STATISTICAL ANALYSIS OF FAST FLUCTUATING RANDOM SIGNALS WITH ARBITRARY-FUNCTION ENVELOPE AND UNKNOWN PARAMETERS STATISTICAL ANALYSIS OF FAST FLUCTUATING RANDOM SIGNALS WITH ARBITRARY-FUNCTION ENVELOPE AND UNKNOWN PARAMETERS

spectral density (SD) of the process t p ^ h . Let us consider that fluctuations t p ^ h are “fast”, that is, the pulse duration 0 x and the characteristic changing time t T of the function f t ^ h essentially exceed the correlation time of the process t p ^ h , so the following conditions are satisfied:


Introduction
In a number of practical appendices of statistical radio engineering and radio physics it is necessary to solve a processing problem of random pulsed signals with unknown parameters against hindrances.In [1 -5, etc.] the detection and measurement tasks of random Gaussian pulses with rectangular or close to the rectangular envelope are considered in the conditions of various parametrical prior uncertainty.However, a received signal waveform can essentially differ from the rectangular often enough.Thereupon, it is of interest to find the optimal receiver structure for random Gaussian pulse with arbitrary-function envelope.

The output signal of the maximum likelihood receiver of a random Gaussian pulse
As a random Gaussian pulse with arbitrary-function envelope, we will understand a multiplicative combination taking a form of: ) Here 0 m is the appearance time, 0 x is the duration of a pulse, t p^h is the stationary Gaussian random process, f t ^h is the determined function (envelope), which is describing the pulse form and is normalized so that max f t 1 =

^h
, and γ is a scaling multiplier.We will designate a t 0 p = ^h as mathematical expectation, Let us consider that fluctuations t p^h are "fast", that is, the pulse duration 0 x and the characteristic changing time t T of the function f t ^h essentially exceed the correlation time of the process t p^h, so the following conditions are satisfied: Here 0 X -bandwidth of the process t p^h.Hindrances and n t ^h registration errors will be approximated by Gaussian white noise with the one-sided SD N0 .

STATISTICAL ANALYSIS OF FAST FLUCTUATING RANDOM SIGNALS WITH ARBITRARY-FUNCTION ENVELOPE AND UNKNOWN PARAMETERS STATISTICAL ANALYSIS OF FAST FLUCTUATING RANDOM SIGNALS WITH ARBITRARY-FUNCTION ENVELOPE AND UNKNOWN PARAMETERS
Oleg V. Chernoyarov -Martin Vaculik -Armen Shirikyan -Alexandra V. Salnikova *

We find a new expression for the solving statistics of fast fluctuating Gaussian pulses with arbitrary-function envelope, based on which we can receive much simpler processing algorithms of random signals with unknown parameters in comparison with available analogues.
For example, the synthesis and analysis of maximum likelihood detector and measurer of a high-frequency random pulse with unknown time parameter is carried out.The asymptotically exact expressions for detection and estimation characteristics including anomalous effects are presented.By methods of statistical computer modeling the adequacy of the considered analytical approach of the statistical analysis of random pulsed signals is corroborated, the working capacity of offered detector and measurer is established, and applicability borders of asymptotically exact formulas for their characteristics are also defined.
Keywords: Fast fluctuating random signal, maximum likelihood method, solving statistics, signal detection and estimation, local Markov approximation method, statistical modeling.
In terms of Eq. ( 9), we specify expressions for the summands entering the logarithm of FLR (3).Using Eqs. ( 7), (9), we can present last summand in Eq. (3) as Further, we set 1 | = in Eq. ( 9) and then substitute Eq. ( 9) into Eq.( 7), and Eq. ( 7) into Eq.( 4), and so, after corresponding transformations for the function V(t), we come to where . Now, after substituting Eq. ( 9) into Eq.( 7) and then Eqs. ( 7), (10), (11) into Eq.( 3), we receive the following expression for the logarithm of FLR for the random Gaussian pulse with arbitrary-function envelope: , , The formula ( 12) is essentially simplified if SD G p ^h allows for rectangular approximation [4]: for low-frequency (LF) process t p^h and for high-frequency (HF) process t p^h with a central frequency 0 o .Here d0 is SD magnitude.Such kind of approximation of the SD form can be used if real SD rapidly decreases outside of the bandwidth 0 X .With the realization of Eq. ( 13) or (14), the first summand in Eq. ( 12) can be written down in a form: For the signal (1) detection and measurement, we will use a maximum likelihood method [1, 6 and 7].According to [6], the maximum likelihood receiver (MLR) should form the logarithm of the functional of likelihood ratio (FLR) L j v ^h as function of current values j v of set of unknown parameters 0 j v (let us accept for further that true value of parameter is designated with the subindex "0", and its current value -without the subindex).Any signal (1) parameters (appearance time, duration, mathematical expectation, etc) can be vector 0 j v components. where is the covariance function of signal s(t).If some signal (1) parameter is known, then in Eqs. ( 3)-( 6) its true value should be used instead of current value.We will try solution of Eq. ( 5) in a form structurally similar to Eq. ( 6), i.e.

, ,
, , Substituting Eqs. ( 6), (7) in Eq. ( 5), we come to the equation of a form Solving Eq. ( 8) by Fourier transformation method [8], with a variable t t y y if process t p^h is LF, and y , if process t p^h is HF.Here, / q d N0 = .

Properties of the output signal of the maximum likelihood receiver of a random Gaussian pulse with unknown appearance time
For the definiteness of the further calculations we will refer to the process t p^h as to a bandpass HF random process ( 14) and we will select pulse appearance time 0 m as unknown parameter (other signal parameters are supposed to be known).
In the presence of a useful signal (1), ( 14), the term depending on the realization of the observable data x t ^h, can be presented as the sum of regular and noise functions [6]: , and the averaging is executed through all possible realizations x t ^h with the fixed 0 m value.While executing the ratios (2), for regular function and covariance function of noise function we receive where where y t x t h t t dt = - .Still we can neglect this difference considering that Eq. (18) enters the logarithm of FLR under integral sign while executing ratios (2).Substituting Eq. (18) into Eq.(15) we have: In view of Eqs. ( 13), ( 14), we will transform summand L3 j v ^h into a form: where 1 f = or 2 f = for LF or HF process t p^h correspondingly.Applying Eqs. ( 19), (20) in Eq. ( 12) for the logarithm of FLR for the Gaussian pulse with the arbitrary-function envelope and band SD, we receive: According to [6], while executing ratio ( 27 of MLE lm , with the allowance for the anomalous errors, can be written down as follows: Here,

@
are, correspondingly, conditional bias, conditional variance and probability of a reliable estimate lm (26).
While determining b l l m 0 0 ^h, V l l m 0 0 ^h and P0 , we will be limited to a condition of a high posterior accuracy when the output signal-to-noise ratio (SNR) z 2 of the algorithm (26) is sufficiently great, i.e.The inequality (29) is valid with ratios (2) executed and when the values of q0 are not too small.
We will presuppose that f t c uu ^h does not vanish in points , i.e. the useful signal (1) is discontinuous (for a continuous signal the results received in [5] can be used at their obvious generalization).Similarly to [6], it can be shown that MLE lm converges in a mean square to a true value of the estimated parameter l0 with the increase of z 2 .Hereupon, for the definition of the characteristics of the reliable estimate lm under z 1 2 22 , it is sufficient to investigate the behavior of the functional l M^h (22) in a small neighborhood of the point l l0 = .We will designate , , l l l l l l max where o T ^h denotes the higher-order infinitesimal terms ! 6 @, possesses drift coefficient If the useful signal ( 1), ( 14) is absent, then, both for regular function and covariance function of noise function of the functional (22), we have

The appearance time estimate of the random Gaussian pulse
Let us state that the signal ( 1), ( 14) is present with the probability 1.Then, according to [1 and 6], the maximum likelihood estimate (MLE) m m of appearance time 0 m will be defined as The block diagram of the measurer (26) is shown in Fig. 1 where the designations are the following: 1 is the switch that is open for time / , / 2 2 1 0 2 0 @; 2 is a filter with transfer function H ~h (14), (17); 3 is the squarer; 4 is the filter matched with the signal / / f t I t q f t 1 @; 5 is the extremator that fixes the location of the greatest maximum of the input signal as an estimate m m .As appears from Fig. 1, synthesized optimal receiver of HF random pulse with arbitrary-function envelope and unknown appearance time has a single-channel structure, unlike alternative multichannel variants presented in the current researches [2, etc.].

Fig. 1 The optimal measurer of appearance time of HF random pulse
Let us find the characteristics of MLE m m .During the analysis process, we find it expedient to divide all the estimates into the two classes: reliable and anomalous [6].The estimate / lm where the regular function (23 , then the estimate and the corresponding estimate error are designated as anomalous [6].It is necessary to consider the anomalous errors if the length of the prior interval , The block diagram of the detection algorithm (32) of the random pulse can be obviously received through the block diagram presented in Fig. 1 by substituting extremator 5 for solving device, which analyzes input realization M m ^h (22) within the interval , and makes a decision about presence or absence of a signal (1), ( 14), comparing an absolute maximum of realization M m ^h with a threshold c.Let us find characteristics of random pulse detector (32).As such characteristics we will use type I (false-alarm) and II (signal missing) error probabilities [1, 2, 4, 6, etc.].The starting point is that the useful signal is absent.Then the false-alarm probability α can be determined as follows Here, As follows from [4], (25), the functional l M u c ^h is asymptotically Gaussian (under " 3 n ) with zero mathematical expectation and unit dispersion.According to [11 and 12], while increasing the threshold c, the distribution of number of outliers over level c by stationary centered Gaussian random process converges to the Poisson's law with threshold c growth.On the basis of results [4 and 9] it allows to write down asymptotically (under 3) exact expression for falsealarm probability (33) as , , , .

Detection of the random Gaussian pulse with unknown appearance time
Now let us consider a detection problem of the HF random pulse (1), ( 14) with unknown appearance time 0 m .As it is well known from [6], MLR makes a decision about the presence of a useful signal in terms of the comparison of the absolute (greatest) maximum value of the logarithm of FLR (21), with a threshold c which is chosen according to the set optimality criterion of detection.Therefore, the magnitude , we receive expressions for false-alarm and missing probabilities at the reception of a HF random pulse (1), ( 14) with rectangular envelope and unknown appearance time [4 and 9].

Results of statistical modeling
For the purpose of an establishment of the applicability borders of the asymptotically exact formulas for the detection and estimation characteristics, we applied the MLR statistical computer modeling.
In the process of modeling, we followed the technique, introduced in [9], and formed the samples of the normalized

Conclusion
For the synthesis of the algorithms of the statistical analysis of random pulsed signals with the arbitrary-function envelope, the approach neglecting the values of an order of correlation time of a pulse random substructure is effective.The given approach makes it possible to receive detectors and measurers of the random pulses with the unknown parameters sufficiently simply realized in practice.
Quality of the synthesized processing algorithms of the pulsed signals with the unknown discontinuous parameters can be theoretically estimated by means of the local Markov approximation method based on the representation of the solving statistics increments by the Markov random process of diffusion type in a small neighborhood of the point of the true value of the unknown parameter [9].As a result, it is possible to write down the closed analytical expressions for detection and estimation characteristics having acceptable accuracy for SNR being more than 1…2, and that was proved by the results of the statistical modeling.
The results of the study are of a general importance.They can be used in radio physical measurements while processing the random pulses of optical, acoustic, electromagnetic and other origins, in the systems information transmission and processing and in the systems of the technological processes monitoring and control, and also in other fields of science and technology, dealing with registration and measurement of random processes.As q 0 decreases when the SNR z≤5...6, the probability P a = 1 -P 0 of the anomalous errors increases significantly and approaches 1.This leads to a jump-like (compared with the case of reliable estimate) increase in the variance of the appearance time estimate (so-called threshold effects occur).As q 0 increases, when z>5...6, the variance V l converges to the variance V 0l , and the estimate l m (26) becomes reliable with the probability close to 1. From Eqs. (28), (31) and Figs. 2 follows that the minimum (threshold) value of the parameter q 0 , at which the influence of the anomalous errors on the accuracy of MLE of the signal (1), ( 14) appearance time can be still neglected, decreases with μ and increases with m. ^h of the reliable estimate of the appearance time is obtained ignoring the errors in estimating the order of correlation time of the random substructure of the pulsed signal (1), (14).Consequently, the error of Eqs.(28), (31) becomes significant, as the relative variance of the MLE decreases to a magnitude of order 2 n -.

2 1 T
=and taking into account the ratios (2), we come to values of appearance time 0 m is much greater than signal (1) duration 0x , i.e. the following condition holds

Using
Eqs. (23)-(25) it is easy to show that if condition (27) is satisfied then values of the absolute maxima of functional l M^h are approximately statistically independent in subareas S C and N C .The specified property allows similarly [4 and 9] to present Eq.(35) in a kind of / F u F u r and g t mn u ^h, G t mn u ^h are defined according to Eqs. (24).Using results [9 and 10], on the basis of regular and noise functions properties (30) of solving statistics (22) for conditional bias l

2
The theoretical and experimental dependences of normalized variance of appearance time estimate a) b) Fig. 3 The theoretical and experimental dependences of false-alarm probability a) b) Fig. 4 The theoretical and experimental dependences of missing probability minimum and maximum values of the function f t ^h.
For z<2...3 the theoretical dependences of variance V 0l of the reliable MLE (26) deviate from experimental ones substantially, as are found without finite length of the interval S C of the reliable MLE like values.Departing of theoretical dependences V l from experimental data is observed under high SNR when q>2...3 too.This is due to the fact that Eq. (31) for the variance V l l m 0 0 the function, the spectrum H ~h of which satisfies the ^h is much less than the signal s t ^h (1) duration and the characteristic changing time t T of the function f t ^h .Then the filter response y t u ^h (16) allows representation: