Correlation analysis of the correlation of two signals. Correlation function of the signal

Correlation functions of signals are used for integral calculation estimates of signal forms and levels of their similarity among themselves.

Autocorrelation functions (ACF) of signals (Correlation function, CF). As far as determination of the signals with the end energy of the ACF, as an integral characteristic of the form of the signal, i is an integral in the form of an addition of two copies of the signal s (t), zsunutih one per hour t:

B s (t) = s (t) s (t + t) dt. (2.25)

How to sing out of this viraz, ACF is a scalar creation of a signal and a copy of the functional fallow size zsuvu t value. Obviously, the ACF can have a physical difference in energy, and at t = 0, the ACF value is without a middle one energy signal:

B s (0) = s (t) 2 dt = E s.

The ACF function is uninterrupted and paired. In the rest, it does not matter to change the change of the change t = t-t in natural expression (2.25):

B s (t) \u003d s (t-t) s (t) dt \u003d s (t) s (t-t) dt \u003d B s (-t). (2.25")

With the improvement of pairing, graphic representation of ACF, only positive values ​​of t are carried out. In practice, the signals are set on the interval of positive values ​​of the arguments in the form 0-T. The sign + t in natural expression (2.25) means that when the value of t is increased, a copy of the signal s (t + t) moves to the left along the t axis and goes beyond 0, which means that the signal continues to move into the region of negative values ​​of the argument. And since when calculating the interval of the task t, as a rule, it is much less than the interval of the task of the signal, then it is more practical to break the copy of the signal to the left along the argument axis, so that it is stuck in the expression (2.25) function s (t-t) instead of s (t + t).

In the world, the value of the value of zsuvu t for the finite signals in the time-hour overlap of the signal with the same copy changes and the scalar twir increases to zero.

Butt. On the interval (0, T) of tasks, a direct-current pulse with amplitude values ​​equal to A. Calculate the autocorrelation function of the pulse.

When sending a copy of the impulse along the t axis to the right, at 0≤t≤T, the signals overlap at intervals from t to T. Scalar TV:

B s (t) \u003d A 2 dt \u003d A 2 (T-t).

When zsuv_ copy_impulse to the left, at -T≤t<0 сигналы перекрываются на интервале от 0 до Т-t. Скалярное произведение:

B s (t) = A 2 dt = A 2 (T + t).

When | t | > T signal and the second copy do not break the break point and the scalar additional signal reaches zero (the signal and the second copy become orthogonal).

Considering the calculation, we can write:

For different periodic signals, the ACF is calculated for one period T, with averaging of the scalar creation and one broken copy in between the period:

B s (t) \u003d (1 / T) s (t) s (t-t) dt.

At t = 0, the value of the ACF in the same period is not one energy, but the average intensity of the signals in the intervals T. The ACF of periodic signals is also a periodic function with the same period T. For a single-tone harmonic signal, it is obvious. The first maximum value of the ACF will be t = 0. When a copy of the signal is given for a quarter of the period, the original integrand functions become orthogonal one to one (cos w o (t-t) = cos (w o t-p / 2) º sin w o t) and give the value of zero ACF. When sounding at t = T / 2, a copy of the signal y directly becomes adjacent to the signal itself and the scalar twir reaches the minimum value. With a slight increase in sound, the reverse process of increasing the value of the scalar creation with a zero overshoot at t = 3T / 2 and repetitions of the maximum value at t = T = 2p / w o (cos w o t-2p copy º cos w o t signal) is initiated. A similar process can be used for periodic signals in a sufficient form (Fig. 2.11).

It is significant that the result cannot be taken away from the cob phase of the harmonic signal, which is typical for any periodic signals and is one of the powers of the ACF.

For signals set on a single interval, the calculation of the ACF is carried out from the normalization for the second interval:

B s (t) = s (t) s (t + t) dt. (2.26)

The autocorrelation of a signal can be estimated by the function of autocorrelation coefficients, which are calculated according to the formula (by signal centering):

r s (t) = cos j (t) = ás (t), s (t + t) ñ / || s (t) || 2.

Mutual correlation function (CCF) of signals (cross-correlation function, CCF) shows how the steps of similarity of the form of two signals, so that they are mutually spread one by one along the coordinate (independent change), for which the same formula (2.25) is used, which is for ACF, ale pіd іntegrаl їtіvіr tvіr dvіrіh zіnіh іnіkі, one zіkіh disrupt tі hour t:

B 12 (t) = s 1 (t) s 2 (t + t) dt. (2.27)

When changing the change t = t-t in the formula (2.4.3), we take:

B 12 (t) \u003d s 1 (t-t) s 2 (t) dt \u003d s 2 (t) s 1 (t-t) dt \u003d B 21 (-t)

Mal. 2.12. Signals and VKF

It is evident that for the VKF there is no umova pairing, and the value of the VKF is not the mother's goiter maximum at t = 0. 2.12, where two identical signals are given with centers at points 0.5 and 1.5. Calculation for the formula (2.27) with incremental increments, the value t means the subsequent destruction of the signal s2 (t) to the left along the axis (for the skin value s1 (t), for the integrand, the values ​​s2 (t + t) are taken).

At t = 0, the signals are orthogonal and the value of B 12 (t) = 0. The maximum of B 12 (t) will be monitored when the signal s2 (t) is sent to the left of the value t = 1; (t + t). When the value of B 21 (-t) is calculated, a similar process is reversed by the last sound signal s1 (t) to the right along the time axis with incremental negative values ​​of t, and the value of B 21 (-t) is mirrored (shodo axis t = 0) values B 12 (t), i navpak. On fig. 2.13 you can bachiti in person.

Mal. 2.13. Signals and VKF

Thus, for the calculation of the new form of the VKF, the numerical whole t is guilty of including negative values, and changing the sign of t in formula (2.27) is equivalent to permuting the signals.

For periodic signals of the understanding of the VKF, the sound does not stop, for a few signals with the same period, for example, signals for the entry and exit of systems when the characteristics of the systems are taken into account.

