公式推导

最新推荐文章于 2021-09-05 20:25:47 发布

jayckler

最新推荐文章于 2021-09-05 20:25:47 发布

阅读量332

点赞数 1

CC 4.0 BY-SA版权

分类专栏：公式

本文链接：https://blog.youkuaiyun.com/jayckler/article/details/79572404

公式专栏收录该内容

2 篇文章

订阅专栏

公式推导

(d n - 1 - w ⃗' n + 1 x ⃗ n - 1) 2 = = = = = = (d n - 1 - (w ⃗ n + 2 μ n e n x ⃗ n)' x ⃗ n - 1) 2 d 2 n - 1 - 2 R [d' n - 1 (w ⃗ n + 2 μ n e n x ⃗ n)' x ⃗ n - 1] + [(w ⃗ n + 2 μ n e n x ⃗ n)' x ⃗ n - 1] 2 d 2 n - 1 - 2 R [d' n - 1 w ⃗' n x ⃗ n - 1] - 4 R [d' n - 1 μ n e' n x ⃗' n x ⃗ n - 1] + (w ⃗' n x ⃗ n - 1 + 2 μ n e' n x ⃗' n x ⃗ n - 1) 2 d 2 n - 1 - 2 R [d' n - 1 w ⃗' n x ⃗ n - 1] - 4 R [d' n - 1 μ n e' n x ⃗' n x ⃗ n - 1] + (w ⃗' n x ⃗ n - 1) 2 + 4 R [x ⃗' n - 1 w ⃗ n μ n e' n x ⃗' n x ⃗ n - 1] + 4 (μ n e' n x ⃗' n x ⃗ n - 1) 2 4 (e' n x ⃗' n x ⃗ n - 1) 2 μ 2 n + 4 R [x ⃗' n - 1 w ⃗ n e' n x ⃗' n x ⃗ n - 1 - d' n - 1 e' n x ⃗' n x ⃗ n - 1] μ n + d 2 n - 1 - 2 R [d' n - 1 w ⃗' n x ⃗ n - 1] + (w ⃗' n x ⃗ n - 1) 2 4 e 2 n x 2 n, n - 1 μ 2 n + 4 R [((w x)' n, n - 1 - d' n - 1) e' n x n, n - 1] μ n + d 2 n - 1 - 2 R [d' n - 1 w x n, n - 1] + (w x n, n - 1) 2 (1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11)

$\begin{align} (d_{n-1}-\vec{w}'_{n+1}\vec{x}_{n-1})^2 =&(d_{n-1}-(\vec{w}_{n}+2\mu_{n}e_n\vec{x}_n)'\vec{x}_{n-1})^2\\ =&d^2_{n-1}-2R[d'_{n-1}(\vec{w}_{n}+2\mu_{n}e_n\vec{x}_n)'\vec{x}_{n-1}]\\&+[(\vec{w}_{n}+2\mu_{n}e_n\vec{x}_n)'\vec{x}_{n-1}]^2\\ =&d^2_{n-1}-2R[d'_{n-1}\vec{w}'_{n}\vec{x}_{n-1}]-4R[d'_{n-1}\mu_{n}e'_n\vec{x}'_n\vec{x}_{n-1}]+\\&(\vec{w}'_{n}\vec{x}_{n-1}+2\mu_{n}e'_n\vec{x}'_n\vec{x}_{n-1})^2\\ =&d^2_{n-1}-2R[d'_{n-1}\vec{w}'_{n}\vec{x}_{n-1}]-4R[d'_{n-1}\mu_{n}e'_n\vec{x}'_n\vec{x}_{n-1}]+(\vec{w}'_{n}\vec{x}_{n-1})^2\\&+4R[\vec{x}'_{n-1}\vec{w}_{n}\mu_{n}e'_n\vec{x}'_n\vec{x}_{n-1}]+4(\mu_{n}e'_n\vec{x}'_n\vec{x}_{n-1})^2\\ =&4(e'_n\vec{x}'_n\vec{x}_{n-1})^2\mu^2_{n}+4R[\vec{x}'_{n-1}\vec{w}_{n}e'_n\vec{x}'_n\vec{x}_{n-1}-d'_{n-1}e'_n\vec{x}'_n\vec{x}_{n-1}]\mu_n\\&+d^2_{n-1}-2R[d'_{n-1}\vec{w}'_{n}\vec{x}_{n-1}]+(\vec{w}'_{n}\vec{x}_{n-1})^2\\ =&4e^2_nx^2_{n,n-1}\mu^2_{n}+4R[((wx)'_{n,n-1}-d'_{n-1})e'_nx_{n,n-1}]\mu_{n}+d^2_{n-1}\\&-2R[d'_{n-1}wx_{n,n-1}]+(wx_{n,n-1})^2 \\ \end{align}$

$x_{n,n-1}=\vec{x}'_n\vec{x}_{n-1}$

$wx_{n,n-1}=\vec{w}'_n\vec{x}_{n-1}$
$' == conj$

$\sum^P_{l=0}=S$

[d f (n - 1) - \sum l = 0 P w f, l (n + 1) ⨀ x f, l (n - 1)] 2 = [d f (n - 1) - \sum l = 0 P (w f, l (n) + 2 μ f (n) x' f, l (n) ⨀ e f (n))) ⨀ x f, l (n - 1)] 2 (34) (35)

$\begin{align} &[{\bf d}_f(n-1)-\sum^P_{l=0}{\bf w}_{f,l}(n+1)\bigodot{\bf x}_{f,l}(n-1)]^2\\& =[{\bf d}_f(n-1)-\sum^P_{l=0}({\bf w}_{f,l}(n)+2\mu_f(n){\bf x}'_{f,l}(n)\bigodot {\bf e}_f(n)))\bigodot{\bf x}_{f,l}(n-1)]^2\\ \end{align}$
如下以一个子带为例：

