随机过程的自相关函数与功率谱

ds(t) jω ( S ω) dt
(1.2.28) (1.2.29) (1.2.30)
d ns(t) ( jω)n S( ) ω n dt t 1 1 s(t)dt S(ω) + S(0)δ (ω) ∫ jω 2 ∞
s(t) 在区间 (∞,+∞)
t
上积分为零，即信号无直流分量，
1 S( ) ω jω
The spectrum density of time function signals --- Fourier Transform
当用复正弦信号作为基本信号时，以时间函数表示的信号 可写成(反傅立叶变换的形式) 1 ∞ s(t) = S( )e jωt d ω ω (1.2.21) ∫ 2 π ∞ 其中
Real signal: The signals expressed with real function of time are ….
特点：具有有限的能量或有限的功率
Features: The energy or power of a real signal is `finite.
（2）能量信号：能量有限的信号
e
2 δ (ω ω0 ) π
（1.2.42）
Prove: According to the frequency shifting feature
s(t)e jω0t S(ω ω0 )
and We have
1 2 δ (ω) π
1 e jω0t 2 δ (ω ω0 ) π
（4）正弦与余弦函数 Sine and Cosine functions
∵
Similarly
ω cos( 0t) =
1 (e jω0t + e jω0t ) 2
∴ cos(ω0t) π[δ (ω ω0) +δ (ω +ω0)]
(1.2.43)
sin( ω0t) jπ[δ (ω +ω0 ) δ (ω ω0 )] (1.2.44)
（5）振幅为A宽度为T、中心位于原点的矩形脉冲函数
<1> 自相关函数 The self-correlation function
R(τ ) = ∫ s(t)s (t τ )dt = ∫ s(t +τ )s* (t)dt (1.2.46)
* ∞ ∞ ∞ ∞
<2> 互相关函数 The mutual correlation function
R (τ ) = ∫ 12
?limt??1tstsrr??2??sr?21r???t??1limt?t??2sspsttr????5随机过程样本函数的功率谱thepowerspectrumofthesamplefunctionsofrps随机过程样本函数是功率型函数但考虑到其频谱的随机性在求其功率谱时还须对作统计平均故随机过程tx?tx2?tx的功率谱密度公式成为其中1
δ (t) 1
1 2 δ (t) π
（2） 单位阶跃函数 Unit jump function <1> Definition:
1 I (t) = 0 t >0 t <0
(1.2.40)
<2> FT:
1 1 I (t) + δ (ω) jω 2
(1.2.41)
（3）指数函数 Exponential function jω0t
Energy signal: The signals with finite energies
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（3）时间函数信号的分解
The decomposition of time function signals
<1> 一个时间函数信号可表示成若干个基本信号的总和或积分
A time function signal can be expressed with the sum or integral of a certain number of basic signals
If
s(t) S(ω) , then the following equality holds: jωt0 s(t t0 ) S( )e
ω
（4）频移性质 Frequency Shift 若 s(t) S(ω) ,则有 s(t)e jω0t S(ω ω0 )(1.2.27)
If s(t) S(ω) , then the following equality holds: jω0t
The rectangular pulse function with the amplitude width T & center at origin
A t s(t) = A rect( ) = T 0
S(ω) = ∫ Ae jωt dt = AT
T 2 T 2
t ≤T 2 其 它
sin(ωT 2) = AT sin c(ωT 2) ωT 2
S( ω) =
s(t)
∞ ∞
∫ s(t)e
jω t
dt
(1.2.22)
S(ω)
s(t)
称为 称为
的频谱密度或
s(t) 的傅立叶变换;
S(ω)
的傅立叶反变换,
并将这种关系记为
s(t) S(ω)
(1.2.23)
When complex sine signals are used as basic signals, a time function signal can be written with the form of Inverse Fourier Transform as
moreover
∞ δ (t) = 0
∞ ∞ ∞
t =0 t ≠0
(1.2.38)
∫ δ (t)dt = 1
<2> Feature: <3> FT:
∞
δ ) ∫ f (t) (t)dt = f (0
∞ ∞ t =0
(ω) = ∫ δ (t)e jωt dt = e jωt
=1
(1.2.39)
or denoted as Inverse FT:
s(t) S(ω)
∞
, 则有
2
（9）频域卷积 Frequency domain convolution 若
s1(t) S1(ω)
∞
∫ s (t)s
1
(t τ )dτ S1(ω)S2 (ω) (1.2.34)
,
s2 (t) S2 (ω)
∞
，则有
s1(t)s2 (t) ∫ S1(ξ )S2 (ω ξ )dξ （1.2.35-1）
i= 1 n
s (t) , 1 s2 (t)
,…
sn (t)
,
∑ai si (t) ∑ai Si (ω)
i= 1
n
Where
n is an integer and ai s are constant coefficients.
