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# 差错检测方法 循环冗余校验 (CRC ， Cyclic Redundancy check) PowerPoint PPT Presentation

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### 循环冗余校验(CRC，Cyclic Redundancy check)

• 码多项式及其算术运算

M的码多项式（称为n-1次多项式）

C(x) = Cn-1 xn-1 + Cn-2xn-2 + …. C2 x2 +C1 x + C0

C(x) =1x6 + 1x5 +0x4+0x3 + 1x2 +0 x + 1

= x6 + x5 +x2 + 1

n位

k位

r位

k+1

k+2

n

1

2

k

### 差错检测循环冗余校验码(CRC，Cyclic Redundancy check)

• 编码

• 每个码多项式的前面k项与待编码的信息多项式相同

• 后面的r=n-k项与校验码元序列对应的校验多项式相同

### 差错检测.循环冗余校验码(CRC，Cyclic Redundancy check)

m(x) = mk-1 xk-1+ mk-2xk-2+ …. +m1 x+m0

xn-k·m(x) = mk-1 xn-1+ mk-2xn-2+ …. +m1 xn-k+1+m0 xn-k = q(x)·g(x) + r(x)

g(x)是(n-k)次多项式

q(x)是商式

r(x)是余式且次数不高于n-k-1

r(x) = rn-k-1 xn-k-1+ rn-k-2xn-k-2+ …. +r1 x+r0

xn-k·m(x) + r(x) = q(x)·g(x)

mk-1xn-1+mk-2xn-2+ ...+m1xn-k+1+m0xn-k+rn-k-1xn-k-1+rn-k-2xn-k-2+... +r1 x+r0

( mk-1, mk-2, ….m1, m0, rn-k-1, rn-k-2, …. ,r1, r0 )

(n-k)个监督位

### 差错检测循环冗余校验码(CRC，Cyclic Redundancy check)

m(x) = x9 + x8 + x6 + x4 + x3 + x + 1, k = 10

(1)假设 P(x) = x4 + x + 1

(2) x4·m(x) = 1101011011，0000

(3) 1101011011.0000 10011

C(x) = xn·m(x) + r(x) =1101011011，1110

### 差错检测循环冗余校验码(CRC，Cyclic Redundancy check)

• 多项式除法

1101011011.0000 10011

1 1 0 0 0 0 1 0 1 0

1 1 0 1 0 1 1 0 1 1,0 0 0 0

1 0 0 1 1

1 0 0 1 1

1 0 0 1 1

1 0 0 1 1

1 0 1 1 0

1 0 0 1 1

1 0 1 0 0

1 0 0 1 1

1 1 1 0

c(x) = xr·m(x) + r(x) = Q(x)·P(x)

CRC-12=x12+x11+x3+x2+x+1

CRC-16=x16+x15+x2+1

CRC-CCITT=x16+x12+x5+1