遞迴最小平方濾波器 （RLS）屬於自适应滤波器 ，會針對和輸入信號有關的加權最小平方 损失函数 ，遞迴尋找可以使其最小化的係數。此方法和想要減少均方误差 的最小均方滤波器 （LMS）不同。在推導遞迴最小平方濾波器時，會假設輸入信號是确定性 的，而最小均方滤波及其他演算法會假設信號隨機
遞迴最小平方濾波器是由卡爾·弗里德里希·高斯 發現，但被遺忘無人使用，直到Plackett在1950年發現高斯1821年的著作之後，才再獲使用。一般來說，遞迴最小平方濾波器可以求得任何可以用自适应滤波器 求解的問題。例如，假設信號
d
(
n
)
{\displaystyle d(n)}
有噪信道 傳輸，接收到的是
x
(
n
)
=
∑
k
=
0
q
b
n
(
k
)
d
(
n
−
k
)
+
v
(
n
)
{\displaystyle x(n)=\sum _{k=0}^{q}b_{n}(k)d(n-k)+v(n)}
其中
v
(
n
)
{\displaystyle v(n)}
加性高斯白噪声 。RLS濾波器的目的是用
p
+
1
{\displaystyle p+1}
有限冲激响应 （FIR）濾波器
w
{\displaystyle \mathbf {w} }
d
(
n
)
{\displaystyle d(n)}
d
(
n
)
≈
∑
k
=
0
p
w
(
k
)
x
(
n
−
k
)
=
w
T
x
n
{\displaystyle d(n)\approx \sum _{k=0}^{p}w(k)x(n-k)=\mathbf {w} ^{\mathit {T}}\mathbf {x} _{n}}
其中
x
n
=
[
x
(
n
)
x
(
n
−
1
)
…
x
(
n
−
p
)
]
T
{\displaystyle \mathbf {x} _{n}=[x(n)\quad x(n-1)\quad \ldots \quad x(n-p)]^{T}}
x
(
n
)
{\displaystyle x(n)}
p
+
1
{\displaystyle p+1}
列向量 。接收到想要訊號的估測值為
d
^
(
n
)
=
∑
k
=
0
p
w
n
(
k
)
x
(
n
−
k
)
=
w
n
T
x
n
{\displaystyle {\hat {d}}(n)=\sum _{k=0}^{p}w_{n}(k)x(n-k)=\mathbf {w} _{n}^{\mathit {T}}\mathbf {x} _{n}}
其目的是估測濾波器
w
{\displaystyle \mathbf {w} }
n
{\displaystyle n}
w
n
{\displaystyle \mathbf {w} _{n}}
w
n
+
1
{\displaystyle \mathbf {w} _{n+1}}
w
n
{\displaystyle \mathbf {w} _{n}}
转置
w
n
T
{\displaystyle \mathbf {w} _{n}^{\mathit {T}}}
矩陣乘法
w
n
T
x
n
{\displaystyle \mathbf {w} _{n}^{\mathit {T}}\mathbf {x} _{n}}
w
n
{\displaystyle \mathbf {w} _{n}}
x
n
{\displaystyle \mathbf {x} _{n}}
點積 ）是
d
^
(
n
)
{\displaystyle {\hat {d}}(n)}
d
^
(
n
)
−
d
(
n
)
{\displaystyle {\hat {d}}(n)-d(n)}
最小二乘法 的概念下，其值是小的，其估測就算是好的。
隨著時間演進，會希望避免完全重做最小方差演算法來找新估測值
w
n
+
1
{\displaystyle \mathbf {w} _{n+1}}
w
n
+
1
{\displaystyle \mathbf {w} _{n+1}}
w
n
{\displaystyle \mathbf {w} _{n}}
遞迴最小平方濾波器的好處是不用進行反矩陣運算，因此可以節省計算時間。另一個好處是在其結果之後，提供了類似卡尔曼滤波 的直覺資訊。
遞迴最小平方濾波器背後的概念是在收到新資料時，適當選擇濾波器係數
w
n
{\displaystyle \mathbf {w} _{n}}
损失函数
C
{\displaystyle C}
e
(
n
)
{\displaystyle e(n)}
d
(
n
)
{\displaystyle d(n)}
负反馈 方塊圖來說明：
誤差會透過估測值
d
^
(
n
)
{\displaystyle {\hat {d}}(n)}
