Calculation the Parameters of the Filters Shaping and the Properties of Random Processes on the Output Linear Filters by Using Mathematical Modeling

Using mathematical modeling of wireless devices and systems, where it was to provide detailed explanations about calculating parameters shaping filters features and characteristics of the operations on the exits

or the design of wireless systems where the signal is distorted due to physical phenomena, it is necessary to characterize the transmitter, channel and receiver using mathematical models. An understanding of random processes is crucial to many engineering fields-including communication theory, digital signal processing in electrical, computer engineering, vibrational theory and stress analysis in mechanical engineering. The filtering, estimation and detection of random processes in noisy environments are critical tasks necessary in the analysis and design of new communications systems and useful signal processing algorithms. Random processes: filtering, estimation and detection clearly explains the basics of probability and random processes and details modern detection and estimation theory to accomplish these tasks. The paper covers four main interrelated topics: Probability and characterizations of random processes, linear filters with random excitations, optimal estimation theory and detection theory to calculate parameters of optimal detectors of radio pulses. The random processes of almost exclusive interest in modeling receiver noise are the Gaussian processes. Gaussian processes are random processes for which the random variables N(t 1 ),N(t 2 ),..,N(t k ) are jointly Gaussian for all t 1 ,..,t k and all k > 0. Filtering in general linear models is perhaps the most widely applied branch of filtering, but in the context of linearity the term "filtering" refers to something that is fundamentally different from the probabilistic models and equations that comprise what mathematicians refer to as filtering theory.

The Model of Discrete White Gaussian Noise
As is known, white noise model is a mathematical abstraction in the form of process, spectrum is uniform for all frequencies and is equal to some constant 2 / 0 N , and the correlation function is a delta function with a weight which is determined by the specified constant. Thus, white noise has infinite variance (power). Any real process has the ultimate power, and, consequently, its power spectral density can only be integral decreasing function of frequency. However, the model of the white noise is used, if the width of the noise spectrum is much larger than the width bandwidth some frequency-selective device. We now consider the model of a discrete white Gaussian noise. Discrete white noise, unlike the white noise has finite capacity. Its correlation function is the identity function with a weight equal to the dispersion process. Any two of the reference process are not correlated. This process can be modeled using computer equipment. However, in reality almost always counts a random process, obtained by sampling a continuous process have finite cross-correlation coefficient, and by using only the frequency characteristic of filters and selecting a special kind sampling in accordance with parameters of adjacent filter samples process may not be correlated [1,2,3]. We will demonstrate this using the example of a discrete process samples in a digital receiver. General functional diagram of a receiver is shown in Fig. (1).