The function of the coefficients of mutual correlation of two signals is calculated according to the formula (by signal centering):

r sv (t) = cos j (t) = ás (t), v (t + t) ñ / || s (t) || || v (t) ||. (2.28)

The value of the coefficients of mutual correlation can change from -1 to 1.

correlation analysis There may be stops for rechecking the presence of the core signal for aphids of present noise and shifting codes, as well as for rechecking the efficiency of robotic digital filters. In the first case, the correlation function is normalized between the fragment of the core signal and the numerical series of the discretized input transition signal. Behind the graph of the correlation function, the presence of a noise signal in a noisy input signal is visually shown.

In another way, with the method of rechecking the efficiency of filtering, the correlation function of the core reference signal, represented by a numerical series, and the filtered signal is reversed. After which the direct discrete transformation of Four'e to the correlation function is taken by way of the correlogram. On the removed schedule, there will be a line of critical equivalence for the improvement of the pardon of filtering against the Student's criterion. The efficiency of filtering is determined visually: more than a critical level of guilt, only a few warehouses are spectral width anchor signal.

For greater precision and objectivity, a vibratory correlation coefficient between the number series of the reference (output core) and filtered signals is considered. The correlation coefficient can take values ​​in the interval -1 ... 1. Negative values ​​indicate that the reference and filtered signals correlate in antiphase, that is, when the filtered signal is inverted. At the same time, the digital filter can be filtered effectively in terms of shifts and noise, the correlation coefficient takes a value close to 1 or -1. The brightness of different digital filters for a hundred of a specific signal can be determined by the way of matching the correlation coefficients.

The analysis of the correlation function of discrete signals is carried out in such a manner. For discrete signals X (i) і Y (i), i = 1 ... N, a fragment of the array is selected Y(i), i=1... N / 2

de - value of zsuvu in discrete.

The correlogram or the spectrum of the correlation function can be taken by way of the direct discrete conversion of Fur'є to the correlation function:

- diyna part of the spectrum

;

- visible part of the spectrum

;

- modulus of spectral width correlation function

Frequencies that match the values ​​of the spectrum,

de - sampling period of the input signal.

Razrahunok correlation coefficient between discrete signals (numeric series) Х (i) і Y (i), i = 1 ... N be carried out in such a rank.

Mean value (mathematical score) for number series X (i) and Y (i):

dispersion

; .

Another change in the central moment

.

Variable correlation coefficient

transcript

1 + 1 SIGNAL AND LINE SYSTEMS Signals and linear systems. Correlation of signals Theme 6. CORRELATION OF SIGNALS Borderline fear and borderline fuse of goodness, however, the boat is turbulent and the drift is called out. Michel Montaigne. French jurist, 16th century Axis number! Two functions may have a hundred-hundred correlation with a third and are orthogonal to one to one. Well, fry the boules of the Almighty at the creation of the World. Anatoly Pishmintsev. Novosibirsk geophysicist of the Ural school, XX century. Edit 1. Autocorrelation functions of signals. Understanding autocorrelation functions (ACF). AKF signals, obmezhenyh at the hour. ACF of periodic signals. Functions of autocovariance (FAK). ACF of discrete signals. ACF of noisy signals. ACF of code signals. 2. Cross-correlation functions of signals (CCF). Mutual correlation function (VKF). Mutual correlation of noisy signals. VKF of discrete signals. Evaluation of periodic signals in noise. The function of mutual correlation coefficients. 3. Spectral width of correlation functions. Spectral width of the ACF. Signal correlation interval. Spectral width of the VKF. Calculation of correlation functions for additional SPF. INTRODUCTION Correlation (correlation), and її okremiya vpadok for zseredzhennyh signals covariance, є method of signal analysis. Let's introduce one of the variants of the method. It is acceptable that it is a signal s (t), in which it can be (and maybe not be) a succession x (t) of the final day T, the time position to call us. In order to know the sequence in the current signal s (t) time intervals T, the scalar creations of the signals s (t) and x (t) are calculated. We ourselves "apply" the noise signal x (t) to the signal s (t), according to the second argument, and by the magnitude of the scalar creation, we evaluate the steps of the similarity of the signals at the points of alignment. Correlation analysis makes it possible to install in the signals (or in the series of digital data signals) the presence of a vocal connection, change the value of the signals by an independent change, so if the values ​​of one signal (when the average values ​​of the signal) are large, then the value of the signal is large. correlation), or, on the other hand, small values ​​of one signal are associated with large values ​​of the other (negative correlation), or two signals are not related in any way (zero correlation). In the functional space of the signal levels, the connection can be expressed in the normalized units of the correlation coefficient, so that in the cosine cut between the signal vectors, i, obviously, if you take the value of the signal 1 (the last fall of the signals) to -1 (the longest life) value (scale) alone vimiryuvan. In the autocorrelation variant, following a similar technique, the scalar creation of the signal s (t) is assigned with a direct copy, in the argument. Autocorrelation allows estimating the average statistical accumulation of flowing signals in the signal in terms of its forward and advancing values ​​(the so-called radius of correlation value of the signal), as well as revealing the presence of periodically repeating elements in the signal. Especially the value of the correlation method may be in the analysis of falling processes for the identification of non-falling storage processes and the assessment of non-falling parameters of these processes. Respectfully, in terms of "correlation" and "covariance" the deuce is a swindler. In the mathematical literature, the term "covariance" reaches the centering of functions, and "correlation" reaches the limit. In the technical literature, and especially in the literature for signals and methods of processing, there is often a direct opposite terminology. I don’t have a fundamental meaning, but if you know the literary sources, you should pay attention to the recognition of these terms in the autocorrelation function of signals. Understanding autocorrelation functions of signals. Autocorrelation function (ACF, CF - correlation function) to the signal s (t), terminal in energy, є kіlkisnoї іntegrаlї characteristic of the form of the signal, manifested in the signal character and parameters in the mutual temporal connection, influencing, so the signal for the interval and step

