Noise proof coding based on orthogonal functions

In article the encoding scheme for messages with use a family of orthogonal functions is suggested. For coding the message, the waiting sum of orthogonal functions is used. Weight coefficients are formed from bits of the transferred message. The received function is transferred on a communication channel as analog signal. For decoding the message, the orthogonality property is used. Integration is carried out numerically. Existence errors of decoding depends on accuracy of a numerical integration. This scheme allows to transfer messages on strongly noisy channels. At increase in noise level it is necessary to increase accuracy of an integration. The computer experiment is made. The dependence a step of a numerical integration on noise level is received.


Introduction
Recently schemes with using dynamic chaos for concealment the transfer the substantial message gained development. The useful signal is somehow mixed to a chaotic signal. For development a chaotic signal chaotic generators are used. Chaotic signal intensity considerably exceeds useful signal intensity. The main problem consists in extraction by host the useful component of the accepted signal. For this purpose, on a communication channel additional information on the chaotic generator is transferred. On the basis of additional information, the host self-contained generates a chaotic component and subtracts it from the received signal. This approach can be characterized as formation two bound identical chaotic generators. Several various approaches gained development: chaotic masking [1], switching the chaotic modes [2], nonlinear mix the transferred message to a chaotic signal [3], modulation the operating parameters of the chaotic generator [4].
The most prime in realization is the method of chaotic masking the message [1]. The transferred message m(t) added with a chaotic signal x(t). The received mixed signal of m'(t)=m(t)+x(t) is transmitted to host. The primal problem consists in synchronization the chaotic generator u(t) with the transferring site u(t)=x(t). The transferred message can be received by a simple subtraction m(t)=m'(t)u(t). In chaotic masking schemes developed now noise level in comparison with the useful signal is 35-65 db [5]. The quality of the received signal strongly decreases if there is a mistiming the operating parameters of noise generators [6][7][8]. One of possible approaches to a subtraction a noise from transmitted signal is the orthogonalization the chaotic signals. Such schemes gained development within the general approach to modulation signals, watered the name DCSK (Differential Chaos Shift Keying) [9]. One of problems these systems is creation the orthogonal chaotic signals. For the solution this problem ordinary noise generators are used with application to them Gilbert transformation [10][11][12], Gram-Schmidt transformations [13][14][15] or Walsh's codes [16][17][18]. The low channel capacity is characteristic of schemes based on orthogonal chaotic signals. Each orthogonal chaotic frame has one bit of information.
In this article the signal encoding scheme steady against the strong noise is suggested. This scheme doesn't demand synchronization the noise generators.

Coding of the message
Let's present the message as the bit's sequence

M=b0b1...bN.
Let's break the message into frames with length n. Quantity of frames is k=[N/n]+1. The message can be submitted as the frames sequence

M=M0M1...Mk.
Transfer of the message is carried out on one frame. It is enough to consider transfer the one frame. Let's consider further that the message consists one frame.

M=b0b1...bn.
Let's choose family of orthogonal functions Let's write down the weighting sum of orthogonal functions The value of the function F(x) is known only in some finite number of points in interval of an orthogonality [a, b]. For calculation the values di only the numerical integration can be used. Results will be received with some error. Therefore, for calculation coefficients ci it is necessary to use the threshold scheme: Accuracy of calculation for integrals depends on the used numerical integration algorithm and sampling step h. In our scheme the sampling step is set at a stage of formation the transferred sequence Fj. The submitted scheme is steady against transmission the messages on strongly noisy channels. Let at information channel there is the uniform noise H(t). The host will receive a signal
Some coefficients will be nonzero because of numerical integration inaccuracy (hi ≠0). If intensity of noise is higher, than it is more than nonzero values hi. Accuracy of calculations can be increased having reduced a sampling step. It will lead to increase the frame period. The message will be longer transferred.

Computer experiment
The computer experiment was made for definition the resistance this scheme to noise of various intensity. Testing was held for family of orthogonal functions ( ) 2 cos( ).  Apparently from figure 2 that at increase in intensity of noise it is necessary to increase frame's transfer period for steady decoding.

Conclusion
The method for coding the messages suggested in this article allows to realize the chaotic masking scheme of a signal without coordination the chaos generators. As a result, resistance to the accidental noise in communication channels significantly increases. The main problem for creation this scheme is ensuring high precision of a numerical integration. Accuracy of an integration affects to correctness of extraction the transferred message. Mistake size at integrals calculation can vary by means of a function sampling step. Decrease the sampling step leads to decrease in number of inaccurately taken bits.