Figure (1): General functional diagram of a receiver
In Fig.(1) denotes: LNA -a low-noise amplifier, the BPF -a band pass filter, LPF -low pass filter, ADC -analog to digital converter, O -Oscillator, DSP -Digital Signal Processing unit.
Suppose that the intrinsic noise of the antenna and the amplifier is much greater than the width bandwidth a band pass filter, amplitude-frequency characteristics of the band pass filter and low pass filter perfectly rectangular. Fig. (2) Shows the power spectral density processes in points 1, 2, 3 and 4, as well as the correlation function of the process in point 4 is shown in Fig. (1). From a consideration of the graphs to the following conclusions, first of all, noise after filtering has a bandwidth defined by the characteristics of the filter, and so the power of the noise is limited. This explains the fundamental difference between the discrete white noise from the white noise ‫ـيـ‬ a discrete white noise has a finite variance, as can be obtained by sampling the noise with a finite bandwidth. Consider now the correlation properties of the analog to digital converter process. From Fig. (2, c, d) can be seen that if the sampling frequency is greater than twice the one-sided strip process, its discrete samples will be correlated [4,5,6]. In order to process the neighboring samples were uncorrelated, the sampling frequency must be chosen from the condition Only in this case the resulting realization of discrete random process will be uncorrelated, as samples of the correlation function of the discrete process will fall into the low-frequency zero correlation function of the process, having the form x x / sin . Ideal filters considered in this example, are unrealizable. Therefore, if the amplitude frequency characteristic of the filters used in the circuit of Fig. (1) has some arbitrary shape (e.g., Gaussian), then the process discrete samples are correlated. However, in practice, can be applied certain special filters whose impulse characteristic autocorrelation function (IC) has equally spaced zeroes as a function x x / sin . A common example of such filters are filters with amplitude -frequency characteristic such as a raised cosine [7,8], which is described by the following expression: Where α -parameter that determines the degree of smoothing of the frequency characteristic, T -the width of the main lobe IC for the first zeros. If α = 0, the amplitude-frequency characteristic of the filter is reduced to a rectangular filter characteristic unrealizable. If α = 1, then the amplitude-frequency characteristic represents one period of a raised cosine. For intermediate values parameter (α) characteristic of the filter has a flat top, and falling slope to the cosine law. IC filter characteristic of the filter with a cosine smoothing has the form , this function reduces to a function x x / sin , and for large values of the parameter (α) has a smaller side-lobe level. To using a filter with a cosine smoothing process samples were not mutually correlated, it is necessary to choose the sampling interval equal to T T s  . Cosine smoothing filters are widely used in digital communication, as provide a minimum level of intersymbol interference. Package MATLAB has built-in means for generating realizations discrete white Gaussian noise. Firstly, among the standard functions, included in the programming language MATLAB, there is a function randn (M, N). This function returns when it is called a two-dimensional array of independent random number size (MxN), distributed according to a normal distribution with parameters (0,1). in the case of absence of one or both of the arguments to the function when it is called, it returns a vector of independent random numbers or scalar random number. In the library of SIMULINK in the sources section, there are two blocks for the formation of the implementation white Gaussian noise: (Random Number) and (Band Limited White Noise). Consider the features of the use of and another block. In the parameter list (Random Number block) contains the following items: the noise variance -Variance, mathematical expectation (ME) -Mean, initial conditions -Initial Seed and sampling interval -Sample time. This noise generator generates a discrete implementation (SD), (PSD) which is uniform in the frequency band     Using one of the considered noise generators simulating Gaussian (SD) usability determined set of parameters or that block [9,10].

Simulation of Gaussian Random Processes
Assignment: build up the correlation function (SD) at the output of the filter with a given pulse IC. Find dispersion and (ME) process to output filter in the steady state at predetermined (ME) and dispersion inlet. To plot depending on (ME) and dispersion process at the filter output, if the zero initial conditions at time (t 0 ) at the inlet becomes effective implementation (WGN) with the specified parameters. Find a normalized cross-correlation coefficient values at the output of the filter process, taken after a specified time interval. As an example, consider the spectral-correlation properties of the process at the output of the filter with a rectangular IC. As noted in [11], the correlation functions of the process at the output of the filter up to a constant factor determined by the autocorrelation function IC of the filter. The autocorrelation function IC rectangular shape has a triangular shape. View IC considered filter and its autocorrelation function is shown in Fig. (4).

Figure (4): Impulse characteristic and its autocorrelation function
As is well known [7,12], the dispersion process at the output of the linear filter upon exposure at the input white noise with power spectral density Where E h -their energy IC. Thus, the process of the correlation function at the output of the filter is identical in form with the autocorrelation function IC of their filter and has a maximum, determined by the value (5). Fig. (5) shows graphs of mathematical expectation (ME) and the dispersion process at the filter output from the time, built on the assumption that at time zero to input filter with zero initial conditions applied to the realization of white noise (ME) m 1 and (PSD) 2 / 0 N . In the graphs according to mathematical expectation (ME) and the dispersion process at the output of the filter has the transition process, whose duration is determined by the duration of the impulse characteristic of the filter. Graph depending on mathematical expectation from time is a transient response of the filter with a weight equal to the expectation input process. It is therefore obvious that if the process in expectation of zero input, then the output process will have zero expectation. [13].

