CN1322063A - Long burst error-correcting decoding method adopting Reed-solomon code - Google Patents

Abstract

The present invention belongs to the field of communication technology. Said method incldues: on the basis of cyclic characteristics of RS code it effectively utilizes RS code to receive relationship of before polynomical cyclic shift and after polynomical cyclic shift and correlation of error position of long burst, and the error position of long burst error is assumed to be lowest 2t bit for receiving polynomical, so that it can simplify the interpretation method into the method of that firstly making cyclic shift of received polynomical, then resolving error image by using a group of linear equation. Said invention uses relative small decoding time-delay and relative high information transmission rate to implement decoding error correction suitable for long burst error.

Description

A kind of long burst error-correcting decoding method that adopts Reed-solomon code

The invention belongs to communication technical field, particularly adopt a kind of employing Reed-solomon code (hereinafter to be referred as RS sign indicating number) of forward error control (FEC) technology in transfer of data and storage system to correct effectively and the fast interpretation method of long burst error.

Data cause burst error through regular meeting in storage and transmission course.The reason that produces this mistake has synchronization loss, the multipath fading in the wireless transmission, the magnetic track in the magnetic storage in line noise, the demodulating process damaged etc.This burst error generally be periodically occur and the duration long.Because the existence of this long burst error, limited the memory capacity of memory under the rate of information throughput under the specific bandwidth and the particular area greatly.Particularly in wireless transmitting system, because the influence of multipath fading, this problem becomes more outstanding.

Reed-solomon code (RS sign indicating number) has the ability of stronger correction burst error, is widely used in transfer of data and the storage system.

The interpretation method principle that adopts the RS sign indicating number to correct long burst error is:

The definition of RS sign indicating number and parameter: the RS sign indicating number is a kind of multi-system BCH code, and it is defined as: establish f (x) and go up the multinomial less than the n rank for galois field GF (q) (q is the arbitrarily time power of arbitrary prime number), α is the primitive element of GF (q), then and if only if α, α ²..., α ^D-1(d is not less than 2 integer) when being all the root of f (x), set { f (x) } is a RS sign indicating number, and this RS sign indicating number is with g (x)=(x-α) (x-α in other words ²) ... (x-α ^D-1) be the polynary BCH code of the rhizosphere consistent with symbol field (all on GF (q)) of generator polynomial and code polynomial.The parameter of entangling the RS sign indicating number of t symbol error has: code length n=q-1, information symbol number k, checking symbol number n-k=2t, and the minimum range d=2t+1 of sign indicating number.

Conventional error-correcting decoding method:

If symbol makes a mistake in transmission or storing process, the mistake multinomial can be expressed as: $E\left(x\right)=e_{i_1}{x}^{i_1}+e_{i_2}{x}^{i_2}+...+e_{i_1}{x}^{i_v}---\left(1\right)$ In the formula, v is the number of errors of actual generation, 0≤v≤t;

Represent r wrong value; i _rRepresent r wrong position.Therefore receiverd polynomial can be expressed as:

R (x)=C (x)+E (x) (2) is C (x)=c wherein _N-1x ^N-1+ c _N-2x ^N-2+ ... + c ₁X+c ₀Be (n, k) code polynomial behind the RS sign indicating number coding.Obtaining syndrome by receiverd polynomial is:

S _j=R (α ^j)=C (α ^j)+E (α ^j)=E α ^j) (definition 1≤j≤2t) (3) for convenience's sake,

(r=1,2 ..., v) be improper value; (r=1,2 ..., v) be errors present, then: $E\left({\mathrm{\α}}^j\right)=e_{i_1}{\left({\mathrm{\α}}^j\right)}^{i_1}+e_{i_2}{\left({\mathrm{\α}}^j\right)}^{j_2}+...+e_{i_v}{\left({\mathrm{\α}}^j\right)}^{j_v}$ $=e_{i_1}{\left({\mathrm{\α}}^{j_1}\right)}^j+e_{i_2}{\left({\mathrm{\α}}^{j_2}\right)}^j+...+e_{i_v}{\left({\mathrm{\α}}^{j_v}\right)}^j---\left(4\right)$ $=Y_1{X}_1^j+Y_2{X}_2^j+...+Y_v{X}_v^j$ Launch with (2) formula substitution (1) formula and with the formula of gained, obtain 2t equation:

