of bandwidth // Proceedings of International Conference CLEO’00. 2000, paper CMB2, R. 7. 13. Matuschek N.,. Kdrtner F. X and Keller U. Exact coupled-mode theories for multilayer interference coatings with arbitrary strong index modulations” IEEE J. Quantum Electron. 1997 Vol. 33, no. 3: R. 295-302.
Received to the editorial board 11/12/2005
Reviewer: Dr. Phys.-Math. sciences, prof. Svich V.A.
Yakushev Sergey Olegovich f-ta ET KNURE. Scientific interests Keywords: systems and methods for the formation of ultrashort pulses and methods for their simulation; semiconductor optical amplifiers of ultrashort optical pulses. Hobbies: sports. Address: Ukraine, 61166, Kharkiv, Lenin Ave., 14.
Shulika Aleksey Vladimirovich, assistant of the Department of Physical Education, KNURE. Research interests: physics of low-dimensional structures, effects of charge carrier transfer in low-dimensional heterostructures, simulation of active and passive photonic components. Hobbies: travel. Address: Ukraine, 61166, Kharkiv, Lenin Ave., 14, [email protected]
UDC621.396.2.: 621.316.2 "
ESTIMATION OF THE IMPULSE RESPONSE OF A COMMUNICATION CHANNEL ON THE BASIS OF HIGHER ORDER STATISTICS
TIKHONOV V.A., SAVCHENKO I.V.___________________
A computationally efficient method for estimating the impulse response of a communication channel using a third-order moment function is proposed. The computational complexity of the proposed method is compared with the method that uses fourth-order cumulants to estimate the impulse response. It is shown that in the presence of Gaussian and non-Gaussian noise, the proposed method provides a higher estimation accuracy.
1. Introduction
Intersymbol interference (ISI) that occurs during high-speed transmission digital signals, is, along with narrow-band interference from similar digital systems operating on adjacent wires of a telephone cable, the main factor that reduces the reliability of information transmission in xDSL systems. The optimal ISI correction method from the point of view of minimizing the error probability, based on the maximum likelihood rule, as well as methods using the Viterbi algorithm for maximum likelihood estimation of sequences, require estimating the impulse response of the communication channel.
For this purpose, higher-order statistics can be used. Thus, the method of blind identification is described by estimating the impulse response of the channel from the received signal using fourth-order cumulants. In the present 3 0
Lysak Vladimir Valerievich, Ph.D. Phys.-Math. Sciences, Art. pr. of the Department of Physical Education for Electronics of KNURE. Scientific interests: fiber - optic data transmission systems, photonic crystals, ultrashort pulse generation systems, methods for modeling the dynamic behavior of semiconductor lasers based on nanoscale structures. Student, member of IEEE LEOS since 2002. Hobbies: sports, traveling. Address: Ukraine, 61166, Kharkiv, Lenin Ave., 14, [email protected]
Sukhoivanov Igor Alexandrovich, Doctor of Phys.-Math. Sciences, Professor of the Department of Physical Education and Ethics of KNURE. Head of the international scientific and educational laboratory "Photonics". Honorary Member and Head of the Ukrainian Branch of the Society for Laser and Optoelectronic Engineering of the International Institute of Electronics Engineers (IEEE LEOS). Scientific interests: fiber-optic technologies, semiconductor quantum-sized lasers and amplifiers, photonic crystals and methods for their modeling. Hobbies: travel. Address: Ukraine, 61166, Kharkiv, Lenin Ave., 14, [email protected]
In this paper, it is proposed to use a third-order moment function for estimating the impulse response. This approach makes it possible to improve the accuracy of estimating the impulse response of a communication channel, and hence the efficiency of intersymbol interference suppression in the presence of additive Gaussian and non-Gaussian noise. The proposed method has less computational complexity compared with while maintaining the accuracy of identification in the presence of Gaussian noise. The condition for applying the proposed method is the non-Gaussianity of the test signals at the input x[t] and output y[t] of the communication channel, which must have a third-order moment function other than zero.
The aim of the study is to develop a method for improving the accuracy of estimating the impulse response of a communication channel in the presence of Gaussian and non-Gaussian noise, reducing computational costs.
The tasks are: substantiation of the possibility of using a third-order moment function to calculate the discrete impulse response of a communication channel; obtaining an expression relating the moment function of the third order with the discrete impulse response; comparison of the effectiveness of using the proposed method and the method based on the use of a fourth-order cumulant for estimating the impulse response.
2. Estimation of the impulse response of a communication channel from a fourth-order cumulant function
It is possible to estimate the characteristics of the communication channel from the received signal using higher-order statistics. In particular, the impulse response of a linear, time-invariant system with
discrete time can be obtained from the fourth-order cumulant function of the received signal, provided that the channel input is non-Gaussian.
3. Estimation of the impulse response of a communication channel from a third-order moment function
Let the signal z[t] be the sum of the transmitted signal y[t] transformed by the channel with discrete time and memory L+1 and the additive white Gaussian noise (AWGN) n[t]:
z[t] = y[t] + n[t] =2hix + n[t].
For AWGN, the kurtosis coefficient and the fourth-order cumulant function are equal to zero. Therefore, the cumulant function of the fourth order of the received signal z[t] is determined only by the cumulant function of the transmitted signal converted by the channel y[t]. The fourth-order cumulant function of a real centered process y[t] is expressed in terms of moment functions
X4y(y[t],y,y,y) =
E(y[t] yy y) -
E(y[t] y)E(y y) - (1)
E(y[t] y)E(yy) -
E(y[t]y)E(yy),
where E(-) is the operation of mathematical averaging.
The first term in (1) is the fourth-order moment function, and the remaining terms are the products correlation functions for some fixed shifts.