e
(
n
)
=
d
(
n
)
−
d
^
(
n
)
{\displaystyle e(n)=d(n)-{\hat {d}}(n)}
加權最小方差函數
C
{\displaystyle C}
e
(
n
)
{\displaystyle e(n)}
C
(
w
n
)
=
∑
i
=
0
n
λ
n
−
i
e
2
(
i
)
{\displaystyle C(\mathbf {w} _{n})=\sum _{i=0}^{n}\lambda ^{n-i}e^{2}(i)}
其中
0
<
λ
≤
1
{\displaystyle 0<\lambda \leq 1}
費用函數最小化的方式，是將係數向量
w
n
{\displaystyle \mathbf {w} _{n}}
∂
C
(
w
n
)
∂
w
n
(
k
)
=
∑
i
=
0
n
2
λ
n
−
i
e
(
i
)
⋅
∂
e
(
i
)
∂
w
n
(
k
)
=
−
∑
i
=
0
n
2
λ
n
−
i
e
(
i
)
x
(
i
−
k
)
=
0
k
=
0
,
1
,
…
,
p
{\displaystyle {\frac {\partial C(\mathbf {w} _{n})}{\partial w_{n}(k)}}=\sum _{i=0}^{n}2\lambda ^{n-i}e(i)\cdot {\frac {\partial e(i)}{\partial w_{n}(k)}}=-\sum _{i=0}^{n}2\lambda ^{n-i}e(i)\,x(i-k)=0\qquad k=0,1,\ldots ,p}
接著，用以下的誤差信號來代替
e
(
n
)
{\displaystyle e(n)}
∑
i
=
0
n
λ
n
−
i
[
d
(
i
)
−
∑
ℓ
=
0
p
w
n
(
ℓ
)
x
(
i
−
ℓ
)
]
x
(
i
−
k
)
=
0
k
=
0
,
1
,
…
,
p
{\displaystyle \sum _{i=0}^{n}\lambda ^{n-i}\left[d(i)-\sum _{\ell =0}^{p}w_{n}(\ell )x(i-\ell )\right]x(i-k)=0\qquad k=0,1,\ldots ,p}
將等式重組如下
∑
ℓ
=
0
p
w
n
(
ℓ
)
[
∑
i
=
0
n
λ
n
−
i
x
(
i
−
ℓ
)
x
(
i
−
k
)
]
=
∑
i
=
0
n
λ
n
−
i
d
(
i
)
x
(
i
−
k
)
k
=
0
,
1
,
…
,
p
{\displaystyle \sum _{\ell =0}^{p}w_{n}(\ell )\left[\sum _{i=0}^{n}\lambda ^{n-i}\,x(i-\ell )x(i-k)\right]=\sum _{i=0}^{n}\lambda ^{n-i}d(i)x(i-k)\qquad k=0,1,\ldots ,p}
可以表示為以下的矩陣
R
x
(
n
)
w
n
=
r
d
x
(
n
)
{\displaystyle \mathbf {R} _{x}(n)\,\mathbf {w} _{n}=\mathbf {r} _{dx}(n)}
其中
R
x
(
n
)
{\displaystyle \mathbf {R} _{x}(n)}
x
(
n
)
{\displaystyle x(n)}
樣本協方差
r
d
x
(
n
)
{\displaystyle \mathbf {r} _{dx}(n)}
d
(
n
)
{\displaystyle d(n)}
x
(
n
)
{\displaystyle x(n)}
互协方差 的等效估計。依照上式可以找到最小化費用函數的係數
w
n
=
R
x
−
1
(
n
)
r
d
x
(
n
)
{\displaystyle \mathbf {w} _{n}=\mathbf {R} _{x}^{-1}(n)\,\mathbf {r} _{dx}(n)}
λ
{\displaystyle \lambda }
λ
=
1
{\displaystyle \lambda =1}
λ
{\displaystyle \lambda }
[ 1]
λ
{\displaystyle \lambda }
[ 2]
以上敘述的結論是可以決定費用函數的參數，使費用函數最小化的方程式。以下則說明如何找到此形式的遞迴解
w
n
=
w
n
−
1
+
Δ
w
n
−
1
{\displaystyle \mathbf {w} _{n}=\mathbf {w} _{n-1}+\Delta \mathbf {w} _{n-1}}