Figure (5): Mathematical expectation & the dispersion process
We define the law change the dispersion process at the filter output depending on the time. For this condition represent the filter with impulse characteristic ) (t h in form of parallel connection K filters with impulse characteristic ) (t h , representing partial portions of the original impulse characteristic, detainees at the time t i , where i -the number of the partial impulse response, t  -its duration. To the parallel connection of partial filters match the original filter, the output process of the partial filters are added. Functional diagram of a system characteristic of the equivalent circuit of parallel channels. In this case, the time interval   t  , 0 there will be a transition process at the output of the first partial filter and at the outputs of the other will zero values. After the end of the transition process to the variance adder output is equal to At the same time begin the transition process at the output of the second partial filter after which the dispersion at the outlet of the second filter is equal to Impulse characteristic of the partial filters are mutually orthogonal functions, since the relative the product of any two partial impulse characteristic equal to zero. This means that processes at the outputs of the partial filters at the same time shall have the meanings that are responsive to these filters staggered in time and therefore independent white noise input samples. Given also the fact responses to the cross-sectional values process do not overlap the input values of the process output taken at the same time, are not correlated, but in the case of Gaussian independent process. Consequently, the dispersion process at the output of the adder will equal the amount of dispersion processes with partial output filters. Then in time moment t k dispersion process at the output of the adder (and at output initial filter) is equal to It is obvious that these arguments are valid only for the case when at the input filter acts white noise. Under the influence of correlated noise law changes the variance of the output process will be more difficult. From the last relation can be seen if the impulse response of the filter is rectangular shape, and all the partial energy of the impulse response is the same, the variance of the process output will increase linearly throughout the duration of the impulse response h (t). In the limit, when Δt → 0, the expression for the dispersion law changes process at the filter output will take the form Thus, in the case of a filter with a rectangular impulse response in the interval time from zero to the duration of the impulse response of the expectation and variance varies linearly. In this range the output process is non-stationary. Therefore, in the simulation of a process with given spectral-correlation properties that the initial portion of the implementation output process typically exclude from consideration.
Consider the process at the output of the discrete shaping filter c rectangular impulse response, which is valid at the entrance of the discrete white Gaussian noise. View of the impulse response of the filter and the correlation function of the output of the process is shown in Fig. (7). The maximum value of the correlation function of the random process at the output of the digital filter is equal to Fig. (8) shows plots of the expectation and variance of the output random process from sample number. . Since in our example all the filter coefficients are equal, transition process changes in the expectation has linear character [14,15]. For building graphics depending on the variance of the output process of the number of reference, consider the following two properties of the dispersion. Firstly, the variance of the product of a random variable by a constant equal to the product dispersion of the initial value by . Therefore, in this example, law changes the variance of the output of the process is also linear.

Figure (9): Block diagram of the FIR digital filter
Assignment: Calculating the digital FIR filter for the formation of realization of a Gaussian process with a given a correlation function or given power spectral density. Provide power rationing process at the output filter to Power process inlet. As an example, consider the design of the shaping filter for modeling a random process with power spectral density of the form: Where a -parameter.
As noted in [11,12], the correlation function of the process at the output of the filter up to a constant factor equal to the autocorrelation function filter impulse response. Therefore, to solve this problem we must design the filter impulse response which represents a signal power spectrum is described by expression (7). As is known, the module Fourier transform of a Gaussian pulse is described by a Gaussian curve. In addition, the autocorrelation function and Gaussian pulse representing a Gaussian curve. In this example, we should mention another important property of the Fourier transform, which is the Fourier transform of an even function, is purely real function. Since the function (7) is even, then to calculate the actual impulse response filter need only calculate its actual amplitude spectrum as the square root function (7) and take the inverse Fourier transform. Using the table of Fourier transforms, we obtain that the autocorrelation impulse response function will have the form , and itself as the impulse response is the inverse Fourier transform, equal to the square root of (7), and is given by Function (9) will be considered as the impulse response of the analog prototype filter. The parameter b in the expression (8) and (9) determines the width of the correlation function of the process being modeled and, therefore, the spectral width of the process. It can be shown that the value of (1 -b) approximately defines the correlation coefficient of two adjacent sampling processes. To obtain digital samples of the impulse response of the shaping filter can use different methods.
In [7,12] described a method for calculating the filter coefficients expansion method the spectral power density in a Fourier series. Practically, this method reduces the frequency window to a method which in the practical implementation is desired sampling samples amplitude characteristics and calculation of the inverse discrete Fourier transform (or, in the special case of an even function, discrete cosine transform). It is possible the subsequent weighting impulse response weighted window to decrease the emission frequency characteristic obtained digital filter caused by the influence of the so-called Gibbs effect [12]. Another common way of designing is method window in the time domain. This method is even easier and more convenient for designing FIR filters in case, if known analytical expression for the impulse response of the analog prototype filter. The method consists in weighting the impulse response of the prototype filter weighting window of limited duration and sampling a subsequent within this window. In this example, using the above approach, we obtain samples of the impulse response of the digital FIR shaping filter in the form (10) ) exp( ) . The value of the sampling interval Δt is chosen according to the bandwidth of the simulated process. The number of filter coefficients is chosen based on the requirements of accuracy of approximation required frequency response. Obviously, the window duration must be chosen so that the edge of the window of the impulse response values was negligible. After receiving the vector of coefficients of the digital filter is necessary to calculate its frequency response, as well as the square modulus of the frequency response and assess the accuracy of the approximation of a given PSD output process resulting function. Amplitude factor  2 a in (7) does not principal values. However, the task we have to fulfill the condition of rationing power output process to the power of the input process. This means that the power (variance) of the process at the filter output should be equal to the power input process. As shown above, the dispersion process at the filter output is equal to is a coefficient through a period of correlation passive interference in a pulsed radar system.. Assignment: calculate the recursive digital filter for simulated random process with exponential correlation function and a given value of the correlation coefficient between adjacent samples of the output process. This example shows model another important special case passive interference in radar, namely exponential interference. To calculate the coefficients of the recursive shaping filter for a given correlation function of the process at the output, you can use the method of factorization filter system function [7]. The system function ) (z k recursive linear filter with constant parameters can be represented as &Tech.Journal, Vol.34,Part (A), No.1,2016