2 + 2 latency penalty value in the current moment and hour in the history of the current moment. The ACF is determined by the integral of adding two copies of the signal s (t), putting one out of one per hour: B s () = s (t) s (t +) dt = s (t), s (t +) = s (t ) s (t +) cos (). (6.1.1) As a scalar creation of the signal and the second copy of the functional fallacy as a change in the value of the sound. Obviously, the ACF has a maximum physical expansion of energy, and at = 0 the value of the ACF is without a middle one energy to the signal i є is the maximum possible (the cosine of the coo modality of the signal with itself is equal to 1): B s (0) = s (t) 2 dt = E s. ACF is brought up to pair functions, why it is not important to change the change of the change t = t- in the expression (6.1.1): B s () = s (t-) s (t) dt = B s (-). The maximum ACF, equal to the energy of the signal at = 0, is always positive, and the ACF modulus does not exceed the energy of the signal at any value of the clock sound. Remain directly vyplyvaє z the power of the scalar creation (like the inconsistency of Koshі-Bunyakovsky): s (t), s (t +) = s (t) s (t + cos (), cos () = 1 at = 0, s ( t) , s (t +) = s (t) s (t) = E s, cos ()< 1 при 0, s(t), s(t+) = s(t) s(t+) cos () < E s. Рис В качестве примера на рис приведены два сигнала прямоугольный импульс и радиоимпульс одинаковой длительности Т, и соответствующие данным сигналам формы их АКФ. Амплитуда колебаний радиоимпульса установлена равной T амплитуды прямоугольного импульса, при этом энергии сигналов также будут одинаковыми, что подтверждается равными значениями центральных максимумов АКФ. При конечной длительности импульсов длительности АКФ также конечны, и равны удвоенным значениям длительности импульсов (при сдвиге копии конечного импульса на интервал его длительности как влево, так и вправо, произведение импульса со своей копией становится равным нулю). Частота колебаний АКФ радиоимпульса равна частоте колебаний заполнения радиоимпульса (боковые минимумы и максимумы АКФ возникают каждый раз при последовательных сдвигах копии радиоимпульса на половину периода колебаний его заполнения). С учетом четности, графическое представление АКФ обычно производится только для положительных значений. На практике сигналы обычно задаются на интервале положительных значений аргументов от 0-Т. Знак + в выражении (6.1.1) означает, что при увеличении значений копия сигнала s(t+) сдвигается влево по оси t и уходит за 0. Для цифровых сигналов это требует соответствующего продления данных в область отрицательных значений аргумента. А так как при вычислениях интервал задания обычно много меньше интервала задания сигнала, то более практичным является сдвиг копии сигнала влево по оси аргументов, т.е. применение в выражении (6.1.1) функции s(t-) вместо s(t+).

3 3 B s () = s (t) s (t-) dt. (6.1.1 ") For finite signals around the world, the value of the value of the zsuve time-hour overlap of the signal with its second copy changes, and, apparently, the cosine of the coota of the interaction and the scalar increase in general to zero: lim Bs (τ) \u003d 0, τ AK calculated for the centered value of the signal s (t), which is an autocovariance function of the signal: C s () = dt, (6.1.2) de s the average value of the signal. = B s () - 2 s. ACF of signals set at different hour intervals. In practice, the signals set at the time interval are heard and analyzed. So, for example, when a signal is specified on the interval: B s () = b 1 s (t) s (t +) dt. (6.1.3) create a scalar signal i yo the second copy when the interval of the specified signal is set to infinite: b TB s () lim s (t) s (t τ) dt TT 1 0. (6.1.4) and the first copy of the functional fallow in the form of a copy. ACF of periodic signals. The energy of periodic signals is not limited, so the ACF of periodic signals is calculated over one period T, with averaging of the scalar create signal and the second broken copy in the intervals of the period: TT 1 0 B s () = (1 / T) T s (t) s (t-) dt. (6.1.5) 0 At = 0, the value normalized for the ACF period is equal to the average signal tension in the intervals of the period. With any ACF of periodic signals - a periodic function with the same period T. So, for a signal s (t) \u003d A cos (0 t + 0) at T \u003d 2/0, it is possible: ω π / ω0 0 B s () \u003d A cos (0 t + 0) A cos (0 (t -) + 0) = (A 2/2) cos (0). (6.1.6) 2π π / ω 0 Subtracting the result does not lie in the cob phase of a harmonic signal, which is typical for any periodic signals and is one of the powers of the ACF. For the help of the autocorrelation function, it is possible to reverse the presence of periodic authorities in any good signals. The application of the autocorrelation function of the periodic signal is shown in Fig. The functions of autocovariance (FAC) are calculated in the same way, by centering the signal values. Miraculous singularity of these functions and their simplicity of dispersion with variance of 2 s signals (square standard - root-mean-square value of the signal from the average value). As you can see, I know.

4 4 values ​​of variance to mean signal tightness, signs following: C s () s 2, C s (0) = s 2 s (t) 2. (6.1.7) : s () = C s () / C s (0) = C s () / s 2 cos). (6.1.8) Another function is called "reference" autocorrelation function. By virtue of the normalization of її, the value does not fall into one (scale) of the value presented to the signal s (t) and characterize the step of the linear connection between the values ​​of the signal in the fallow in the form of the value of zsuvu between the signals of the signal. The value of s () cos () can change from 1 (reversed correlation) to -1 (reverse correlation). Fig. In fig. the application of signals s () і s1 () = s () + noise with similar signals by FAK coefficients - s і s1. As can be seen on the graphs, the FAC clearly showed the presence of periodic surges in the signals. Noise in the signal s1 () lowering the amplitude of periodic knocks without changing the period. It confirms the graph of the curve C s / s1, so that the FAC of the signal s () is normalized (for setting) on ​​the value of the variance of the signal s1 (), it is precisely possible to check that the noise pulses with the total statistical independence of their values ) in relation to the value C s (0) and sprat "split" the function of the autocovariance coefficients. Therefore, it was noted that the value of s () of noise signals in the range up to 1 at 0 and fluctuates to zero at 0, with which amplitude the fluctuations are statistically independent and lie in the number of vibrations of the signal (go to zero with an increase in the number of signals). ACF of discrete signals. At the data discretization interval t = const, the calculation of the ACF is counted at intervals = t and the sound is recorded, like discrete function numbers n zsuvu replies n: B s (nt) = t s s -n. (6.1.9) Discrete signals are set in the form of numerical arrays singing dozhini with the numbering of the variables k = 0.1, K at t = 1, and the calculation of the discrete ACF in units of energy is counted in a one-sided version with the adjustment of the number of arrays. If the whole array is counted for the signal and the number of ACF responses is equal to the number of responses in the array, then the calculation is calculated according to the formula: B s (n) = K-n K K n s s -n. (6.1.10) The multiplier K / (K-n) in this function of the correction coefficients on the step of changing the number is multiplied by the summed values ​​in the world of the increase in the number n. Without any correction for the off-centered signal, the ACF values ​​show a trend of subsuming the average values. When the signals are measured in units of tension, the multiplier K / (K-n) is replaced by the multiplier 1 / (K-n). The formula (6.1.10) rarely occurs, mainly for deterministic signals with a small number of responses. For vipadkovy and noisy signals of a change in the standard (K-n) and numbers, multiply in the world the increase in sound to produce an increase in statistical fluctuations in the calculation of the ACF. Great confidence in their minds for the safety of calculating the ACF in units of intensity of the signal for the formula: 0

