3
4.3 设有一窄带信号 x (t ) = xc (t ) cos(ω0t ) xs (t ) sin(ω0t ), 其中的 xc (t )与xs (t )的带宽远小于 ω0 .设X c (ω )和X s (ω ) 分别为 xc (t )与xs (t )的傅里叶变换 , Z (ω )为x (t )的解析 函数 z (t ) = x (t ) + jx (t )的傅里叶变换 , 试证 : 1 X c (ω ) = [ Z (ω + ω0 ) + Z * ( ω + ω0 )] 2 1 X s (ω ) = [ Z (ω + ω0 ) Z * ( ω + ω0 )] 2j
RX (τ ) = E[ X * (t ) X (t + τ )]
= E[{ A(t ) jB(t )}{ A(t + τ ) + jB(t + τ )}]
= RA (τ ) + RB (τ ) + j[ RAB (τ ) RBA (τ )]
1
4.2 设复随机过程 X (t )是广义平稳的 , 试证明 :
4.2 设复随机过程 X (t )是广义平稳的 , 试证明 : R X (τ ) = R ( τ )
* X
并证明功率谱密度 S X (ω )是实函数 .
解 : 设复随机过程X (t ) = A(t ) + jB (t ), 其中A(t )和B (t ) 都是实平稳随机过程, 且是联合平稳的. ,且是联合平稳的
数学期望为零的窄带平 稳随机过程 X (t ) = AC (t ) cos(ω0t ) AS (t ) sin(ω0t )
AC (t)与AS (t)正交的条件是: SX (ω)的单边谱关于ω0对称
本题中的 S X (ω )的单边谱是关于 ω0 对称的
AC (t)与AS (t)正交
9
x(t ) = xc (t ) cos(ω0t ) xs (t ) sin(ω0t )
x(t ) = xc (t ) sin(ω0t ) + xs (t ) cos(ω0t )
z (t ) = x(t ) + jx(t ) = [ xc (t ) + jxs (t )]e
jω 0 t
4
1 4.3 X c (ω ) = [ Z (ω + ω0 ) + Z * (ω + ω0 )] 2 1 X s (ω ) = [ Z (ω + ω0 ) Z * (ω + ω0 )] 2j
* RX (τ ) = RA (τ ) + RB (τ ) j[ RBA (τ ) RAB (τ )]
RX (τ ) = R (τ ) = RA (τ ) + RB (τ ) + j[RAB (τ ) RBA(τ )] 2
* X
4.2 设复随机过程 X (t )是广义平稳的 , 试证明 :
* R X (τ ) = R X ( τ )
并证明功率谱密度 S X (ω )是实函数 .
RX (τ ) = RA(τ ) + RB (τ ) + j[RAB (τ ) RBA(τ )] RX (τ ) = RA(τ ) + RB (τ ) + j[RAB (τ ) RAB (τ )] SX (ω) = FT[RX (τ )] = SA(ω) + SB (ω) 2Im SAB (ω)] [ = SA(ω) + SB (ω) + j[SAB (ω) SAB (ω)] = SA(ω) + SB (ω) + j 2 j Im SAB (ω)] [
*
5
4.4 数学期望为零的窄带平 稳随机过程 X (t ) = AC (t ) cos(ω0t ) AS (t ) sin(ω 0t ), 其功率谱密度为 : a cos[π (ω ω0 ) ω ] ω 2 ≤ ω ω0 ≤ ω 2 S X (ω ) = a cos[π (ω + ω 0 ) ω ] ω 2 ≤ ω + ω0 ≤ ω 2 0 其它 式中, a, ω , ω0皆为正常数 , 且ω0 >> ω. 试求 : (1) AC (t ), AS (t )的功率谱密度和平均功 率. ( 2) AC (t )和AS (t )是否正交 ?
* R X (τ ) = R X ( τ )
并证明功率谱密 RA (τ ) + RB (τ ) + j[ RAB (τ ) RBA (τ )]
RX (τ ) = RA (τ ) + RB (τ ) + j[ RAB (τ ) RBA (τ )]
RX (τ ) = RA (τ ) + RB (τ ) + j[ RBA (τ ) RAB (τ )]
S X (ω + ω0 ) + S X (ω ω0 ) (1) S AC (ω ) = S As (ω ) = 0
ω < ω 2
其它
6
a cos[π (ω ω0 ) ω ] ω 2 ≤ ω ω0 ≤ ω 2 S X (ω ) = a cos[π (ω + ω0 ) ω ] ω 2 ≤ ω + ω0 ≤ ω 2 0 其它 试求 : (1) AC (t ), AS (t )的功率谱密度和平均功 率.
z (t ) = [ xc (t ) + jxs (t )]e
jω 0 t
z(t)e
jω0t
= xc (t) + jxs (t)
Z (ω + ω 0 ) = X c (ω ) + jX s (ω )
Z (ω + ω0 ) = X c (ω ) + jX s (ω )
Z (ω + ω0 ) = X c (ω ) jX s (ω )
1 RAC (0) = RAS (0) = 2π
∫
∞
∞
S AC (ω )dω
1 = 2π
ω π 2a cos( ω )dω = 2a 2 cos( π ω )dω ∫ω ω ω π ∫0
ω 2
2
RAC (0) = RAS (0) =
2a
π
2
ω
8
a cos[π (ω ω0 ) ω ] ω 2 ≤ ω ω0 ≤ ω 2 S X (ω ) = a cos[π (ω + ω0 ) ω ] ω 2 ≤ ω + ω0 ≤ ω 2 0 其它 试求 : (2) AC (t )和AS (t )是否正交 ?
π π ω ) + a cos( ω) a cos( S AC (ω ) = S As (ω ) = ω ω 0
ω ω< 2 其它
ω π ω) ω < 2a cos( SAC (ω) = SAs (ω) = 2 ω 0 其它
7
π ω ω) ω < 2a cos( SAC (ω) = SAs (ω) = ω 2 0 其它