Spectral density. The power of the spectral tightness of the strain Spectral pressure of the signal

Mayuchi on uvazi by a vypadkovy process without lich (ensemble) functions for an hour check the form, show the versatile spectral characteristics. The averaging of the complex spectral power, introduced in § 2.6 or 2.1, for all functions, to bring the process to a zero spectrum (at) through the density and indifference of the phases of spectral warehouses in new implementations.

It is possible, however, to introduce an understanding of the spectral power of the middle square of the vipadkovo function, although the value of the middle square does not lie in the relative phase of the harmonies, so it’s going to be lost. As far as the vipadkovy function is based on the respect of the electric power or the strum, then the middle square of the whole function can be seen as the middle pull, which can be seen in the support of 1 Ohm. The toughness of rozpodіlen for frequencies in the process smoothies, as to lay down in the mechanism of setting up the decay process. The spectral intensity of the middle pulling є the middle pulling, which falls by 1 Hz at a given frequency. Functional range,

The spectral density of the lipid process can be known as the mechanism of the establishment of the lipid process. There will be noises tied to the atomistic structure of the material and the electricity, the design will be looked at in § 7.3. Here we are intertwined with decilkom and character.

Having seen from the ensemble, be it a realization and encircling the triviality of the Kintsev interval T, it is possible to stagnate until it is not normal for Fur'є re-creation and in terms of spectral competence (s). The current energy of the developed implementation can be calculated according to the additional formula (2.66):

Raised energy to the middle the urgency of the k-th implementation on the basis of T

With an increase in T energy of growth, a protest against a pragmatic boundary. Zdіysnivshi border crossing otrimaєmo

є the spectral power of the average effort of the analyzed implementation.

In the case of a drop, the value of ma buty is averaged over the absence of realizations. Interconnected at the sight of a stationary and annual process, take into account, which is known to averaged over the implementation of the function, characterizing the whole process of burning.

The index k is lowered, the residual viraz can be dispensed for an average tension in the process

If you can see a kind of process with non-zero mean values, then the spectral power of the tax in the form

The value that characterizes the power dissipation over the spectrum of the signal is called the energy spectral power, which is less for the signals, for some of the energy for an infinitely long interval for an hour of the end of the signal, it will be stagnant to them.

For non-extinguishing signals, the energy is infinitely large and the integral (1.54) diverges. Zavdannya spectrum of amplitudes is uncomfortable. However, the average strain of the Psr, how to start

vyavlyayetsya kintsevoyu. This will be stuck in a broader sense of the "spectral power of the willingness". Visually, I will lose the average effort to the signal for the frequency that is meaningfully Ck (u):

The index k is used to identify the spectral intensity of the exertion as a characteristic of the deterministic function u (t), which describes the implementation of the signal.

The characteristic of the signal is less significant, the spectral amplitude of the amplitudes is lower, the fragments are replaced by the phase information [div. (1.38)]. It is unfortunate for that to unambiguously update on the new signal implementation. Protect the visibility of the phase information, allowing the understanding of the understanding to be heard before signals, as the phase is not assigned.

To establish a connection between the spectral gap Ck (u) and the spectrum of the amplitudes of the velocity with the signal u (t), wisnu at the adjacent interval hour (-T<. t

de - the spectral density of the pressure to the signal, enclosed in hours.

Nadal will be shown (div. § 1.11), well, the average characteristic for the impossibility of implementation, it is possible to deduct the spectral density of the strain for the great class of vypadkovyh processes.

Function of auto-correlation of the determined signal

Now the frequency area has two characteristics: the spectral characteristic and the spectral intensity of the tightness. The spectral characteristic, to reveal more information about the signal u (t), is shown as a re-implementation of Fur in the view of the clock function. Z'yasuєmo, which appears in the clock area of ​​the spectral density of the tightness, is replaced by the phase information.

As a result, one of the same type of spectral strength of the effort appears to be without clock functions, which are transformed into phases. Radyansky students L.Ya. Hinchin and the American clerk N. Winer practically immediately found out that Fur’s revival of the spectral power of exertion was found out:

I use the clock function r (), but do not take revenge on the phase information, called the clock auto-correlation function. It will show the steps of the link the value of the function u (t), distributed at intervals of an hour, and can be taken out of the statistical theory by way of development of the understanding of the correlation. Significantly, in the team-hour function, averaging is carried out for an hour at the boundaries of the realization to achieve the great triviality.

