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

通信学报 ›› 2018, Vol. 39 ›› Issue (12): 1-9.doi: 10.11959/j.issn.1000-436x.2018219

• 学术论文 • 下一篇

信号特征谱表示与平稳随机信号谱分解统一的研究

王宏禹,邱天爽()

大连理工大学电子信息与电气工程学部，辽宁大连 116024

修回日期:2018-11-14 出版日期:2018-12-01 发布日期:2018-11-23
作者简介:王宏禹（1929–），男，北京人，大连理工大学教授、博士生导师，主要研究方向为随机与时变信号处理。|邱天爽（1954–），男，江苏海门人，大连理工大学教授、博士生导师，主要研究方向为射频与医学信号处理。
基金资助:
国家自然科学基金资助项目(No.61671105);国家自然科学基金资助项目(No.61172108);国家自然科学基金资助项目(No.61139001)

Unified study on characteristic spectrum representation of signals and spectral decomposition for stationary random signals

Hongyu WANG,Tianshuang QIU()

Faculty of Electronic Information and Electrical Engineering, Dalian University of Technology, Dalian 116024, China

Revised:2018-11-14 Online:2018-12-01 Published:2018-11-23
Supported by:
The National Natural Science Foundation of China(No.61671105);The National Natural Science Foundation of China(No.61172108);The National Natural Science Foundation of China(No.61139001)

摘要/Abstract

摘要：

深入研究了信号特征谱表示与平稳随机信号谱分解统一的关系。利用微分算子、积分算子与特征微分方程格林函数的关系，分析了厄尔密特微分算子与厄尔密特积分算子的互逆关系，指出了特征微分方程存在与其对应的特征积分方程，并给出了厄尔密特微分算子与厄尔密特积分算子的谱表示式，以及将这2种算子的谱表示统一的一般表示式。利用对平稳随机信号谱分解的随机振幅简谐振动叠加法及希尔伯特空间酉算子法进行分析研究，明确了信号特征谱表示与平稳随机信号谱分解的联系及数学上的统一。

关键词: 格林函数, 厄尔密特算子, 微分算子, 积分算子, 正交投影算子, 谱表示, 酉算子, 谱分解

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

中图分类号:

TN911.7

王宏禹,邱天爽. 信号特征谱表示与平稳随机信号谱分解统一的研究[J]. 通信学报, 2018, 39(12): 1-9.

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.

图/表 4

图1

表1

f(x)特征谱表示与x(t)谱分解式相对应的关系"

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)

表1

图2

表2

2种信号特征展开与特征谱表示的关系"

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， $E [y_{i}^{2}] = λ_{i}$

表2

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信号特征谱表示与平稳随机信号谱分解统一的研究

Unified study on characteristic spectrum representation of signals and spectral decomposition for stationary random signals

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