JPS6329450B2 - - Google Patents

Description

[Detailed description of the invention]

The present invention relates to a novel error correction encoding method suitable for application to a transmission system with many errors. The present invention enables error detection and error correction on the receiving side using an error correction code. Conventional error correction codes have been considered for use in cases where both the transmission path and the information are of high quality, such as in the main memory of an electronic computer, or where both are of low quality, such as in space communications. As an example of such an error correcting code, there is a known one that forms an error correcting code by performing bit-by-bit operations in the row and column directions from bit information arranged in a matrix, as shown in Japanese Patent Application Laid-Open No. 147925/1982. ing. However, when recording and reproducing audio signals using the PCM method using a magnetic recording and reproducing device, it is required to transmit high quality information over a low quality transmission path. Furthermore, even if correction is impossible, if the frequency of errors is low, it is permissible to detect errors and correct them (previous value hold or average value interpolation). Conventional error-correcting codes have insufficient error-correcting ability for such uses, and if an error that exceeds that ability occurs, the correction will result in a correction error that will result in a new error. Error detection codes had to be used in combination, resulting in a problem of lower overall efficiency. Furthermore, in bit-by-bit correction coding, it is necessary to know the position of each error bit when performing correction decoding, and since each bit is corrected one by one, the computation time required for correction is long, making high-speed processing difficult. The present invention aims to provide an error correction encoding method using a novel error correction code suitable for use in recording and reproducing audio signals using the PCM system as described above. That is, the error correction code according to the present invention has the features listed below. First, it has high error correction ability and is efficient. Second, it is possible to reliably detect errors that exceed the correction ability without causing correction errors. Third, when an error is detected that exceeds the correction ability,
It is not the entire block that is moved to correction, but the minimum number of words necessary, and when applied to PCM recording, it is possible to reduce the auditory unnaturalness caused by correction. Fourth, it is effective against both burst errors and random errors. Fifth, the configurations of the encoder and decoder are relatively simple. An error correction code (referred to as a crossword code) having such features will be described below.
First, let the transmission matrix be V, the syndrome matrix S, the parity check matrix H, the error matrix E, and the reception matrix U. For example, when there are 8 transmission words and 4 information words, one block cross An example of a word code is shown in the formula below. HV=O…(1) HU=HE=S…(2) U=V+E…(3)

【formula】

【formula】

【formula】

【formula】

However, in the above equation, M ₁ to M ₈ are information words of one word each, R ₅ to R ₈ are test words of one word each, and s ₁ to _{s 4} are syndrome words of one word each, which is the word length. are all equally n-bits and are represented as n-dimensional horizontal vectors. In this specification, + means addition according to the arithmetic method modulo 2 (mod.2), and when all the elements of a matrix or vector are 0, it is treated as 0, and when even one element is 1, it is treated as 1. do. Furthermore, if one block of the above-mentioned crossword code is expressed in another form,