其中
Δ
w
n
−
1
{\displaystyle \Delta \mathbf {w} _{n-1}}
n
−
1
{\displaystyle {n-1}}
r
d
x
(
n
)
{\displaystyle \mathbf {r} _{dx}(n)}
r
d
x
(
n
−
1
)
{\displaystyle \mathbf {r} _{dx}(n-1)}
r
d
x
(
n
)
{\displaystyle \mathbf {r} _{dx}(n)}
=
∑
i
=
0
n
λ
n
−
i
d
(
i
)
x
(
i
)
{\displaystyle =\sum _{i=0}^{n}\lambda ^{n-i}d(i)\mathbf {x} (i)}
=
∑
i
=
0
n
−
1
λ
n
−
i
d
(
i
)
x
(
i
)
+
λ
0
d
(
n
)
x
(
n
)
{\displaystyle =\sum _{i=0}^{n-1}\lambda ^{n-i}d(i)\mathbf {x} (i)+\lambda ^{0}d(n)\mathbf {x} (n)}
=
λ
r
d
x
(
n
−
1
)
+
d
(
n
)
x
(
n
)
{\displaystyle =\lambda \mathbf {r} _{dx}(n-1)+d(n)\mathbf {x} (n)}
其中
x
(
i
)
{\displaystyle \mathbf {x} (i)}
p
+
1
{\displaystyle {p+1}}
x
(
i
)
=
[
x
(
i
)
,
x
(
i
−
1
)
,
…
,
x
(
i
−
p
)
]
T
{\displaystyle \mathbf {x} (i)=[x(i),x(i-1),\dots ,x(i-p)]^{T}}
接下來以相似的方式，用
R
x
(
n
−
1
)
{\displaystyle \mathbf {R} _{x}(n-1)}
R
x
(
n
)
{\displaystyle \mathbf {R} _{x}(n)}
R
x
(
n
)
{\displaystyle \mathbf {R} _{x}(n)}
=
∑
i
=
0
n
λ
n
−
i
x
(
i
)
x
T
(
i
)
{\displaystyle =\sum _{i=0}^{n}\lambda ^{n-i}\mathbf {x} (i)\mathbf {x} ^{T}(i)}
=
λ
R
x
(
n
−
1
)
+
x
(
n
)
x
T
(
n
)
{\displaystyle =\lambda \mathbf {R} _{x}(n-1)+\mathbf {x} (n)\mathbf {x} ^{T}(n)}
為了要找到其係數向量，接下來要關注的是決定性自協方差矩陣的反矩陣。這問題可以使用伍德伯里矩陣恆等式
A
{\displaystyle A}
=
λ
R
x
(
n
−
1
)
{\displaystyle =\lambda \mathbf {R} _{x}(n-1)}
(
p
+
1
)
×
(
p
+
1
)
{\displaystyle (p+1)\times (p+1)}
U
{\displaystyle U}
=
x
(
n
)
{\displaystyle =\mathbf {x} (n)}
(
p
+
1
)
×
1
{\displaystyle (p+1)\times 1}
V
{\displaystyle V}
=
x
T
(
n
)
{\displaystyle =\mathbf {x} ^{T}(n)}
1
×
(
p
+
1
)
{\displaystyle 1\times (p+1)}
C
{\displaystyle C}
=
I
1
{\displaystyle =\mathbf {I} _{1}}
1
×
1
{\displaystyle 1\times 1}
單位矩陣
依照伍德伯里矩陣恆等式，可得到下式
R
x
−
1
(
n
)
{\displaystyle \mathbf {R} _{x}^{-1}(n)}
=
{\displaystyle =}
[
λ
R
x
(
n
−
1
)
+
x
(
n
)
x
T
(
n
)
]
−
1
{\displaystyle \left[\lambda \mathbf {R} _{x}(n-1)+\mathbf {x} (n)\mathbf {x} ^{T}(n)\right]^{-1}}
=
{\displaystyle =}
1
λ
{
R
x
−
1
(
n
−
1
)
−
R
x
−
1
(
n
−
1
)
x
(
n
