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International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences, Vol XXXV, Part Bl. Istanbul 2004 
  
and f(x) in Figurel.The meanings of the digital sequences and 
the process of CRCC are as follows. 
P(x)=1+x> +x +x +x*. In Figurel, p(x) is the digital 
sequence polynomial that is sent out, G(x) is another digital 
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
: ia sequence polynomial that is usually called 
Any binary digital sequence can be expressed as a code generation 
polynomial of variable x. For instance, the binary digital 
sequence 101 100 101 is expressed as the code polynomial 
Transmitter Spatial Channel | Receiver 
; oa : F(x)= 
Input Symbols Right Shift Adding GRC : RC (heck Judgement Output Symbols 
P(x) io eS d RS 
| nuuc x)+ Fesi due i due =0 
Rx) > P(x) [| P(x)+ residue EIS > Tm 
dx) Resi due; U 
Answering Signal 
GC 
(qu 
[FP (x)/ dx)] 
[P (x)/@x)] Fesi due ig i 
Fesi due Correct Signal£ Resi due =0£ © Y 
Incorrect Signalf Resi duej OE © 
  
  
  
  
  
Figure 1. The principle of cyclic redundancy checking codes 
polynomial. It is proven that G(x) is one and only polynomial 
whose power is n-k in the group of 2% codewords. Let G(x) 
divide code polynomial P'(x) here p'(x)= P(x)" [the term 
with the highest power of G(x) J( equal to P(x) ’s right 
shift)» x'^* P(x) the quotient be Q(x) and the residue be R(x), 
then we can get: 
AUT P). RG) (1) 
„ea 
where x" P(x) = Q(x)G(x) - R(x) 
Because the result of addition operation of Mod 2 
polynomial is the same as that of subtraction operation! ‘the 
above formula can also be expressed as: 
x" P(x) + R(x) = QG)G(x) (2) 
The residue of y'"*P(x)/G(x) 18 called checkout code 
polynomial (viz. checkout code CRC lt is shown that the 
residue is 0 when the new code polynomial 
F(x) = P + residue = x" * P(x) + R(x) is divided by checkout 
code generating polynomial G(x). 
G(x) is used on the receiver the same as on the transmitter in 
the process of error detection. With the received code 
polynomial (x)= F(x) divided by the code polynomial G(x), 
transmission is right if the residue is 0; otherwise transmission 
is wrong. If necessary, the judgement results will be returned to 
the transmitter. According to the judgement results of ground 
receiver, the satellite transmitter will send the data which was 
wrongly received again until the judgement results becomes 
right. 
2.3 Convolutional Codes 
Convolutional Codes, denoted by (n, k, m), are a type of Trellis 
Codes. In the representation, n denotes code length, k denotes 
information bit and m denotes coding storage. In the process of 
coding, information sequence is cut into segments of k code 
symbols too. After being coded, each segment is transferred to 
a codeword of n code symbols (n>k), called sub-group . 
Usually, n and k are smaller integers. Their remarkable 
character is that the n code symbols outputted in each time unit 
are not only concerned with the k code symbols inputted at the 
moment, but also concerned with the code symbols which were 
inputted in a long period before. The whole coding and 
decoding procedure is progressed step by step, so the 
convolution codes are also called interlink codes, in terms of 
mathematics it is called convolution operation!“], shown as 
  
  
  
  
  
  
  
  
  
Figure2. 
oll) 
Input Sequence RN Output Sequence 
i + 9011101 i 41100110000111 
> 
Figure 2 One kind of coder for convolutional codes 
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