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[转帖]布尔函数的WALSH谱
发表于: 2011-12-6 18:36 5546

[转帖]布尔函数的WALSH谱

lilianjie 活跃值
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2011-12-6 18:36
5546
059K9s2c8@1M7q4)9K6b7g2)9J5c8W2)9J5c8Y4N6%4N6#2)9J5k6h3c8G2j5$3W2F1i4K6u0W2j5$3!0E0i4K6u0r3M7q4)9J5k6o6j5&6y4e0V1&6z5o6l9I4i4K6u0W2K9s2c8E0L8l9`.`.

e71K9s2c8@1M7q4)9K6b7g2)9J5c8W2)9J5c8Y4N6%4N6#2)9J5k6h3y4F1K9$3W2Q4x3X3g2U0L8$3#2Q4x3X3g2U0L8W2)9J5c8V1q4J5N6r3W2U0L8r3g2Q4x3V1k6o6d9V1k6p5g2r3!0@1j5h3I4Q4x3X3c8s2c8p5A6j5y4o6l9I4i4K6u0W2x3o6l9%4i4K6u0W2K9s2c8E0

[培训]内核驱动高级班，冲击BAT一流互联网大厂工作，每周日13:00-18:00直播授课

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哈哈，这本书我有，讲很多布尔函数的。
2011-12-6 21:49
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lilianjie
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这书难，还是杨晓元的好懂点

Z:=Matrix(GF(2), 1 ,[1]);
Z;
X := RandomMatrix(GF(2), 3, 3);
> X;
Y := RandomMatrix(GF(2), 4, 4);
> Y;

KZX:=KroneckerProduct(Z, X);
KZX;
KZX1:=KroneckerProduct(Z, KZX);
KZX1;
KZX2:=KroneckerProduct(Z, KZX1);
KZX2;

KZX3:=KroneckerProduct(Z, KZX2);
KZX3;

KZX4:=KroneckerProduct(Z, KZX3);
KZX4;

[1]
[0 1 1]
[0 1 1]
[0 1 0]
[1 0 1 0]
[1 1 0 1]
[1 1 0 0]
[0 0 1 0]
[0 1 1]
[0 1 1]
[0 1 0]
[0 1 1]
[0 1 1]
[0 1 0]
[0 1 1]
[0 1 1]
[0 1 0]
[0 1 1]
[0 1 1]
[0 1 0]
[0 1 1]
[0 1 1]
[0 1 0]

Z:=Matrix(GF(2), 1 ,[1]);
Z;
X := RandomMatrix(GF(2), 2, 2);
> X;
Y := RandomMatrix(GF(2), 4, 4);
> Y;

沃尔什转换矩阵：

KXY:=KroneckerProduct(X, Y);
KXY;
KXY1:=KroneckerProduct(X, KXY);
KXY1;
KXY2:=KroneckerProduct(X, KXY1);
KXY2;