)
x
T
(
n
)
R
x
−
1
(
n
−
1
)
λ
+
x
T
(
n
)
R
x
−
1
(
n
−
1
)
x
(
n
)
}
{\displaystyle {\dfrac {1}{\lambda }}\left\lbrace \mathbf {R} _{x}^{-1}(n-1)-{\dfrac {\mathbf {R} _{x}^{-1}(n-1)\mathbf {x} (n)\mathbf {x} ^{T}(n)\mathbf {R} _{x}^{-1}(n-1)}{\lambda +\mathbf {x} ^{T}(n)\mathbf {R} _{x}^{-1}(n-1)\mathbf {x} (n)}}\right\rbrace }
{\displaystyle }
為了和標準的文獻一致，定義
P
(
n
)
{\displaystyle \mathbf {P} (n)}
=
R
x
−
1
(
n
)
{\displaystyle =\mathbf {R} _{x}^{-1}(n)}
=
λ
−
1
P
(
n
−
1
)
−
g
(
n
)
x
T
(
n
)
λ
−
1
P
(
n
−
1
)
{\displaystyle =\lambda ^{-1}\mathbf {P} (n-1)-\mathbf {g} (n)\mathbf {x} ^{T}(n)\lambda ^{-1}\mathbf {P} (n-1)}
其中的增益向量
g
(
n
)
{\displaystyle g(n)}
g
(
n
)
{\displaystyle \mathbf {g} (n)}
=
λ
−
1
P
(
n
−
1
)
x
(
n
)
{
1
+
x
T
(
n
)
λ
−
1
P
(
n
−
1
)
x
(
n
)
}
−
1
{\displaystyle =\lambda ^{-1}\mathbf {P} (n-1)\mathbf {x} (n)\left\{1+\mathbf {x} ^{T}(n)\lambda ^{-1}\mathbf {P} (n-1)\mathbf {x} (n)\right\}^{-1}}
=
P
(
n
−
1
)
x
(
n
)
{
λ
+
x
T
(
n
)
P
(
n
−
1
)
x
(
n
)
}
−
1
{\displaystyle =\mathbf {P} (n-1)\mathbf {x} (n)\left\{\lambda +\mathbf {x} ^{T}(n)\mathbf {P} (n-1)\mathbf {x} (n)\right\}^{-1}}
在往下推導之前，需要將
g
(
n
)
{\displaystyle \mathbf {g} (n)}
g
(
n
)
{
1
+
x
T
(
n
)
λ
−
1
P
(
n
−
1
)
x
(
n
)
}
{\displaystyle \mathbf {g} (n)\left\{1+\mathbf {x} ^{T}(n)\lambda ^{-1}\mathbf {P} (n-1)\mathbf {x} (n)\right\}}
=
λ
−
1
P
(
n
−
1
)
x
(
n
)
{\displaystyle =\lambda ^{-1}\mathbf {P} (n-1)\mathbf {x} (n)}
g
(
n
)
+
g
(
n
)
x
T
(
n
)
λ
−
1
P
(
n
−
1
)
x
(
n
)
{\displaystyle \mathbf {g} (n)+\mathbf {g} (n)\mathbf {x} ^{T}(n)\lambda ^{-1}\mathbf {P} (n-1)\mathbf {x} (n)}
=
λ
−
1
P
(
n
−
1
)
x
(
n
)
{\displaystyle =\lambda ^{-1}\mathbf {P} (n-1)\mathbf {x} (n)}
等式兩側減去左邊的第二項，得到
g
(
n
)
{\displaystyle \mathbf {g} (n)}
=
λ
−
1
P
(
n
−
1
)
x
(
n
)
−
g
(
n
)
x
T
(
n
)
λ
−
1
P
(
n
−
1
)
x
(
n
)
{\displaystyle =\lambda ^{-1}\mathbf {P} (n-1)\mathbf {x} (n)-\mathbf {g} (n)\mathbf {x} ^{T}(n)\lambda ^{-1}\mathbf {P} (n-1)\mathbf {x} (n)}
=
λ
−
1
[
P
(
n
−
1
)
−
g
(
n
)
x
T
(
n
)
P
(
n