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N -the number of taps of the filter is not recursive, M -the number of taps (order) recursive part. Procedure for the application of this method is as follows.
1. Located z -transform ) (z F of the correlation function R(k) of the simulated process: The resulting function of the complex variable z at points located at the unit circle, belonging to the complex plane, it makes sense to the power spectral density of the simulated process. As mentioned above, in order to find the impulse response filter must obtain the function equal to the square root from spectrum density power. However, the function ) (z F is a function of the complex variable z , which implies the following paragraph.
2. Implemented factorization function ) (z F : Considering the expression (13) determines the function of the filter system. 3. The system function K (z) is converted to the form (12) to find the coefficients of the recursive filter. In this example, the discrete exponential correlation function of the simulated process is given by Where  -parameter that determines the rate of decay of the correlation function. In accordance with the described method of calculation, find two-sided ztransform of the correlation function (15). For this we use following known relation: According to The table z-transform we find: If we substitute this expression in (16), after the transformation can be obtained: Comparing the expressions (14) and (18) it is easy to see that the numerator and denominator of the complex system function of the designed filter is defined Comparing the expressions obtained for the numerator and denominator of the system function with the general view of (11), write down the expressions for the coefficients of direct and recursive part of the filter: . As a result of solving the problem, we come to the model recursive filter of first -order, whose work is described by the difference equation of the form: n n n (19) Where ξ [n] -implementation of the input process, ζ [n] -implementation of the output process. Block diagram of the designed filter is shown in Fig . (10). As follows from (19), feedback coefficient is equal to the correlation coefficient of two adjacent sampling processes ] [n  at the output of the filter. In accordance with the assignment, simulation process with a given correlation coefficient between adjacent samples of the output process is reduced to the corresponding assignment of the feedback coefficient the circuit in Fig. (10). Fig. (11) shows a circuit consisting of a white Gaussian noise generator, and Dual-channel filtering schemes, and also frequency characteristics appropriate filters.

Assignment:
Assignment: In Fig. (12) shows the functional circuit consisting from generator discrete white Gaussian noise, digital low pass filter with impulse response shown in same figure, delay lines at intervals time equal to the duration of the impulse response filter and half of this duration, adder, mutually -correlation device, device evaluation standard deviations, and also devices multiplication and division.
The sampling frequency in the system is equal to Hz f s 100  , level power spectral density of a discrete white Gaussian noise is equal to The number of samples of the impulse response of the filter 16 

N
. It is necessary to find dispersion process in points 1, 2 and 3 of scheme, and the true value of a parameter estimation, which is calculated in point 4 of scheme. The Optimal Detector Simple Radio Pulses Model optimal detector simple radio pulse will be considered, based on the mathematical apparatus of the complex envelope. Block diagram of the model optimal detector radio pulse signal is shown in Fig. (13). In the model detector consists of a matched filter (MF), the amplitude detector (AD), a threshold device (TD) and a unit for calculating the adaptive threshold detection. At the input of the detector receives the value of real and imaginary parts of the complex envelope of the radio pulse. The output signal of the detector is a binary signal λ = {0,1}. In this case, the zero level corresponds to the solution of the absence of the signal at the input, the unit -decision on availability signal [16,17,18].