[1]
[1 1]
[0 1]
[0 0 1 1]
[1 0 0 1]
[0 0 0 0]
[1 1 0 0]
[0 0 1 1 0 0 1 1]
[1 0 0 1 1 0 0 1]
[0 0 0 0 0 0 0 0]
[1 1 0 0 1 1 0 0]
[0 0 0 0 0 0 1 1]
[0 0 0 0 1 0 0 1]
[0 0 0 0 0 0 0 0]
[0 0 0 0 1 1 0 0][0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1]
[1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1]
[0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0]
[1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0]
[0 0 0 0 0 0 1 1 0 0 0 0 0 0 1 1]
[0 0 0 0 1 0 0 1 0 0 0 0 1 0 0 1]
[0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0]
[0 0 0 0 1 1 0 0 0 0 0 0 1 1 0 0]
[0 0 0 0 0 0 0 0 0 0 1 1 0 0 1 1]
[0 0 0 0 0 0 0 0 1 0 0 1 1 0 0 1]
[0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0]
[0 0 0 0 0 0 0 0 1 1 0 0 1 1 0 0]
[0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1]
[0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1]
[0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0]
[0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0]
[0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1]
[1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1]
[0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0]
[1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0]
[0 0 0 0 0 0 1 1 0 0 0 0 0 0 1 1 0 0 0 0 0 0 1 1 0 0 0 0 0 0 1 1]
[0 0 0 0 1 0 0 1 0 0 0 0 1 0 0 1 0 0 0 0 1 0 0 1 0 0 0 0 1 0 0 1]
[0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0]
[0 0 0 0 1 1 0 0 0 0 0 0 1 1 0 0 0 0 0 0 1 1 0 0 0 0 0 0 1 1 0 0]
[0 0 0 0 0 0 0 0 0 0 1 1 0 0 1 1 0 0 0 0 0 0 0 0 0 0 1 1 0 0 1 1]
[0 0 0 0 0 0 0 0 1 0 0 1 1 0 0 1 0 0 0 0 0 0 0 0 1 0 0 1 1 0 0 1]
[0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0]
[0 0 0 0 0 0 0 0 1 1 0 0 1 1 0 0 0 0 0 0 0 0 0 0 1 1 0 0 1 1 0 0]
[0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1]
[0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1]
[0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0]
[0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0]
[0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1]
[0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1]
[0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0]
[0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0]
[0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 1 1]
[0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 1 0 0 1]
[0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0]
[0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 1 1 0 0]
[0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 1 1]
[0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 1 0 0 1]
[0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0]
[0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 1 1 0 0]
[0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1]
[0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1]
[0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0]
[0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0]

逆针

[1 0 0 0 1 0 0 0]
[0 1 1 0 0 1 1 0]
[1 0 1 1 1 0 1 1]
[1 1 1 1 1 1 1 1]
[0 0 0 0 1 0 0 0]
[0 0 0 0 0 1 1 0]
[0 0 0 0 1 0 1 1]
[0 0 0 0 1 1 1 1][1 0 0 0 1 0 0 0 1 0 0 0 1 0 0 0]
[0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0]
[1 0 1 1 1 0 1 1 1 0 1 1 1 0 1 1]
[1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1]
[0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0]
[0 0 0 0 0 1 1 0 0 0 0 0 0 1 1 0]
[0 0 0 0 1 0 1 1 0 0 0 0 1 0 1 1]
[0 0 0 0 1 1 1 1 0 0 0 0 1 1 1 1]
[0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0]
[0 0 0 0 0 0 0 0 0 1 1 0 0 1 1 0]
[0 0 0 0 0 0 0 0 1 0 1 1 1 0 1 1]
[0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1]
[0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0]
[0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0]
[0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 1]
[0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1]
[1 0 0 0 1 0 0 0 1 0 0 0 1 0 0 0 1 0 0 0 1 0 0 0 1 0 0 0 1 0 0 0]
[0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0]
[1 0 1 1 1 0 1 1 1 0 1 1 1 0 1 1 1 0 1 1 1 0 1 1 1 0 1 1 1 0 1 1]
[1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1]
[0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0]
[0 0 0 0 0 1 1 0 0 0 0 0 0 1 1 0 0 0 0 0 0 1 1 0 0 0 0 0 0 1 1 0]
[0 0 0 0 1 0 1 1 0 0 0 0 1 0 1 1 0 0 0 0 1 0 1 1 0 0 0 0 1 0 1 1]
[0 0 0 0 1 1 1 1 0 0 0 0 1 1 1 1 0 0 0 0 1 1 1 1 0 0 0 0 1 1 1 1]
[0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0]
[0 0 0 0 0 0 0 0 0 1 1 0 0 1 1 0 0 0 0 0 0 0 0 0 0 1 1 0 0 1 1 0]
[0 0 0 0 0 0 0 0 1 0 1 1 1 0 1 1 0 0 0 0 0 0 0 0 1 0 1 1 1 0 1 1]
[0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1]
[0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0]
[0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0]
[0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 1]
[0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1]
[0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 1 0 0 0 1 0 0 0]
[0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0]
[0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 1 1 0 1 1 1 0 1 1 1 0 1 1]
[0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1]
[0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0]
[0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 1 1 0]
[0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 1 0 0 0 0 1 0 1 1]
[0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 0 0 0 0 1 1 1 1]
[0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0]
[0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 1 1 0]
[0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 1 1 0 1 1]
[0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1]
[0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0]
[0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0]
[0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 1]
[0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1]