−
1
)
]
x
(
n
)
{\displaystyle =\lambda ^{-1}\left[\mathbf {P} (n-1)-\mathbf {g} (n)\mathbf {x} ^{T}(n)\mathbf {P} (n-1)\right]\mathbf {x} (n)}
配合
P
(
n
)
{\displaystyle \mathbf {P} (n)}
g
(
n
)
=
P
(
n
)
x
(
n
)
{\displaystyle \mathbf {g} (n)=\mathbf {P} (n)\mathbf {x} (n)}
此時就可以完成遞迴，如以上討論
w
n
{\displaystyle \mathbf {w} _{n}}
=
P
(
n
)
r
d
x
(
n
)
{\displaystyle =\mathbf {P} (n)\,\mathbf {r} _{dx}(n)}
=
λ
P
(
n
)
r
d
x
(
n
−
1
)
+
d
(
n
)
P
(
n
)
x
(
n
)
{\displaystyle =\lambda \mathbf {P} (n)\,\mathbf {r} _{dx}(n-1)+d(n)\mathbf {P} (n)\,\mathbf {x} (n)}
第二步是從
r
d
x
(
n
)
{\displaystyle \mathbf {r} _{dx}(n)}
P
(
n
)
{\displaystyle \mathbf {P} (n)}
g
(
n
)
{\displaystyle \mathbf {g} (n)}
w
n
{\displaystyle \mathbf {w} _{n}}
=
λ
[
λ
−
1
P
(
n
−
1
)
−
g
(
n
)
x
T
(
n
)
λ
−
1
P
(
n
−
1
)
]
r
d
x
(
n
−
1
)
+
d
(
n
)
g
(
n
)
{\displaystyle =\lambda \left[\lambda ^{-1}\mathbf {P} (n-1)-\mathbf {g} (n)\mathbf {x} ^{T}(n)\lambda ^{-1}\mathbf {P} (n-1)\right]\mathbf {r} _{dx}(n-1)+d(n)\mathbf {g} (n)}
=
P
(
n
−
1
)
r
d
x
(
n
−
1
)
−
g
(
n
)
x
T
(
n
)
P
(
n
−
1
)
r
d
x
(
n
−
1
)
+
d
(
n
)
g
(
n
)
{\displaystyle =\mathbf {P} (n-1)\mathbf {r} _{dx}(n-1)-\mathbf {g} (n)\mathbf {x} ^{T}(n)\mathbf {P} (n-1)\mathbf {r} _{dx}(n-1)+d(n)\mathbf {g} (n)}
=
P
(
n
−
1
)
r
d
x
(
n
−
1
)
+
g
(
n
)
[
d
(
n
)
−
x
T
(
n
)
P
(
n
−
1
)
r
d
x
(
n
−
1
)
]
{\displaystyle =\mathbf {P} (n-1)\mathbf {r} _{dx}(n-1)+\mathbf {g} (n)\left[d(n)-\mathbf {x} ^{T}(n)\mathbf {P} (n-1)\mathbf {r} _{dx}(n-1)\right]}
配合
w
n
−
1
=
P
(
n
−
1
)
r
d
x
(
n
−
1
)
{\displaystyle \mathbf {w} _{n-1}=\mathbf {P} (n-1)\mathbf {r} _{dx}(n-1)}
w
n
{\displaystyle \mathbf {w} _{n}}
=
w
n
−
1
+
g
(
n
)
[
d
(
n
)
−
x
T
(
n
)
w
n
−
1
]
{\displaystyle =\mathbf {w} _{n-1}+\mathbf {g} (n)\left[d(n)-\mathbf {x} ^{T}(n)\mathbf {w} _{n-1}\right]}
=
w
n
−
1
+
g
(
n
)
α
(
n
)
{\displaystyle =\mathbf {w} _{n-1}+\mathbf {g} (n)\alpha (n)}
其中
α
(
n
)
=
d
(
n
)
−
x
T
(
n
)
w
n
−
1
{\displaystyle \alpha (n)=d(n)-\mathbf {x} ^{T}(n)\mathbf {w} _{n-1}}
