信号特征谱表示与平稳随机信号谱分解统一的研究

doi:10.11959/j.issn.1000-436x.2018219

Abstract

Abstract:

The unified relationship between the signal characteristic spectrum representation and the spectral decomposition for the stationary random signal was deeply studied. By using the relations among the differential operators, the integral operator and the Green's function of the characteristic differential equation, the inverse relationship between the Hermitian differential operator and the Hermitian integral operator were given, the characteristic differential equation and corresponding characteristic integral equation were demonstrated, and the spectral representations of both Hermitian differential and integral operators and the general spectral representations for both operators were provided. Based on the superposition method of the stochastic simple harmonic vibration and the Hilbert space unitary operator method for the stationary random signal spectral decomposition, the connection and unification on mathematics of the signal characteristic spectral representation and the stationary random signal spectral decomposition are revealed.

Key words: Green function, Hermitian operator, differential operator, integral operator, orthogonal projection operator, spectral representation, unitary operator, spectral decomposition

CLC Number:

TN911.7

Hongyu WANG,Tianshuang QIU. Unified study on characteristic spectrum representation of signals and spectral decomposition for stationary random signals[J]. Journal on Communications, 2018, 39(12): 1-9.

Figures/Tables 4

序号	确定性信号f(x)	对应关系	平稳随机信号x(t)
1	信号 f(x)特征谱表示式 $f (x) = \int_{- \infty}^{\infty} λ d E (λ)$	?	平稳随机信号x(t)谱分解式 $x (t) = \int_{- \infty}^{\infty} e^{j n ω_{0} t} d Z (ω)$ (
2	特征谱λ	?	正弦频谱 $e^{j n ω_{0} t}$ t
3	正交投影算子P_i	?	正交展开系数ξ_i
4	正交投影算子函数 $E (λ_{n}) = \sum_{i = 1}^{n} P_{i}$	?	正交投影系数函数 $Z (n ω_{0}) = \sum_{i = 1}^{\infty} ξ_{i}$ n
5	E 函数增量的极限(λ_n?λ_n?1)→0, E(λ_n)?E(λ_n?1)=dE(λ)	?	正交展开系数函数增量的极限ω₀→0, Z(nω₀)?Z[(n?1)ω₀]=dZ(ω)

序号	名称	确定性信号 f(x)	?	平稳随机信号x(t)
1	特征展开	$f (x) = \sum_{i = 1}^{\infty} c_{i} ϕ_{i} (x)$	?	$x (t) = \sum_{i = 1}^{\infty} y_{i} ϕ_{i} (t), 0 \leq t \leq T$
2	展开系数	$c_{i} = \int_{0}^{\infty} f (x) ϕ_{i} (x) d x$	?	$y_{i} = \int_{0}^{T} x (t) ϕ_{i} (t) d t$
3	归一化正交特征函数	$\int_{0}^{\infty} ϕ_{i} (x) ϕ_{j} (x) d x = δ_{i j}$	?	$\int_{0}^{T} ϕ_{i} (t) ϕ_{j} (t) d t = δ_{i j}$
4	特征积分方程	$λ_{i} ϕ_{i} (x) = \int_{0}^{\infty} K (x, ξ) ϕ_{j} (x) d ξ$	?	$λ_{i} ϕ_{i} (t_{1}) = \int_{0}^{T} K (t_{1}, t_{2}) ϕ_{j} (t_{2}) d t_{2}$
5	核函数	K=g (x,ξ), g(x,ξ)为格林函数	?	k=r_xx(t ₁,t₂),r_xx(t₁,t₂)为自相关函数
6	谱表示	$g (x, ξ) = \sum_{i = 1}^{\infty} λ_{i} ϕ_{i} (x) ϕ_{j} (x)$	?	$r_{x x} (t_{1}, t_{2}) = \sum_{i = 1}^{\infty} λ_{i} ϕ_{i} (t_{1}) ϕ_{j} (t_{2})$
注：y_i(i=1,2,…)为随机变量， $E [y_{i}^{2}] = λ_{i}$

References 10

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