Information Technology Reference
In-Depth Information
M
(
x
)
Z
D
1
D
2
D
3
Figure D.3
CRC coder
Now to prove that this circuit does generate the polynomial. The output
Z
(
x
) will be
[
]
−
1
−
2
−
2
−
1
Zx
()
=
Zxx
()
+
Mxx
()
+
Zxx
()
x
(
)
3
1
3
−
−
−
=
Zx x
()
+
x
+
Mxx
()
Thus
(
)
−
1
−
3
Z
(
x
)
1
+
x
+
x
M
(
x
)
=
−
3
x
giving
Mx
Zx
()
()
3
2
Px
()
=
=
xx
+
+
1
Question B
If the previous CRC system uses a message of 1 +
x
2
+
x
4
+
x
5
then determine the sequence of
events that occur and hence determine the encoded message as a polynomial
T
(
x
). Synthesise
the same code algebraically using modulo-2 division.
Answer B
First prescale the input polynomial of
M
(
x
) by
x
3
, the highest power of
G
(
x
):
M
´(
x
)=
x
3
M
(
x
)=
x
3
+
x
5
+
x
7
+
x
8
The input is thus
x
3
+
x
5
+
x
7
+
x
8
(000101011), and the generated states are:
Time
M
´(
x
)
D
1
D
2
D
3
D
4
1
000101011
0
0
0
0
MSD
2
00010101
1
0
0
0
3
0001010
1
1
0
0
4
000101
0
1
1
1
5
00010
0
0
0
0
6
0001
0
0
0
0
7
000
1
0
0
0
8
00
0
1
0
0
LSD
9
0
0
0
1
1
10
1
0
1