It is true that other integral relations for the betting of Fur'є are reconfigured:

Appendix 1.6 Visibility of the time function of the auto-correlation of the harmonic signal u (t) = u0 cos (t-c). Up to (1.64)

Having spent awkward re-making

residual maєmo

Yak і next ochіkuvati, ru () does not lie in і c і, from the same, (1.66) is true for the powerless harmonics, which occur in phases.

International Health Corporation

Faculty of Applied Sciences

abstract

on the topic of"The spectrum of the tension of this connection and the function of the correlation"

Discipline"Theory of electrical communication »

Viconala: student group

FPN-PEIT (s) -4C *

Dzhumageldin D

Reversed: Glukhova N.V.

Almaty, 2015

I Entry

II Main part

1. Spectral intensity of discomfort

1.1 Vypadkovі values

1.2 The strength of the quality of the function of the size

2. Vypadkovy process

3. The method of measuring the spectral strength of the strain for the correlation function

ІІІ Visnovok

IV List of Victory Literature

Entry

The theory of the definitions of the magnitude of this characteristic in the "statistic". I will describe in advance that vyvchennya vypadkovykh signals "in dynamics", as vіvchennya vypadkovykh appearances, which develops in an hour, for be-what kind of vipadkovyh signals, circulating the theory of vypadkovyh processes.

In the capacity of the universal coordinate for the rise in the size of the independent winter, we will be victorious, as a rule, I will change "t" and interpret it, purely for sake of convenience, as the time coordinate. The rise in values ​​for the hour, as well as the signals that are displayed in any mathematical form, are called vypadkovy processes. In the technical literature of the term, a “vivid signal” and a “vivid process” become synonymous.

In the process of processing and the analysis of physical and technical data, it is necessary to inform the mother about three types of signals, which are described by the methods of statistics. First, there are information signals that represent physical processes that are related to their nature, such as, for example, the restoration of particles of ionizing radiation In other words, informational signals, stale parameters of physical processes and objects, meaning those that are not known in the past, as they are waiting for the value of the given informational signals. First of all, there are noises and noises, how chaotically fluctuating in the hour, when supervising information signals, but, as a rule, statistically independent ones like for their values, and changes in the hour.

Spectral intensity of discomfort

The spectral density of the exertion allows one to judge the frequency of the power of the decay process. Vona characterizes the intensity of the lower frequencies of chi, іnkshe, middling, but a fit to one of the smog frequencies.

The picture of a rise in average exertion for frequencies is called the spectrum of exertion. The prilad, who is aware of the spectrum of exertion, is called the analyzer of the spectrum. The knowledge of the result in the spectrum is called the hardware spectrum.

The spectrum analyzer robot is runting on the offensive methods of vimiryuvan:

· Methods of filtering;

· Methods of re-implementation of the Winer-Hinchen theorem;

· Methods of Fur'є-transformation;

· Method of vicarious functions of sign functions;

· Method of device storage of orthogonal functions.

The peculiarity of the spectrum of the toughness of the field in terms of the triviality of the experiment. Frequently I am changing the triviality of realización, for an hour, in a matter of hours, it takes an hour to achieve the staunchness of the process. Evaluations of the spectrum of effort, obtained by the implementation of the stationary year-old process, which is usually accepted. It is often reported that the number of times is clear, and it is necessary to average the implementation of each hour, and in the ensemble. In many cases, in the implementation of pre-lingering processes, it is forgotten in front of it, which allows the experiment to be repeated in a bagatorazny way from analysis to analysis, to the development of new algorithms for processing and apparatus.

At the time of recording the implementation of the hardware abduction, which is related to the process of hardware, there may be a change to a value that has been zoomed in by the endless triviality of implementation and non-stationary behavior.

The memory of the implementation of the implementation allows you to speed up the hardware analysis and automation.

Vypadkovy values

Vipadkov value is described by imovirnіsnyh laws. Imovirn_st of the fact that there is no interruption in the value X at vimiri have a drink in the interval x 1<х <х 2 , Virazom:

, de p (x)- The strength of the quality, moreover. For a discrete vypadkovo value x i P (x = x i) = Pi, de P i- ymovіrnіst, wіdpovіdaє i-u equal to the value X.

