# Design of a Baseband Signal for the 406 MHz Satellite Emergency Radio Transmitter Based on STM32
## 3 Baseband Signal Generation METHOD
### The Principle of BCH Check Bits Coding  
BCH--named after its discoverers, [PERSON], [PERSON] and [PERSON]--is a cyclic code capable of correcting multiple random errors and is widely used because of its strong error-correcting power and easy construction. The procedure of error correcting coding is adding some redundant detection codes in addition to the signal number during transmission. The detection codes, carrying no information, are only used to detect errors if there is an error in the information code during the transmission [19]. This kind of code has a strict algebraic structure, high coding efficiency, and can be performed by linear shift registers. Its coding algorithm has been one of the most important topics in the coding theory field. BCH coding has been widely employed in practice, such as satellite and deep space communications, data storage and data transmission, mobile communications, digital audio and video transmission [20]. Triple error correcting coding of BCH (127, 106) codes and BCH (63, 51) codes are used in the baseband signal coding of Cospas-SARSAT transmitters.  
The BCH code is defined as follows:  
Given any finite field \(GF(q)\) and its extension \(GF(q^{m})\), where \(q\) is a prime number or a power of a prime number and \(m\) is a positive integer. If any code element is taken from the cyclic code (\(n\),\(k\)) of \(GF(q^{m})\), \(n=2^{m}-1\), its generating polynomial \(g(x)\) has \(2t\) continuous roots \(\left\{a^{1},a^{2},\cdots,a^{(2-1)},a^{2t}\right\}\), \(t\) is the number of corrected bits and the cyclic code generated by the generating polynomial \(g(x)\), marked as \((n,k,t)\) is called a \(q\)-ary BCH code [21].  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Type** & **System Bits** & **Data Bits** & **Error Correcting Codes** & **Emergency Codes** & **Additional Selection Code** \\ \hline \multirow{4}{*}{Short message} & B1–B15 bits & B16–B24 & B25–B85 & B86–B106 & B107–B112 & / \\  & 15 bits & 9 bits & 61 bits & 21 bits & 6 bits & / \\  & & & 1 (Frame length flag) + & & & \\  & & & 1 (User Logo) & & & \\  & & & + 10 Country & & & \\  & & & & (Indicator Code) & & \\ \hline \multirow{4}{*}{Long message} & B1–B15 & B16–B24 & B25–B85 & B86–B106 & B107–B132 & B133–B144 \\  & 15 bits & 9 bits & 61 bits & 21 bits & 26 bits & 12 bits \\  & & & 1 (Frame length flag) + & & & \\ \cline{1-1}  & Bit & Frame & 1 (User Logo) + & & & \\ \cline{1-1}  & synchronization & synchronization & 10 (Country code) + 49 & & & \\ \cline{1-1}  & & & & (Indicator Code) & & \\ \hline \hline \end{tabular}
\end{table}
Table 1: The format of the message frame.  
Figure 1: The structure of the 406 MHz Cospas-SARSAT transmitter.  
In this paper, the first protection field of the 406 MHz baseband signal message uses the BCH (127, 106) binary error correcting code, and the second protection field uses the BCH (63, 51) codes. In the first protection field, the BCH (127, 106) codes can be expressed as BCH (127, 106, 3) code in which the number of error correcting bits \(t=3\), the code length is \(n=127\), the message length \(k=106\), and the check digital length \(m-k=m\times t=21\). In the second protection field, the check bits is 12, \(m\times t=12\), \(n=26\), so \(m=6\), \(t=2\), and the message length \(k=51\). Hence in the BCH code of this paper, BCH (38, 26) code is also BCH (63, 51) code, which can correct two errors, so the BCH (63, 51) code is also BCH (63, 51, 2) code.