Figure (13): Block diagram of the model optimal detector radio pulse signal
The main parameters consideration of the detector, subject to evaluation includes the following values: 1. The signal-to-noise ratio at the output of a matched filter (MF).
2. The value of the detection threshold 0 U , supplied to the lower input a threshold device (TD). 3. The probability of correct detection signal (D). To calculate these parameters you must set the following set of input data: 1. The signal parameters -the shape, duration, a priori data about amplitude and initial phase. Based on the parameters used in the Signal System, in accordance with the Nyquist theorem is given by the sampling frequency s f . 2. The parameters of the noise at the input of a matched filter. In general, any random process is completely characterized by multidimensional density distribution probabilities. In the case of stationary discrete white Gaussian noise is enough to know only two parameters of its onedimensional normal distribution law -the mathematical expectation (ME) and a dispersion, usually believed the mathematical expectation zero. 3. The required level of probability of false alarm. After a matched filter and amplitude detector included threshold device, whose task is to decision -making of a signal in the input process according to the Neyman-Pearson.
Decision -making rule, optimal according to the Neyman-Pearson provides maximizing the probability of correct detection of a signal at a fixed level of probability of false alarm. Because the noise power is unknown a priori, in the model included block estimates mean square deviations own noise. As an estimate of the mean square deviations this block calculates the sample mean square deviations: Where N -size averaging window. Threshold value 0 U formed as follows: Where the constant k determined based on the acceptable level of false alarm probability.
Assuming that own noise receive path in each quadrature channel is independent Gaussian Eng. &Tech.Journal, Vol.34,Part (A), No.1,2016 Calculation the Parameters of the Filters Shaping and the Properties of Random Processes on the Output Linear Filters by Using Mathematical Modeling 641 spectral density, noise output amplitude detector in the absence of the desired signal is distributed Rayleigh. Then the probability of false alarm F is given by: Where n   -noise variance at the output a matched filter. Given that k U n    0 , from formula (22) you can easily find the value of k . Calculate the probability of correct detection of a signal at a given false alarm probability F (i.e. established threshold 0 U ). We give a detailed derivation of the relations in the case of detection of a video on a background of white Gaussian noise, and then present the final design relations for the case detection quadratic deviations radio pulse with an unknown initial phase. The decision about the detection of a video signal taken by signal value at output matched filter [19,20]. The maximum value at output matched filter is observed in expiration time of the input signal and the greatest probability of correct detection D this will correspond exactly to the time. At the output of the matched filter at each time point will be observed random variable (section spectral density) distributed according to a Gaussian law. Since the filter is linear, and the mixture signal and noise at the input additive, then the variance of this distribution is always determined only by the input noise variance and filter parameters. With regard to mathematical expectation, it depends on the presence or absence of the signal, and is the parameter for which to be decided.
Mathematical expectation noise is usually equal zero. On the other hand, mathematical expectation process reference at the output matched filter at the time of expiration signal the inlet is defined by a maximum output value, i.e. its energy. On Fig. (14. a), and shows graphs of the probability density function reference in the presence and absence of the desired signal in the input implementation.
Expression (23) can be regarded as a definition of probability correct detection for arbitrary distribution law. In the case of normal distribution upon detection of a video signal condition, U determined by the energy of the signal, however, to calculate this amplitude needs amplitude detector ‫ــ‬ non-linear device, changing law of distribution of noise. It is known that the modulus Gaussian spectral density in the presence of a useful signal of constant amplitude is distributed according to the law Rayleigh-Rice, and in his absence -in law Rayleigh [21,22]. Graphs corresponding probability densities are shown in Fig. (14.b). Substituting into (23) probability density of the corresponding distributions Rayleigh-Rice can obtain the following expression for the probability of correct detection: