Succeeded in releasing a microsatellite jointly developed by Muroran Institute of Technology and Osaka Prefectural University into outer space
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Conventionally, ultra-small satellites used the communication speed of 1200 bps or 9600 bps (AX.25 protocol) in UHF / VHF, but in the first mission, "Demonstration of high-speed communication technology in the amateur radio band", By adopting a protocol that uses Reed-Solomon coding and convolution processing, we are aiming to expand the range of amateur radio by enabling higher-speed communication at 1 bps GMSK and 13600 bps 19200FSK.
Ultra-small jointly developed by Muroran Institute of Technology and Osaka Prefecture University from the International Space Station around 3:14 pm on March 8th ... → Continue reading
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Reed-Solomon code1960To (British)When(British) Is an error correction code developed by.Since code generation and decoding are complicated, it is not often used in fields where processing speed is required, but on the other hand, it has high error correction capability.Digital terrestrial broadcasting,Satellite communication,ADSLOr, in a familiar placeCD,DVD,BD,QR codeIt is applied to error correction of.
As a feature, the Galois field (Galois field) is used for the code generation method.Finite field) Is used.This is multiplebitIs regarded as one block (called a symbol or word), and a code word is represented by a group of symbols, and an error is detected and corrected for each symbol.No matter how many bits in one symbol contain errors, they are regarded as "one symbol error" as a whole, so they are particularly resistant to consecutive bit errors (burst errors).The number of bits of one symbol of the Reed-Solomon code is not usually specified, but it is 1 bits (1) due to computer specifications.Part-Time Job) Is often implemented in the system.
Here, the basic theory and implementation method of the Reed-Solomon code will be described.The addition symbols that appear below as the basic concept of coding theory () Are not direct sums, but for each bitExclusive ORRepresents
In Reed-Solomon code r A contiguous mass of bits is regarded as one symbol, and N symbols, that is, r × The sequence of N bits is regarded as one codeword.
At this time, K symbols are the information actually sent, and the remaining (NK) symbols are redundant symbols generated by the coding described later.However, r , N, K satisfy the following conditions.
Here (NK) / 2 t If, the Reed-Solomon code is t It is possible to correct errors in up to one symbol.
First in Reed-Solomon r × The sequence of N bits is expressed in the form of the following polynomial with the symbol as a coefficient (N-1).Assume that each 8-bit string is converted into the following symbol as shown in the figure.
At this time, this bit string is the Reed-Solomon code.
It is expressed in the form of a polynomial.
Conversion to symbols is performed as follows.First continuous r Since the bit is one symbol, the symbols are all 2 r There will be types.Therefore 2 r Define an expanded Galois field composed of individual elements.Specifically, first r nextPrimitive polynomialSelect one from the following, for example, set r = 8 and use the following.
At this time, if the root of this equation is α
Can be expressed as.Therefore, using this relationship, the power of α is defined as follows to correspond to each bit string.
· · ·
· · ·
256 = 2 in total by adding 8The elements of are aligned, and the correspondence with each bit string can be taken.
The coding is performed by the following procedure.Information to send first K × r Bits are symbolized by the method described above to generate a K-1 degree polynomial, which is called an information polynomial. I (x) It is represented by.Next, prepare a generation polynomial expressed by the following equation.
In the above formula, b is an appropriate integer.As an example, b = 0 generates a code that corrects a two-symbol error.At this time, t = 2, and the generated polynomial is
Will be.At this time, the following operations are performed using the information polynomial and the generation polynomial.
Is.The code to which the bit string corresponding to C (x) generated here is transmitted.
The process for decoding from the transmitted data is performed according to the following procedure.
- Error detection, syndrome calculation
- Detects the number of symbols containing errors
- Detects the position of a symbol containing an error
- Detects incorrect values
- Error correction
There are several types of Reed-Solomon code decoding methods, and the typical ones are known.Here, the Peterson method is described. Although this method is simple in processing, it has a drawback that it cannot be used for large-scale decoding because the amount of calculation increases when the number of symbols to be corrected is large.
First, error detection is performed, this method uses a polynomial generated from the received data. Y (x) It is represented by.At this time Y (x) Is expressed by the following formula.
here E (x) Is an error that was mixed in during communication.What is a syndrome that calculates the syndrome first?
It is a 2t number represented by.
The aforementioned C (x) と G (x) From the relationship of C (x) G (x) It is clear that it is divisible by.Therefore, whether there is an error G (x) Each root ofb, ab + 1, ..., αb + 2t-1Can be detected by substituting
IfY (x) ≠ C (x) AndY (x) There will be an error in.
If an error is present, then the number of symbols containing the error is detected.For detection, the number of errors is first assumed, and a determinant is generated from the above syndrome.For example, if the assumed number of errors is k, the following relational expression holds.
If the number of errors is not k, then the above equation will always be 0.As a result, the number of errors is assumed to be t first, and if the above determinant becomes 0, then the number of errors is assumed to be t-1. It can be seen that the number of errors assumed in is the actual number of errors.
After detecting the number of errors, the position of the error is detected.This can be obtained by generating the following error position polynomial when the number of errors is k.
However, σ1, p2,…, σk Is a variable that satisfies the following equation.
At this time, α in σ (x)0~ ΑN-2When is substituted sequentially, there is always a place where σ (x) = 0 at k points.At this time, the inverse element of the root is the position where there is an error. σ (αm) = 0, then x of Y (x)255-mThe coefficient of is containing an error.
When the position can be detected, the error value is finally detected and decrypted. Since Y (x) = C (x) + E (x), the above-mentioned syndrome is
You can say that.So from this formulaSimultaneous linear equationsIs used to calculate the error value.As an example, the result of error position detection m1, ..., mk Assuming that there is an error in the order of, then the following equation is generated.
From this formula e1, ..., ek Seeking,
Can recover the error.