the input of the lancer is injected with a lot of noise, in which the width of the spectrum is not limited, and the lancer is not allowed to pass through the smog, then the outflow process will be gausic.

Passage of vypadkovyh vuzkosmugovyh signals through the lines of swarthy lantsyugi. It was meant in § 2.4 that the process of high-frequency processes (that is, for those, the width of the spectrum is much higher than the middle frequency) can be manually represented in the quasi-harmonic form (2.68). How about the middle frequency? That is, the spectrum occupies the frequency range of the lower ones, below the spectrum of the signal itself. This is also the case in bagatio vipadkah, at the stages of synthesis and analysis of signal transmission systems (occasionally), even corisne. So, for the submission (2.72) on the interval T of Kotelnikov's order, 2T (f 0 + F) is known, for the submission on the same interval T of two independent low-frequency speech functions AC (t) and AS (t) (or one complex t)), reach 4FT views, which is approximately f 0 / 2F times less.

Zauvazhimo takozh, scho at neobhіdnostі modelyuvati vuzkosmugovі signal that system zv'yazku of such signals on obchislyuvalnіy mashinі abo at neobhіdnostі realіzatsії rіznih peretvoren such signalіv on osnovі suchasnoї mіkroelektronnoї bazi, vinikayut trudnoschі, naychastіshe practical neperebornі through obmezhenu shvidkodіyu Tsikh machines abo vіdpovіdnі mіkroskhemi. ... Naturally, it is simpler in terms of operating with low-frequency signal equivalents, such as warehouses.

Viraz for the low-frequency equivalent x (t) of the vuzkosmugovy signal (2.72), which starts from (2.70, a):

A X (t) = X (t) exp [-i 0 t]

we read (2.32) the spectrum according to Fur'є

S? X (iω) = Sx.

Figure 3.1 The instrument of spectral performance for the speech vuzkosmugovogo signal X * (t) (Fig. 3.1, a), which is analogous to the signal X (t) (Fig. 3.1.6) and the th low-frequency equivalent A? X (t) (Fig. 3.1, c)).

* (It is clueless that the spectrum SX (iω) of the speech signal X (t) is symmetric to the cob of coordinates, S * X (-iω) = SX (iω) (that is, the amplitude spectrum is a paired function of the frequency, and the phase is unpaired, or the speech part SX (iω) is a paired function of frequency, and explicit is unpaired).)

The main part of real, uninterrupted channels, the call is carried to the line and the dark, so the signals on these outputs can be seen as a reaction to the high-dark signal X (t) of the dark filter with a transmission (figurative) module. 3.1, a. Overshoot of the signal supply for the additional low-frequency equivalent (complex low-frequency) is caused by the fact that the dark-dark signal can be interpreted as low-frequency complex low-frequency complex signals.

The visibility of the vuzkosmugovy signal X (t) through the vuzkosmugovy canal (darker filter) with the same parameters and the transfer function k (iω) (Fig. 3.2, a).

Vuzkosmugovy input signal (2.72)

In front of the front wine, it is not important to show, but the spectrum of the tied complex fire A * X (t) = AC (t) - iA S (t) road S * X (-iω), de (iω) - spectrum according to Fur'є from AX (t). Oscillations of multiple functions per hour on e ±?

Similarly, if the middle frequency of the input signal 0 is set to the center frequency of the filter, you can use the transmitting function of the dark filter (reimplementing the Fur'є pulse response of the filter g (t) *

de Γ -Four spectrum of a complex (analytical) signal? (t) = g (t) + ig? (t) =? (t) e it? 0 approved by g (t). The value Γ (iω) is the spectral characteristic of the complex fire γ̇ (t) impulse reaction of the filter g (t), which is a low-frequency equivalent of the vuzkosmugov canal.

* (Significantly, the function Γ і Γ * [- i (ω + ω 0)], being symmetric behind the module, the ordinate axes for the dark filter do not overlap, so it’s practical to lie in the area of ​​positive frequencies, and the other negative. The analogous firmness is valid for the functions S and S * [-i (ω + ω 0)] in the narrow-cropped signal.)