[d f (n = . . . = = = - 1) - \sum l = 0 P (w f, l (n) + 2 μ f (n) x' f, l (n) e f (n)) x f, l (n - 1)] 2 d 2 f (n - 1) - 2 R {d' f (n - 1) \sum l = 0 P [w f, l (n) x f, l (n - 1) + 2 μ f (n) x' f, l (n) e f (n) x f, l (n - 1)]} + {\sum l = 0 P [w f, l (n) x f, l (n - 1) + 2 μ f (n) x' f, l (n) e f (n) x f, l (n - 1)]} 2 d 2 f (n - 1) - 2 R {d' f (n - 1) \sum l = 0 P [w f, l (n) x f, l (n - 1)]} - 4 R {d' f (n - 1) \sum l = 0 P [x' f, l (n) e f (n) x f, l (n - 1)]} μ f (n) + [\sum l = 0 P w f, l (n) x f, l (n - 1)] 2 + 4 [\sum l = 0 P x' f, l (n) e f (n) x f, l (n - 1)] 2 μ 2 f (n) + 4 R {[\sum l = 0 P w f, l (n) x f, l (n - 1)] [\sum l = 0 P x' f, l (n) e f (n) x f, l (n - 1)]} μ f (n) 4 [\sum l = 0 P x' f, l (n) e f (n) x f, l (n - 1)] 2 μ 2 f (n) + 4 R {[\sum l = 0 P w f, l (n) x f, l (n - 1)] [\sum l = 0 P x' f, l (n) e f (n) x f, l (n - 1)] - d' f (n - 1) \sum l = 0 P [x' f, l (n) e f (n) x f, l (n - 1)]} μ f (n) + d 2 f (n - 1) - 2 R {d' f (n - 1) \sum l = 0 P [w f, l (n) x f, l (n - 1)]} + [\sum l = 0 P w f, l (n) x f, l (n - 1)] 2 4 [\sum l = 0 P x' f, l (n) x f, l (n - 1)] 2 e 2 f (n) μ 2 f (n) + 4 R {\sum l = 0 P [w f, l (n) x f, l (n - 1) - d' f (n - 1)] \sum l = 0 P [x' f, l (n) x f, l (n - 1)] e f (n)} μ f (n) + d 2 f (n - 1) - 2 R {d' f (n - 1) \sum l = 0 P [w f, l (n) x f, l (n - 1)]} + [\sum l = 0 P w f, l (n) x f, l (n - 1)] 2 (36) (37) (38) (39) (40) (41) (42) (43) (44) (45) (46) (47) (48) (49)

$\begin{align} [d_f(n&-1)-\sum^P_{l=0}(w_{f,l}(n)+2\mu_f(n)x'_{f,l}(n)e_f(n))x_{f,l}(n-1)]^2\\ =&d^2_f(n-1)-2R\{d'_f(n-1)\sum^P_{l=0}[w_{f,l}(n)x_{f,l}(n-1)+2\mu_f(n)x'_{f,l}(n)e_f(n)x_{f,l}(n-1)]\}\\&+\{\sum^P_{l=0}[w_{f,l}(n)x_{f,l}(n-1)+2\mu_f(n)x'_{f,l}(n)e_f(n)x_{f,l}(n-1)]\}^2\\ ...\\ =&d^2_f(n-1)-2R\{d'_f(n-1)\sum^P_{l=0}[w_{f,l}(n)x_{f,l}(n-1)]\}\\ &-4R\{d'_f(n-1)\sum^P_{l=0}[x'_{f,l}(n)e_f(n)x_{f,l}(n-1)]\}\mu_f(n)\\&+[\sum^P_{l=0}w_{f,l}(n)x_{f,l}(n-1)]^2 +4[\sum^P_{l=0}x'_{f,l}(n)e_f(n)x_{f,l}(n-1)]^2\mu^2_f(n)+\\&4R\{[\sum^P_{l=0}w_{f,l}(n)x_{f,l}(n-1)][\sum^P_{l=0}x'_{f,l}(n)e_f(n)x_{f,l}(n-1)]\}\mu_f(n)\\ =&4[\sum^P_{l=0}x'_{f,l}(n)e_f(n)x_{f,l}(n-1)]^2\mu^2_f(n)+\\&4R\{[\sum^P_{l=0}w_{f,l}(n)x_{f,l}(n-1)][\sum^P_{l=0}x'_{f,l}(n)e_f(n)x_{f,l}(n-1)]-d'_f(n-1)\sum^P_{l=0}[x'_{f,l}(n)e_f(n)x_{f,l}(n-1)]\}\mu_f(n)\\& +d^2_f(n-1)-2R\{d'_f(n-1)\sum^P_{l=0}[w_{f,l}(n)x_{f,l}(n-1)]\}+[\sum^P_{l=0}w_{f,l}(n)x_{f,l}(n-1)]^2\\ =&4[\sum^P_{l=0}x'_{f,l}(n)x_{f,l}(n-1)]^2e^2_f(n)\mu^2_f(n)+\\&4R\{\sum^P_{l=0}[w_{f,l}(n)x_{f,l}(n-1)-d'_f(n-1)]\sum^P_{l=0}[x'_{f,l}(n)x_{f,l}(n-1)]e_f(n)\}\mu_f(n)\\& +d^2_f(n-1)-2R\{d'_f(n-1)\sum^P_{l=0}[w_{f,l}(n)x_{f,l}(n-1)]\}+[\sum^P_{l=0}w_{f,l}(n)x_{f,l}(n-1)]^2\\ \end{align}$
$$