Using of Potential Functions Method for Recognition of Two Channels in Additive Noise Igor I. Troickiy Rustam Z. Yakubov Information Security Department Information Security Department Bauman Moscow State Technical University Bauman Moscow State Technical University Moscow, Russia Moscow, Russia iitroickiy@mail.ru yakubov_rustam@inbox.ru This article represents results of research of potential functions II. MATHEMATICAL REPRESENTATION OF THE method in case of discrete signal with additive noise. Noise in CHANNELS MODEL transmission channel has Gaussian distribution, i.e. it is a white noise. Besides, noises in the channels have a static interrelation. The main aim of the paper is researching of applicability of the The main idea of using potential function technique consists method of potential functions for recognition of the binary signal of finding a decision function, which can determine the class of from it mix with additive noise. Number of researches have been every object. conducted to assess the possibility of using the described method. There are two classes of objects, w0 and w1 (“0” and “1”, The results were presented in a series of graphics. Tests were made respectively). Objects of w0 have a negative potential, and for different numbers of parameters of the model, which had described in this paper. There was done comparative analysis of objects of w1 - a positive one. The value of a potential is working of potential functions and k-nearest neighbors algorithm, computed by potential of the point charge, which is determined which had considered in previous author’s works. The results of as follows:  this research can be used as the base for further research and can be applicable in organization of transport channels. g(x) = ∑ qi K(x, xi) (1) Keywords—signal-to-noise ratio, computer modeling, statistics, where K(x, xi) – potential function, it is inversely potential functions, correlation coefficient, white noise, probability proportional to value || x - xi ||; xi – i-th element of the training of recognition sample. I. INTRODUCTION qi – charge of the i-the elements «ones» and «zeros» q1 = +1, The paper is the logical continuation of [12] and [13], which q0 = -1. are devoted to the challenge of recognition of a discrete signal, For every new element is computed it's potential relatively when there are additive noises in the channels. In that works we received earlier: showed the results of analytical working on the model of two binary channels. It had suggested the method of linear filtration if g(x) > 0, then x w1 ; for noise compensation. It allowed to increase ratio of signal and if g(x) < 0, then x w0; noise ratio, besides we had made the set of computed experiments to show the influence of different parameters of the case g(x) = 0 is impossible from the point of view of the transmitted channels [4, 9, 10]. We had used k-nearest theory of probability, as K(x, xi) is continuous random value. neighbors’ algorithm for this aim, and compare it results in different conditions [6, 11]. But there are able to use a number There are most commonly used potential function, which has of methods for signal recognition, like parametric (including peak at x=xi and decreases monotonically to null during || x - xi || qualifier of a Bayes and working with the conditional probability → . It’s convenient to present potential function as function of densities), nonparametric techniques (like density function distance between entry and other elements of the set. assessment, assessment by means of parzen’s windows, In this paper the potential function K(x, xi) is: assessment by means of posterior probabilities, the linear discriminant of the Phisher), wavelet transformation, fast K(x, xi) = exp( -a || x - xi ||2) (2) Fourier transform, etc. [1-5, 8, 14, 15]. The parameter a > 0 – constant. In this paper we propose potential functions for solving the Evaluation of the successful recognition of the binary signal challenge. Signal is transmitted on two channels, noises in the in the channel looks like: channels are statistically connected. One channel is contained noise only, the second – noise and useful signal. If the second P* = m / N, where (3) channel doesn’t exist, we can introduce it in some cases. 131 m – an amount of successfully determined classes of the Correlation coefficient of the noises  and  took message; values from -1 to 1 in 0.1 steps. Standard deviation of  more 10 N – the length of the test set. times than standard deviations of  and  . Signal-to-noise ratio [further – ratio] is considerably less, than 1. Matlab 2011a was There are using model of the channels y1, y2, which described, used for experiments. as The results of the algorithm work in case, where isn’t y1 = a * / 2 +  + ; y2 = b *  / 2 +  + h * , common noise  (it’s dispersion is equal to zero), are shown if Figure 1. In that case parameter b are changing. where , , , – random variables, The behavior of the system is predictable in this case, the  takes the values 1 and -1 with probability P = 1/2; probability of recognition of the signal grows in the power   N(0, 2) - the noise in the first channel;   N(0, 2) – increasing. Mention must be made of the high level of the noise in the second channel,  and  are statistically recognition in case b = 1. interrelated, r, it is correlation coefficient of them; As demonstration of the algorithm working are presented   N(0,  ) – common noise in the channels, h – constant; 2 results of working for two parameter sets: (a=1, b=1; a=2, b=1). Experiments were made for different values of h: h=0,  and  – random values, which are independent from  and h=|0,5|, h=|1|, h=|1,5, h=|5|. , and each other; Results of case (a = 1, b = 1) are shown in Fig.2 and  and  – natural noises, attending in the channels; Fig.3. It’s demonstrates, that the quality of recognition depends on parameter h. Fig.3 shows, that the probability of recognition  – noise, appearing in the channels with some technical strives to 1 for correlation coefficient with value -1. This is due conditions; to the fact that ratio of the model of the signals is much more, a = |m11 - m-11| – discrete signal in the channel y1; than 1. In some cases in ratio signal grow and noise decreased. When it’s happening, you can notice the situation like in the b = |m12 - m-12| – discrete signal in the channel y2; Fig.3, Fig.4, Fig.5. In this work N(m, 2) – normal distribution law with Results for a = 1, b = 2 are shown in Fig.4 and Fig.5. Fig.4 parameters (m, 2) shows, that signal recognition significantly improves with h = 1. As a whole, we note, that recognition probability with b=2 Conditional probability densities P(y1 |  = 1) и P(y1 |  = -1) larger, than with b=1. You can see, that we found some have a normal law with parameters N(m1, 2+2 ) и N(m-1, excludes: when correlation coefficient is going to -1, P→1. It is 2+2). For 𝑦2 that parameters are defined similarly (so, going because mathematic expression of ratio grows very fast in normal law has parameters N(m±1, 2+2 )). some point: where fraction signal/noise strives to infinity. III. EXPERIMENTS For investigations of method of potential functions there had 0,59 been conducted the set of experiments, which allowed to valuate 0,58 Probability of signal recognition, P recognition probability of the signal in different cases. 0,57 1 Probability of signal recognition, P 0,56 0,95 0,55 0,9 0,85 0,54 0,8 0,53 0,75 0,52 0,7 0,51 0,65 0,5 0,6 -1,0 -0,8 -0,6 -0,4 -0,2 0,0 0,2 0,4 0,6 0,8 1,0 0,55 Correlation coefficient of noises ε and 𝛿, rε,𝛿 -1,0 -0,8 -0,6 -0,4 -0,2 0,0 0,2 0,4 0,6 0,8 1,0 Correlation coefficient of noises ε and 𝛿, rε,𝛿 h=0 h=0,5 h=1 h=1.5 h=5 P, b=0 P, b=1 P, b=3 P, b=5 Fig. 2 Signal recognition probability for a=1, b=1, h ≥ 0 Fig. 1 Dependence of probability P on parameter’s value h 132 1 0,7 Probability of signal recognition, P 0,95 Probability of signal recognition, P 0,9 0,65 0,85 0,8 0,6 0,75 0,55 0,7 0,65 0,5 0,6 0,55 0,45 0,5 -1,0 -0,8 -0,6 -0,4 -0,2 0,0 0,2 0,4 0,6 0,8 1,0 -1,0 -0,8 -0,6 -0,4 -0,2 0,0 0,2 0,4 0,6 0,8 1,0 Correlation coefficient of noises ε and 𝛿, rε,𝛿 Correlation coefficient of noises ε and 𝛿, rε,𝛿 PF, h=0 PF, h=1 PF, h=5 h=0 h=-0,5 h=-1 h=-1.5 h=-5 K, h=0 K, h=1 K, h=5 Fig. 3 Signal recognition probability for a=1, b=1, h ≤ 0 Fig. 6 Comparison of the k-nearest and potential functions in case a = b 0,95 IV. COMPARISION WITH K-NEAREST NEIGHBORS ALGORITHM 0,9 Probability of signal recognition, P Previous work of authors [12] focused on using of the 0,85 method of k-nearest neighbors algorithm. Compare the results of 0,8 the computing experiments. 0,75 A description of k-nearest neighbors is presented further (in 0,7 the context of the work). There are two classes of elements: w0 and w1. Elements with label of the class w0 are “0”, and with 0,65 label of w1 are “1”, respectively. There are generated sample, 0,6 which size is large enough (in the work the sample’s size is 0,55 around 10000 [13]). As with potential functions method, we designate xi as i-th element of training set. 0,5 -1,0 -0,8 -0,6 -0,4 -0,2 0,0 0,2 0,4 0,6 0,8 1,0 A new classified two-dimensional object x comes to the Correlation coefficient of noises ε and 𝛿, rε,𝛿 input of the qualifier. The distance || x - xi || between x and h=0 h=0,5 h=1 elements of the training set is computed. k nearest are selected. h=1.5 h=5 x will get the label, which had the most of the k choosed Fig. 4 Signal recognition probability for a = 1, b = 2, h ≥ 0 elements. k is always ood in order to avoid collisions. 1 1 0,95 Probability of signal recognition, P 0,95 Probability of signal recognition, P 0,9 0,9 0,85 0,85 0,8 0,8 0,75 0,75 0,7 0,65 0,7 0,6 0,65 0,55 0,6 0,5 0,55 -1,0 -0,8 -0,6 -0,4 -0,2 0,0 0,2 0,4 0,6 0,8 1,0 -1,0 -0,8 -0,6 -0,4 -0,2 0,0 0,2 0,4 0,6 0,8 1,0 Correlation coefficient of noises ε and 𝛿, rε,𝛿 Correlation coefficient of noises ε and 𝛿, rε,𝛿 PF, h=0 PF, h=1 PF, h=5 h=0 h=-0,5 h=-1 h=-1.5 h=-5 K, h=0 K, h=1 K, h=5 Fig. 7 Comparison of the k-nearest and potential functions in case a=2*b Fig. 5 Signal recognition probability for a = 1, b = 2, h ≤ 0 133 REFERENCES 0,95 [1] Fazilov Kh. 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