4.3 窄带随机过程的基本特点

S AC (ω ) = S AS (ω ) = FT [ R X (τ ) cos(ω0τ ) + RX (τ ) sin(ω0τ )]
1 j = [ S X (ω + ω0 ) + S X (ω ω0 )] + [ S X (ω + ω0 ) S X (ω ω0 )] 2 2
S X (ω ) = j sgn( ω ) S X (ω )
AC (t)与AS (t)的互相关函数是奇函数
当τ = 0时, 有 : RAC AS (0) = 0
在同一时刻 AC (t)与AS (t)之间是正交的 , .
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RAC AS (τ ) = RAS AC (τ ) SAC AS (ω) = SAS AC (ω) = FT[RAC AS (τ )]
RAC AS (τ ) = RX (τ ) sin( ω0τ ) + RX (τ ) cos(ω0τ )
1 SAC (ω) = SAS (ω) = {SX (ω +ω0 )[1+ sgn( ω +ω0 )] 2 + SX (ω ω0 )[1sgn( ω ω0 )]}
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ω
SX (ω ω0 ) + SX (ω +ω0 )
1 ω 2
偶函数
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ω SX (ω +ω0 ) + SX (ω ω0 ) ω < SAC (ω) = SAS (ω) = 2 0 其它
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E[ AC (t)] = E[ AS (t)] = 0
AC (t)和AS (t)都是平稳过程
RAC (τ ) = RAS (τ ) = RX (τ ) cos(ω0τ ) + RX (τ ) sin( ω0τ )
当τ = 0时, 有 : RAC (0) = RAS (0) = RX (0)
AC (t)和AS (t)与X (t)具有相同的平均功率和方差
AS (t) Φ(t) = arctan AC (t)
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4.3.2 窄带过程的特点
本节所讨论的随机过程X(t)是任意的宽 本节所讨论的随机过程X(t)是任意的宽 X(t)是任意的 平稳、数学期望为零的窄带随机过程 的窄带随机过程。 平稳、数学期望为零的窄带随机过程。
X (t) = A(t) cos[ω0t + Φ(t)]
令ω′ = ω ω0
ω 1 2 = ∫ ω 2 S X (ω ′ + ω 0 ) cos( ω ′τ + ω 0τ ) d ω ′ 2π 2
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1 RX (τ ) = ∫ 2π
ω 2 ω 2
2S X (ω + ω0 ) cos(ωτ + ω0τ )dω
1 = 2π
∫
ω 2 ω 2
2 S X (ω + ω0 ) cos(ωτ ) cos(ω0τ )dω
4.3 窄带随机过程的基本特点
4.3.1 窄带随机过程的表达式
如果信号的频谱集中在以ω0为中心频率的 有限带宽ω内, 且有ω0 >> ω , 则信号被称 之为窄带信号.如果系统的频率响应也具有 上述特点, 则系统被称之为窄带系统.
型的确定 性的窄带 号可表 信 示为: 幅度调制或包 典 络调制信号 x(t) = a(t) cos[ω0t +(t)] 相位调制信号
ω 2 ω 2
0
1 2π
∫
2 S X (ω + ω0 ) sin(ωτ ) sin(ω0τ )dω
ω 称的 如 果窄 带过 X (t)的 程 单边 功率 谱是 关于 0对 ,
ω ω 则 X (ω +ω0 )在 ( , )区 间与 X (ω ω0 )相 , 等 S S 2 2 且 ω的 是 偶函 . 数
= RX (τ ) cos(ω0t ) sin(ω0t + ω0τ ) + RX (τ ) sin(ω0t ) cos(ω0t + ω0τ ) + RXX (τ ) cos(ω0t ) cos(ω0t + ω0τ ) RXX (τ ) sin(ω0t ) sin(ω0t + ω0τ )
RX (τ ) = RX (τ ) RXX (τ ) = RXX (τ ) = RXX (τ ) = RX (τ )
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1 RX (τ ) = 2π
∫
ω 2 ω 2
2 S X (ω + ω0 ) cos(ωτ ) cos(ω0τ )dω
1 = cos(ω0τ ) 2π 1 = cos(ω0τ ) 2π
∫
ω 2 ω 2 ω 2 ω 2
S AC (ω ) cos(ωτ )dω S AC (ω )e
jωτ
∫
j SAC AS (ω) = SAS AC (ω) = [SX (ω ω0 ) SX (ω +ω0 )] 2 1 + [SX (ω +ω0 ) + SX (ω ω0 )] 2 SAC AS (ω) = SAS AC (ω) = SX (ω) = j sgn( ω)SX (ω)
j [SX (ω ω0 ) sgn( ω ω0 )SX (ω ω0 ) 2 SX (ω +ω0 ) sgn( ω +ω0 )SX (ω +ω0 )]
S 0 0
R AC AS (t , t + τ ) = E [ AC (t ) AS (t + τ )]
R AC AS (t , t + τ ) = E[{ X (t ) cos(ω0t ) + X (t ) sin(ω0t )} { X (t + τ ) sin(ω0t + ω0τ ) + X (t + τ ) cos(ω0t + ω0τ )}]
dω
= cos(ω0τ ) RAC (τ )
RX (τ ) = RAC (τ ) cos(ω0τ ) = RAS (τ ) cos(ω0τ )
前提条件: SX (ω)单边功率谱关于ω0对称
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A (t) = X (t) cos(ω0t) + X (t) sin( ω0t) C A (t) = X (t) sin( ω t) + X (t) cos(ω t)
如果窄带过程 X (t )的单边功率谱是关于 (t
ω 0 对称 , 则有 : S A
C
AS
(ω ) = S AS AC (ω ) = 0
A (t) = A(t) cos[Φ(t)] C
AS (t) = A(t) sin[ Φ(t)]
X (t) = AC (t) cos(ω0t) AS (t) sin( ω0t)
H[a(t) cos(ω0t)] = a(t) sin( ω0t) H[a(t) sin( ω0t)] = a(t) cos(ω0t)