[Table] becomes. Expressed in this form, test word R ₅ is for information words M ₁ and M ₂ , test word R ₆ is for information words M ₃ and M ₄ , and test word R ₇ is for information words M _{1 and M 4} . M ₃ and the test word R ₈ is the information word M ₂
and M ₄ may be intuitively understood. In PCM recording, etc., the order in which one block of codes is transmitted serially is starting from M ₁ and then M ₃ , R ₇ , M ₂ , M ₄ , R ₈ , R ₅ , R ₆ or M ₁ , M ₂ , _R5 , _M3 , _M4 , _R6 , _R7 , _R8 . Now, the important assumptions in constructing the crossword code are shown below. Assumption: ei+ej=0 (i≠j) because ei=
0, only when ej=0. The probability that (ei + ej = 0) when (ei≠0, ej≠0) is 2 ^{- n} (n = word length), so if the word length n is set large enough for the error rate, etc.
The above assumption holds approximately true. In other words,
In a crossword code, the word length n is selected so that the probability that two erroneous words (ei, ej) coincidentally become the same is negligibly small. Decoding of the above-mentioned crossword code involves logical operations for detecting error positions, syndrome words,
It is divided into the execution of correction by addition of received words. FIG. 1 is a flowchart showing logical operations for detecting error positions in this decoding,
The correction execution method is represented by a return code, and the number of cases for each of the number of error words from 0 to 8 is also shown. In FIG. 1, Y means affirmation and N means negation. Further, the return code has four values, 0, 1, 2, and x, and each value has the following meaning regarding execution of correction. 0: Since the word is correct, neither correction nor correction is performed. 1: Correct the error word (e ₁ or e ₂ ) by adding the syndrome word s ₁ to the word u ₁ or u ₂ of the received sequence, respectively, and correct the information word M ₁ or M ₂ of the transmitted sequence. seek. Correct the error word (e ₃ or e ₄ ) by adding the syndrome word s ₂ to the word u ₃ or u ₄ of the received sequence, respectively, to obtain the information word M ₃ or M ₄ of the transmitted sequence. . 2: Correcting the error word (e ₁ or e ₃ ) by adding the syndrome word s ₃ to the word u ₁ or u ₃ of the received sequence, respectively, to the information word M ₁ or M ₃ of the transmitted sequence seek. Correct the error word (e ₂ or e ₄ ) by adding the syndrome word s ₄ to the word u ₂ or u ₄ of the received sequence, respectively, to obtain the information word M ₂ or M ₄ of the transmitted sequence. . ×: Correction is not possible, so correction (previous value hold or average value interpolation) is performed for that word. In addition, syndrome word s ₁ for word u ₁
or s ₃ , the syndrome word ( _{s 1 or s 4 )} (s ₂ or _{s 3 )} (s ₂ or
If the error can be corrected by adding any of s ₄ ), the return code is set to 1. To explain some cases of decoding shown in FIG. 1, from the above-mentioned equations (2), (5), (7), and (8), becomes. If decoding is started and (s ₁ = 0) and (s ₂ = 0), then (e ₁ e ₂ e ₃ e ₄ e ₅ e ₆ ) can be obtained from the above assumption and equation (9). All are 0, each of u ₁ u ₂ u ₃ u ₄ is correct, and these are the information words of the transmission sequence
It is said that M ₁ M ₂ M ₃ M ₄ . Here, e ₇ e ₈ is 0
It is unclear whether this is the case. Therefore (e ₇ = e ₈ =
0), when all the information words and check words of the transmission sequence are correct, that is, when the number of error words is 0, it is 1.
There are two cases where either e ₇ or e ₈ exists, that is, the number of error words is 1, e ₇ and e ₈
There is one case where both exist, that is, the number of error words is 2. However, the test word R ₇ or R ₈
There is no problem even if there is an error in the information, so there is no need to make corrections or amendments. If no error occurs regarding the information word in this way, (s ₁ = 0)