（2）尺度性质 Scale transformation 若 s(t) S(ω) ,则对实常数 a 有
s(t)e
S(ω ω0 )
（5）时域微分与积分 Differential and integral in time domain 若 s(t) S(ω) , 则下列各式成立 If s(t) S(ω) , then following equalities hold:
<1> <2> <3> <4> 若
∞
∞
∞ 1 ∞ * jω(t +τ ) = dωdt ∫ sx (t) ∫ S y (ω)e π 2 ∞ ∞
1 ∞ ∞ * jωt jωτ = ∫ ∫ sx (t)e dtSy (ω)e dω 2π ∞ ∞
1 = π 2
∞ ∞
ω ω ∫ Sx ( )Sy ( )e
则上式化简为
∞
∫ s(t)dt
(1.2.31)
（6） 时间倒置 Time Reverse 若
s(t) S(ω)
t ω , 则有 s( ) S( )
, 则有 S(t) 2 s( ) π ω
(1.2.32)
（7）对偶性 Duality 若
s(t) S(ω)
(1.2.33)
（8）时域卷积 Time domain convolution 若
∞
1 s(t) = π 2
∞
t S( )e jω d ω ω ∫
Where
S( ω) =
∞
∞
∫ s(t)e
jω t
d t
is called the spectrum density, or the Fourier Transform of the s(t) and , the s(t) the Inverse Fourier Transform of the S(ω) This relation is is . denoted as s(t) S(ω) . The two functions are called a FT pair.
1 s(at) S( ) ω a 1 s(at) S( ) ω a
(1.2.25)
If s(t) S(ω) , then for a real constant a , the following equality holds:
（3）时延性质 Time Delay 若 s(t) S(ω) ,则有 s(t t0 ) S(ω)e jωt0 (1.2.26)
i.e.
t A rect() AT sin c(ωT 2) T
(1.2.45)
二 相关函数和功率 The Correlation Functions & Power
1、相关函数的普遍定义（应以遍历过程为条件）
The general definition of correlation function (condition: ergodic process)
或记为
∞
s1(t)s2 (t) S1(ω) * S2 (ω) （10）复共轭特性 Complex conjugation
若
s(t) S(ω)
(1.2.35-2)
, 则有 (1.2.36) (1.2.37)
s*(t) S*( ) ω s*(t) S*(ω)
3、典型函数的傅立叶变换 The FT of typical functions （1）单位脉冲函数（ δ 函数）Unit pulse function ( δ function) <1> Definition:
2、傅立叶变换的重要特性 The important properties of FT （1）线性性质 Linearity 若函数
s (t) 1
、s2 (t) …
sn (t)
所对应的傅立叶变换分别是
S1(ω) 、 S2 (ω)
n
…， Sn (ω) ， 则下列变换对成立：
n
式中 n 为有限正整数, ai 为常系数。
i= 1
∑ai si (t) ∑ai Si (ω)
i= 1
(1.2.24)
The following equality will hold if S1(ω) , S2 (ω) , … Sn (ω) are the corresponding Fourier transforms of respectively:
2 、相关函数的傅立叶变换 The FT of correlation functions <1> 互相关函数的傅立叶变换 The FT of self-correlation functions
* Rxy (τ ) S y (ω)Sx (ω)
(1.2.48)
Deriving:
∵
Rxy (τ ) = ∫ sy (t +τ )sx*(t)dt
∞ ∞
s2 (t)s *(t 1
τ )dt = ∫ s2 (t +τ )s *(t)dt 1
∞
∞
(1.2.47)
物理含义: 两个信号之间的交迭程度(相关程度): 两信号完全 不交迭时积分为零；完全交迭时积分值最大；部分交迭时积分值 介于零与最大值之间。
Physical meaning: Describing the extent of the overlapping (correlated) between two signals: the integral will be zero when the two signals are not overlapped thoroughly; maximum when they are overlapped thoroughly; between zero and maximum when they are overlapped partly.
2.2 随机过程的自相关函数与功率谱
The self-correlation functions & power spectra of RPs
一 信号的频谱和傅立叶变换
The spectra & Fourier transforms of signals
1 、基本概念 Basic concepts （1）实信号：可用时间的实函数表示的信号
<2> 常用基本信号：复正弦信号、 δ 函数、sinc 函数等
The basic signals frequently used: complex sine signal, sinc function (sample function) etc.
δ
function,
（4）时间函数信号的频谱密度---傅立叶变换