Estimation of the spectral density of tightness is apparently a problem for vypadkovyh processes. Buttons of video processes can be noise, as well as signals that carry information. Name it is necessary to know the statistic rating. Signal analysis in the lecture course "Digital signal processing". Cob houses

For signals with visible statistical characteristics, the spectral storage can be used as a signal interval. If the statistical characteristics are unacceptable, the signal versus the signal can be deduced from the evaluation of its spectrum. Rіznі methods vikoristovuyut ізні stews, and to that give rise to estimates.

When vibrating an assessment, the signal goes into the signal analysis as a process. It is necessary to vibrate an unbiased estimate, since there is a small dispersion, so that the spectrum of the signal can be averaged. The substitutions are called the difference between the average values ​​of the estimate and the reference values ​​of the value. An unrepentant assessment is called an assessment of zero usunions. Estimation of small dispersion of good localization of shukan values, tobto. The scope of immobility is concentrated close to the middle value. Bazhano mother I will help the assessment, tobto. assessment, as for the increase in size, the vibration is equal to the reference value (replacement and variance should be removed from zero). Parametric evaluations are developed, so that the information about the signal and non-parametric information is lost, but the statistical model for the drop signal, and the parameters of the whole model are changed.

When evaluating vypadkovyh processes broadening the definition of correlation functions.

For a year-to-year process, the value of statistical parameters in the process is averaged over one implementation.

For stationary in the process The correlation function R x (t) is stored for the interval of the hour that it starts. Tsya value characterizes the links between the values ​​of x (t), separated by the interval t. The more R (t) changes, the more progress is made, by stretching which to prevent the statistical connection between the values ​​of the fall process.

de - Mathematical ochіkuvannya x (t).

The relationship between the correlation function R (t) and the spectral force W (w) for the process is based on the Winer-Khinchin theorem

For discrete processes, the Winer-Khinchin theorem establishes links between the spectrum of a discrete process W (w) and its correlation function R x (n)

W (w) = R x (n) exp (-j w n T)

To assess the energy of the signal at the clock and frequency regions, the parseval's parity should be examined

One of the broader ways of rejecting the assessment of spectral distribution is by using the periodogram method.

Periodogram In this method, a discrete transformation of the Fur'є is carried out for the signal x (n), given at discrete points of the vibration by the number of N signals, and this is statistically averaged. In fact, the calculation of the spectrum X (k), it will be less than the number of frequency points N. The spectral density of the pulling is calculated, when the fit is on one vidlik vibirka:

P xx (X k) = | X (k) | 2 / N, X (k) =, k = 0,1, ..., N-1.

To reject statistically stable assessments, the obvious data are broken into vibrations, which are crooked, with a wider averaged spectrum, removed by skin vibrations. Set the number of hits per vibrator N and the ear of an onset of cutaneous vibrations to the ear in front of the front N t. The less the number of results in the vibration, the less the number of vibrations and the less variance in the estimates. Alle the oscilka of the frequency distribution of the vibration N is tied to the frequency distribution of the building (2.4), then the change in the distribution of the vibration to the reduction of the frequency distribution of the building.

In such a rank, the signal is peeped through the window, and the dan, which one does not use at the window, will be taken to be equal to zero. Kintsev signal x (n), which can be stored in N signals, you want to represent the result of multiplying the signal that is not terminated for an hour (n) on a straight line with a donut w R (n):

x (n) = (n)∙ w R (n),

and the uninterrupted spectrum of X N (f) signals, where x (n) is predicted to appear as a star of Fur'є-images X (f), W R (f) of an uninterrupted signal after an hour (n)∙ і вікна w R (n)

X N (f) = X (f) * W R (f) =

The spectrum of the intermittent rectangular window (rect) is the shape of the integral sinus sinc (x) = sin (x) / x. Take revenge on the head "pelust" and on the beetles, the largest ones by about 13 dB lower than the main peak (div. Fig. 15).

The fur'є-image (spectrum) of discrete after-effects, obsessed with N-point discretization of the non-intermittent rectangular window, is shown in Fig. 32. You can calculate the sums for the replacement of integral sinuses (2.9), as a result you can enter the Dirichle kernel

Small. 32. Spectrum of a discrete rectangular window

At that hour, there is a signal with an uninterrupted concentration of pressure exactly in a discrete frequency f k a direct vibration to a signal of low pressure, a spectrum of pressure. Chim is shorter than the vibration, moreover, the range is more flexible.

In the case of spectral analysis, a well-respected tribute for additional window functions is carried out, in order to increase the change in the inflow of bird dusters to spectral assessments.

If two harmonics f 1 and f 2 can be detected with close frequencies, it is necessary for the timing window T, the width of the head "peel" is Df -3 ≈ Df L = 0 = 1 / T, which should start at a value of -3 dB, which is less than the difference in frequencies , scho joke around.

Df = f 1 -f 2> Df -3

The width of the clock window T is tied from the sampling frequency f s and the number of vibrating signals by the formula (2.4).

Instrumental for harmonic analysis... Signal Processing is used for signaling signals in the MATLAB package.

Modified by periodograms vikoristovuyt oblique vіkonnі funktsії, sho to change the effect of Gibbs. Applied can be the victorian of Vikna Hemming (Hamming). Ale at the same time, one hour to see an approximate increase in the width of the cephalic dandruff by spectrograms. More optimized by the Kaiser window. Increasing the width of the main pelusts from the hour of opening the filters of the lower frequencies until the increasing transition line (between the smugs passing through and clipping).

Welch assessment function... The method is based on the distribution of the last hour in a segment (possibly with re-lines), the skin segment is far from being processed, and then the spectrum is estimated at a pace by averaging the results of the segments being processed. For a reduced assessment, you can use indirect window functions, for example, Hemming's window. The increase in the number of segments decreases the variance, while the number of segments decreases with the frequency. The method gives non-fouling results when there is a small change in the signal over the noise and is often practical.

In Fig. 33, estimates of the harmonious warehouse for tributes are provided, for the purpose of revealing the high-pitched signals and loud noise, in case of low vibrations (N = 100, N = 67), and on the basis of new methods.

Small. 33. Estimation of signal harmonics for 1024 point FFT-conversion

Parametric methods vikoristovuyt autoregression models (AR). The methods will have models of filters and additional evaluation of the signal spectrum. Effortless methods for the presence of signal noise give compensation estimates. Signal processing methods are used to reduce the harmony of storage and noise. The order of the method (filter) is set in two more, lower number of harmonics, but є at the signal. A number of parametric methods have been propagated.

The Burg method gives a high frequency distribution for short vibrations. With a high order of the filter, the spectral peaks are split. The position of the spectral peaks lie in the cob harmonic phases.

The insidious method allows the estimation of the spectrum of the signal, so that the sum of the harmonic components is revealed.

The Yule-Walker method is good for all vibrators and is not recommended for short vibrations.

Corel methods... The MISIC method (Multiple Signal Classification) and EV (eigenvectors) see the results in the form of the pseudospectrum. The methods are based on the analysis of vectors in the correlation matrix of the signal. Tsi methods give a cheap short call for frequency, lower autocorrelation methods.

Below is a short description of the signals that begin to show the spectral gustini. When the spectral density of signals is determined, I am satisfied with the absolute integration, I do not use the formula (4.41).

Spectral density of a number of signals in the induction table. 4.2.

1) The impulse of the rectangular form (table 4.2, pos. 4). Kolivannya, pictured in fig. (4.28, a), can be written in the viewer

Yogo spectral proficiency

Graph of spectral intensity (Fig. 4.28 a) of inducements on the basis of an analysis of the spectrum of periodic after-effects of unipolar, direct-current impulses carried out earlier (4.14). Yak is seen from (Fig. 4.28 b), the function turns to zero for the values ​​of the argument = n, de P - 1, 2, 3, ... - be it a whole number. At the same kutovі frequencies and rіvnі =.

Small. 4.28. Direct current waveform pulse (a) and spectral power (b)

The spectral intensity of the impulse at a numerically large area, tobto G(0)=A. The price is true for the impulse s(t) dovіlnoї form. Really, vvazhayuchi zagalom (4.41) = 0, otrimaєmo

tobto the area of ​​an impulse s(t).

Table 4.3.

When the pulses are stretched out, the function becomes faster with zeros, so that the grip is shown to the spectrum. The value of your own growth. Navpaki, when the pulse is squeezed, the expanded spectrum grows, and the value changes. On (Fig. 4.29, a, b) graphs of the amplitude and phase spectra of the direct-current pulse are shown.

Small. 4.29. Amplitude graphs (a) Small. 4.30. Pulse of rectangular form, and phase (b) spectrum of damage per hour

When the impulse is broken to the right (lost) by the hour (Fig. 4.30), the phase spectrum changes by the value indicated by the argument of the multiplier exp () (Table 4.2, pos. 9). The resulting phase spectrum of the pulse, which can be remembered, of the images in Fig. 4.29 with a dotted line.