先驗 誤差。將此和後驗誤差（在濾波器更新後計算的誤差）比較
e
(
n
)
=
d
(
n
)
−
x
T
(
n
)
w
n
{\displaystyle e(n)=d(n)-\mathbf {x} ^{T}(n)\mathbf {w} _{n}}
這就找到了修正因子
Δ
w
n
−
1
=
g
(
n
)
α
(
n
)
{\displaystyle \Delta \mathbf {w} _{n-1}=\mathbf {g} (n)\alpha (n)}
這個結論指出了修正係數直接和誤差和增益向量成正比，增益向量會透過加權因子
λ
{\displaystyle \lambda }
p 階RLS濾波器的演算法可以摘要如下
參數：
p
=
{\displaystyle p=}
λ
=
{\displaystyle \lambda =}
δ
=
P
(
0
)
{\displaystyle \delta =\mathbf {P} (0)}
開始：
w
(
0
)
=
0
{\displaystyle \mathbf {w} (0)=0}
x
(
k
)
=
0
,
k
=
−
p
,
…
,
−
1
{\displaystyle x(k)=0,k=-p,\dots ,-1}
d
(
k
)
=
0
,
k
=
−
p
,
…
,
−
1
{\displaystyle d(k)=0,k=-p,\dots ,-1}
P
(
0
)
=
δ
I
{\displaystyle \mathbf {P} (0)=\delta I}
I
{\displaystyle I}
p
+
1
{\displaystyle p+1}
單位矩陣
計算：
針對
n
=
1
,
2
,
…
{\displaystyle n=1,2,\dots }
x
(
n
)
=
[
x
(
n
)
x
(
n
−
1
)
⋮
x
(
n
−
p
)
]
{\displaystyle \mathbf {x} (n)=\left[{\begin{matrix}x(n)\\x(n-1)\\\vdots \\x(n-p)\end{matrix}}\right]}
α
(
n
)
=
d
(
n
)
−
x
T
(
n
)
w
(
n
−
1
)
{\displaystyle \alpha (n)=d(n)-\mathbf {x} ^{T}(n)\mathbf {w} (n-1)}
g
(
n
)
=
P
(
n
−
1
)
x
(
n
)
{
λ
+
x
T
(
n
)
P
(
n
−
1
)
x
(
n
)
}
−
1
{\displaystyle \mathbf {g} (n)=\mathbf {P} (n-1)\mathbf {x} (n)\left\{\lambda +\mathbf {x} ^{T}(n)\mathbf {P} (n-1)\mathbf {x} (n)\right\}^{-1}}
P
(
n
)
=
λ
−
1
P
(
n
−
1
)
−
g
(
n
)
x
T
(
n
)
λ
−
1
P
(
n
−
1
)
{\displaystyle \mathbf {P} (n)=\lambda ^{-1}\mathbf {P} (n-1)-\mathbf {g} (n)\mathbf {x} ^{T}(n)\lambda ^{-1}\mathbf {P} (n-1)}
w
(
n
)
=
w
(
n
−
1
)
+
α
(
n
)
g
(
n
)
{\displaystyle \mathbf {w} (n)=\mathbf {w} (n-1)+\,\alpha (n)\mathbf {g} (n)}
P
{\displaystyle P}
代數Riccati方程 ，也類似卡尔曼滤波 的結果[ 3]
^ Emannual C. Ifeacor, Barrie W. Jervis. Digital signal processing: a practical approach, second edition. Indianapolis: Pearson Education Limited, 2002, p. 718
^ Steven Van Vaerenbergh, Ignacio Santamaría, Miguel Lázaro-Gredilla "Estimation of the forgetting factor in kernel recursive least squares" , 2012 IEEE International Workshop on Machine Learning for Signal Processing, 2012, accessed June 23, 2016.