5 K 5 B s (n) = K 1 s s -n, s -n = 0 at -n< 0, (6.1.11) 0 т.е. с нормированием на постоянный множитель 1/K и с продлением сигнала нулевыми значениями (в левую сторону при сдвигах -n или в правую сторону при использовании сдвигов +n). Эта оценка является смещенной и имеет несколько меньшую дисперсию, чем по формуле (6.1.10). Разницу между нормировками по формулам (6.1.10) и (6.1.11) можно наглядно видеть на рис Рис Формулу (6.1.11) можно рассматривать, как усреднение суммы произведений, т.е. как оценку mathematical refinement: B s (n) = M (s s -n) s s. (6.1.12) n Practically, a discrete ACF can be as powerful as an ACF without interruption. Vaughn is also є steam room, and її value at n = 0 more energy, or the intensity of the discrete signal in the fallow time is normalized. ACF of noisy signals. A noisy signal is recorded in the visual sum v () = s () + q (). In general, the noise is not necessarily to blame for the mother zero mean value, and the autocorrelation function of the digital signal is normalized by tightness, which avenges N variables, is recorded in the offensive view: B v (n) = (1 / N) s () + q (), s(-n) + q(-n) == (1/N) == B s(n) + M(sq-n) + M(qs-n) + M(qq-n). B v (n) = B s (n) + s q n + q s n + q q n. (6.1.13) If the correlation signal s () і noise q () is statistically independent, then the following formula can be used: Figure B v (n) = B s (n) + 2 sq + q. (6.1.13 ") The butt of the signal crossing and the second ACF in the presence of a non-noisy signal is shown in Fig. 3 of the formulas (6.1.13) showing that the ACF of the signal crossing is added to the ACF of the signal component of the corre- sponding signal with the superimposed fading value up to the value 2s q + q 2 Noise function At large values ​​of K, if q 0, then B v (n) B s (n). also, with high accuracy, it is necessary to determine their period and shape in between the period, and for single-frequency harmonic signals, their amplitude with variable frequency (6.1.6).

6 Table 6.1. M Barker signal ACF signal 2 1, -1 2, 1, -1 3, 0, 1, 1, -1 4, 1, 0, -1 1, 1, -1, 1 4, -1, 0, 1 5 1, 1, 1, -1, 1 5, 0, 1, 0, 1 7 1, 1, 1, -1, -1, 1, -1 7, 0, -1, 0, -1, 0 , 1.1, -1, -1, -1.1, -1, -1.1, -1 11.0, 1.0, 1.0, 1.0, 1.0, 1.1, 1,1, -1, -1,1,1-1,1, -1,1 13,0,1,0,1,0,1,0,1,0,1,0, 1 6 Code signals є variety of discrete signals. On the sing interval of the code word Mt stink, there can only be two amplitude values: 0 and 1 or 1 and 1. When the codes are seen on the absolute noise level, the form of the ACF of the code word may be especially significant. From this position, the most important are such codes, the value of the ACF pellusts, which are minimal for the entire length of the code word interval at the maximum value of the central peak. Before the number of such codes, the Barker code is added, guidance in tables 6.1. As can be seen from the table, the amplitude of the central peak of the code is numerically more important than the value of M, with which the amplitude of the binary oscillations at n 0 does not outweigh the mutually correlative functions of the signal. The cross-correlation function (CCF) of different signals (CCF) describes both the steps of the similarity of the formation of two signals, and they are mutually spreading one by one along the coordinate (independent change). Using the formula (6.1.1) of the autocorrelation function for two different signals s (t) and u (t), it is necessary to advance the scalar twir signals: B su () = s (t) u (t +) dt. (6.2.1) Cross-correlation of signals characterizes the same correlation of phenomena and physical processes that are reflected by these signals, and can serve as a peace of mind of this interrelationship during separate processing of signals in outbuildings. For end-of-energy signals, the VKF is also finite, with which: B su () s (t) u (t), which is due to the unevenness of Kosh-Bunyakovsky and the independence of the signal norms in the sound of the coordinates. When replacing change t = t- in formula (6.2.1), it is necessary: ​​B su () = s (t-) u (t) dt = u (t) s (t-) dt = B us (-). Fig Signali i VKF. It is clear that for the VKF there is no mutual parity, B su () B su (-), and the value of the VKF does not goitre the mother at maximum at = 0. 0.5 and 1.5. The calculation for the formula (6.2.1) with incremental increments means the subsequent destruction of the signal s2 (t) to the left along the axis (for the skin value s1 (t), for the integrand, the values ​​s2 (t +) are taken). At = 0, the signals are orthogonal and the value B 12 () = 0. The maximum B 12 () will be monitored when the signal s2 (t) is applied to the left on the value = 1, with which the signals s1 (t) and s2 (t +) are lowered . One and the same values ​​of the VKF for the formulas (6.2.1) and (6.2.1 ") are guarded at one and the same mutual position of the signals: when the signal u (t) is extended to the interval, it is visible s (t) to the right along the y-axis and the signal s (t) according to the signal u (t) to the left, then B su () = B us (-