2) Delta-function (table 4.3, pos. 9). Spectral power - the function is known by the formula (4.41), δ -functions:

In such a rank, the amplitude spectrum is equal to the area δ -functions [= 1], and the phase spectrum to zero [= 0].

To the vicious transformations of Fur'є from the function = 1 to repent with one of the values δ -functions:

Corrode the power of the hourly zsuvu (table 4.2, pos. 9), due to the spectral power of the function , stay for an hour okay :

Amplitude and phase spectra of the function are shown in table. 4.3, pos. 10. The revolt of the transformation of Fur'є from the function of the ma viglyad

3) Harmonious communication (Table 4.3, pos. 12). Harmonious communication є absolutely integrated signal. Protect the value of the th spectral power is fixed directly by the transformation of Fur'є, we write down the formula (4.41) as:

Todi z urahuvannyam (4.47) otrimumo

δ(ω) - Delta functions, shifting along the frequency axis to the frequency, both to the right and to the left. Yak can be seen from (4.48), the spectral power of the harmonic vibration with the endurance amplitude of an infinitely great value at discrete frequencies.

Similarly, analogous re-implementation, the spectral power of the number can be recognized (table 4.3, pos. 13)

4) Function type (table 4.3, pos. 11)

Spectral power of the signal A start with the formula (4.48), with the key = 0:

5) A single function (or a single strip) (table 4.3, pos. 8). The function is not absolutely integrated. Yaksho uyaviti yak between the exponential impulse , tobto.

then the spectral power of the function can be significant as between the spectral power of the exponential pulse (Table 4.3, pos. 1) when:

It is better to add to the right part of the viraz to zero at all frequencies, crim = 0, it is almost impossible to convert to non-inconsistency, but the area will go to the function of transporting permanent values.

Tom will be the first to use the function. Mezheyu other dodanku є function. Residual mo

Recognition of two downgrades at the viraz (4.51) at viglyadі 1/2 + 1 / 2sign ( t). Permanent warehouses 1/2 free of charge (4.50) based on spectral performance, but unpaired functions - Clearly significant spectral power.

When analyzing the injection of a single streak onto a lance, the transfer function of those at = 0 is zero (i.e., to a lance, but do not skip a continuous stream), in formula (4.51), it is possible to use only other pre-streak, representing

6) Complex exponential signal (table 4.3, pos. 16). Yaksho to show the function of the viglyad

then, on the basis of the linearity of the transformation of the Fur'є and the parameters of the viraz (4.48) and (4.49), the spectral power of the complex exponential signal

Also, the complex signal is a asymmetric spectrum, representations of a single delta-function, substituted by a frequency, is just right.

7) Reliable periodic function. Guided by a periodic function (Fig. 4.31, a) by the complex series Fur'є

de - Frequency of passing impulses.

Coefficiencies for a number of Fur'є

swing through the value of the spectral density of a single pulse s(t) at frequencies ( n=0, ± 1, ± 2, ...). Periodic (4.55) (4.54) and perceptible to children (4.53), due to the spectral power of periodic functions:

According to (4.56), the spectral power of the same periodic function can be seen from the last functions, which change one to one, to the frequency (Fig. 4.31, b). Coefficiencies at δ -functions vary according to the spectral power of a single pulse s(t) (dashed curve in Fig.4.31, b).

8) Periodic sequence of δ-functions (table 4.3, pos. 17). Spectral density of periodic post-functions

start with the formula (4.56) as an approximation of the spectral density of the periodic function at = 1:

Figure 4.31. Satisfactory pulse rate (a) and spectral power (b)

Small. 4.32. Radio signal (a), spectral power of the radio signal (c) of that fire (b)

and I can see the periodical last δ -function, multiplied by the performance.

9) A radio signal from a straight-sided fire. The radio signal supplied to (Fig. 4.32, a), you can record yak

Up to pos. 11 of Table 4.2 the spectral power of the radio signal travels along the path of the spectral power of the straight-line wave along the frequency axis to the right and also changes from the ordinates to the right, so that.

Tsei viraz go out from (4.42) by way of changing the frequency to the frequency - zsuv right-handed and zsuv livoruch. The transformation of the spectrum is shown in (Fig. 4.32, b, c).

Attach a spectrum of non-periodic signals to the target as well.