这是窄带过程通用的表达式
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X (t) = A(t) cos[ω0t + Φ(t)]
X (t) = AC (t) cos(ω0t) AS (t) sin( ω0t)
AC (t) = A(t) cos[Φ(t)]
A(t) = A (t) + A (t)
2 C 2 S
AS (t) = A(t) sin[ Φ(t)]
RX (τ ) = RX (τ ) RXX (τ ) = RXX (τ ) = RXX (τ ) = RX (τ )
RAC (τ ) = RX (τ ) cos(ω0τ ) + RXX (τ ) sin(ω0τ ) A (t)平稳 C
RAC (τ ) = RX (τ ) cos(ω0τ ) + RX (τ ) sin( ω0τ )= RAS (τ )
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RAC AS (τ ) = RX (τ ) sin( ω0τ ) + RX (τ ) cos(ω0τ )
AC (t)与AS (t)联合平稳
RAC AS (τ ) = RX (τ ) sin( ω0τ ) RX (τ ) cos(ω0τ )
RAC AS (τ ) = RAC AS (τ ) = RAS AC (τ ) = RAS AC (τ )
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RAC (t , t + τ ) = E[{ X (t ) cos(ω0t ) + X (t ) sin(ω0t )} { X (t + τ ) cos(ω0t + ω0τ ) + X (t + τ ) sin(ω0t + ω0τ )}]
= R X (τ ) cos(ω0t ) cos(ω0t + ω0τ ) + R X (τ ) sin(ω0t ) sin(ω0t + ω0τ ) + R XX (τ ) cos(ω0t ) sin(ω0t + ω0τ ) + R XX (τ ) sin(ω0t ) cos(ω0t + ω0τ )
Байду номын сангаас
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SX (ω) = j sgn( ω)SX (ω)
1 S Ac (ω ) = S As (ω ) = [ S X (ω + ω0 ) + S X (ω ω0 )] 2 j + [ S X (ω + ω0 ) S X (ω ω0 )] 2
1 SAC (ω) = SAS (ω) = [SX (ω +ω0 ) + SX (ω ω0 ) + 2 sgn( ω +ω0 )SX (ω +ω0 ) sgn( ω ω0 )SX (ω ω0 )]
X (t) = A (t)sin( ω0t) + AS (t) cos(ω0t) C
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X (t) = AC (t) cos(ω0t) AS (t) sin( ω0t)
X (t) = A (t) sin( ω0t) + AS (t) cos(ω0t) C
AC (t) = X (t) cos(ω0t) + X (t) sin( ω0t) A (t) = X (t) sin( ω t) + X (t) cos(ω t)
如果一个随机过程的功率谱是集中在 以ω0为中心频率的有限带宽ω内, 并满 足ω0 >> ω , 则称它为窄带随机过程.
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ω0 >> ω
X (t) = A(t) cos[ω0t + Φ(t)]
令: AC (t) = A(t) cos[Φ(t)] AS (t) = A(t) sin[ Φ(t)]
A(t) = A (t) + A (t)
S 0 0
E[ X (t )] = E[ X (t )] = 0
E[ AC (t)] = E[ AS (t)] = 0
RAC (t , t + τ ) = E[ AC (t ) AC (t + τ )]
= E[{ X (t ) cos(ω0t ) + X (t ) sin(ω0t )} { X (t + τ ) cos(ω0t + ω0τ ) + X (t + τ ) sin(ω0t + ω0τ )}]
2 C 2 S
AS (t) Φ(t) = arctan AC (t)
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AC (t) = A(t) cos[Φ(t)]
AS (t) = A(t) sin[ Φ(t)]
cos(α + β ) = cosα cos β sin α sin β
X (t) = A(t) cos[ω0t + Φ(t)] = A(t) cos(ω0t) cos[Φ(t)] A(t) sin( ω0t) sin[ Φ(t)] = A(t) cos[Φ(t)]cos(ω0t) A(t) sin[ Φ(t)]sin( ω0t) = AC (t) cos(ω0t) AS (t) sin( ω0t)
a(t)和(t)都是时间的函数 相对载频ω0而言都是慢变的 , .
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典型的确定性窄带信号: x(t) = a(t) cos[ω0t +(t)]
窄带随机过程可表示为: X (t) = A(t) cos[ω0t + Φ(t)]
A(t)是 带 窄 过程 的包 , Φ(t)是 络 窄带 过程 的 相位 它 , 们相 ω0都 对 是慢 变随 机过 . 程
ω AC (t)和AS (t)都是低频过程 它们的功率谱集中在ω < , 内 . 2
1 RX (τ ) = 2π
∫
∞
∞
S X (ω )e
jωτ
∞ 1 2∫ S X (ω ) cos(ωτ )dω dω = 0 2π
1 ω 0 + 2ω = ∫ ω 2 S X (ω ) cos( ωτ ) dω 2π ω 0 2
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SX (ω ω0 ) SX (ω +ω0 )
1 ω 2
奇函数
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ω 互 密 S 中 内 , A (t)与 S (t)的 谱 度 AC AS (ω)与 AS AC (ω)集 在ω < A S C 2 且 奇 数其 达 为: 为 函 . 表 式 ω j[SX (ω ω0 ) SX (ω +ω0 )] ω < SAC AS (ω) = SAS AC (ω) = 2 0 其 它