Now we know the spectrum of the signal Fur'є on the output channel y (t). On one side, the splintered signal of the voskosmugovaya from the middle frequency to the spectrum ω 0 can be similarly recorded (3.11)

de S? y is the spectrum of FUR'є of the complex (analytical) signal? (t) = y (t) +? (t) =? ye itω 0, for any S? ... On the other side, for a linear system with fixed parameters, the spectral characteristics of signals at the inputs and outputs are related to social relations

Sy (iω) - Sx (iω) k (iω). (3.14)

Provided in (3.14) for the delivery of (3.11) and (3.12) and vrahoyuchi wine on the side. 78

S (3.13) and (3.15)

As a legacy of a complex signal at the input of a vuzkosmugovy channel A y (t), a signal at the output of a complex signal of an input signal A x (t) and a complex flame reaction of a filter γ̇ (t)

Since I don’t use the filter, so Γ (iω) = γe -it 0 ω or ġ (t) = γδ (t-t 0), then, I’ll filter the power of the b-function, s (3.17) is rejected mo

Written complex obginal through inphase and quadrature components:

? X (t) = A X, C (t) + iA X, S (t);

γ̇ (t) = γ C (t) + iγ S (t);

? y (t) = A Y, C (t) + iA Y, S (t), (3.18)

Todi s (3.17)

In a private room for sports (3.19), you can fill in the following way:

Otzhe, smugova filtration with the transfer function k (iω) vuzkosmugovogo

to the process x (t) is equivalent to the low-frequency filter with the transfer function Γ (iω) of the complex low-frequency process x (t) (div. Fig. 3.2).

The processes A X, Z і A X, S can be trimmed out from x (t) at the attachment, the functional diagram of which is shown in Fig. 3.3, a. Anyway, multiplying x (t) by 2cos 0 t we can

[AX, С (t) cos ω 0 t + AX, S (t) sin ω 0 t] 2 cos ω 0 t = AX, C (t) + AX, C (t) cos 2 ω 0 t + AX, S (t) sin 2ω 0 t, (3.21)

and the low-pass filter skip only the first low-frequency sieve of the two members є high-frequency and will be a filter. Similarly, in the other hallway, there is a quadratic warehouse A X, S (t).

It is now understandable, as it is possible to realize a complex low-frequency filter (3.19) or (3.20) three additional help of real low-frequency filters (such a filter has an output to the speech signal, but also the transmission function of the rear storage side). It’s going to work right away from (3.19) or (3.20) two-channel filtering of speech low-frequency in-phase and quadrature components (Fig. 3.3.6).

Passage of vipadkovy signals through non-lantsyugi. Interconnected by the look of deprivation of non-inertial non-linear systems with regular parameters, some inputs and outputs are associated with a non-linear fallowness, which is called a characteristic of the system:

y (t) = φ, (3.22)

Spivvidnoshennyam (3.22) to achieve exactly what the robot can be characterized by a number of lanes of real channels in the connection, for example, to enter the warehouse of demodulators, interconnecting, modulators only. The transformation x (t) → y (t), as a rule, is unambiguous, but it is not possible to say about the reversion y (t) → x (t) (for example, a quadratic lance with the characteristic y = kx 2). Due to the lack of a superposition to non-linear systems, the view of a foldable injection (for example, a sum of a detergent and drop-shaped dodank) cannot be brought to a view of the skin of a warehouse environment.

In case of non-linear reconfigurations, the transformation of the spectrum of the input into the inflow is transformed. So, even at the input of a nonlinear system, the sum of a regular signal and additive noise X (t) = u (t) + N (t) at a university smoothie of frequencies F c is injected, so that it should be grouped close to the middle frequency and f 0 then in the outward warehouses of combined frequencies of three types, which are grouped close to the frequencies nf 0 (n = 0, 1, ...), the products of the warehouse input signal between itself (s × s), the products of the warehouse input noise (w × w); beating signal and noise (s × w). Growth at the gateway of the system, and you don't care.

As well as the characteristic y = φ (x) of a non-linear system and two-dimensional function of the input flow rate w (x 1, x 2, t 1, t 2), the main statistical characteristics of the input process, in principle, can always be significant. So, mathematically speaking,

and its correlation function

To the rotten re-incarnations of Fur'є one can know (3.24) and the energy spectrum.

Vikoristovuchi rules of knowledge of laws of distribution for functions of different values ​​(of different processes), it is possible, in principle, that it is possible, in principle, that it is possible to generate an input process, in any order, as in the case of an input process. However, the value of the imotional characteristics of the non-linear systems (lantsyugiv) to navigate to the stationary inputs to the inflow appears to be even bulky and foldable, unimportant to those who, for the purpose of making a cost-effective reduction of the special design. In bagatories, especially for high-school signals, it is significant to say goodbye to the victorious quasi-harmonious submission to the process.

Yak butt can be clearly seen through the quadratic detector sumi harmonic signal s (t) = U 0 cos ω 0 t і stationary quasi-blue vuzkosmugov noise n (t) = X cn (t) × X cos ω 0 t + X sn sin ω 0 t, de X cn (t), X sn (t) are not correlated quadrature Gaussian components of noise, for which m X cn = m X sn = 0, X cn (τ) = X sn (τ) = B (τ), and The energy spectrum is equal to that of the darkening of frequencies F n