2t equation arranged in the top equation group, only less than 2t unknown number, and X ₁, X ₂..., X _vIn equation, constitute a reversible all the time Fan Demeng matrix, so equation is separated.Conventional interpretation method adopts separates such as the Berlekamp alternative manner that this equation group at first obtains errors present and then substitution (5) formula is tried to achieve the error map sample value, carries out error correction at last.These class methods are for correcting random error and very effective with the short mixing mistake that happens suddenly at random.But it is merely able to correct the burst that maximum length is t, and equation group (5) or a nonlinear equation, and the interpretation method complexity is very big, therefore is not suitable for correcting long burst error.

The method that existing employing RS sign indicating number is corrected long burst error mainly contains two kinds: method one is to adopt the code interleaving technology, and it mainly is by interweaving a plurality of short codes to be formed a long code.For example, to (n, k) linear code is carried out the j level and interweaved, and can obtain (jn, jk) sign indicating number.If originally (n, k) sign indicating number burst-error-correction ability be t, obtain after then interweaving (jn, jk) sign indicating number burst-error-correction ability be jt.This way is actually burst error is dispersed as random error and makes it to be evenly distributed in each short code and goes.Its advantage is to keep improving under the constant situation of information rate the ability that the RS sign indicating number entangles long burst error.But, owing to before code vector is encoded back and decoding, will interweave respectively and deinterleaving, therefore can bring bigger transmission delay, and the complexity of increase system realization.Method two is to adopt the method that reduces information rate to improve the error correcting capability of code vector.Such as in the length that keeps increasing by the length that reduces information bit under the constant prerequisite of code length check bit, perhaps keeping increasing the length of check bit by increasing code length under the constant prerequisite of information bit length.This method makes the minimum range in code vector space strengthen by sacrificing the rate of information throughput guaranteeing not increase under the prerequisite of transmission delay, thereby has improved error correcting capability.

The objective of the invention is to overcome the weak point of prior art, a kind of long burst error-correcting decoding method that adopts Reed-solomon code is proposed, by utilizing the characteristic of burst error, with with respect to the much smaller decoding delay of said method one with the rate of information throughput much higher, realize being suitable for the error-correcting decoding of long burst error with respect to said method two.

A kind of long burst error-correcting decoding method that adopts Reed-solomon code that the present invention proposes may further comprise the steps: 1) carry out initialization: the burst length l that the error pattern of current maximum possible is set _mValue be 2t, burst pattern Y _mValue be 0, minimum burst position N _mValue be 0; While shift count variable N _cValue be set to 0; 2) begin to receive transmission information series R (x), and calculate its corresponding syndrome S _j(1≤j≤2t); 3) syndrome of previous step gained is judged: if syndrome is not equal to zero, then next step transfers to step 4, otherwise next step jumps to step 9; 4) calculate burst pattern Y _c, S ' here _j=S _j(1≤j≤2t) also calculates corresponding burst length l _c5) compare l _cWith l _mSize: if l _c＜l _m, next step transfers to step 6, otherwise next step jumps to step 7; 6) Y _c, l _c, N _cBe the parameter of the burst pattern of current maximum possible, the value with them overrides Y respectively _m, l _m, N _mValue; 7) judge N _cValue whether greater than code length n: if then next step transfers to step 9; Otherwise transfer to step 8; 8) receiverd polynomial ring shift right 2t-v _Max+ 1 code word size also calculates corresponding syndrome, promptly calculates $S_j^{\′}=\frac{S_j}{{\left({\mathrm{\α}}^j\right)}^{{2t-v}_{\max }+1}}$ (method that also can adopt ring shift left to be in decode procedure is at every turn with receiverd polynomial ring shift left 2t-v _Max+ 1 code word size promptly calculates $S_j^{\′}=S_j\·{\left({\mathrm{\α}}^j\right)}^{{2t-v}_{\max }+1})$ , the value with the shift count variable increases 2t-v simultaneously _Max+ 1, next step turns back to step 4 then; 9) according to error pattern Y _mWith errors present N _mValue carry out error correction, export the information series after the error correction simultaneously; After this step is finished, transfer to step 10, judge whether decode procedure finishes: if finish decode procedure; Otherwise next step jumps back to step 1, restarts the decoding of next code vector.