In the method of blind identification, for estimating the impulse response of a communication channel, a useful binary signal is processed, which has no statistical connections. It has a uniform distribution with non-zero one-time fourth-order cumulant % 4X. Then the transformation of the fourth-order cumulant function by a linear system with a discrete impulse response ht is given by
Х4x Z htht+jht+vht+u
It can be shown that in this case the impulse response of the communication channel is determined through the values of the cumulant function of the output signal z[t] 6:
where p = 1,.., L . Here, the values of the fourth-order cumulative function % 4z are estimated from the samples of the received signal sequence z[t] according to (1).
Let us consider the case when there is an additive non-Gaussian noise at the channel output with a uniform distribution of the probability density. The fourth-order cumulant function of such interference is not equal to zero. Therefore, the fourth-order cumulative function of the received useful signal z[t] will contain an interference component. Therefore, when estimating the impulse response of a communication channel using expression (2) for small signal-to-noise ratios, it will not be possible to achieve high accuracy of estimates.
To improve the accuracy of estimating the discrete impulse response of a communication channel in the presence of non-Gaussian noise, in this paper, it is proposed to calculate the values of the impulse response readings from the third-order moment function. The third-order moment function of the real process y[t] is defined as
m3y=shzu=
E(y[t]yy). W
The transformation of the third-order moment function by a linear system with a discrete impulse response ht, according to , is determined by the expression
m3y = Z Z Z (hkhlhn x
k=-w 1=-something n=-something
x Wx ).
If the test signal x[t] is non-Gaussian white noise with non-zero skewness, then
m3x=
Ш3Х 55, (5)
where m3x is the central moment of the third order of the signal at the channel input.
Substituting expression (5) into expression (4), we obtain
m3y = Z Z Zhkh1hn х k=-<х 1=-<х n=-<х)
x m3x5 5 =
M3x Zhkhk+jhk+v.
Taking into account that the third-order moment function of a non-Gaussian interference with a uniform distribution is equal to zero, we obtain
m3z=m3y=
M3x Z hkhk+jhk+v (6)
Let shifts j = v = -L. Then, under the sum sign in (6), the product of the impulse response coefficients of the physically realized filter will differ from zero only for k = L , i.e.
m3z[-L,-L] = m3xhLh0 . (7)
With shifts j = L, v = p under the sum sign in (6), the product of the impulse response coefficients will differ from zero only at k = 0. Therefore,
m3z = m3xh0hLhp. (8)
Using expression (8), taking into account (7), we obtain the samples of the discrete impulse response through the values of the moment function:
m3z _ m3x h0hLhp _ m3z[_L,_L] m3xhLh° h0
The samples of the third-order moment function m3z are estimated by averaging over the samples of the received signal sequence z[t] according to (3).
Methods for estimating the impulse response of a communication channel, based on the calculation of the third-order moment function and the fourth-order cumulant function , can be used when a non-Gaussian test signal with non-zero kurtosis and skewness coefficients is used. It is advisable to use them in the case of Gaussian noise, in which the third-order moment function and the fourth-order cumulant function are equal to zero. However, the method proposed in the article has a much lower computational complexity. This is explained by the fact that in order to estimate one value of the fourth-order cumulant function according to (1), it is required to perform 3N + 6N + 13 operations of multiplication and addition. At the same time, in order to estimate one value of the third-order moment function, according to (3), it will be necessary to perform only 2N + 1 multiplication and addition operations. Here N is the number of samples of the test signal. The remaining calculations performed according to (2) and (9) will require the same number of operations for both methods.
4. Analysis of simulation results
The advantages of the proposed method for estimating the impulse response of a communication channel in the presence of Gaussian and non-Gaussian interference are confirmed by the results of experiments that were carried out by the method of statistical modeling. The inefficiency of the blind equalization method in the presence of Gaussian noise is explained by the fact that when
blind identification uses an equiprobably distributed signal. The two-level pseudo-random sequence has a kurtosis factor of 1 and a fourth-order cumulant of -2. After filtering by a narrow-band communication channel, the signal is partially normalized; its kurtosis coefficient approaches that of Gaussian noise, which is zero. The value of the cumulant of the fourth order approaches the value of the cumulant of the fourth order of the Gaussian signal, which is also equal to zero. Therefore, at low signal/(Gaussian noise) ratios and in cases where the fourth-order cumulants of the signal and noise differ slightly, accurate identification is not possible.
Experiments have confirmed that at low signal-to-noise ratios, the blind identification method is ineffective. Through the model of the communication channel with a given discrete impulse response, the coefficients of which were 0.2000, 0.1485, 0.0584, 0.0104, a signal was passed in the form of a two-level pseudo-random sequence with a length of 1024 samples. Correlated Gaussian noise and AWGN were added to the channel output signal. The amplitude-frequency characteristic (AFC, Amplitude response characteristic - ARC) of the communication channel model is represented by curve 1 in fig. 1.
Rice. 1. True frequency response and estimates of the frequency response of the communication channel model, PSD of Gaussian noise
Here and below, the abscissa axis shows the values of the normalized frequency f" = (2f) / ^, where ^ is the sampling frequency. The power spectral density (PSD) of the correlated noise obtained using the shaping autoregressive filter is shown in Fig. 1 curve 2 According to (2), the discrete impulse response of the communication channel was estimated at large signal-to-noise and signal-to-noise ratios equal to 15 dB, as well as at lower signal-to-noise and signal-to-noise ratios, equal to 10 dB and 3, respectively. dB Noise and interference were Gaussian Estimates of the frequency response of the communication channel corresponding to the found discrete impulse responses are shown in Fig. 1 (curves 3 and 4).
In this paper, it is shown that to identify a communication channel using fourth-order cumulants at low signal-to-noise ratios, test non-Gaussian signals can be used, the coefficient of kurtosis of which, even after normalization by the communication channel, is noticeably different from zero. When modeling, a test signal with a gamma distribution with a shape parameter c=0.8 and a scale parameter b=2 was used. The coefficient of signal kurtosis at the channel input was 7.48, and at the channel output it was 3.72.
On fig. Curves 1 and 2 in Fig. 2 show the frequency response of the communication channel model and the PSD of the correlated interference. The signal/noise and signal/noise ratios were 10 dB and 3 dB, respectively. Noise and interference were Gaussian. The estimate of the frequency response of the communication channel, found from the estimate of the discrete impulse response (2), is shown in fig. 2 (curve 3).
Rice. 2. True frequency response and estimates of the frequency response of the communication channel model, PSD of Gaussian noise
In the presence of Gaussian interference and AWGN in the communication channel, it is proposed to use a more computationally efficient identification method based on the use of a third-order moment function. In this case, it is necessary that the asymmetry coefficient of the test signal at the output of the communication channel be non-zero, i.e. differed from the skewness coefficient of the Gaussian noise. For statistical experiments, a test signal with a gamma distribution with a shape parameter c=0.1 and a scale parameter b=2 was used. The signal asymmetry coefficient at the channel input was 6.55, and at the channel output it was 4.46.
The estimate of the frequency response of the communication channel model, found from the estimate (9) of the discrete impulse response, is shown in fig. 2 (curve 4). Analysis of the graphs in fig. 2 shows that the accuracy of the frequency response estimation using fourth-order cumulant functions and third-order moment functions is approximately the same.
We also considered the case of simultaneous presence of white noise with Gaussian and non-Gaussian distribution in the communication channel. In statistical modeling, a test signal with gamma
distribution, with shape parameter c=1 and scale parameter b=2. The signal kurtosis coefficient at the channel output was 2.9, while the interference kurtosis coefficient with a uniform probability density distribution was -1.2. The signal asymmetry coefficient at the channel output was equal to 1.38, and the estimate of the interference asymmetry coefficient was close to zero.
Curve 1 in fig. 3 shows the frequency response of the communication channel model, and curves 2 and 3 show estimates of the communication channel frequency response using fourth order cumulants (2) and third order moment function (9). The signal-to-noise ratio was 10 dB, and the signal-to-noise ratio was 3 dB.
Rice. 3. True frequency response and estimates of the frequency response of the communication channel model
As can be seen from the graphs presented in Fig. 3, when using a method based on the calculation of fourth-order cumulants to identify a communication channel, interference with a non-zero kurtosis coefficient at small signal-to-noise ratios significantly reduces the identification accuracy. At the same time, when a third-order moment function is used to identify a communication channel, interference with a zero asymmetry coefficient will not significantly affect the accuracy of impulse response estimation at low signal-to-noise ratios.
5. Conclusion
For the first time, a method for estimating the impulse response of a communication channel using a third-order moment function is proposed. It is shown that the use of the proposed identification method can significantly reduce the influence of non-Gaussian interference on the accuracy of estimating the impulse response of the channel. With Gaussian noise in the communication channel, the proposed method, in comparison with the method for estimating the impulse response by fourth-order cumulants, has a much lower computational complexity and can be used in the case of using a non-Gaussian test signal.
The scientific novelty of the research, the results of which are presented in the article, lies in the fact that for the first time
expressions for calculating the coefficients of the discrete impulse response of the communication channel from the values of the third-order moment function are given.
The practical significance of the results obtained lies in the fact that the proposed identification method provides an increase in the accuracy of estimating the impulse response of a communication channel in the presence of interference, as well as a more effective suppression of intersymbol interference using the Viterbi algorithm and other methods that require a preliminary assessment of the characteristics of the communication channel. channel.
References: 1. R. Fischer, W. Gerstacker, and J. Huber. Dynamics Limited Precoding, Shaping, and Blind Equalization for Fast Digital Transmission over Twisted Pair Lines. IEEE Journal on Selected Areas in Communications, SAC-13: 1622-1633, December, 1995. 2. G.D. Forney. Maximum Likelihood Sequence Estimation of Digital Sequences in the Presence of Intersymbol Interference. IEEE Tr. IT, 363-378, 1972. 3. Forney G.D. The Viterbi Algorithm. Proceedings of the IEEE, vol. 61, no. 3, March, 1978, pp. 268-278. 4. Omura J. Optimal Receiver Design for Convolutions Codes and Channels with Memory Via Control Theoretical Concepts,
inform. Sc., Vol. 3. P. 243-266. 5. Prokis J. Digital communication: TRANS. from English. / Ed. D.D. Klovsky. M: Radio and communication, 2000. 797 p. 6. Malakhov A.N. Cumulant analysis of random non-Gaussian processes and their transformations. M.: Sov. radio, 1978. 376 p. 7. Tikhonov V.A., Netrebenko K.V. Parametric Estimation of Higher-Order Spectra of Non-Gaussian Processes // ACS and Automation Instruments. 2004. Issue. 127. S. 68-73.
Received to the editorial board 06/27/2005
Reviewer: Dr. tech. Sciences Velichko A.F.
Tikhonov Vyacheslav Anatolievich, Ph.D. tech. Sciences, Associate Professor of the Department of RES KNURE. Research interests: radar, pattern recognition, statistical models. Address: Ukraine, 61726, Kharkiv, Lenin Ave., 14, tel. 70215-87.
Savchenko Igor Vasilievich, post-graduate student, assistant of the department of RES KNURE. Scientific interests: methods of intersymbol interference correction, higher-order spectra, non-Gaussian processes, linear prediction theory, error-correcting coding. Address: Ukraine, 61726, Kharkiv, Lenin Ave., 14, tel. 70-215-87.
