Channel coding Hamming(7,4)
mapping in example x. parity of red, green, , blue circles even.
suppose want transmit data (1011) on noisy communications channel. specifically, binary symmetric channel meaning error corruption not favor either 0 or 1 (it symmetric in causing errors). furthermore, source vectors assumed equiprobable. take product of g , p, entries modulo 2, determine transmitted codeword x:
x
=
g
p
=
(
1
1
0
1
1
0
1
1
1
0
0
0
0
1
1
1
0
1
0
0
0
0
1
0
0
0
0
1
)
(
1
0
1
1
)
=
(
2
3
1
2
0
1
1
)
=
(
0
1
1
0
0
1
1
)
{\displaystyle \mathbf {x} =\mathbf {g} \mathbf {p} ={\begin{pmatrix}1&1&0&1\\1&0&1&1\\1&0&0&0\\0&1&1&1\\0&1&0&0\\0&0&1&0\\0&0&0&1\\\end{pmatrix}}{\begin{pmatrix}1\\0\\1\\1\end{pmatrix}}={\begin{pmatrix}2\\3\\1\\2\\0\\1\\1\end{pmatrix}}={\begin{pmatrix}0\\1\\1\\0\\0\\1\\1\end{pmatrix}}}
this means 0110011 transmitted instead of transmitting 1011.
programmers concerned multiplication should observe each row of result least significant bit of population count of set bits resulting row , column being bitwise anded rather multiplied.
in adjacent diagram, 7 bits of encoded word inserted respective locations; inspection clear parity of red, green, , blue circles even:
red circle has 2 1 s
green circle has 2 1 s
blue circle has 4 1 s
what shown shortly if, during transmission, bit flipped parity of 2 or 3 circles incorrect , errored bit can determined (even if 1 of parity bits) knowing parity of 3 of these circles should even.
Comments
Post a Comment