The principle of the method for the invention and the algorithmic descriptions of employing are as follows:

At first consider long burst error, its multinomial can be expressed as: $E\left(x\right)=e_{i_1}{x}^{i_1}+e_{i_2}{x}^{i_1+1}+...+e_{i_v}{x}^{i_1+v-1}---\left(6\right)$

I wherein ₁Be the extreme lower position of long burst error in code vector, v is a burst length.Because the RS sign indicating number has the character of cyclic code, the code vector cyclic shift that obtains after being about to encode, the new vector that obtains remains a code vector.Therefore receive vector $R\left(x\right)=C\left(x\right)+E\left(x\right)=c_{H-1}{x}^{n-1}+...+(c_{i_1+v}+e_v)x^{i_1+v}+...+(c_{i_1}+e_1)x^{i_1}+...+c_0---\left(7\right)$

Process is to the n-i that shifts left ₁Individual Baud Length (also cyclic shift i to the right ₁Individual Baud Length), the lowest order that can obtain happening suddenly is x ⁰=1 receiverd polynomial: $R^{\′}\left(x\right)=R\left(x\right)\·x^{{n-i}_1}=C\left(x\right)\·x^{{n-i}_1}+E\left(x\right)\·x^{{n-i}_1}=C^{\′}\left(x\right)+E^{\′}\left(x\right)---\left(8\right)$ Because C ＇ (x) remains a code vector, therefore can use (3), (8) formula to calculate the syndrome of R ＇ (x).By calculating the corresponding syndrome of R ＇ (x) be:

S′ _j＝S _j·(α ^j) ^n-i??(1≤j≤2t)?????????(9)。

Above-mentioned v is expanded to 2t, if actual burst length v less than 2t, then the value of all the other 2t-v the unit that happen suddenly is 0.In conjunction with (6) formula, (9) formula can be write as like this:

The lowest order of pattern of wherein happening suddenly is x ⁰, i.e. X ₁=α ⁰=1.(10) formula can be rewritten as thus:

Coefficient matrix in the formula is a nonsingular Fan Demeng matrix and α ⁰, α ¹..., α ^(2t-1)Be mutually different, this coefficient matrix is reversible.Thereby, have:

Notice that the length of burst pattern has been increased to 2t by the t of routine decoding equation in the equation group in the following formula; Solving equation becomes linear equation (12) by nonlinear equation (5) and coefficient matrix is a non-singular matrix, and is corresponding one by one between syndrome and the burst pattern.This means that if include burst error in the receiverd polynomial then the lowest order through burst error after the suitable cyclic shift necessarily can move on to x ⁰, the pattern that happens suddenly this moment can be calculated with (12) formula.

Suppose that the Maximum Burst Size that occurs in actual applications is v _Max, when syndrome was not equal to zero, mobile 2t-v circulated _Max+ 1 sign bit calculates corresponding burst pattern then, so repeats $\left[\frac{n+2t-v_{\max }}{2t-v_{\max }+1}\right]$ Inferior, the then actual burst pattern that takes place necessarily is included in resulting $\left[\frac{n+2t-v_{\max }}{2t-v_{\max }+1}\right]$ In the individual burst pattern.According to the maximum-likelihood decoding criterion, can choose the burst pattern that the shortest conduct decoding of burst length is tried to achieve in these burst patterns, and come information series is carried out error correction with it.

