2016 International Conference on Mathematical, Computational and Statistical Sciences and Engineering (MCSSE 2016) ISBN: 978-1-60595-396-0
The Decision Statistics of the Gaussian Signal against
Correlated Gaussian Interferences
Oleg V. CHERNOYAROV
1,2,3,*, Mahdi M. SHAHMORADIAN
1and
Konstantin S. KALASHNIKOV
31National Research University “MPEI”, 14, Krasnokazarmennaya st., 111250 Moscow, Russia 2
International Laboratory of Statistics of Stochastic Processes and Quantitative Finance of National Research Tomsk State University, 36, Lenin Avenue, 634050 Tomsk, Russia
3
Voronezh State University, 1, Universitetskaya sq., 394018 Voronezh, Russia *Corresponding author
Keywords: Gaussian random process, Functional of the likelihood ratio, Maximum likelihood Processing algorithm, Correlated interference.
Abstract. We introduce an approach to obtaining the approximations of the decision statistics of the fast-fluctuating Gaussian random signals. By means of the presented procedure, there can be found a new much simpler expression for the logarithm of the functional of the likelihood ratio of the Gaussian random process against white noise and correlated Gaussian interference. As a result, new simpler circuits for the receivers of random information process can be synthesized.
Introduction
When we are to solve tasks concerning the signals detection and the measurement of their unknown signal parameters, the procedure should be suggested describing how to form the decision statistics. Under decision statistics we here understand the functional of the likelihood ratio (FLR), or its logarithm. For quasi-deterministic (i.e., specified within the finite number of unknown parameters) useful signal, the corresponding approaches to obtaining the logarithm of FLR are already considered in detail in many papers [1,2 etc.]. However, if the useful signal is a random (Gaussian) process, then the expression for the logarithm of FLR in continuous time can be relatively easy written down only for the case of the fractional rational spectral densities of the signal and its correlated distortions. When the signal spectral density is of an arbitrary form, the known approximations for the logarithm of FLR are rather complex, even if background interferences are described by white noise. In case of the correlated random distortions, the structure of the logarithm of FLR tends to become increasingly complicated.
Below we demonstrate the procedure for obtaining the simplified expressions for the logarithm of FLR of Gaussian random processes against correlated Gaussian interferences, specifically for the case of the fast fluctuations of the observed data. Referring to the found approximations for the logarithm of FLR, the simpler circuits for the receivers of low-frequency and high-frequency random signals can be synthesized under varied parametrical prior uncertainty.
The Logarithm of the Functional of the Likelihood Ratio of the Fast-fluctuating Gaussian Signal against Fast-fluctuating Gaussian Interferences
Thus, we start with considering the logarithm of FLR for the hypothesis
:
1
H x
t st t nt , t
0,T (1) against the alternative:
0
Here s
t is the random useful signal, n
t is Gaussian white noise with one-sided spectral density0
N ,
t is the correlated interference.We presuppose that the signal s
t is the segment of the realization of the stationary Gaussian random process
t beginning at the point of time t1 2 and ending at the point of time2
2
t . The parameters λ and τ determine the time of appearance and the duration of the signal
ts . At the same time, the conditions t10, t2 T are satisfied, i.e. for all the values of λ, τ the signal s
t is always situated within the observation interval
0,T .We designate a
t as the mathematical expectation,
t2 t1
t1
t1
t2
t2
B
as the covariance function,
B e d
G j as the spectral density of the process
t , and then we write down the signal s
t the following way:
t I t
t
s ,
. 2 1 , 0
, 2 1 , 1
x x x
I
Let us assume that process
t fluctuations are “fast”, i.e. the following condition is satisfied:1 2
1
. (3) Here 1 is the bandwidth of the process
t which is determined as
2 2
1 G d supG , if the process
t is low-frequency, or as
0
2 2
1 G d supG , if the process
t is high-frequency, correspondently [3].As an interference
t , we are to consider the fast-fluctuating stationary centered (
t 0)Gaussian random process with the covariance function B
t2 t1
t1 t2 and the spectral density G
, for which the condition analogous to Eq. (3) is satisfied [3-5].The problem of the hypothesis test H1 (1) against alternative H0 (2) for the Gaussian stochastic signal is considered in a number of papers [2,3 etc.]. In these papers the various formulas for the logarithm of FLR are found. Following [3], we write down the logarithm of FLR for Gaussian signal in the form of
T Tx t xt Qt t dtdt TxtV t dt Tas t V t dt d TQ t t dt N
L
0 1
0 0
0 0 0
2 1 2 1 2 1 0
, , ~ 2
1 2
1 ,
1
. (4)
Here
t s t t aI t
as ,
T s
s t a t Qt t dt
a N t V
0 0
, 2
, (5)
t1,t2
Q~
t1,t2,1
Q , and the function Q~
t1,t2,
is determined as the decision for the following integral equation:
1 2
0
2 1
2 1 0
, ,
, , ~ ,
, ~
2 Q t t Qt t B t t dt B t t
N T
where
21 1
1
1
1
22
2
22 1
2 1, t t B t I t I t t B t a t t s t a t t s t t
B s s
(7)
is the covariance function of the additive mix s
t t .We will now seek the solution for the equation (6) in the form which is structurally similar to its right-hand member:
,
1 ~
,
1 1 .~ 1 , ~ , ~ , , ~ 2 1 1 2 4 2 1 1 2 3 2 1 1 2 2 2 1 1 2 1 2 1 t I t I t t Q t I t I t t Q t I t I t t Q t I t I t t Q t t Q (8)
In this case, as in case of [3], we can show that the functions Q~i
t2 t1,
, i1,4 are nonzero within the domain t2t1 c and close to zero within the domain t2 t1 c. Here c is the maximum from the correlation times of the processes
t and
t . While the inequality (3) holds, the value c is small, and thus the values of the order c, or less, can be neglected in Eq. (8). Then, the expression (8) is transformed into:
1 2
1 2 4 2 1 1 2 1 2
1 , 1 1
~ , ~ , , ~ t I t I t t Q t I t I t t Q t t
Q , (9)
where Q~1
t2 t1,
, Q~4
t2t1,
are determined as the solutions for the following integral equations:
,
1 ~
,
.~ 2 , , ~ , ~ 2 1 2 0 2 1 4 1 2 4 0 1 2 2 2 2 1 1 1 2 1 0 t t B dt t t B t t Q t I t t Q N t t B dt t t B t t Q t t Q N T
(10)Here B
t2t1
B
t2t1
B
t2t1
.The solutions for Eqs. (10) can be found by Fourier transformation method, as described, for example, in paper [3]. Then, following [3], from Eq. (10) we find
. 2 , ~ , ~ , 2 , ~ , ~ 0 4 4 0 1 1 G N G dt e t Q Q G G N G G dt e t Q Q t j t j
(11).Having referred to Eqs. (9), (11), we can now specify the expressions for the members included into the logarithm of FLR (4). The last summand in Eq. (4), when Eqs. (9), (11) are taken into account, can be presented in the form of
d G N G N G T dt t t Q d T 2 1 ln 2 1 ln 4 1 , , ~ 2 1 0 0 0 1 0. (12)
0 1
0
1 , ~ 1 2
I t Q
N a t
V . (13)
Substituting now Eqs. (9), (11)-(13) into Eq. (4), we obtain the following expression for the logarithm of FLR of Gaussian signal against white noise and correlated interference:
3 2
1 L L
L
L , (14)
T
dt t d t t h t x N
dt t d t t h t x N
L
0
2
2 0
2
2
2
1 0
1
1 1
,
2
1 , ~ 1
2 2
2 0
1 0
2
a dt t x Q
N a
L ,
d
G N
G
N G T
L
2 1
ln 2
1 ln 4
1
0 0
3 .
Here hi
t , i1,2 are the functions, the spectra of which satisfy the conditions:
, 22 2 1
, ~ 1 , ~
0 0
0 4
1 2
1
G N G
N
G N Q
Q H
G N
G Q
H
2 2 1
, ~
0 4
2
2 ,
where G
G
G
. If G
0, then from Eq. (14) we get the expression for the logarithm of FLR of the fast-fluctuating Gaussian signal against white noise [3].The Logarithm of the Functional of the Likelihood Ratio of the Band Gaussian Signal against Band Gaussian Interferences
The formula (14) is simplified significantly, if the spectral densities G
, G
allow for the rectangular approximation [3-5]
2 1
d I
G , G
2 I 2
(15) for the low-frequency processes
t ,
t , and
1 1 1
1
2 I I
d
G ,
2 2 2
2
2 I I
G (16)
for the high-frequency processes
t ,
t with band centers 1, 2, correspondently. Here d, γ are the magnitudes of spectral densities G
, G
.
0, ~
, max 1 ln 1 ln 1 ln 1 ln 4 1 ln 4 , 2 1 1 0 0 0 0 1 0 1 2 0 2 0 0 0 2 2 0 0 2 2 0 2 2 2 12 0 0 0 0 2 2 2 1 0 0
N N d N d N d N T d N a L L dt t y N N dt t x d N a dt t y N d N d N N d dt t y d N N d L T (17)If the processes
t ,
t are low-frequency ones, and
max0, ~ ~ 2 max0, ~ ~ 2
,1 ln 1 ln 1 ln 1 ln 2 1 ln 2 , 1 1 0 0 0 0 1 0 1 2 0 0 0 2 2 0 0 2 2 2 12 0 0 0 0 2 2 2 1 0 0
N N d N d N d N T L L dt t y N N dt t y N d N d N N d dt t y d N N d L T (18)If the processes
t ,
t are high-frequency ones. Here y1
t , y12
t , y2
t are the responses ofthe filters, the transfer functions of which satisfy to the conditions: H1
2 2G
d ,
G
G dH12 2 4 , H2
2 2G
, and ~
12
1, ~
12
1. In a special case, under 1 2, 2 1, the expressions (17), (18) agree with the ones presented in [4,5].Conclusion
Thus, if the realization of the observable data can be described by the model in the form of a fast-fluctuating Gaussian random process, then the simplified expression for the logarithm of the functional of the likelihood ratio can be obtained by means of the introduced procedure, based on neglecting the values of the order of the random process correlation time. By means of the resulting approximation for the decision statistics, it is possible to implement the synthesis and the analysis of the simpler optimal and quasi-optimal processing (detection and parameters estimation) algorithms for the information Gaussian signals against white noise and correlated Gaussian interferences, in the conditions of varied parametrical prior uncertainty.
Acknowledgement
This research was financially supported by the Russian Science Foundation (research project No. 15-11-10022).
References
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[3] A.P. Trifonov, E.P. Nechaev, V.I. Parfenov, Detection of Stochastic Signals with Unknown Parameters (in Russian), Voronezh State University, Voronezh, 1991.
[4] O.V. Chernoyarov, A.V. Salnikova, S. Dachian, D.N. Shepelev, Adaptive Detection and Time and Power Parameters Estimation of the Low-frequency Random Pulse with Inexactly Known Durationagainst White andCorrelated Hindrances,AppliedMathematical Sciences,2(2015),83-105.