Identification of the impulse response of the communication channel. Mathematical model of a linear communication channel with memory based on characteristic functions and a probabilistic mixture of signal distributions
UDC 621.391.8
A. G. BOGACHEV
MATHEMATICAL MODEL OF A LINEAR COMMUNICATION CHANNEL WITH MEMORY BASED ON CHARACTERISTIC FUNCTIONS AND A PROBABILISTIC MIXTURE OF SIGNAL DISTRIBUTIONS
MATHEMATICAL MODEL OF LINEAR COMMUNICATION CHANNEL WITH MEMORY BASED ON CHARACTERISTIC FUNCTIONS AND PROBABILISTIC MIXTURE DISTRIBUTION OF SIGNALS
The article describes an approach to building a model of a linear communication channel with memory based on characteristic functions and a probabilistic mixture of signal distributions
Keywords: communication channel, communication channel identification
The article describes an approach to the construction of a model of linear communication channel with memory based on characteristic functions and probabilistic mixture distribution of signals
keywords: channel, channel identification
In the works of most authors, the instantaneous characteristics of the communication channel and the signals expected at the reception are assumed to be known exactly. However, in reality there is some error in the channel characteristics, which directly affects the reference signals at the receiver, resulting in a significant reduction in the quality of the demodulation. In the works of a number of authors, estimates are given that show that with an increase in the mean square of the error in estimating the channel parameters by 1–2 dB, the probability of a coherent demodulation error increases by about an order of magnitude. In the last 10-15 years, a scientific direction has been actively developing, associated with the evaluation of the characteristics of communication channels without transmitting a test sequence. In modern radio communication systems, the time spent on testing the communication channel reaches 18% (for the GSM standard), which makes it attractive to use this time resource for upgrading radio communication systems. For shortwave communication systems, the share of the test sequence can reach 50% of the total transmission time over the radio channel.
There are two main types of blind signal processing tasks: blind channel identification (estimation of an unknown impulse response or transfer function), blind alignment (or correction) of the channel (direct assessment of the information signal) . In both cases, only implementations of the input signal of the receiving device are available for processing. The first task is the most general, since it can have various practical applications that differ from applications related to the transmission of information signals (for example: radar systems for monitoring outer space; distortion compensation in imaging and processing systems, including in medical technology). Note that the second problem of blind signal processing can be solved based on the solution of the first one. In connection with the indicated circumstances, we dwell on the problem of blind estimation of the impulse response.
Blind processing tasks involve a wide class of models for describing the observed signals. In the most general case the continuous model is described as a system of systems with multiple input and multiple output (in the English literature Multiple-Input Multiple-Output or MIMO). The novelty and complexity of the proposed model does not allow using the MIMO system as an object of study, therefore, we restrict ourselves to considering a particular case with one input and one output. This corresponds to the case of a stationary scalar channel, which can be described by the input-output relation:
where https://pandia.ru/text/79/208/images/image003_3.png" width="31" height="23 src="> is the unknown impulse response of the communication channel;
https://pandia.ru/text/79/208/images/image005_2.png" width="12" height="13">-th input signal () at the -th time interval ;
https://pandia.ru/text/79/208/images/image010_0.png" width="15" height="17 src="> – time interval.
The blind identifiability of a system is understood as the possibility of restoring the impulse response of the system with an accuracy up to a complex factor only from the output signals.
The papers present key theorems on the basis of which the necessary and sufficient conditions for blind identifiability are formulated. The essence of these conditions is to fulfill the following requirements:
– all channels in the system must be different from each other, for example they cannot be identical;
– the input sequence must be quite complex. It cannot be zero, a constant, or a single sine wave;
– there must be enough output readings available.
Conditions of blind identifiability determine the class of models used in the problem under consideration. Common properties for this class of models are:
1) formation of a vector channel:
1a) using a multi-channel model (one input-many outputs or SIMO in English), which corresponds to the methods of diversity reception in space;
1b) by high-speed processing (multirate) of signals at the reception, which corresponds to the induction of the vector channel by oversampling ;
2) the presence of a random impact at the input of the model with given statistical characteristics, which forms an information sequence.
The class of models for the situation under consideration should be chosen so that their main property of the model is the explicit dependence of the output on the channel impulse response. In this case, the specific implementation of the information sequence that is fed to the input of the system is, of course, insignificant. Therefore, when modeling, it is possible to apply averaging over all possible information sequences using the probability of their occurrence. Then the model can be defined as a system that specifies the channel response in this moment time per clock interval depending on the impulse response when averaged over the input sequences. Here, averaging is understood as the restoration of the probability density of the channel response over a given number of initial moments (a model of averaging the channel response over sequences of transmitted symbols). Such a model is presented in . Here we consider an option in which one sample of the output signal () is taken at the clock interval:
where https://pandia.ru/text/79/208/images/image014_1.png" width="139" height="29">;
https://pandia.ru/text/79/208/images/image016_1.png" width="15" height="17 src="> – channel impulse response duration;
https://pandia.ru/text/79/208/images/image018_1.png" width="117" height="29 src=">– data vector;
https://pandia.ru/text/79/208/images/image020_1.png" width="51" height="28 src=">.png" width="96" height="28">;
– size of the symbolic constellation (positional modulation).
Analysis of model (2) shows that the likelihood function for the impulse response is multimodal, which makes it difficult to find an effective estimate. Therefore, in practice, such a multimodal probability density is approximated using the first and second order moments by some Gaussian distribution. This significantly reduces the computational complexity of obtaining an estimate, but at the same time reduces its accuracy.
With a significant depth of inter-symbol interference (which corresponds to a fairly extended impulse response), even with a small amount of the alphabet of transmitted characters, the number of possible input character sequences grows exponentially https://pandia.ru/text/79/208/images/image025_0.png" width=" 13" height="17"> probabilities.
A significant simplification of the description of sequences of input symbols in formula (2) can be achieved by using the apparatus of homogeneous Markov chains:
, (3)
where https://pandia.ru/text/79/208/images/image028_0.png" width="13" height="15">-dimensional probability;
https://pandia.ru/text/79/208/images/image030_0.png" width="83" height="29 src=">.png" width="16" height="17">.
We formulate the mathematical model as a likelihood function of the observed response of the communication channel to the sequence of states of the Markov chain and their transformations in the modulator for a given impulse response. It is important that in the case of identification of the impulse response of a communication channel by a test sequence, it is possible to use the mathematical apparatus of non-stationary homogeneous Markov chains. Under these conditions, changes mathematical model will be insignificant.
Let us set a mathematical model by composing operators that describe signal transformations and the formation of observations.
1) We will find the response at the output of a stationary linear system (linear communication channel) using the signal-system duality principle.
We represent the transformation in the modulator as:
where https://pandia.ru/text/79/208/images/image035.png" height="17 src=">.png" width="106" height="23 src="> is the dimension of the observed signal on the length of the ISI segment (the number of signal samples in the ISI interval);
https://pandia.ru/text/79/208/images/image039_0.png" width="16" height="19 src=">- ISI depth, measured in clock intervals.
The transformation in the linear part of the communication channel is defined as:
where https://pandia.ru/text/79/208/images/image042.png" width="14" height="26 src="> is the impulse response vector of the communication channel at clock intervals from https://pandia. ru/text/79/208/images/image044_0.png" width="14" height="25 src="> – ), equidistant discretization.
Let's transform the vector into a matrix of reactions 
.
2) The operator for forming observations on the clock interval is set using the assignment matrix :
where https://pandia.ru/text/79/208/images/image047_0.png" width="36" height="24 src="> is the assignment matrix for highlighting significant readings;
https://pandia.ru/text/79/208/images/image051_0.png" width="250 height=112" height="112">.
In the rows of the assignment matrix, all elements are equal to zero except for one equal to one..png" width="62" height="23 src=">-th column,…, in the -th row one in the -th column.
3) To form a randomized mixture of signals at the output of the communication channel, we use the apparatus of characteristic functions , which allows us to represent the probability density of the sum of independent random variables through the product of their characteristic functions, and the mixture itself through the sum of probability densities. This approach allows one to find an analytical assignment of the likelihood function based on multi-step transition probabilities (3).
In the theory of generalized functions, it is considered that the Fourier transform of the delta function (impulse function, Dirac function) is:
where https://pandia.ru/text/79/208/images/image057.png" width="16" height="24 src="> is the value of a random variable (the value of the signal count at the output of the communication channel).
Then the description of the reaction at the output of the communication channel is from https://pandia.ru/text/79/208/images/image050.png" width="14" height="18 src=">th sample of the -th clock interval
where https://pandia.ru/text/79/208/images/image061.png" height="19 src=">.png" width="42" height="33 src=">.png" width= "14" height="20">.png" width="49" height="26 src=">, .
4) We connect the matrix of transition probabilities with the characteristic function of the state of the Markov chain (an element of the information sequence). Then, on the set of states of the characteristic functions, we define a one-step matrix of transition probabilities:
,
where https://pandia.ru/text/79/208/images/image036_0.png" width="16" height="16 src="> states;
https://pandia.ru/text/79/208/images/image070.png" width="13 height=19" height="19">th clock interval;
https://pandia.ru/text/79/208/images/image072.png" width="112" height="56">.
6) The resulting randomized mixture can be formed as the sum of the characteristic functions of the final state probabilities:
,
https://pandia.ru/text/79/208/images/image075.png" width="36" height="16 src=">;
https://pandia.ru/text/79/208/images/image075.png" width="36 height=16" height="16">.
7) Let's supplement the mathematical model with additive white Gaussian noise of observations. – characteristic function of the normal law with zero mathematical expectation and standard deviation –
.
Therefore, the desired observation likelihood function https://pandia.ru/text/79/208/images/image042.png" width="14" height="26 src="> is found by the inverse Fourier transform:
.
We form a set of observations in the form of the number of time channels , which corresponds to the number of analyzed clock intervals, and selected th samples at each clock interval
where -https://pandia.ru/text/79/208/images/image053.png" width="45" height="23">,
https://pandia.ru/text/79/208/images/image085.png" width="311" height="53">, (4)
where https://pandia.ru/text/79/208/images/image087.png" width="73 height=48" height="48">.png" width="41" height="19">, low frequency impulse response duration of 24 samples, 8 samples at each clock interval.
Simulation results in the form of frequency histograms (https://pandia.ru/text/79/208/images/image091.png" width="13" height="15 src=">) of channel responses for selected samples on a clock interval ( Figures 1 and 2. A sample containing 3000 clock intervals was used.
Figure 1 - Histogram of the frequencies of hitting a random variable in the range of channel response values for the third sample on the clock interval
Figure 2 - Histogram of the frequencies of hitting a random variable in the range of channel response values for the eighth sample on a clock interval
Next, the likelihood function was built based on the proposed mathematical model (1-4)..png" width="13" height="15 src=">) for the selected samples on the clock interval are shown in Figures 3 and 4.
Figure 3 - Probability density of the random variable of the channel response for the third sample on the clock interval
Figure 4 - Probability density of the random variable of the channel response for the eighth sample on the clock interval
The results of the simulation are histograms of hit frequencies of the random variable of the channel response for the selected samples on the clock interval. A sample containing 3000 clock intervals was used. Next, the likelihood function was built on the basis of the proposed mathematical model (1-4). It was found that with an increase in the volume of the statistical sample in terms of the number of clock intervals, the histogram (Fig. 1, 2) becomes more and more similar to the formed mathematical model (Fig. 3, 4).
1. A direct description of the model is necessary to develop a simulation model of the channel.
2. The developed model of a channel with intersymbol interference is given as an indirect description, which can later be used to search for an effective estimate of the impulse response from the maximum likelihood.
3. Mathematical and statistical models have a pronounced multimodal structure, the number of modes of which depends on the channel memory. However, on some samples on a selected clock interval, individual extrema become visually indistinguishable. This can be due to: low signal-to-noise ratio, a large number of points in the signal constellation, a large depth of inter-symbol interference, a large number of samples per clock interval.
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Academy of the FSO of Russia, Orel
Researcher
In a multipath channel, it is necessary to reduce the influence of delayed beams, for example, using the following scheme:
Each line element delays the signal by time Δ. Let us assume that when transmitting a single pulse, the receiver receives 3 pulses with an amplitude ratio of 1: 0.5: 0.2, following at equal time intervals Δ. This signal x(t) is described by readings: X 0 = 1, X 1 = 0.5, X 2 = 0.2.
The signal at the filter output is obtained by summation, with weighting coefficients b 0 , b 1 , b 2 , signal x(t) and its delayed copies:
Options b i must be chosen so that at the output of the filter to receive readings y 0 = 1, y 1 = y 2 = 0 with input samples 1, 0.5, 0.2:
Solution b 0 = 1, b 1 = – 0.5, b 2 = 0.05. With these weighting factors
In the considered example, the equalizer parameters are calculated from the known impulse response of the channel. This characteristic is determined by the response of the channel to the "training" (tuning) sequence known to the receiver. With a large excess delay and a high level of multipath signal components, the length of the training sequence, the number of delay elements in the filter, and the signal sampling rate should be sufficiently large. Because the real channel is not stationary, the determination of its characteristics and the correction of the filter parameters have to be repeated periodically. As the filter becomes more complex, its adaptation time increases.
Channel characteristics identification
Correlation Method for Impulse Response Identification
Filter output 
Let the impulse response be described by three samples:
The model adequacy criterion is the minimum error variance
Minimum variance conditions
or
|
|
|
This system, written in general form
is a discrete form of writing the Wiener–Hopf equation
For a signal x(t) of the white noise type R x(τ) ≈ 0.5 N 0 δ(τ),
and the estimation of the impulse response is reduced to the determination of the correlation function R zx (τ).
Equalizer with inverse channel response
Knowledge of the channel characteristic is not necessary for its equalization. Filter parameters can be selected according to the criterion of minimum dispersion D e mistakes e(t) = x(t) – x*(t), where x(t) is the training sequence transmitted over the communication channel and generated in the receiver.
Ideal equalization of the channel response (when H k (ω) H f (ω) = 1) may be undesirable if the frequency response of the channel has deep dips: a correction filter will require a very large gain at frequencies corresponding to zeros of the channel transfer function, noise will increase.
The principle of operation of the Viterbi equalizer
Signal z(t) received when transmitting the training sequence x(t) is applied to the filter matched to the tuning sequence. The output of the matched filter can be thought of as an estimate of the channel's impulse response.
A signal is detected representing a sequence of n bit. All 2 n possible binary sequences that could be transmitted are formed in the receiver and passed through the filter - the channel model. A sequence is selected for which the filter response differs the least from the received signal.
through which echo pulses are transmitted
The adaptive receiver contains a communication channel mathematical model identification system that implements the algorithm (4.2.6), (4.2.8)-(4.2.12).
With the help of this system, simulation modeling of the process of identification of the mathematical model of the communication channel was performed using the developed method in the conditions of MSI. The total additive noise level was 15–5 dB. The identification of the parameter vector of the communication channel model was carried out in the process of transmission by subscriber modems of a sequence of service (adjustment) characters known on the receiving side. The number of service pulses used to identify the impulse function of the communication channel was changed in the range from 200 to 2000.
Figure 4.7 shows the signal received by the receiver modem with a signal-to-echo ratio of 5 dB. In addition, the same figure shows the echo contained in the received sum signal.
Rice. 4.7. Signal received by receiver modem (1) and echo (2)
The results of identification of the impulse function of the communication channel model according to the algorithm (4.1.10), (4.1.12) - (4.1.16) using this message containing 600 characters are shown in Figure 4.8. Figure 4.8 shows the real impulse function of the radio interception channel (line 1) and its estimate (line 2), calculated by the algorithm (4.2.6), (4.2.8) - (4.2.12). Here is an estimate of this impulse function (line 3), calculated from the same sample using recurrent LSM (according to the Kalman filter algorithm).
Rice. 4.8. The results of identification of the impulse function of the communication channel with a signal/echo signal ratio of 5dB:
1 – impulse function of the communication channel; 2 – estimate of the impulse function calculated according to the algorithm (4.1.10), (4.1.12)-(4.1.16); 3 – impulse function estimate calculated by the Kalman filter algorithm
Figure 4.8 shows that the algorithm (4.2.6), (4.2.8) - (4.2.12) provides the accuracy of the impulse function identification sufficient for high-quality demodulation of the received message. At the same time, the developed algorithm provides a higher accuracy of identification of the parameters of the communication channel model compared to the Kalman filter algorithm when using the same sample. The developed algorithm provides an average error in the identification of the impulse function, equal to 0.5%, when using a sample obtained by transmitting 400 service pulses over a communication channel with a signal-to-echo-signal ratio of 7 decibels. The total additive noise level was 5 decibels. With the help of the Kalman filter, this error in the identification of the impulse function was achieved using the information contained in the sample obtained during the transmission of 1500 service pulses. Similar results were also obtained for other combinations of information signal, echo signal, and Gaussian noise during transmission of QAM signals over the communication channel.
Thus, in Section 4.2, an algorithm for identifying the mathematical model of a communication channel for multi-position QAM signals is developed, which does not require knowledge of the noise probability distribution functions. This algorithm provides the minimum value of the generalized performance indicator (4.1.11), which is an additive convolution of the error signal, the moving average of the error signal in time, and the time average of the squared deviation of the current values of the error signals from their moving average values calculated in the moving time window.
4.3. Adaptive QAM demodulation system,
received over a communication channel with an unknown mathematical model
To derive the demodulation algorithm for QAM signals, we transform the mathematical model of the communication channel (4.1.1) - (4.1.3) as follows.
In a sliding time window with number , which has the value
at the point in time where 
; , form the vector of information parameters
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: 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 cores of a telephone cable, the main factor that reduces the reliability of information transmission in xDSL systems. Optimal from the point of view of minimizing the error probability, the ISI correction method 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, systems for the formation of ultrashort pulses, 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 suppressing intersymbol interference in the presence of additive Gaussian and non-Gaussian noise. The proposed method has a lower computational complexity compared to while maintaining the identification accuracy 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, and 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 a 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 a fourth-order moment function, and the remaining terms are products of 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 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. (eight)
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 calculating 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. A signal in the form of a two-level pseudo-random sequence 1024 samples long was passed through the communication channel model with a given discrete impulse response, the coefficients of which were 0.2000, 0.1485, 0.0584, 0.0104. 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. one.
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 low 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.
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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.