7 7 Go to fig. All signals may have the same trivality T, at which the signal v (t) breaks forward by the interval T / 2. The signals s (t) and u (t) are the same in terms of time difference and the "overshoot" area of ​​the signal is maximum at = 0, which signal functions. and fixed by the function B su. At the same time, the function B su is sharply asymmetric, so that with an asymmetric signal u (t) for a symmetrical signal s (t) (to the center of the signals), the area of ​​"overlap" of the signals changes according to the different zero). When the position of the signal u (t) is shifted to the left along the ordinate axis (on the forward signal s (t) - the signal v (t)) the VKF shape is overwritten without changing and shifts to the right on the same value of the value of the shift function B sv in Fig. expression of functions in (6.2.1), then new function B vs will be a mirror-image = 0 function B sv. Z rahuvannyam tsikh features outside the VKF are calculated, as a rule, for positive and negative entries: B su () = s (t) u (t +) dt. B us () = u (t) s (t +) dt. (6.2.1 ") Mutual correlation of noisy signals. For two noisy signals u (t) = s1 (t) + q1 (t) і v (t) = s2 (t) + q2 (t), using the technique of deriving formulas ( 6.1.13) by replacing a copy of the signal s (t) with the signal s2 (t), it is not important to enter the formula of mutual correlation in the forward view: B uv () = B s1s2 () + B s1q2 () + B q1s2 () + B q1q2 (). (6.2.2) The remaining three terms in the right part (6.2.2) fade to zero when increased. With large intervals for setting signals, it can be written in the offensive form: B uv () = B s1s2 () + s1 ( ) q2 () + q1 () s2 () + q1 () q2 () (6.2.3) With zero average values ​​of noise and statistical independence in the signals in the month: B uv () B s1s2 (). all authorities of the VKF analog signals useful for VKF of discrete signals, at the same time for them the specificity of discrete signals is useful for discrete ACF (formulas). Zocrema, at t = const = 1 for signals x () і y () with the number of responses Before: B xy (n) = When normalized in units of tightness: K K n K K-n 0 x y -n. (6.2.4) B xy (n) = K 1 x y -n x y n. (6.2.5) 0 Evaluation of periodic signals in noise. The noisy signal can be estimated by the mutual correlation with the "reference" signal by the method of trials and pardons by adjusting the mutual correlation function to the maximum value. For the signal u () = s () + q () with statistical independence of the noise i q 0, the cross-correlation function (6.2.2) with the template signal p () with q2 () = 0 looks like: B up () = B sp () + B qp () = B sp () + q p. And the spikes q 0 with the increase in N, then B up () B sp (). Obviously, the function B up () will have a maximum if p () = s (). By changing the shape of the template p () and maximizing the function B up (), you can subtract the score s () from the seemingly optimal shape p (). The function of mutual correlation coefficients (VKF) is a measure of the degree of similarity of signals s (t) and u (t). Similarly, the functions of autocorrelation coefficients

8 8 ficient, it is calculated through the centering of the function values ​​(to calculate the mutual covariance, it is possible to center only one of the functions), and it is normalized to the additional value of the standard functions s (t) and v (t): su () = C su () / s v. (6.2.6) The interval for changing the value of correlation coefficients in case of failures can change from 1 (exactly reverse correlation) to 1 (exact similarity or hundredth correlation). In case of failures, on which zero values ​​of su () are expected, the signals are independent of one type of one (uncorrelated). Mutual correlation coefficient makes it possible to establish the presence of a connection between signals independently of the physical powers of the signals and their magnitudes. When calculating the VKF of noisy discrete signals in the interleaved period with the varying formulas (6.2.4) є ymovіrnіst value su (n)> 1 appears. studying the characteristics of systems. The spectral width of the ACF can be determined from the onset of simple microscopy. Vіdpovіdno up to virase (6.1.1) ACF є scalar function create a signal i th copy, collapsed into an interval at -< < : B s () = s(t), s(t-). Скалярное произведение может быть определено через спектральные плотности сигнала и его копии, произведение которых представляет собой спектральную плотность взаимной мощности: s(t), s(t-) = (1/2) S() S *() d Смещение сигнала по оси абсцисс на интервал отображается в спектральном представлении умножением спектра сигнала на exp(-j), а для сопряженного спектра на множитель exp(j): S *() = S*() exp(j). С учетом этого получаем: s ()= (1/2) S() S*() exp(j) d = (1/2) S() 2 exp(j) d (6.3.1) Но последнее выражение представляет собой обратное преобразование Фурье энергетического спектра сигнала (спектральной плотности энергии). Следовательно, энергетический спектр сигнала и его автокорреляционная функция связаны преобразованием Фурье: B s () S() 2 = W s (). (6.3.2) Таким образом, спектральная плотность АКФ есть не что иное, как спектральная плотность мощности сигнала, которая, в свою очередь, может определяться прямым преобразованием Фурье через АКФ: S() 2 = B s () exp(-j) d. (6.3.3) Последние выражение накладывает определенные ограничения на форму АКФ и методику их ограничения по длительности. Энергетический спектр сигналов всегда положителен, мощность сигналов не может быть отрицательной. Следовательно, АКФ не может иметь формы прямоугольного импульса, т.к. преобразование Фурье прямоугольного импульса знакопеременный интегральный синус. На АКФ не должно быть и разрывов Рис Спектр несуществующей АКФ первого рода (скачков), т.к. с учетом четности АКФ любой симметричный скачек по координате по-

