最小二乘自适应滤波(Least-Squares Adaptive Filters)

J min = ε y − zT H = ε y − zT Φ −1z
2006/11/24
11
MMVCLAB
6.2
标准RLS自适应滤波器 (Standard RLS Adaptive Filters)
2006/11/24
12
MMVCLAB
J (H) =
x ( n)
Z-1
n = n2 n = n1
ΦH = z ⇒ H LS = Φ −1z
2006/11/24
8
MMVCLAB
6.1
最小二乘法
n = n2 n = n1
Φ (m, k ) = z (−k ) =
∑ x(n − k ) x(n − m), m, k = 0,1,..., N − 1
n = n2 n = n1
∑ x(n − k ) y(n), k = 0,1,..., N − 1
一种解决途径：以时间均值代替统计均值
2006/11/24
2
MMVCLAB
•采用最小二乘准则; •按时间进行准确迭代
n = n2 n = n1 n = n2 n = n1
J (H) =
2 e ∑ ( n) =
2 ~ [( y ( n ) − y ( n )] = min ∑
J (H ) = E[e 2 (n)] = min
1)n1 =1, n2=L+N-1,前后补零N-1(自相关法) 2)n1 =1, n2=L,前补零N-1,后不补零(前加窗法) 3)n1 =N, n2=L+N-1,前不补零后补零N-1 (后加窗法) 4)n1 =N, n2=L,前后不补零(协方差法) 2006/11/24
10
MMVCLAB
6.1
最小二乘法
2006/11/24
3
MMVCLAB
6.1
最小二乘法
(Method of Least-Squares)
2006/11/24
4
MMVCLAB
6.1
最小二乘法
一 线性LS估计问题 设:(1)观察信号 x(1), x(2),..., x( L); (2)期望信号 y (1), y (2),..., y ( L)
J [H (n)] = ∑ λn −i e 2 (i ) = min
i =1
λ → 遗忘因子
e(i ) = y (i ) − ~ y (i )
x(i ) = [ x(i ), x(i − 1),..., x(i − N + 1)]T
H (n) = [h0 (n), h1 (n),..., hN −1 (n)]T
第六章 最小二乘自适应滤波 (Least-Squares Adaptive Filters)
2006/11/24
1
MMVCLAB
Motivation • LMS失调较大；. • 非平稳时存在较大的跟踪误差，因此要求信号平稳或在 较长的时间内具有平稳性
--------》不要用误差的样本值代替误差的统计均值
~ y (n) = ∑ hk x(n − k )
正交原理:LS滤波器的输入x(n-k)和误差e(n)正交, k = 0,1,…,N-1 推论1:滤波器的输出和误差e(n)正交
n = n2 n = n1
~ ∑ y ( n )e( n ) = 0
推论2:LS滤波等价于将期望信号y(n)进行正交分解
( n ) + e( n ) y ( n) = y
A = λΦ(n − 1) B = x(n);C = 1, D = xT (n) P(n) = λ −1P(n − 1) − λ −1P(n − 1)x(n)[1 + xT (n)λ −1P(n − 1)x(n)]−1 xT (n)λ −1P(n − 1)
−2 T λ P ( n 1 ) x ( n ) x (n)P (n − 1) − −1 P (n) = λ P (n − 1) − [1 + λ−1xT (n)P (n − 1)x(n)]
Φ = [Φ (m, k )]N × N z = [ z (0), z (−1),..., z (− N + 1)]T
2006/11/24
9
MMVCLAB
6.1
最小二乘法
J (H) =
n = n2 n = n1
2 e ∑ ( n) =
n = n2 n = n1
2 ~ [( y ( n ) − y ( n )] = min ∑
n =1
J min (n) = ε y (n) − zT (n)H (n)
{H RLS (n)}
ε y = λε y (n − 1) + y 2 (n)
