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Additive White Gaussian Noise (AWGN) is common to every communication channel. It is statistically random radio noise characterized by a wide frequency range with regards to a signal in communication channels. In this paper, AWGN signal is generated through design an analogue circuit method, and then the multiple recursive method is also used to generate random data signal that is used for testing by Lyapunov exponent. Furthermore an algorithm for software generating of Additive White Gaussian Noise is presented. Lyapunov exponent test for chaos is used to distinguish between regular and chaotic dynamics of the generated data by the two methods. Simulation results are enhanced with the use of Microcontroller chip, since the hardware of the application is implemented by microcontroller-embedded system to obtain computerized noise generator. The results show that the generated AWGN signal by the analogue method and the multiple recursive method is chaotic which implies the random like-noise behavior.

White noise is a random signal (or process) with a flat power spectral density. In other words, it is the signal contains equal power within a fixed bandwidth at any center frequency. A wide band communication circuit can be measured and tested by using this noise. AWGN signal generators are hardware cost-effective, thus this ar- ticle presents a simple and inexpensive way to build white noise generators. Continuous dynamical of AWGN equations are very common in many applied sciences and engineering. Regrettably, most of these equations are nonlinear whereas most the methods of solution are linear [

· The noise is additive, i.e., the received signal equals the transmit signal plus some noise, where the noise is statistically independent of the signal.

· The noise is white, i.e., the power spectral density is flat, so the autocorrelation of the noise in time domain is zero for any non-zero time offset.

· The noise samples have a Gaussian distribution. The operation of the analogue system is based on the noise generated by the Zener breakdown phenomenon in an inversely polarized diode as shown in

Since many statistical methods rely on random samples, applied statisticians often need a source of “random numbers”. The use of random numbers in statistics has expanded beyond random sampling or random assign- ment of treatments to experimental units [

It is the most useful type of generator of pseudorandom processes updates a current sequence of numbers in a manner that appears to be random. Such a deterministic generator, f, yields numbers recursively, in a fixed se- quence. The previous k numbers (often just the single previous number) determine(s) the next number [

The number of previous numbers used, k, is called the “order” of the generator. The set of values at the start of the recursion is called the seed. Each time the recursion is begun with the same seed, the same sequence is generated. The length of the sequence prior to beginning to repeat is called the period or cycle length. The stan- dard methods of generating pseudorandom numbers use modular reduction in congruential relationships. There are currently two basic techniques in common use for generating uniform random numbers: congruential me- thods and feedback shift register methods. The basic relation of modular arithmetic is equivalence modulo m, where m is some integer. This is also called congruence modulo m. Two numbers are said to be equivalent, or

congruent, modulo m if their difference is an integer evenly divisible by m. For a and b, this relation is written as [

A simple extension of the multiplicative congruential generator is to use multiples of the previous k values to generate the next one:

When k > 1, this is sometimes called a “multiple recursive” multiplicative congruential generator. The number of previous numbers used, k, is called the “order” of the generator. (If k = 1, it is just a multiplicative congruen- tial generator). The period of a multiple recursive generator can be much longer than that of a simple multiplica- tive generator.

In every communications channel noise is always present. Furthermore, other disturbances which cause the transmitted signal to change, such as fading, may also be present. In most cases, AWGN is used to evaluate the performance of a communication system in a noisy channel [

The greatest power of science lies in its ability to relate causes to effects. The evolution of a dynamical system may occur either in continuous time or in discrete time. The former is called flow, and the latter is called map. For nonlinear systems, continuous flow and discrete map are also two mathematical concepts used to model chaotic behavior [

The Lyapunov exponents of a system under consideration characterise the nature of that particular system. They are perhaps the most powerful diagnostic in determining whether the system is chaotic or not. Furthermore, Lyapunov exponents are not only used to determine whether the system is chaotic or not, but also to determine how chaotic it is. The Lyapunov exponents characterise the system in the following manner. Suppose that do is a measure of the distance among two initial conditions of the two structurally identical chaotic systems. Then, af- ter some small amount of time the new distance is [

where

For chaotic maps, Equation (2), is rewritten in the form of Equation (3):

where

where

where

A dynamical system is said to be chaotic if the Lyapunov exponent

has a one positive Lyapunov exponent. The Hénon map (two dimensional maps) of has two Lyapunov expo- nents, one positive and the other negative. The Lorenz chaotic flow of (three dimensional maps) has three Lya- punov exponents, one positive, one negative and one equal to zero [

If all of the relevant information in the system is well known, the calculation of the theoretical Lyapunov expo- nents can be based on the equations of that system. This method includes repeatedly using equation linearization. In reality, the equations in a given system are not easy to obtain. However, time series data sets can easily be acquired. When only time series data are recorded, the calculation method introduced above is impossible to use. Alan Wolf [

This procedure can briefly be summarized as follows:

1) Choose and put the initial values of the multiple recursive method.

2) Calculate the next step of the multiple recursive method.

3) Calculate the derivative of the multiple recursive equation.

4) Calculate Lyapunov exponent value.

5) Calculate the Largest value of Lyapunov exponent,

6) Test the

7) If > 0, then the system behavior is chaotic, and if <0, then the system behavior is periodic (i.e., non-chao- tic).

The flow chart for calculating Lyapunov exponent values is shown in

In this section, the simulation results are given to verify the theoretical results by implementing the AWGN ge- nerator system of

The noise amplitude versus the number of samples is obtained by applying the multiple recursive method as shown in

In order to test the AWGN signal that is generated by the PIC microcontroller, so Lyapunov exponent chaos test is applied. When the time series data, i.e., AWGN signal, is entered as vector of values, then an embedding lag of state space reconstruction is initialized such that an embedding dimension must be calculated. If embed- ding dimension be selected correctly, then there would have smooth part (or fairly horizontal) on the Lyapunov exponent curve. So if there is no smooth section on the curve, it is better to try with other embedding dimensions. When there is not any information about proper value of embedding dimension, then should let it zero (0). In this case code automatically selects proper m by False Nearest Neighbors (or FNN) method, if this method fails due to high noise in data, the code will use another method named symplectic geometry. This method is a gra- phical in nature however it use test for selection of vector based on variance change of eigenvalues. The sym- plectic geometry method for determination embedded dimension is shown in

Chaos is a developing research topic. In this paper, Lyapunov exponent test for Multiple Recursive Method and additive white Gaussian noise generator is implemented. The application of Lyapunov exponent in real world dynamical systems is rarely developed. So, the method of testing by using Lyapunov exponent is proposed in this paper work. In order to obtain Lyapunov exponent from the experimental AWGN time series data, a method mainly based on phase space reconstruction is demonstrated. The phase space reconstruction requires a time de- lay and an embedding dimension. The digital software implementation of AWGN generators achieved; reliabil- ity and accuracy, where digital circuits have more reliability than analogue circuits, and wide number generation range, so in digital circuits, the range of random numbers is more wide than the range of random numbers ob- tained by analogue circuits, since there are various algebraic methods can be implemented by software.

Lyapunov exponent test for chaos is used for the analysis of nonlinear discrete and continues deterministic dynamical systems, the test distinguish between regular and chaotic dynamics. This distinction is extremely clear by means of the diagnostic variable