随机信号分析
第五章 随机信号通过线性系统
5.1
1.低通白噪声
若h(t)是单位增益的理想低通滤波器，带宽为BHz (或 W = 2πB rad/s),则白噪声通过该滤波器的输出噪声为Y(t),被称为带宽为BHz的低通（低频带限）白噪声。容易得出：
S Y ( ω ) = { N 0 2 0 ≤ ∣ ω ∣ ≤ W 0 ∣ ω ∣ > W S_Y\left( \omega \right) =\left\{ \begin{array}{l} \begin{matrix} \frac{N_0}{2}& 0\le \left| \omega \right|\le W\\ \end{matrix}\\ \begin{matrix} 0& \left| \omega \right|>W\\ \end{matrix}\\ \end{array} \right. SY(ω)={ 2N00≤∣ω∣≤W0∣ω∣>W
R Y ( τ ) = C Y ( τ ) = N 0 sin W t 2 π τ R_Y\left( \tau \right) =C_Y\left( \tau \right) =\frac{N_0\sin Wt}{2\pi \tau} RY(τ)=CY(τ)=2πτN0sinWt
ρ Y ( τ ) = C Y ( τ ) C Y ( 0 ) = sin W t W τ \rho _Y\left( \tau \right) =\frac{C_Y\left( \tau \right)}{C_Y\left( 0 \right)}=\frac{\sin Wt}{W\tau} ρY(τ)=CY(0)CY(τ)=WτsinWt
τ c = ∫ 0 + ∞ ρ Y ( τ ) d τ = ∫ 0 + ∞ sin W τ W τ d τ = π 2 W = 1 4 B \tau _c=\int_0^{+\infty}{\rho _Y\left( \tau \right) d\tau =}\int_0^{+\infty}{\frac{\sin W\tau}{W\tau}}d\tau =\frac{\pi}{2W}=\frac{1}{4B} τc=∫0+∞ρY(τ)dτ=∫0+∞WτsinWτdτ=2Wπ=4B1
P Y = N 0 2 ⋅ 2 W ⋅ 1 2 π = N 0 B P_Y=\frac{N_0}{2}\cdot 2W\cdot \frac{1}{2\pi}=N_0B PY=2N0⋅2W⋅2π1=N0B
第六章 带通信号
6.1希尔伯特变换
对于信号x(t)的希尔伯特变化为：
x ( t ) = H ( x ( t ) ) = x ( t ) ∗ 1 π t {x\left( t \right)}=H\left( x\left( t \right) \right) =x\left( t \right) *\frac{1}{\pi t} x(t)=H(x(t))=x(t)∗πt1
x ( t ) = z ( t ) + z ∗ ( t ) 2 x(t)=\frac{z(t)+z^*(t)}{2} x(t)=2z(t)+z∗(t)
x(t)是
x ( t ) x(t) x(t)
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