J min (n) = ε y (n) − zT (n)H RLS (n) = λε y (n − 1) + y 2 (n) − [λ z (n − 1) + x(n) y (n)]T [H (n − 1) + k (n)ξ (n)] = λ[ε y (n − 1) − zT (n − 1)H (n − 1)] + y (n)[ y (n) − xT (n)H (n − 1)] − zT (n)k (n)ξ (n)
i =1 n −1 i =1
Φ(n) = λ Φ(n − 1) + x(n)xT (n)
= λ ∑ λn −1−i x(i )xT (i ) + x(n)xT (n)
2006/11/24
14
MMVCLAB
z (n) = ∑ λ y (i )x(i ) = λ ∑ λn −1−i x(i ) y (i ) + x(n) y (n)
7
MMVCLAB
6.1
最小二乘法
三 正则方程(Normal Equation)
e( n ) = y ( n ) − ~ y (n) = y (n) − ∑ hk x(n − k )
n = n2 n = n1 N −1
N −1 k =0
∑ x(n − k )e(n) = 0, k = 0,1,..., N − 1
2006/11/24
17
MMVCLAB
z (n) = λz (n − 1) + x(n) y (n) H (n) = H (n − 1) + k (n)ξ (n)
4) J [ H (n)] = min = J RLS 的递推计算
n
J min = ε y − z H LS
T
ε y = ∑ λ n −i y 2 ( n)
五 Minimum Sum of Error Squares
ε~ y =
n = n2 n = n1
2 ~ y ∑ ( n) =
n = n2 n = n 2 n = n1 n = n1
2 [ h x ( n − k )] ∑∑ k
= H T ΦH = H T z = z T H
H LS = Φ −1z
J (H ) =
n = n2 n = n1
∑e
2
( n) =
n = n2 n = n1
y (n)] ∑ [( y(n) − ~
N −1 k =0
2
= min
二 正交原理(Principle of Orthogonality)
n = n2
∂J (H ) = 0 ⇒ ∑ x(n − k )e(n) = 0, k = 0,1,..., N − 1 ∂H n = n1
P (n) = λ−1P (n − 1) − λ−1k (n)xT (n)P(n − 1)
k ( n) = P ( n) x( n)
3) H (n)的递推计算
H (n) = Φ −1 (n)z (n) = P(n)z (n)
验前误差
H (n) = H (n − 1) + k (n)ξ (n) ξ (n) = y (n) − H T (n − 1)x(n)
n ⎤ ⎡ n n −i T n −i λ x ( i ) x ( i ) H ( n ) = λ y (i )x(i ) ∑ ⎢∑ ⎥ i =1 ⎦ ⎣ i =1
Φ( n) H ( n) = z ( n)
1) Φ(n), z (n)的递推计算
n
Φ(n) = ∑ λn −i x(i )xT (i )
x ( n)
Z-1
H = [h0 , h1 ,..., hN −1 ]T
Z-1
x( n − 1)
Z-1
x(n − N + 1)
e( n)பைடு நூலகம்
h0
⊗
h1
⊗
⊗
hN −1
⊗
~ y ( n)
⊕ −⊕ + 线性FIR自适应滤波器原理框图
线性FIR自适应滤波器的输出:
⊕
⊕
y ( n)
~ y (n) = ∑ hk x(n − k )
四 selection of n1 and n2(Data windowing)
(1) (2)
N N
L
N
(4)
n = n2 n = n1
(3)
Φ (m, k ) = z (−k ) =
∑ x(n − k ) x(n − m), m, k = 0,1,..., N − 1
n = n2 n = n1
∑ x(n − k ) y(n), k = 0,1,..., N − 1
2006/11/24
6
MMVCLAB
6.1
最小二乘法
n = n2 n = n1 2 y ∑ ( n) = n = n2 n = n1 2 2 ~ ( ) y n + e ∑ ∑ ( n) n = n1 n = n2
εy = ε~ y + J min ⇒ J min = ε y − ε ~ y