^ Welch, Greg and Bishop, Gary "An Introduction to the Kalman Filter" , Department of Computer Science, University of North Carolina at Chapel Hill, September 17, 1997, accessed July 19, 2011.
Hayes, Monson H. 9.4: Recursive Least Squares. Statistical Digital Signal Processing and Modeling. Wiley. 1996: 541. ISBN 0-471-59431-8 Simon Haykin, Adaptive Filter Theory , Prentice Hall, 2002, ISBN 0-13-048434-2
M.H.A Davis, R.B. Vinter, Stochastic Modelling and Control , Springer, 1985, ISBN 0-412-16200-8
Weifeng Liu, Jose Principe and Simon Haykin, Kernel Adaptive Filtering: A Comprehensive Introduction , John Wiley, 2010, ISBN 0-470-44753-2
R.L.Plackett, Some Theorems in Least Squares , Biometrika, 1950, 37, 149–157, ISSN 0006-3444
C.F.Gauss, Theoria combinationis observationum erroribus minimis obnoxiae , 1821, Werke, 4. Gottinge 
![{\displaystyle \mathbf {x} _{n}=[x(n)\quad x(n-1)\quad \ldots \quad x(n-p)]^{T}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/09eb921b307dabe5f3f396085912215d09c02114)
![{\displaystyle \sum _{i=0}^{n}\lambda ^{n-i}\left[d(i)-\sum _{\ell =0}^{p}w_{n}(\ell )x(i-\ell )\right]x(i-k)=0\qquad k=0,1,\ldots ,p}](https://wikimedia.org/api/rest_v1/media/math/render/svg/19ddcbc3bdad707cdb76a5d5371bc1bee9872174)
![{\displaystyle \sum _{\ell =0}^{p}w_{n}(\ell )\left[\sum _{i=0}^{n}\lambda ^{n-i}\,x(i-\ell )x(i-k)\right]=\sum _{i=0}^{n}\lambda ^{n-i}d(i)x(i-k)\qquad k=0,1,\ldots ,p}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e5289d4da73b7a149fdf9a53cbd8c80a11c0dfe6)
![{\displaystyle \mathbf {x} (i)=[x(i),x(i-1),\dots ,x(i-p)]^{T}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/dab2ea44a54d82de74ac9f6d26e3f5f6c2a44d8a)
![{\displaystyle \left[\lambda \mathbf {R} _{x}(n-1)+\mathbf {x} (n)\mathbf {x} ^{T}(n)\right]^{-1}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f12c798517a6c530e616e5915429e3e16370785e)
![{\displaystyle =\lambda ^{-1}\left[\mathbf {P} (n-1)-\mathbf {g} (n)\mathbf {x} ^{T}(n)\mathbf {P} (n-1)\right]\mathbf {x} (n)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3baaff4ae27f9cfc51e54ada85a9de0999978262)
![{\displaystyle =\lambda \left[\lambda ^{-1}\mathbf {P} (n-1)-\mathbf {g} (n)\mathbf {x} ^{T}(n)\lambda ^{-1}\mathbf {P} (n-1)\right]\mathbf {r} _{dx}(n-1)+d(n)\mathbf {g} (n)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d1dfdfb80807912183bd47c28d76f65ea9d4c553)
![{\displaystyle =\mathbf {P} (n-1)\mathbf {r} _{dx}(n-1)+\mathbf {g} (n)\left[d(n)-\mathbf {x} ^{T}(n)\mathbf {P} (n-1)\mathbf {r} _{dx}(n-1)\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9800c914543c33d7b8ebd7f295efc9fdc51b57e7)
![{\displaystyle =\mathbf {w} _{n-1}+\mathbf {g} (n)\left[d(n)-\mathbf {x} ^{T}(n)\mathbf {w} _{n-1}\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/78ebe76268d69116b2d676aab45f0272977a6df2)
![{\displaystyle \mathbf {x} (n)=\left[{\begin{matrix}x(n)\\x(n-1)\\\vdots \\x(n-p)\end{matrix}}\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d85a8706d78172d40a299b2bb60ae26b22a21f6c)