Because (n, k) the intersymbol minimum range of RS sign indicating number is d=n-k+1=2t+1, therefore when being less than or equal to t, burst length adopt this interpretation method not have decoding error, when burst length is less than or equal to 2t greater than t, then might find the burst shorter than actual burst length in decode procedure, decoding error will take place in this moment.But adopt its error probability of interpretation method of the present invention to remain very low.As an example, the probability of erroneous decoding under the particular burst of (255,223) RS sign indicating number is as shown in table 1, wherein v _Max=2t=32.Here need to prove when actual burst length equals 2t, because the burst pattern calculated all is not more than 2t in the decode procedure, so can't correct decoding, probability of erroneous decoding is 1.By table 1 as seen, when burst length was 28 symbols (equaling 224 bits), the error probability of this interpretation method can be low to moderate 10 ^-8Magnitude.

The probability of erroneous decoding of table 1 (255,223) RS sign indicating number under different burst lengths.

Burst length (symbol)	Probability of erroneous decoding	Burst length (symbol)	Probability of erroneous decoding
				17	?8.4×10 -35	?25	?1.4×10 -15
18	?2.1×10 -32	?26	?3.4×10 -13
				19	?5.3×10 -30	?27	?8.6×10 -11
20	?1.3×10 -27	?28	?2.2×10 -8
				21	?3.4×10 -25	?29	?5.4×10 -6
22	?8.5×10 -23	?30	?1.4×10 -3

?23	?2.1×10 -20	?31	?0.34
				?24	?5.4×10 -18	?32	?1.0

As seen, adopt this method to correct the long burst error of length, therefore improved the ability of the long burst error of RS sign indicating number correction greatly with very high accuracy near 2t.

Effect of the present invention is, by utilizing the characteristics of burst error, makes the length of the long burst error that the RS sign indicating number can be corrected be increased near 2t by the t of traditional interpretation method, and its error correcting capability has improved one times nearly.Simultaneously, compare with existent method one, it has reduced and has interweaved and the time-delay that brought of deinterleaving, compares with existent method two, and it does not have the loss of information transmission rate; This method also is simplified to the interpretation method of understanding system of linear equations to the complicated approach of separating Nonlinear System of Equations that conventional method adopted in addition, both reduced the complexity of interpretation method, make it to be convenient to hardware and realize, help adopting the way that improves circuit work frequency further to reduce decoding delay again.Therefore, when the RS sign indicating number was applied to correct long burst error, the performance of this interpretation method obviously was better than additive method.

Brief Description Of Drawings:

Fig. 1 is a software realization flow block diagram of the present invention.

Fig. 2 is the block diagram according to the decoder of the inventive method design.

Fig. 3 is the communication system figure of the long burst error of correction of application the inventive method.

Fig. 4 is the RS sign indicating number of this interpretation method of employing is applied to compound channel as the outer yard cascaded code of forming as ISN with the employing convolution code a communication system block diagram.

Below, explain the present invention in further detail with two embodiment with reference to the accompanying drawings:

Embodiment one: present embodiment is realized the error-correcting decoding method that the present invention proposes for adopting software, as shown in Figure 1.Present embodiment adopts (255,223) RS sign indicating number, maximum can be corrected the decoding algorithm that burst length is 28 code words, and decode procedure may further comprise the steps: after the decoding beginning, decoder is transferred to step 1b from step 1a, carries out initialization: the burst length l that the error pattern of current maximum possible is set _mValue be 32, burst pattern Y _mValue be 0, minimum burst position N _mValue be 0; While shift count variable N _cValue be set to 0; Decoder is transferred to step 1c then, begins to receive transmission information series R (x), and calculates its corresponding syndrome S with (3) formula _j(1≤j≤32); Decoder forwards step 1d to after finishing this step, and the syndrome of previous step gained is judged: if syndrome is not equal to zero, then next step transfers to step 1e, otherwise next step jumps to step 1j.At step 1e, calculate burst pattern Y with (12) formula _c(S ' here _j=S _j(1≤j≤32)) and calculate corresponding burst length l _cDecoder is transferred to step 1f by step 1e then, relatively l _cWith l _mSize: if l _c＜l _m, next step transfers to step 1g, otherwise next step jumps to step 1h.In step 1g, Y _c, l _c, N _cBe the parameter of the burst pattern of current maximum possible, the value with them overrides Y respectively _m, l _m, N _mValue.Subsequently, decoder is transferred to step 1h, judges N _cValue whether greater than 255: if then next step transfers to step 1j; Otherwise transfer to step 1i.At step 1i, decoder is with 5 code word sizes of receiverd polynomial ring shift right and calculate corresponding syndrome, promptly calculates $S_j^{\′}=\frac{S_j}{{\left({\mathrm{\α}}^j\right)}^5}(1\≤j\≤32)$ (method that also can adopt ring shift left to be in decode procedure with 5 code word sizes of receiverd polynomial ring shift left, is promptly calculated S ' at every turn _j=S _j(α ^j) ⁵(1≤j≤32)), the value with the shift count variable increases by 5 simultaneously, and next step turns back to step 1e then.At step 1j, decoder is according to error pattern Y _mWith errors present N _mValue carry out error correction, export the information series after the error correction simultaneously.After the operation of completing steps 1j, decoder is transferred to step 1k, judges whether decode procedure finishes: if then next step transfers to step 1l, finish decode procedure; Otherwise next step jumps back to step 1b, restarts the decoding of next code vector.

Embodiment two: present embodiment is realized interpretation method of the present invention for the method that adopts hardware.Present embodiment still adopts (255,223) RS sign indicating number, maximum can be corrected the decoding algorithm that burst length is 28 code words, the composition of this hardware circuit and work decoding process: as shown in Figure 2, in the input process of code stream 21,32 syndromes calculate and shift register circuit 22-24 is in the syndrome state that calculates, the message part of input code flow by synchronously temporary be in the memory 25 of 223 code words to capacity.Decoding circuit of right-hand component is finished initialization in during this: burst pattern register 29 is put 0 value, it is 32 that register among burst length calculating and the comparison circuit 2a is put initial value, the value of burst lowest order memory 2b is changed to 0, and shift counter 2c is reset to 0 state.Finish when a code polynomial is received, decoder switches to the burst pattern state that calculates.This moment, 32 syndromes calculating and shift circuit 22-24 were operated in displaced condition, everyly come a clock pulse syndrome circuit to finish a shift operation, simultaneously shift counter adds 5, the computing that the value of the new syndrome of gained is finished (12) formula by the matrix multiplication circuit 26 that is made of the combinational logic circuit pattern 27,28 that happened suddenly accordingly.The pattern 28 that wherein happens suddenly outputs to that burst length is calculated and comparison circuit 2a carries out burst length and calculates, and compares with register value in this part circuit.If calculate burst length less than register value, make comparative result output high level when then burst length value being replaced register value, make burst pattern register override original figure sample value with burst pattern 27, the extreme lower position that happens suddenly simultaneously memory is written into the currency of shift counter 2c as corresponding burst extreme lower position.So greater than code length 255, decoding circuit left-hand component 21-25 changes accepting state over to subsequently up to the value of shift counter 2c in periodic duty; The right circuit 25-2d changes the error correction output state over to simultaneously: message queue is exported from memory 25 according to timeticks, burst extreme lower position memory 2b enable to happen suddenly on burst position pattern register 29 and alternative selector 2d make output of burst pattern and the addition of information series finish error correction.

From top embodiment as seen, this interpretation method compares with two kinds of conventional error-correcting decoding methods that method adopted that the front was introduced not only that error correcting capability is strong, time-delay is little, and realizes also simple.Illustrate that by two examples the present invention is in Application in Communication Systems more below.

Example one, with reference to Fig. 3, the communication system of correcting long burst error comprises an information source 31 that produces digital information flow, RS encoder 33, Burst Channel 35, and correction burst error decoder 37 as shown in Figure 2.In this example, the data symbol stream 32 of the information of carrying that information source 31 produces is sent to RS code coder 33, and 33 pairs of information of RS code coder are carried out chnnel coding.RS code stream 34 behind the coding is interfered in long Burst Channel 35 transmission courses and produces long burst error, and the RS code decoder 37 that the code stream 36 that comprises long burst error is repaired long burst error receives.Process is corrected the error-correcting decoding of the RS code decoder 37 of long burst error, and the code stream 38 of output is correct digital information flow.

This interpretation method is not only applicable to pure Burst Channel, can also and other yards form cascaded code together and be used for correcting the mixing mistake that existing random error that compound channel occurs has long burst error again.Provide an example below.

Example two: as shown in Figure 4, the data flow that digital source 41 produces is sent to the coding that RS code coder 43 and encoder for convolution codes 45 carry out outer sign indicating number and ISN successively, and the cascade code stream 46 of output subsequently transmits by compound channel 47.Owing in transmission course, be subjected to influence of various factors, the cascade code stream 48 that receives will comprise the compound mistake of random error and burst error, and these information flows that comprise compound mistake at first are sent to the error-correcting decoding that Convolutional Decoder Assembly is primarily aimed at random error.When only containing the random error in convolution code error correcting capability scope in a segment encode flows, can come these random errors correction by the decoding of convolution code, its output 4a is one section correct information series; The number that has comprised long bursts or random error in this segment encode stream is greater than the error correction scope of convolution code, convolution code decoding failure then, and its output 4a can think a long burst error.After burst-error-correction decoder 4b by the RS sign indicating number carried out error-correcting decoding, long burst error obtained correcting.Obviously, choose the length and the error correcting capability of inside and outside sign indicating number and adopt the interpretation method of this method according to the characteristic of channel is suitable as outer sign indicating number, can obtain the stronger cascaded code of error correcting capability, thereby under the identical rate of information throughput, obtain the data of lower error rate, perhaps under the condition of same error rate, can adopt higher data transfer rate to transmit.

Because the RS sign indicating number is a subclass of BCH code, method of the present invention is applicable to BCH code fully, goes but also can be generalized in all cyclic codes.

Claims (1)

1, a kind of long burst error-correcting decoding method that adopts Reed-solomon code may further comprise the steps:

1) carries out initialization: the burst length l that the error pattern of current maximum possible is set _mValue be 2t, burst pattern Y _mValue be 0, minimum burst position N _mValue be 0; While shift count variable N _cValue be set to 0;

2) begin to receive transmission information series R (x), and calculate its corresponding syndrome S _j(1≤j≤2t);

3) syndrome of previous step gained is judged: if syndrome is not equal to zero, then next step transfers to step 4, otherwise next step jumps to step 9;

4) calculate burst pattern Y _c, S ＇ here _j=S _j(1≤j≤2t) also calculates corresponding burst length l _c

5) compare l _cWith l _mSize: if l _c＜l _m, next step transfers to step 6, otherwise next step jumps to step 7;

6) Y _c, l _c, N _cBe the parameter of the burst pattern of current maximum possible, the value with them overrides Y respectively _m, l _m, N _mValue;

7) judge N _cValue whether greater than code length n: if then next step transfers to step 9; Otherwise transfer to step 8;

8) receiverd polynomial ring shift right 2t-v _Max+ 1 code word size also calculates corresponding syndrome, promptly calculates $S_j^{\′}=\frac{S_j}{{\left(a^j\right)}^{{2t-v}_{\max }+1}}$ , the value with the shift count variable increases 2t-v simultaneously _Max+ 1, next step turns back to step 4 then;

9) according to error pattern Y _mWith errors present N _mValue carry out error correction, export the information series after the error correction simultaneously; After this step is finished, transfer to step 10, judge whether decode procedure finishes: if finish decode procedure; Otherwise next step jumps back to step 1, restarts the decoding of next code vector.

Images