最佳检测和最优估计的参数的包络孤子

a
Kotel’nikov Institute of Radio Engineering and Electronics, Russian Academy of Sciences, building 7, ul. Mokhovaya 11, GSP 9, Moscow K 9, 125009 Russia b Kharkov National University, pl. Svobody 4, Kharkov, 61077 Ukraine e mail: potapov@cplire.ru 73
cyclic frequency. Then, the dimensionless period is 2π 4π T= = . ω u1 u2 As the envelope soliton duration ts, take the dis tance between two points on the time axis where the envelope decreases by a factor of n as compared to its maximum value. Then, 2κ ts = 2 ν T = . α κ , where ν is the number of periods along νT half the envelope soliton duration: As α = κ = ln ( n + n – 1 ) . In this study, it was assumed for certainty that n = e. Then, κ ≈ 1.66. DETECTION OF THE ENVELOPE SOLITON As is known (see, for example, [9]), when the con dition L = Ws [ x ( t ) ] > L th Ww [ x ( t ) ] (3)
ISSN 1028 3358, Doklady Physics, 2011, Vol. 56, No. 2, pp. 73–77. © Pleiades Publishing, Ltd., 2011. Original Russian Text © Yu.V. Gulyaev, S.V. Panasenko, A.A. Potapov, L.F. Chernogor, 2011, published in Doklady Akademii Nauk, 2011, Vol. 436, No. 5, pp. 606–610.
ϕ0 =
u1 ( x – φ0 ) – u2 ( x – x0 ) . 2
Similarly to narrowband radio signals, we call ϕ0 a dimensionless initial phase and ω, a dimensionless
74
GULYAEV et al.
u × exp i 1 ( x – u 2 t – φ 0 ) . 2
Here b = ( u 1 – 2 u 2 ) u 1 and u1, u2 are the envelope and phase propagation velocities, and x0 and φ0 are the initial coordinate and phase, respectively. For further analysis, the real part of function (2) was used presented for a fixed coordinate in the form v(t) = where A = and b , 2β α = bu 1 , 2 τ = x – x0 , u1 ω = u1 u2 , 2 A cos [ ω ( t – τ ) – ϕ 0 ] , cosh [ α ( t – τ ) ]
u 2
is satisfied, where Ww[x(t)] and Ws[x(t)] are the plausi bility functionals in the absence and presence of the detected signal s(t) in the received signal x(t), respec tively, and Lth is the threshold value of detection crite rion L, the solution on the presence of the detected signal is made. If the interfering signal w(t) is normal white noise, expression (3) can be written as [9]
PHYSICS
Optimal Detection and Optimal Estimation of Parameters of the Envelope Soliton
Academician Yu. V. Gulyaeva, S. V. Panasenkob, A. A. Potapova, and L. F. Chernogorb
k k
X1 + Y1 ,
k k k
2
2
∑ a [ X cos ( ω τ ) + Y sin ( ω τ ) ] ,
k
Y1 = Here,
∑ b [ Y cos ( ω τ ) – X sin ( ω τ ) ] .
k k k k k k
π ( ω – ωk ) π ( ω + ωk ) a k = π sech + sech , 2α 2α α t0 π ( ω – ωk ) π ( ω + ωk ) b k = π sech – sech , α t0 2α 2α and Xk and Yk are the estimates of amplitudes of the quadrature components of the signal x(t). Since ak ≈ bk, Xk and Yk have equal dispersions and the characteristic Z1 is distributed according to the Rayleigh–Rice law. Then, the expressions for proba bilities of false alarm p f . a . and signal skip ps.o. will acquire the form k p f . a . = exp ⎛ A ⎞ , ⎝ 2 κμ⎠ where
2
Upon detection of the envelope soliton with an unknown initial phase, expression (3) will be equiva lent to the condition Z 1 > Z 1 th , where Z1 = X1 =
k mq⎞ , ps .o . = Θ ⎛ a , ⎝ κμ 2 μ ⎠
L =
∫
λ
exp ⎛ ⎝
2 Z ( λ ) – E ( λ )⎞ W ( λ ) dλ > L th . ⎠ 0 N0
Θ ( u, v ) =
∫
0
ξ exp ⎛ – ⎝
ξ + v ⎞I 0 ( ξ v ) dξ , 2 ⎠
2
2
Here, N0 is the spectral density of the white noise power
Received October 18, 2010
DOI: 10.1134/S102833581102008X
INTRODUCTION At present, much attention is paid to generation, transmission, detection, and receiving of ultrawide band (UWB) signals (see, for example, [1]). These sig nals have considerable advantages over traditional nar rowband signals [1, 2]. One of the ultrawideband signal class is the nonlin ear UWB signals introduced by the authors of [2]. These signals are governed by solutions of the nonlin ear differential equations (Burgers, Korteweg–de Vries, Schrödinger, and other equations, [3, 4]) and, therefore, retain their form upon propagation in non linear medium. The aim of this study is to solve the problems of optimal detection against the additive white Gaussian noise background and optimal estimation of the parameters of the envelope soliton which is nonlinear UWB signal. The theory of optimal signal detection and optimal estimation of their parameters was developed by some authors [1, 5–7]. They solved the problems of optimal detection and optimal estimation of parameters mainly of narrowband and wideband signals. In study [8], the problem of optimal detection of UWB quasi radio signals was solved. Practical application of the result obtained is limited because of the necessity of numerical solution of integral equations and the com plexity and inconvenience of the calculations. THEORETICAL MODEL The nonlinear Schrödinger equation has the form (see, for example, [3])