It is only necessary to determine that (s ₂ =0). In most cases, the data will pass through this path, so the overall time required for decoding is extremely short. Next, if (s ₁ ≠ 0) → (s ₃ = 0) → (s ₄ = 0), it is determined that all of (e ₁ e ₂ e ₃ e ₄ e ₇ e ₈ ) are 0. , u ₁ u ₂ u ₃ u ₄ are determined to be correct and are information words M ₁ M ₂ M ₃ M ₄ , and no corrections or corrections are made. Regarding test words R ₅ and R ₆ , in the case of a one-word error where only test word R ₅ contains an error (e ₅ ) (because s ₁ ≠ 0)
and both test words R ₅ and R ₆ are incorrect 2
There may be cases of word errors. Next, if (s ₁ = 0) → (s ₂ ≠ 0) → (s ₂ = s ₃ ), then from the condition (s ₁ = 0), (e ₁ , e ₂ , e ₅ = 0) It can be determined that there is. Also, (s ₂ = s ₃ ) becomes e ₃ +e ₄ +e ₆ =e ₁ +e ₃ +e ₇ , which becomes e ₄ +e ₆ =e ₁ +e ₇ . Here, since (e ₁ =0), e ₄ +e ₆ +e ₇ =0. Therefore, based on the assumption, it can be determined that (e ₄ , e ₆ , e ₇ =0). Although it is unclear whether e ₈ is 0, it follows from the condition (s ₂ ≠ 0) that there is an error in the received word u ₃ (e ₃ ≠ 0). In other words,
In this case, the error can be corrected by u ₃ +s ₃ =(M ₃ +e ₃ )+e ₃ =M ₃ , and the return code becomes 2. There is one case when the number of error words is 1 (e ₃ ≠0), and there is one case when the number of error words is 2 (e ₃ , e ₈ ≠0). Next, (s ₁ ≠ 0) → (s ₃ ≠ 0) → (s ₃ = s ₁ + s ₂ )
The case will be explained below. In this case, e ₁ + e ₃ + e ₇ = e ₁ + e ₂ + e ₅ + e ₃ + e ₄ + e ₆ e ₂ + e ₄ + e ₅ + e ₆ + e ₇ = 0, and (e ₂ , e ₄ , e ₅ , e ₆ , e ₇ =0). Therefore, the received words u ₁ and u ₃ are incorrect, respectively u ₁ +s ₁ =(M ₁ +e ₁ )+e ₁ =M ₁ u ₃ +s ₂ =(M ₃ +e ₃ )+e ₃ =M ₃ The error can be corrected by , and the return code for u ₁ and u ₃ is 1. Since it is unknown whether e ₈ is 0 or not, there is one case of a 2-word error and one case of a 3-word error. Next, (s ₁ ≠ 0) → (s ₃ ≠ 0) → (s ₃ ≠ s ₁ + s ₂ )
→ (s ₁ = s ₃ ) → (s ₂ ≠ 0) → (s ₄ ≠ 0) → (s ₂ ≠
Let's discuss the case of _s4 ). In this case, since e ₁ +e ₂ +e ₅ =e ₁ +e ₃ +e ₇ , (e ₂ , e ₃ , e ₅ , e ₇ =0). (s ₁
≠0) or (s ₃ ≠0), so (e ₁ ≠0). Therefore, for received word u ₁ , (u ₁ +s ₁ )
Or the error can be corrected by adding (u ₁ + s ₃ ),
The return code for u ₁ is 1. Also, (s ₂
≠0) or (s ₄ ≠0) is (e ₄ +e ₆ ≠0)
Or, since it is rewritten as (e ₄ +e ₈ ≠0), it is assumed that (e ₄ ≠0), that is, the received word u ₄ may be incorrect. The above condition holds true for (e ₄ =0) only when (e ₆ , e ₈ ≠0). However, it is impossible to detect this from syndrome words. Since error correction is impossible when the received word u ₄ is erroneous in this way, it must be corrected. The return code for u ₄ at this time is ×. If we consider the same way below, the entire process of decoding will be as shown in FIG. FIG. 2 shows the error correction ability of the crossword code given in the above example, which has 8 transmission words and 4 information words (50% redundancy). In FIG. 2, Pw is the word error rate, and when errors are random, the probability of i-fold word errors is Pw ⁱ (1-Pw) ^8-i .
Therefore, the performance of the crossword code against random errors can be expressed from FIG. 2 as follows. Probability of no error: (1-Pw) Probability of ⁸ correction: 8Pw (1-Pw) ⁷ +16Pw ² (1-
Pw) ⁶ +4Pw ³ (1-Pw) ⁵ Probability of 1-word correction: 12Pw ² (1-Pw) ⁶ +4Pw ³
(1-Pw) ⁵ Probability of 2-word correction: 44Pw ³ (1-Pw) ⁵ +24Pw ⁴
(1-Pw) ⁴ +4Pw ⁵ (1-Pw) ³ Probability of 4-word correction: 4Pw ³ (1-Pw) ⁵ +46Pw ⁴
(1-Pw) ⁴ +52Pw ⁵ (1-Pw) ³ +28Pw ⁶ (1-
Pw) ² +8Pw ⁷ (1-Pw) +Pw ⁸ Note that in the case of 4-word correction, since not even a single correct information word can be obtained for one block, it can be said that correction is impossible. Therefore, only one word error can be corrected.
100 [%], 57 [%] of 2-word errors, and 7 [%] of 3-word errors. Also, 1-word correction is required for 43% of 2-word errors and 7% of 3-word errors.
It is. Note that the probabilities of detection errors and correction errors are both 2 ^-n (n=word length). Such detection errors and correction errors occur because a two-word error is regarded as no error, or a three-word error is regarded as a one-word error, but the probability of such a detection error occurring is determined by the number of bits in one word. It may be set to a length that is so small that it can be practically ignored. Furthermore, regarding burst errors, it has the ability to correct burst errors of two words in one block. FIG. 3 shows a communication system to which the present invention using the above-mentioned crossword code is applied. In Figure 3, 1
is an input terminal to which n-bit information words _M1 to _M4 , which are obtained by PCM modulating an audio signal, are applied in series, for example. These information words are stored in buffer memory 2. 3 is a test word forming circuit which creates test words R ₅ to _{R 8} from the information words stored in the buffer memory 2; For example, the test word R ₅ is formed by (M ₁ +M ₂ ) and can therefore be obtained by feeding each bit of the information words M ₁ and M ₂ in parallel to n exclusive OR circuits. Reference numeral 4 designates a switching circuit for selecting and serializing the information word from the buffer memory 2 and the test word from the test word forming circuit 3. Transmission words (information words and check words) from this switching circuit 4 are supplied to a transmitting circuit 6. The buffer memory 2, test word forming circuit 3 and switching circuit 4 are controlled by a timing control circuit 5. The transmission circuit 6 is for converting the transmission word into a signal in a form suitable for the transmission line 7. The transmitting circuit 6 may be a simple amplifier or may be a modulation (amplitude modulation, frequency modulation, etc.) circuit. Furthermore, when the transmission line 7 is a VTR (video tape recorder), the transmission word can be converted into the same signal form as a television signal, without making any changes to the VTR itself.
It is possible to record and play back PCM signals. Reference numeral 8 indicates a receiving circuit, and a received sequence obtained as an output thereof is stored in a buffer memory 9. A syndrome word is formed by a syndrome forming circuit 10 from the received sequence stored in the buffer memory 9. This syndrome word is supplied to the correction logic operation circuit 11, which performs the logic operation shown in the flowchart of FIG. 1, and as a result, the aforementioned return code is generated. An error correction circuit 12 is provided which performs a predetermined error correction operation using this return code. The error correction circuit 12 generates an information word that is inherently correct or correct with errors corrected, or a corrected information word, which is supplied to a buffer memory 13. The buffer memory 13 compensates for the fact that the time required for correction or correction varies depending on the type of error, and information words with constant bit timing and word timing are obtained at the output terminal 14. Note that 15 is a timing control circuit for the buffer memories 9 and 13, the syndrome forming circuit 10, the correction logic operation circuit 11, and the error correction circuit 12. The crossword code according to the present invention may have various modified configurations other than the eight transmission words and four information words described above that satisfy the assumptions and equations (1) to (3). Other configurations of crossword codes are shown below. One of them has a configuration in which the transmission words are 10 words and the information words are 4 words, and the transmission matrix V, parity check matrix H, and syndrome matrix S are expressed by the following formula.