Goryachkin O.V.
The article deals with the actual problem of blind identification of a communication channel. To solve the problem
polynomial representations of cumulants of random sequences of finite length are used.
This approach makes it possible to use the methods of algebraic geometry and commutative algebra to construct blind identification algorithms. A number of blind identification algorithms are described that use the properties of manifolds of a given correlation value. The results of modeling and comparative analysis of the effectiveness of the proposed algorithms are presented. It is shown that the algorithm based on the use of the non-zero correlation transformation provides better noise immunity characteristics than the known spectral factorization algorithm.
BLI D IDE TIFICATIO OF TELECOMMU ICATIO CHA ELS WITH USE AFFI E
VARIETIES OF POLY OMIAL CUMULA TS
Oleg V. Goriachkin In the paper a blind identification problem of telecommunication channels are discussed. For solution of the blind identification problem the equations connecting with polynomial moments are used. In the case we can use the powerful methods of commutative algebra. In the paper some blind identification algorithms based on the analysis of independence affine varieties of polynomial cumulants are proposed.1. Introduction In recent years, there has been great interest in the so-called"blind problem". In general, the task of blind processing can be formulated as digital processing of unknown signals that have passed through a linear channel or medium with unknown characteristics against the background of additive noise. Blind identification is the opposite of the problems of classical system identification, where both the observed signal and the input signals are considered given. The increase in research activity in the “blind problem” seems to be due to the potential application in mobile radio communication systems, which are currently being intensively developed. In these systems, the distortion caused by multipath interference affects both the transmission quality and their throughput. Usually, receivers of such systems require either knowledge of the channel parameters or the transmission of some test signal to compensate for distortion.
For channels with variable parameters, the efficiency loss can be significant. For example, in cellular communication systems, the time used to transmit a test signal can take up to 30% of the total transmission time. Another example is computer networks, where the connection between the terminals and the central computer is established in an asynchronous manner, so that in some cases, learning the receiver is not possible. Outside the field of communication, blind channel estimation is applied in various areas:
compensation of distortions caused by propagation effects in radar and radio navigation systems, linear distortion correction in imaging systems, seismic signal processing in geophysics, distortion compensation in speech recognition systems.
An important issue in solving problems of blind identification is the identifiability of the system. Blind system identifiability is understood as the possibility of restoring the transfer function and/or impulse response (IR) of the system with an accuracy up to a complex factor only from the output signals. For channels with one input and one output, the identifiability conditions are formulated in the context of statistical identification. Statistical identification assumes the presence of a certain set of output signal implementations, during the formation of which the IR of the channel is constant. In this case, the system is identifiable if there is a non-stationary or non-Gaussian random process at the input.
For the first time, an algorithm for direct blind equalization of a communication channel, using the non-Gaussian nature of information signals in digital systems with amplitude modulation, was apparently proposed by Sato in 1975. . Sato's algorithm was subsequently generalized by Godard in 1980. for the case of combined amplitude-phase modulation (also known as "algorithm of constant modules"). To date, a large number of algorithms for blind identification and correction of communication channels are known that use various adaptation criteria for linear equalizers, which are combined in the literature into the class of stochastic gradient algorithms or Basgang algorithms. The basic limitations of these algorithms are relatively slow convergence, the requirement for reliable initial conditions, high computational complexity due to the presence of a procedure for nonlinear optimization of the equalizer coefficients, and low noise immunity.
Another class of blind identification algorithms, developed relatively recently, are algorithms that use the maximum likelihood rule. These algorithms provide asymptotic efficiency and consistency of the obtained estimates, have higher noise immunity, however, the computational complexity and local maxima are their two main problems.
Very tempting for the development of blind estimators is the method of moments, the essence of which is to replace the equations relating the signals at the input and output of the system with equations relating the corresponding moment functions. The estimates obtained within the method of moments are not the best among all estimates in terms of their asymptotic efficiency, however, this approach, as a rule, allows one to obtain an explicit channel estimate, bypassing the nonlinear optimization procedure. An important advantage of these methods in the context of the "blind problem" is the absence of requirements for a priori knowledge of the probability distributions of information signals and noise. It is well known that the covariance functions of a stationary process at the output of a linear system do not contain information about the phase of its transfer function, and identification is possible only for a narrow class of systems with a minimum phase. Historically, this has led to interest, first of all, in high-order statistics and, accordingly, in non-Gaussian models of input signals. The use of 2nd order statistics for blind channel identification, possibly for a non-stationary model of input or output signals and in the particular case of a periodically correlated (cyclostationary) signal. The possibility of such identification for telecommunication channels in the general case for a non-stationary input is shown in. As a rule, cumulative spectra (or “polyspectra”) are used to construct estimates within the method of moments, since in this case the equations for the unknown channel can be written in a simple algebraic form. In this paper, we develop a new approach to the synthesis of algorithms for statistical blind identification, based on the polynomial representation of the moments of random sequences.
For systems with a passive pause, the communication channel model can be described by a linear combination of polynomials of positive degree. Consider random polynomials as complex random fields defined on the complex plane. In this case, one can define the moment and cumulant functions of these random fields, which will be polynomials in many variables. Let x C n be a complex random vector described by the probability density f x (x1,..., xn) defined in k=k1+k2+…+kR, m=m1+m2+…+mR random vector x polynomial R variables as follows :
Obviously, the set of polynomial moments (2) defined in this way, taking into account the well-known problem of moments, completely determines the probability density function and the characteristic function of the complex random vector formed by the R values of the random polynomial x(z) C at the points (z1,..., z R ).