9 9 people gave the ACF to the sum of a continuous non-stop function and a straight-line pulse to trivality 2 with a significant appearance of negative values ​​in the energy spectrum. The butt of the rest is pointed at rice (the graphs of the functions are drawn, as is customary for pair functions, only with its right side). ACF to reach long-drawn-out signals are interleaved after measurements (there are additional intervening intervals of correlation of data from T / 2 to T / 2). However, the enhancement of the ACF, the multiplication of the ACF for a recto-current selective pulse, the trivality T, which in the frequency domain appears as a convolution of the actual spectrum of tension with the sign function of the integral sine sinc (t / 2). From one side, it calls for a flattening of the spectrum of tightness, which often becomes darker, for example, with consecutive signals at a significant level of noise. Ale, from the other side, it may be possible that there is an underestimation of the magnitude of the energy peaks, as in the signal there are some harmonic warehouses, as well as the appearance of negative values ​​of tension on the marginal parts of the peaks and stribkiv. An example of the manifestation of these factors is shown in Fig. Fig. Calculation of the energy spectrum of a signal according to the ACF of a different period. Apparently, the spectra of the intensity of the signals do not change the phase characteristics and it is impossible to identify signals from them. Also, the ACF of signals, as well as the timing of the manifestation of the spectra of sweating, so there is no information about the phase characteristics of the signals and the correction of signals by the ACF is impossible. Signals of the same form, sent in hours, may have the same ACF. Moreover, the signals of different shapes can be similar to ACF, which may have a similar spectrum of tension. Rewrite equality (6.3.1) in offensive form s (t) s (t-) dt = (1/2) S () S * () exp (j) d well known and is called parseval's equivalence s 2 (t) dt = (1/2) S () 2 d. It allows you to calculate the energy of the signal, both in terms of the clock, and in the frequency domain of the description of the signals. Correlation interval for the signal is a numerical parameter for estimating the width of the ACF and the level of significant correlation is the value of the signal by the argument. Let's assume that the signal s (t) can be approximately equal energy spectrum with the values ​​of W 0 and with the upper limiting frequency up to in (the shape of a centered rectilinear pulse, for example, signal 1 in Fig. s f in \u003d 50 Hz in a one-sided file), then the ACF of the signal is indicated by the viraz: Fig ω B s () = (W o /) at 0 cos () d = (Wo in /) sin (c) / (y). The signal correlation interval to takes into account the value of the width of the central peak of the ACF in

10 10 maximum to the first cross of the zero line. In this way for rectilinear spectrum with the upper limiting frequency to the first limit of zero, sinc (v) = 0 at v =, stars: k = / v = 1 / 2f v. (6.3.4) The correlation interval is smaller than the upper limit frequency of the signal spectrum. For signals with a smooth peak at the upper cutoff frequency, the role of the parameter in the graph is the average width of the spectrum (signal 2 in Fig. 1). Spectral amplitude of the intensity of the statistical noise with a single variance of the depression function W q () from the average values ​​of W q () q 2, de q 2 dispersion of noise. At the boundary, with an even spectral spread, the noise level was 0 to, the ACF noise was up to the value of B q () q 2 at 0, B q () 0 at 0, so the statistical noise is not correlated (up to 0). Practical calculations of the ACF of financial signals are interspersed with an interval of failures = (0, (3-5)), in which, as a rule, the main information on the autocorrelation of signals is collected. The spectral width of the VKF can be taken away on the basis of the same spectroscopy, as for ROS, or directly from the formula (6.3.1) by replacing the spectral width of the signal S () with the spectral width of another signal U (): su () = (1/2 ) S * () U () exp (j) d (6.3.5) Otherwise, when changing the order of signals: us () = (1/2) U * () S () exp (j) d (6.3.5 ") Tver S * () U () is a mutual energy spectrum W su () signals s (t) and u (t). Obviously, U * () S () = W us (). Also, yak i ACF , cross-correlation function and spectral width of the mutual tension of the signals connected between themselves by Fur'є transformations: B su () W su () W * us (). (6.3.6) B us () W us () W * su ( ). (6.3 .6 ") In a more general way, behind the spectrum of pairing functions, consider the lack of pairing for the functions of the VKF slid, which reciprocates the energy spectra with complex functions: U () = A u () + j B u (), V () = A v () + j B v (). W uv = A u A v + B u B v + j (bu A v - A u B v) = Re W uv (w) + j Im W uv (), the destruction of the VKF maximum is formed. In the figure, one can visually show the peculiarities of the VKF molding on the butt of two of the same for the shape of the signals, which are visible one of the other. Rice Forming VKF. The shape of the signals and their reciprocal induction on visual A. The modulus and argument of the spectrum of the signal s (t) on visual B. The modulus of the spectrum u (t) is the same as the modulus S (). On this view, the modulus of the spectrum of the mutual tension of the signals S () U * () is induced. Apparently, when complex spectra are multiplied, the spectra modules are multiplied, and the phases are added together, with which for the resulting spectrum U * () the phase cod changes sign. Like perchim in shape -

11 11 If the calculation of the VKF (6.2.1) is a varto signal s (t), and the signal u (t-) on the y-axis is shifted in front of s (t), then the phases of the cut S () with increasing frequencies increase in the direction of negative values ​​of the cuts ( without adjusting the periodic drop value by 2), and the phase cuts U * () behind the absolute values ​​are less than the phase cuts s (t) і increase (for the gain rate) towards positive values. As a result of the multiplication of the spectra (as can be seen in Fig, view C) and the phase cuts S () the value of the cuts U * (), at the same phase of the spectrum S () U * () are overfilled in the region of visible values, which is safe destruction of all functions of the VKF (and її peak values) to the right from zero along the axis by a single value (for the same signals by the value of the difference between the signals along the ordinate axis). When the cob position of the signal u (t) is shifted to the side of the signal s (t), the phase coils S () U * () change, in between to zero values ​​​​at the total signal input, with which the function B su (t) shifts to zero values, in the interval between the two in the ACF (for the same signals s (t) and u (t)). As it seems to determine the signals, as the spectra of two signals do not overlap and, obviously, the mutual energy of the signal is equal to zero, so the signals are orthogonal to one to one. The connection between energy spectra and correlation functions of signals shows one more side of signal interaction. If the spectra of the signals do not overlap and their mutual energy spectrum is equal to zero at all frequencies, then in case of any time disturbances, one or more of the VKFs is also equal to zero. And tse means that such signals are uncorrelated. It is useful both for determinants and for vipadical signals and processes. Calculation of correlation functions for additional SPF є, especially for long numerical series, in tens and hundreds of times by the same method, lower by the last breakdowns in the temporal region with large correlation intervals. The essence of the method is based on formulas (6.3.2) for the ACF and (6.3.6) for the VKF. Looking at how the ACF can be considered as a number of variations of the VKF for one and the same signal, the calculation process can be seen in the application of the VKF for signals x () and y () with the number of variables K. Vіn includes: 1. Calculation of the FFT spectra of signals x ( ) X () and y () Y (). With a different quantity, a larger short row is added with zeros to the size of a larger row. 2. Calculation of the spectrum of tightness of tension W xy () = X * () Y (). 3. Zvorotne FFT W xy () B xy (). It is significant that the method is special. With a reverse FFT, apparently, a cyclic cluster of functions x () 3 y () is calculated. Even though the number of example functions is one K, the number of complex example spectra of functions is also one K, so the same number of example creates W xy (). Obviously, the number of responses B xy () in the reverse FFT is also the same. double-sided expansion becomes a 2K point. Also, in case of reversed FFT with the improvement of the cyclicity of the fold, there will be an overlay on the head period of the VKF of the її bіchnyh periods, like in the case of the primary cyclical fold of two functions. Figure B1 is a linear sequence, B2 FFT without signal continuation with zeros, B3 FFT with signal continuation with zeros. VKF, calculated by linear convolution (B1xy) and cyclic convolution through FFT (B2xy). To turn off the effect of superposition of the last periods, it is necessary to add the signals with zeros, in between, before the number of indications is reduced, with which the result of the FFT (chart В3ху on the little 6.3.5) will completely repeat the result of the linear reduction (with the correction of the normalization in the output). In practice, the number of zeros for the continuation of signals is determined by the nature of the correlation function. The minimum number of zeros is taken equal to the significant informational part of the functions, so that the order (3-5) of correlation intervals.

12 12 p. LITERATURE 1. Baskakov S.I. Radiotechnical lancers and signals Pdruchnik for universities. - M. Vishcha school, R. Enokson, L. Applied analysis of watch series. M .: Mir, p. 25. Sergienko A.B. digital processing signaling / A tutor for universities. St. Petersburg: Peter, p. 33. Eifcher E., Jervis B. Digital signal processing. Practical pidkhid. / M., "Williams", 2004, 992 Author's site ~ Lectures ~ Workshop

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Chapter 8 Functions and graphs Changes and fallows between them. Two values ​​\u200b\u200ba are called directly proportional, as if they are constant, i.e. Yaksho \u003d, de constant number that does not change with change

Federal state lighting budget setting higher professional education Volga State University of Telecommunications and Informatics Department of SARS

Lecture 10. Schrodinger's algorithm for assigning terms and orbitals of stationary stations

Methodical instructions of the DISCIPLINE "VIPADKOV processes in radio engineering" FOR STUDENTS IN GROUP VDBV-6-14 References 1. Statistical analysis and synthesis of radio engineering devices and systems:

Topic 8 Linear Discrete System Concept Discrete System Methods for describing linear discrete systems: impulse response, Frequency transfer function

Lecture 6 (page 358-36) Discrete Four's Reversal (DFT) Direct Z Reversal Designation of direct and reverse discrete Four's Reversal Let's take a look at the calculation algorithm of Four's Reversal

Option 8 Know the scope of the function: y sin The scope of the function is defined by two irregularities: i sin

Spectral analysis and synthesis Digital sound and video Lecture 2 + 2 Analysis and synthesis

Correlation is a mathematical operation, similar to convolution, allowing you to take two signals out of the third. Buvay: autocorrelation (autocorrelation function), cross-correlation (cross-correlation function, cross-correlation function). butt:

[Cross correlation function]

[Autocorrelation function]

Correlation - the same technique of revealing behind a range of signals on aphids of noise, also called optimal filtering. Although the correlation is more similar to a bunch, but the stench is counted in a different way. The areas of congestion are also different (c (t) = a (t) * b (t) - a bunch of two functions, d (t) = a (t) * b (-t) - mutual correlation).

Correlation - the same fold, only one of the signals is inverted to the right. Autocorrelation (autocorrelation function) characterizes the degree of connection between the signal and its destruction by τ copy. The cross-correlation function characterizes the degree of connection between 2 different signals.

The power of the autocorrelation function:

• 1) R (τ) = R (-τ). The function R (τ) is paired.
• 2) If x (t) is a sinusoidal function of the hour, then її autocorrelation function is a cosinusoidal tiєї frequency. Information about the bud phase is included. If x (t) = A * sin (ωt + φ), then R (τ) = A 2/2 * cos (ωτ).
• 3) The function of autocorrelation and the spectrum of tension associated with Four's transformations.
• 4) If x (t) is a periodic function, then R (τ) for it can be represented in the form of a sum of autocorrelation functions in the form of a constant warehouse and a sinusoidally changing warehouse.
• 5) The function R (τ) did not carry any information about the cob phases of the harmonic warehouse signal.
• 6) For the vypadkovy function, the hour R (τ) changes rapidly with the increase in τ. An hour interval after which R (τ) becomes equal to 0 is called the autocorrelation interval.
• 7) For a given x (t) it can be used as a whole R (τ), but for one and the same R (τ) can be different functions x (t)

Output signal with noise:

Autocorrelation function of the output signal:

Power of Mutual Correlation Function (VKF):

• 1) VKF is not a paired or unpaired function, so R xy (τ) is not equal to R xy (-τ).
• 2) VKF becomes unchanged when the function is changed and the sign of the argument is changed, so R xy (τ) = R xy (-τ).
• 3) If the variable functions x (t) and y (t) do not replace the permanent warehouses and are created by independent pockets, then for them R xy (τ) is right up to 0. Such functions are called uncorrelated.

Output signal with noise:

Square wave w frequency:

Correlation of output signal and meander:

Respect! Leather electronic summary of lectures and intellectual authority of its author and publications on the site, exclusively for educational purposes.

Mutual correlation function (CCF) of different signals (cross-correlation function, CCF) describes how the steps of the similarity of the formation of two signals, so they are mutually spreading one at a time along the coordinate (independent change). Using the formula (6.1.1) of the autocorrelation function for two different signals s (t) and u (t), it is necessary to advance the scalar wave of signals:

B su () = s (t) u (t + ) dt. (6.2.1)

Mutual correlation of signals characterizes the same correlation of phenomena and physical processes that are reflected by these signals, and can serve as a peace of "stability" given mutual connection with different processing of signals in different outbuildings. For terminal signals in terms of energy, the VKF is also finite, with which:

| B su () |  || s (t) ||  || u (t) ||,

scho vyplivaє z nerіvnostі Koshі-Bunyakovskyi and nezalezhnostі norms singlіv vіd zsuvu for coordinates.

When changing the change t = t- in the formula (6.2.1), we take:

B su () = s (t-) u (t) dt = u (t) s (t-) dt = B us (-).

It is clear that for the VKF the mind parity does not count, B su ()  B su (-), and the value of the VKF is not the goiter of the mother at a maximum at  = 0.

Mal. 6.2.1. Signals and VKF.

Tse can be nachalno bachiti in fig. 6.2.1, where two identical signals are given with centers at points 0.5 and 1.5. Calculation for the formula (6.2.1) with incremental increments, the value  means the successive destruction of the signal s2 (t) to the left along the clock axis (for the skin value s1 (t), for the integrand, the values ​​s2 (t + ) are taken). When  = 0, the signals are orthogonal and the value B 12 () = 0. The maximum B 12 () will be alarmed when the signal s2 (t) is sent to the left on the value  = 1, when it is taken outside the signal s1 (t) and s2 (t + ).

One and the same values ​​of the VKF for the formulas (6.2.1) i (6.2.1 ") are predicted for one and the same mutual position of the signals: when zsuvі on the interval , the signal u (t) is visible s (t) to the right along the y-axis i signal s (t) as well as signal u (t) to the left, then B su () = B us (-

Mal. 6.2.2. Mutual covariance functions of signals.

On fig. 6.2.2 pointing the application of the VKF for a direct signal s (t) and two identical tricurrent signals u (t) and v (t). All signals may have the same trivality T, with which the signal v (t) breaks forward by the interval T / 2.

The signals s (t) and u (t) are the same in terms of the time difference and the area of ​​"overshoot" of the signal is maximum at  = 0, which is fixed by the function B su. At the same time, the function B su is sharply asymmetric, so that with an asymmetric shape of the signal u (t) for a symmetrical shape s (t) (to the center of the signals) the area of ​​"overshoot" of the signals changes according to the difference in  as zero). When the position of the signal u (t) is shifted to the left along the ordinate axis (on the forward signal s (t) - the signal v (t)) the VKF shape is lost without change and shifts to the right on the same value of the sound value - the function B sv in Fig. 6.2.2. Keeping in mind the expressions of the functions in (6.2.1), the new function B vs will be a mirror-rotated function B sv = 0.

With the improvement of these features outside the VKF, they are calculated, as a rule, for positive and negative signs:

B su () = s (t) u (t + ) dt. B us () = u (t) s (t + ) dt. (6.2.1")

Mutual correlation of noisy signals . For two noisy signals u (t) = s1 (t) + q1 (t) і v (t) = s2 (t) + q2 (t), stopping the method of derivation of formulas (6.1.13) with a replacement copy of the signal s (t ) to the signal s2 (t), it is not important to enter the formula of mutual correlation in the offensive form:

B uv () = B s1s2 () + B s1q2 () + B q1s2 () + B q1q2 (). (6.2.2)

The remaining three terms in the right part (6.2.2) fade to zero when  is increased. At great intervals, the task of signaling viraz can be written in the offensive form:

B uv () = B s 1 s 2 () +
+
+
. (6.2.3)

At zero average values ​​of noise and statistical independence of signals in May:

B uv () → B s 1 s 2 ().

VKF of discrete signals. All the power of the VKF of analog signals is valid for the VKF of discrete signals, while for them the validity and singularity of discrete signals is greater for discrete ACF (formulas 6.1.9-6.1.12). Zocrema, with t ​​= const = 1 for signals x (k) і y (k) with the number of responses Up to:

B xy (n) =
x k y k-n. (6.2.4)

When normalized in units of tightness:

B xy (n) = x k y k-n 
. (6.2.5)

Evaluation of periodic signals in noise . The noisy signal can be estimated by the mutual correlation with the "reference" signal by the method of trials and pardons by adjusting the mutual correlation function to the maximum value.

For signal u (k) = s (k) + q (k) with statistical independence of noise i → 0 the cross-correlation function (6.2.2) with the template signal p (k) at q2 (k) = 0 looks like:

B up (k) = B sp (k) + B qp (k) = B sp (k) + .

And the shards → 0 as N increases, then B up (k) → B sp (k). Obviously, the function B up (k) will have a maximum if p (k) = s (k). By changing the shape of the template p (k) and maximizing the function B up (k), we can take the estimate s (k) from the seemingly optimal shape p (k).

The function of mutual correlation coefficients (VKF) is a measure of the degree of similarity of signals s (t) and u (t). Similarly to the function of autocorrelation coefficients, it is calculated through the centering of the function values ​​(to calculate the mutual covariance, center only one of the functions), and normalize to the additional value of the standard functions s (t) and v (t):

 su () = C su () /  s  v. (6.2.6)

The interval for changing the value of the correlation coefficients in case of collapses  can change from -1 (permanently reversed correlation) to 1 (permanently similarity or stovidsotkovy correlation). In case of failures , on which zero values ​​ su () are expected, the signals are independent of one type of one (uncorrelated). Mutual correlation coefficient makes it possible to establish the presence of a connection between signals independently of the physical powers of the signals and their magnitudes.

When calculating the VKF of noisy discrete signals in the interchangeable time with different formulas (6.2.4) the value  su (n) | > 1.

For periodic signals of the understanding of the VKF, the sound does not stop, for a few signals with the same period, for example, signals for entry and exit when the characteristics of systems are changed.