【formula】

[Formula] The crossword code shown in these formulas (10) to (12) consists of the test word R ₉ set to (M ₁ +M ₄ +R ₉ =0) and (M ₂ +M ₃ +R ₁₀ =0). ) is added with the test word _R10 . Although the details of the decoding of this crossword code are omitted, its performance is shown in FIG. The redundancy of this crossword code is 60%, but the 4th
As is clear from the figure, the performance is improved so that not only 1-word errors but also 2-word errors can be corrected to 100%. Similarly, there are 10 transmission words and 4 information words.
Although the redundancy is 60%, test words _R9 and _R10 may be added to the test words _R5 to _R8 . That is, the transmission matrix V and the syndrome matrix S are the same as equations (11) and (12), but the parity check matrix H has the structure shown in the following equation. FIG. 5 shows the performance of the crossword code having the check matrix H of equation (13). However, the probability is the same as in FIG. 4 and is omitted. Furthermore, the crossword code has 12 transmission words and 6 information words (redundancy 50%), and the transmission matrix V, error matrix E, reception matrix U, syndrome matrix S, and check matrix H are as follows: I will explain about it.

【formula】

【formula】

【formula】 【formula】

FIG. 6 shows the performance of the crossword code configured as shown in equations (14) to (18). As is clear from Figure 6, in this example, errors of up to two words can be corrected, and errors of up to three words can be corrected by 46.4
[%] are correctable, 21.8 [%] require correction of only one word, and the remaining 31.8 [%] require correction of two or more words. Regarding 4-word errors and 5-word errors, the correctable rate is 15.8 [%] each.
and 0.8 [%]. Furthermore, correction of words related to only one of the bottom three rows of equation (18) can be performed with an extremely simple decoder. That is, it is sufficient to determine the error line based on s ₄ to _{s 6} and add s ₁ to _{s 3} . This method is useful in conjunction with interleaving every four words (reordering the words) or in a multi-track fixed head system. There is also a crossword code with 15 transmission words and 9 information words (redundancy: 40%), where the transmission matrix V, error matrix E, reception matrix U, syndrome matrix S, and check matrix H are as follows: Can be configured.

【formula】

【formula】

【formula】 【formula】

The performance of the crossword code expressed by these equations (19) to (23) is shown in FIG. The performance of each example of the crossword code according to the present invention described above is summarized as shown in Table 1.

【table】

[Table] Regarding the BCH code, which is one of the random error correction codes that has already been proposed, we also show the performance when it has similar information words, redundancy, etc., corresponding to each example of the crossword code mentioned above. Table 2).

[Table] Furthermore, regarding the blocked Iwadare code, which is one of the burst error correction codes that have already been proposed,
Table 3 shows the performance of each example of the crossword code described above (excluding those having the performance shown in FIG. 4).

[Table] Comparing these tables, it is found that BCH codes and Iwadare codes cause correction errors when they cannot be corrected, so other error detection codes such as CRCC (Cyclic
In contrast, the crossword code according to the present invention corrects the minimum number of words necessary when correction is impossible without using other codes. can be moved to
Therefore, it has the feature of high efficiency. Furthermore, the crossword code according to the present invention has lower performance than the BCH code when it comes to only random errors, and it has lower performance than the blocked Iwadare code when it comes to burst errors only. In this case, when burst errors of several bits occur randomly within a block, the crossword code according to the present invention constructs a product code using each information word composed of multiple bits as a unit, and performs each error check. Since error correction encoding is performed so that the code is generated by word-by-word calculations, if the position of a word containing an error bit is known during correction decoding, multiple erroneous bits can be corrected word by word at once. By processing,
This has the effect that if it is a bit within the word, it can be corrected without knowing the location of each error bit, and even if multiple bit errors occur in a burst, they are included in each word of the number of words that can be corrected. If it is a bit, it can be corrected. Therefore, not only one-word errors but also two-word and three-word errors can be corrected in some cases. In addition, since the first test word and the second test word are configured with the same number of bits, the calculation method and circuit can be unified to those that handle data with the same number of bits, and in some cases, some parts may be shared. There are advantages such as being able to By transmitting information words encoded by the error correction encoding method according to the present invention, errors occurring in received information words can be efficiently corrected with high correction ability. The present invention having such features is effective when applied to the case where high quality information is transmitted over a low quality transmission path such as converting an audio signal into PCM and recording and reproducing it by a magnetic recording and reproducing device. .

[Brief explanation of drawings]

FIG. 1 is a flowchart used to explain decoding according to the present invention, FIG. 2 is a diagram used to explain an example of the present invention, FIG. 3 is a block diagram of a communication system to which the present invention is applied, and FIGS. FIG. 7 is a diagram used to explain other modifications of the present invention. 3 is a check word forming circuit, 7 is a transmission path, 10 is a syndrome word forming circuit, 11 is a correction logic operation circuit, and 12 is an error correction circuit.

Claims (1)

[Claims] 1 Each information word is composed of a predetermined number of bits such that the probability that the sum of the modulo-2 addition of two error words occurring when transmitting the information word is negligibly small. , extract m information words from the input data series of a plurality of information words (m is a positive integer),
Each of the m information words is subjected to a word-by-word operation to generate each first check word consisting of the predetermined number of bits. A plurality of first test blocks are constructed so as to include the following test words, and n information words are taken out from each different block of each of the plurality of first test blocks (n is a positive integer). Each second check word consisting of the predetermined number of bits is generated by performing a word-by-word operation on each of the n information words, and each of the n information words and the corresponding second check word are generated. 1. An error correction encoding method characterized in that each of a plurality of second check blocks is configured to include a check word.