Polynomial moments do not commute the sum of independent random polynomials, so it is often more convenient to use generalized correlations or cumulants of random polynomial values. The polynomial cumulants of a random polynomial will be denoted by the letter "K". The equation relating the polynomial cumulants at the input and output of the identified system with a passive pause (3) can be written in the following form 2. Identification of an IR channel by manifolds of a given correlation.
This article discusses approaches to solving the problem of blind identification of systems with a passive pause. Note that, in contrast to systems with a test pulse, a passive pause takes 2 times less time.
Let x R n be a random vector described by the probability density f x (x1,..., xn) in R n. Let x(z) to the ring C be a random polynomial of degree n 1 given by a random vector x R n. Let x(z1) and x(z 2) be two different values of a random polynomial x(z).
Let us determine all possible values z1 z 2 for which x(z1) and x(z 2) have a given value of the correlation function by solving the system a polynomial equation of the form The affine manifold V2x,0 (t) defined in this way for each t in C 2 will be called the manifold a given (non-zero) correlation of a random polynomial x(z), and in the case of t = 0, a decorrelation manifold, or a manifold of zero correlation. If we choose m different complex numbers (c0,..., cm 1 ), so that any pair of these numbers V2x,0 (t), then we can define the corresponding linear mapping of the vector x R n into the vector y C m. The definition of a decorrelating manifold (4) can easily be generalized to the generalized sense . Let x1 (z), x2 (z),..., xn (z) be a collection of independent random polynomials.
Let Vkx,1m (t1),Vkx,2 (t 2),...,Vk,n (t n) corresponding to them be manifolds of given correlation.
Then the varieties resulting from the product of the corresponding polynomials are described by the following expressions If there are only very general decorrelating varieties about the statistics of the information sequence. Since the noise statistics are known, expression (3) can be written in the form It is a known fact, which is a consequence of Hilbert's theorem on the finite generation of an ideal, that any variety can be represented as a union of a finite number of irreducible varieties, and moreover, such a representation is unique if Vkh, m (0) Vkx, m (0) and vice versa. Obviously, if representation (6) is unique, then the manifold Vkh, m (0) completely characterizes the impulse response of the channel and can manifolds, and we do not need a priori knowledge of the moments of the information sequence. However, such a decomposition is an extremely difficult problem in the field of complex numbers. Therefore, we will use the difference in the dimensions of the manifolds generated by the IR channel and the information sequence. It is obvious that the manifold is a zero manifold, the manifold Vkx, m (0) usually has dimension 1, and in the particular case of independent, identically distributed samples of the information sequence, it is a bundle of curves in C R. Analyzing decomposition (6), taking into account their dimension, we can separate unknown manifolds by choosing different sections. That. blind identification algorithm (A1) at R=2 is reduced to the following sequence of actions:
1. Based on M implementations of the output signal, we estimate their polynomial covariance 2. Calculate the vectors containing the roots of polynomials in one variable 3. Form the vector rh containing the L closest roots in the plane C by If we have a priori information about the statistics of the input signal, then to build the algorithm blind identification, we can directly use the manifold structure of a given correlation of a random polynomial. Let x(z) to the ring C be a random polynomial of degree n 1, given by a random Gaussian vector x C n with zero mathematical expectation, independent components and variance of the components 2, then the manifold of the given correlation of the values of the random polynomial Consider now the case when the points are chosen so that the pairwise correlations of the components are not equal to zero, but they are not equal to each other, i.e. may belong to different manifolds of given correlations. Let the coordinates be ( 1,..., n 1) roots of the polynomial P (x). If t 0, then it can be shown that any pair combination of these roots is V1,x (0). This means that the value of the second mixed cumulant has the form Thus, we can construct a linear mapping of the vector x C n into the off-diagonal component vector. This means that the channel estimation algorithm is an algorithm for finding the eigenvector corresponding to the maximum eigenvalue .
That. blind identification algorithm (A2) is reduced to the following sequence of actions:
1. Transformation of pair correlations of the observed signal where: Vn1 (1,..., n1) - (n 1) n Vandermonde matrix; y k - k-th vector of observed signal samples.
2. Estimation of the sample covariance matrix 3. Calculation of the eigenvector of the matrix R = ri, j ti, j, 4. Calculation of the impulse response of the channel where the symbol "#" is the Moore-Penrose inversion.
3. Results of mathematical modeling To evaluate the effectiveness of the proposed approach, let us consider the characteristics of the algorithms in comparison with the well-known approach based on polyspectra. As was shown, the algebraic equation for the spectral moments of the 2nd order where H (m) is the transfer function of the channel, n = 0,..., the moments of the second order in (19) is determined in the form of a sequence and noise, and the spectral moment of the sequence of readings on the output of the channel is evaluated directly by the observed realizations. Algorithms for solving equation (13) with respect to the unknown channel transfer function can be obtained from the assumption that this equation is valid for the estimate Fyy (n, m). The spectral factorization algorithm (A3) minimizes the mean square of the error between the analytical and sample solution of equation (13) under the condition that the energy of the transfer function is normalized to unity and, of course, subject to the condition Fxx (m) 0. It is known that the solution in this case is the eigenvector of the Hermitian matrix corresponding to the maximum eigenvalue. Figure 1 shows the results of modeling the operation of the A3 algorithm. The relative error was calculated using the formula Q = E h h h. The impulse response is taken the same for all experiments h = (0.7,1.0,0.7). Figure 2 shows the results of mathematical modeling of the algorithm for blind identification of channel A1 over two sections of the decorrelating manifold V2y0 v (0) C 2. The sections are taken on planes in C. The noise immunity of this algorithm is lower than that of A3 for small signal-to-noise ratios, but tends to zero with a fixed sample. An important advantage of this algorithm is the absence of requirements for knowing the statistics of the information sequence, as well as the high rate of convergence. So, with a high value of the signal-to-noise ratio A, it gives an acceptable error when using only a few implementations (=3…5).
Figure 3 shows the results of modeling the algorithm A2. The noise immunity of this algorithm is higher than that of A3 at approximately the same convergence rate. Higher noise immunity is achieved here by using a non-zero correlation transformation, which ensures good conditionality of the matrix R, in contrast to the spectral factorization algorithm, where the condition Fxx (m) 0 is generally not satisfied in the case under consideration. In terms of computational complexity, all the considered algorithms are in principle equivalent.
4. Conclusion The use of polynomial representations of random vectors in blind identification problems has made it possible to find a number of new algorithms for blind identification of a communication channel based on the use of commutative algebra and algebraic geometry methods.
It is shown that the manifolds generated by polynomial cumulants have a number of unique properties. For example, zero-correlation manifolds generated by a random sequence and a deterministic channel can be separated by their dimension, i.e. it is possible to blindly identify the channel in the absence of a priori information about the statistics of the information sequence. It is shown that the algorithm based on the use of the non-zero correlation transformation provides better noise immunity characteristics than the spectral factorization algorithm.
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Fig.1. Relative identification error Q, algorithm A4, depending on the signal-to-noise ratio, for a different number of implementations =20 ("+"), = Fig.2. Relative identification error Q of the A1 algorithm depending on the signal-to-noise ratio, for various =0.01 (“+”), =0.03 (“o”), =0. Fig.3. Relative identification error Q of the A2 algorithm, depending on the signal-to-noise ratio, for a different number of implementations = 20 ("+"), = Goryachkin Oleg Valerievich, born in 1965, Doctor of Technical Sciences, Head of the Department of Theoretical Fundamentals of Radio Engineering and Communications PGATI Author more 90 scientific papers. Research interests: digital signal processing in radio engineering and communication systems, radiophysical methods of remote sensing of the Earth, radar with antenna aperture synthesis, blind identification of systems, applied statistics.
^ 3.7. Channel characteristics identification
Identification of the characteristics of any object is obtaining its mathematical model from an experimentally recorded response to a known input action. A linear filter is often used as a model, which is described in different ways: by the transfer function H(s), impulse response h(t), differential or difference equation in ordinary or matrix form. The filter parameters are determined by selection or as a result of solving equations based on experimental data. The criterion for the adequacy of the model is most often the minimum error variance e(t) = z(t) – y*(t), Where z(t) And y*(t) - signals at the outputs of the channel and filter (Fig. 17).
Consider the correlation method for identifying the impulse response of a filter simulating a channel. Output signal y*(t) of the filter is the convolution of the input signal x(t) and impulse response h(t):
Suppose, for simplicity, that the impulse response is described by three samples, i.e. filter output
Rice. 17 illustrates the formation of this signal by summation, with weights equal to the values of the input signal samples, shifted in time by the filter's discrete impulse responses. Components highlighted k th sample of the output variable. Error variance
Minimum variance conditions
Can be presented as follows
 |
Where
System (), written in general form linking the impulse response of the channel with the autocorrelation function of the input signal and the cross-correlation function of the input and output signals.
To obtain an adequate model of the object, the signal x(t) must be wideband and must not correlate with interference n(t). A pseudo-random sequence is used as such a signal. Its autocorrelation function has the form of a short pulse and, like the autocorrelation function of white noise, can be approximately represented as R x(τ) ≈ 0.5 N 0 δ(τ). In this case, equation (17) is simplified:
 | (18) |
and the estimation of the impulse response is reduced to the determination of the correlation function R zx (τ).
The solution of system (16) is complicated by the fact that it is often “ill-conditioned”: some equations turn out to be almost linearly dependent. In this case, slight changes in the experimentally found coefficients of the equations - discrete values of the correlation functions lead to fundamentally different solutions, including those devoid of physical meaning. This situation is typical for "inverse" problems, when the mathematical model of an object is determined by its input and output signals ("direct" problem - determining the reaction of an object with known characteristics to a given input signal is solved without any complications). To obtain a practically realizable model, the form of the equations of dynamics or characteristics of the model is set on the basis of physical considerations, and the numerical values of the model parameters, at which it is most adequate to the object, are selected in different ways, comparing the behavior of the object and the model. This identification is called "parametric". The considered "non-parametric" method of identification does not use any a priori information about the type of characteristics of the object.
1. What are the main indicators of the quality of a data transmission channel. What is channel volume.
2. How does the use of error-correcting coding affect the spectral and energy efficiency of the channel.
3. What the Nyquist and Kotelnikov theorems state.
4. Imagine the response to a square wave of a channel that is a low-pass filter, a wide band filter, and a narrow band filter.
5. How does the smoothing coefficient of the Nyquist filter affect the impulse response of the channel.
6. What factors determine the probability of a symbolic error.
7. What is the relationship between the signal-to-noise ratio and specific energy costs.
8. How does an increase in the volume of the alphabet of channel symbols affect the dependence of the probability of a symbol error on the signal-to-noise ratio and on the specific energy costs in amplitude-phase and frequency manipulation.
9. What is the difference between the concepts of technical and information speed of a data transmission channel
10. What is the bandwidth of the channel
11. What is the relationship between the maximum possible spectral efficiency of the channel and specific energy costs.
12. What is the theoretical value of the lower limit of specific energy costs.
13. Is it possible to correctly transmit messages with a high probability of errors in determining channel symbols
14. How is the amount of information per one character of the source alphabet estimated?
15. What is efficient coding, what are its advantages and disadvantages
16. How is the loss of signal power during transmission in free space
17. How the noise figure and effective noise temperature are determined
18. What phenomena are observed in a multipath channel
19. What parameters characterize a multipath channel
20. What is the relationship between time spread and channel frequency response
21. Explain the concepts of amplitude and frequency selective fading, Doppler shift and scattering.
22. Under what conditions does spreading the spectrum increase the noise immunity of a multipath channel
23. Explain the concept of parametric identification
Methods of multichannel data transmission
Multichannel data transmission is the simultaneous transmission of data from many sources of information over one communication line, also called multiple, or multiple, channel access, multiplexing, multiplexing, channel separation.
The main ways to separate channels are as follows.
frequency division (frequency division multiply access, FDMA): each subscriber has its own frequency range.
temporary separation (time division multiply access, TDMA): the subscriber is periodically allocated time slots to transmit a message.
Code division (code division multiply access, CDMA): each subscriber of a spread spectrum communication system is assigned a pseudo-random (pseudonoise - PN) code.
In the same system, different methods of distributing communication channels between subscribers can be used simultaneously. Separate communication channels can be permanently assigned to certain subscribers, or provided on request. The use of public channels provided for communication as needed (the trunking principle) dramatically increases the system throughput with an increase in the number of channels. Systems with dynamic channel assignment are called demand-assignment multiple access (DAMA) systems. To reduce the likelihood of conflicts that occur when several subscribers access the channel at the same time, special channel access control algorithms are used.
We will consider the principles of channel separation in digital systems using specific examples.
^ 4.1. Temporary channel separation
in a wired communication system
In systems with time multiplexing, sources and recipients of information are connected in turn to the communication channel (group path) by switches on the transmitting and receiving sides. One period of the switch is a cycle (frame, frame), in which all sources are connected to the channel once. Source data is transmitted during a “time slot”, a “window”. Part of the windows in the cycle is reserved for the transmission of service information and synchronization signals for the operation of the switches.
For example, in the European digital telephone system, data from 30 subscribers constitutes the primary digital data stream divided into frames. One frame with a duration of 125 µs contains 32 time windows, of which 30 windows are reserved for transmitting subscriber messages, 2 windows are used for transmitting control signals (Fig. 18, A). In one window, 8 bits of the message are transmitted. At an audio signal sampling rate of 8 kHz (sampling period 125 µs), the data rate in the primary stream is 8000 ∙ 8 ∙ 32 = 2.048 Mbps.
Four primary digital streams are combined into one secondary stream, 4 secondary streams are combined into a stream at a speed of 34 Mbps, etc. up to 560 Mbps for fiber transmission. The equipment that provides the combination of streams and their separation at the receiving end is called "muldex" (multiplexer - demultiplexer).
Digital streams are transmitted over communication lines by channel codes that do not have a constant component and provide self-synchronization. To group multiple threads, multiplex performs the following operations:
Translation of channel codes in each input stream into the BVN code with the representation of binary symbols by unipolar signals,
Sequential polling of all input channels during one bit and the formation of a combined stream of binary symbols in a unipolar BVN code (Fig. 18, b, the moments of the poll are marked with dots),
The binary symbol representation of the combined stream in the channel code. In addition, framing words are introduced into the combined stream.
Transmission speeds in different streams are slightly different. To match the speeds, intermediate storage of the data of each stream is performed until the moment of reading by synchronized pulses. The frequency of reading data in the stream is somewhat higher than the frequency of their arrival. Similar systems with the union of non-synchronous streams is called the plesiochronous digital hierarchy. There are more complex systems with a synchronous digital hierarchy.
^ 4.2. Frequency-time division of channels in the GSM communication system
In a cellular communication system of the GSM standard, subscribers (mobile stations MS) exchange messages through base stations (BS). The system uses frequency and time division of channels. The frequency range and the number of frequency channels depend on the modification of the system. The channel separation scheme in the GSM-900 system is shown in fig. 19.
Transmission from the BS to the MS on the "direct" (downlink, forward, downlink, fall) channel and from the MS to the BS on the "reverse" (uplink, reverse, uplink, rise) channel is carried out at different frequencies, separated by an interval of 45 MHz. Each frequency channel occupies a 200 kHz bandwidth. The system is assigned the ranges 890-915 MHz (124 return channels) and 935-960 MHz (124 direct channels). At the same frequency, 8 time-multiplexed channels operate in turn, each within one time window with a duration of 576.9 μs. The windows form frames, multiframes, superframes, and hyperframes.
The long duration of the hyperframe (3.5 hours) is determined by the requirements of cryptographic protection. Superframes are of equal duration and contain either 26 multiframes (26×51 frames) for sync transmission or 51 multiframes (51×26 frames) for voice and data transmission. All frames contain 8 windows and have the same duration (about 4.6 ms). The system uses windows of several types with the same duration.
Transmission in all windows of one frame is carried out at the same frequency. When moving to another frame, the frequency may change abruptly. This is done to improve noise immunity.
All transmitted information, depending on the type (speech, data, control and synchronization commands), is distributed over different logical channels and transmitted in separate "portions" in different windows - physical channels. Data from different logical channels can be transmitted in one window. Different types of windows are used to convey different types of information. Guard intervals are introduced between windows to eliminate signal overlap from different subscribers. The length of the guard interval determines the maximum cell size (cell).
Logical channels are divided into communication and control channels.
Channels of connection (TCH - traffic channels) transmit speech and data at speeds from 2.4 to 22.8 kbps. The system uses a PRE-LPC type source encoder (linear encoder with regular pulsed predictor). Its standard voice rate of 13 kbps is increased to 22.8 kbps as a result of channel coding.
Control channels are divided into 4 types.
Broadcast Control Channels transmit from the BS clock signals and control commands necessary for all MS for normal operation. Each MS receives from the BS:
Synchronization signals for setting the carrier frequency via the FCCH channel (frequency correction channel - carrier synchronization channel),
Number of the current frame on the SCH channel (synchronization channel - MS synchronization channel in time),
The identification number of the BS and the code that determines the sequence of carrier frequency hops over the BCCH channel (broadcast control channel - the channel for transmitting commands to control the messaging process).
General control channels (CCCH - common control channels) are used when establishing communication between the BS and the MS in the following order:
The BS notifies the MS of the call via the PCH - paging channel,
The MS requests from the BS, via the RACH channel (random access channel - random parallel access channel), the number of the physical channel for connecting to the network,
The BS issues to the MS, via the AGCH (access grant channel), permission to use the communication channel (TCH) or a dedicated individual control channel.
Dedicated individual control channels (SDCCH - stand-alone dedicated control channels) are used to transmit from the MS to the BS a request for a type of service and to transmit from the BS to the MS the number of the physical channel assigned to the MS and the initial phase of the pseudo-random sequence that determines the frequency hopping program for this MS.
Combined control channels (ACCH - associated control channels) are used to transmit control commands when the MS moves to another cell (FACCH channel - fast associated control channel) and to send information about the level of the received signal from the MS to the BS (via the SACCH channel - slow associated control channel).
In "normal" NB type windows, the transmitted information is placed -114 bits. A 26-bit training sequence known to the receiver is used to estimate the impulse response of the communication channel in order to tune the receiver's equalizer,
Equalizing the characteristics of the communication channel, as well as for assessing the quality of communication and determining the time delay of the signal. End combinations TB (tail bits) are placed at the window borders, at the end of the window there is a guard interval GP (guard period) of duration 30.46 μs. The SF (steering flag) bits indicate the type of information.
The FB type windows are designed to adjust the MC frequency. The 142 zero bits are transmitted on an unmodulated carrier wave. Repeating windows of this type constitute the frequency setting logical channel of the FCCH.
SB type windows are designed to synchronize MS and BS in time. Repeating windows form a logical synchronization channel SCH. 78 information bits contain the frame number and the BS identification code.
Windows of type AB are designed to obtain permission to access the MS to the BS. The sync bit sequence transmitted by the MS configures the BS to properly read the next 36 bit sequence containing the service request. The guard interval in the AB window has been increased to accommodate the large cell size.
^ 4.3. Code division of channels
in an IS-95 communication system.
The system is allocated frequency bands 869-894 MHz for signal transmission over the forward channel and 824-849 MHz for reverse transmission. The frequency interval between the forward and reverse channels is 45 MHz. The operation of the direct channel on one carrier frequency during speech transmission is illustrated in Fig. 21.
The binary character sequence from the channel encoder is converted as follows:
– “scrambled” – summed modulo 2 with the individual code of the subscriber to whom the message is transmitted (“long” PSP),
– sums up with the Walsh sequence. Orthogonal Walsh sequences, the same for all BSs, divide one frequency channel into 64 independent channels,
– is divided by a commutator (CM) into two quadrature streams I And Q.
The symbols in these streams modulate the quadrature components of the carrier wave. To separate signals from different stations, the symbols in the quadrature streams are summed with "short" PRS- I and PSP- Q– BS identifiers.
The system uses unified data encoding equipment. GPS receivers are used to synchronize all BSs in time. The elementary symbols of the PSP follow at a rate of 1.2288 MSym/s. Long PRP with a period of 41 days is formed by a register containing 42 bits. Subscribers' individual codes are fragments of a long PRS that differ in initial phases. Short PRPs of duration 2/75s are formed by shift registers containing 15 bits, and differ in different BSs by an individual shift relative to the moments of the beginning of two-second time intervals.
When summing with the output sequence of the encoder, which has a frequency of 19.2 kbit / s, the long PRS is thinned out to equalize the speeds of the summed sequences: every 64th symbol is taken from it. When summing the received sequence with the Walsh codeword, one symbol of the sequence is converted into 64 Walsh chips, so that the switch receives a digital stream at a rate of 1.2288 MS/s. Short PSPs have the same symbol rate. Therefore, for the most efficient use of the frequency range, according to the Nyquist and Kotelnikov theorems, the spectrum of the symbol sequence at the input of the bandpass modulator in the transmitter should be limited to a frequency of 1.2288/2 MHz. For this purpose, a low-pass filter is installed at the input of the modulator with the boundaries of the pass and delay bands of 590 kHz and 740 kHz.
Each BS modulates a short SRP signal, which is output via a special "pilot" channel. The MS, shifting the short PRS in time, finds the BS with the strongest pilot signal and receives from the BS via the synchronization channel the data necessary for communication, in particular, the system time value to set its long code. After setting the long code, the MS can receive messages sent to it or start the procedure for accessing the BS on its own initiative. During operation, the MS monitors the level of the pilot signal and, when a stronger signal is detected, switches to another BS.
Data that needs to be transmitted at high speed is divided into packets and transmitted simultaneously over different frequency channels.
In the reverse channel (Fig. 22), the transmitter power and signal-to-noise ratio are lower than in the forward channel. To improve noise immunity, the speed of the convolutional encoder is reduced to k/n= 1/3, the encoder outputs data at 28.8 kbps. The spectrum of this digital stream is expanded: each 6-bit data packet is replaced by one of the 64 Walsh symbols repeated 4 times. The symbol number is determined by the content of the data packet.
After expansion, the symbol sequence is summed modulo 2 with the subscriber's long bandwidth and is divided by the switch into two sequences: in-phase ( I) and quadrature ( Q), which after summation with short PSP- I and PSP- Q, modulate the in-phase and quadrature carrier oscillations. To reduce phase hops, the quadrature modulating sequence is shifted in time by half the duration of the elementary symbol.
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