转换式与反转换式只差了一个常数，这是由于沃尔什转换矩阵的反矩阵就是自己的转置矩阵乘上一个常数

Transpose(KXY);

[1 1 1 1 0 0 0 0]
[0 1 0 0 0 0 0 0]
[1 0 1 1 0 0 0 0]
[1 0 0 1 0 0 0 0]
[0 0 0 0 0 0 0 0]
[0 0 0 0 0 0 0 0]
[0 0 0 0 0 0 0 0]
[0 0 0 0 0 0 0 0]

序数顺序(沃尔什顺序) 双积顺序(培力顺序) 自然顺序(哈德码得顺序) W[m,n]
Sequency Ordering(Walsh Ordering) Dyadic Ordering(Paley Ordering) Natural Ordering (Hadamard Ordering)

双积顺序的二进制编号是序数顺序的格雷码编码
G：格雷码 B：二进制码
G(N) = B(n+1) XOR B(n)

自然顺序的二进制编号是双积顺序的位元反转。

沃尔什转换矩阵的每个列是互相正交的
SwapColumns(KXY, 1, 2) ;

SwapColumns(KXY, 2, 3) ;
SwapColumns(KXY, 3, 4) ;

[0 0 0 0 0 0 0 0]
[0 0 0 0 0 0 0 0]
[0 0 0 0 0 0 0 0]
[0 0 0 0 0 0 0 0]
[0 0 0 0 0 0 0 0]
[0 0 0 0 0 0 0 0]
[0 0 0 0 0 0 0 0]
[0 0 0 0 0 0 0 0]
[0 0 0 0 0 0 0 0]
[0 0 0 0 0 0 0 0]
[0 0 0 0 0 0 0 0]
[0 0 0 0 0 0 0 0]
[0 0 0 0 0 0 0 0]
[0 0 0 0 0 0 0 0]
[0 0 0 0 0 0 0 0]
[0 0 0 0 0 0 0 0]
[0 0 0 0 0 0 0 0]
[0 0 0 0 0 0 0 0]
[0 0 0 0 0 0 0 0]
[0 0 0 0 0 0 0 0]
[0 0 0 0 0 0 0 0]
[0 0 0 0 0 0 0 0]
[0 0 0 0 0 0 0 0]
[0 0 0 0 0 0 0 0]
[0 0 0 0 0 0 0 0]
[0 0 0 0 0 0 0 0]
[0 0 0 0 0 0 0 0]
[0 0 0 0 0 0 0 0]
[0 0 0 0 0 0 0 0]
[0 0 0 0 0 0 0 0]
[0 0 0 0 0 0 0 0]
[0 0 0 0 0 0 0 0]
2011-12-28 14:17
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lilianjie
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零交(zero-crossing)性质

奇偶性质

沃尔什转换矩阵(沃尔什顺序)中，编号为偶数的列是偶对称，编号为奇数的列是奇对称。(有第0列)

[0 0 1 1 0 0 1 1]
[1 0 0 1 1 0 0 1]
[0 0 0 0 0 0 0 0]
[1 1 0 0 1 1 0 0]
[0 0 0 0 0 0 1 1]
[0 0 0 0 1 0 0 1]
[0 0 0 0 0 0 0 0]
[0 0 0 0 1 1 0 0

================
chrestenson谱：

扩沃尔什从2^n到k^n,3进制以上计算机？
上传的附件：
2011-12-28 15:51
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