WO2008111824A2

WO2008111824A2 - Method of encoding and decoding data using ldpc code

Info

Publication number: WO2008111824A2
Application number: PCT/KR2008/001495
Authority: WO
Inventors: Ki Hyoung Cho; Jong Sun No; Beom Kyu Shin; Woo Myung Park
Original assignee: Lg Electronics Inc.
Priority date: 2007-03-15
Filing date: 2008-03-17
Publication date: 2008-09-18
Also published as: WO2008111824A3; KR101227514B1; KR20080084179A

Abstract

A method of encoding and decoding data using a low density parity check (LDPC) code is disclosed. The method of encoding data using a low density parity check (LDPC) code includes providing an information bit stream for encoding; and encoding the information bit stream using a parity check matrix which includes a plurality of sub-blocks, wherein at least one of the sub-blocks of the parity check matrix is generated by overlapping at least two sub-blocks.

Description

METHOD OF ENCODINGAND DECODING DATA USING LDPC CODE

[DESCRIPTION] TECHNICAL FIELD

The present invention relates to a method of encoding and decoding data, and more particularly, to a method of encoding data using a low density parity check (LDPC) code.

BACKGROUND ART FIG. 1 illustrates a structure of a mobile communication channel to which embodiments in accordance with the present invention and the related art can be applied. Hereinafter, the structure of the mobile communication channel will be described with reference to FIG. 1. A transmitter undergoes a channel coding procedure to transmit data without loss or distortion through a wireless channel. Examples of the channel coding include convolutional coding, turbo coding, LDPC coding, etc. The data which has undergone the channel coding procedure can be transmitted to the wireless channel as a single symbol which includes several bits. At this time, a scheme of mapping several bits to a single symbol is referred to as "modulation."

The modulated data is converted into a signal for multiple transmission through a multiplexing procedure or a multiple access scheme. Examples of the multiplexing procedure include CDM, TDM, FDM, etc. The signal which has undergone the multiplexing block is changed to a structure suitable for transmission to one or more multi-antennas, and then is transferred to a receiver through the wireless channel. Fading and thermal noise occur in the transmitted data when the data passes through a wireless channel. For this reason, distortion may occur in the received data.

The modulated data is transferred to the receiver through the wireless channel. In this case, fading and thermal noise occur in the received data, whereby distortion may occur therein. The receiver performs a series of procedures of the transmitter in reverse order after receiving the distorted data. The receiver performs demodulation to convert the data mapped to the symbol into a bit stream, undergoes channel decoding, and recovers the distorted data to the original data.

An apparatus of performing the channel coding stores a matrix H or a generation matrix G, wherein the matrix H is a parity check matrix used to generate parity bits to be added to input data (information bits or systematic bits), and the generation matrix G is derived from the matrix H. In other words, the transmitter includes an encoder which generates parity bits through the input data and the matrix H or G. An apparatus of performing channel decoding checks whether the systematic bits are recovered well, through operation of the received data (distorted systematic bits + parity bits) with the matrix H, and performs operation again if the systematic bits are failed to be recovered.

Examples of the modulation include BPSK (Binary Phase Shift Keying), QPSK

(Quadrature Phase Shift Keying), 16-QAM (Quadrature Amplitude Modulation), 64-QAM,

256-QAM, etc. For example, 16-QAM maps the data stream which has undergone channel encoding during modulation to a single symbol in a unit of 4 bits. 16-QAM de-maps the single symbol received through the wireless channel during demodulation to four bits.

Hereinafter, a data retransmission scheme that can be used along with the embodiments of the present invention will be described. There are provided various examples of the data retransmission scheme. Of them, HARQ (Hybrid Automatic Repeat reQuest) scheme will be described below. The HARQ scheme is obtained by combination of FEC (Forward Error Correction) symbol and ARQ (Automatic Repeat reQuest) which is a retransmission scheme in a mobile communication system. According to the ARQ scheme, if an error is detected from data received by the receiver, the receiver requests the transmitter to perform retransmission. Examples of the ARQ scheme include Stop-And-Wait, Selective Repeat, Go-Back-N, etc. depending on the retransmission method.

Hereinafter, LDPC coding will be described. A concept of LDPC coding is as follows.

A linear code can be described with the generation matrix G or the parity check

T matrix H. A characteristic of the linear code is that the Equation of "^c is satisfied for every bit of a codeword 'c'. As one of the linear code, the LDPC code which is recently paid attention to was proposed by Gallager in 1962 for the first time. One of the characteristics of the LDPC coding is that most of elements of the parity check matrix H are

'0' and the number of elements which are not '0' is small compared to the codeword, so that repetitive decoding based on probability is possible. A parity check matrix H for the first proposed LDPC code was defined in a non-systematic form and each row and column of the parity check matrix were designed to equally have a small weight.

In this case, the weight means the number of ' 1 ' included in each row and column.

The LDPC coding scheme has low decoding complexity since a density of elements which are not '0' in the parity check matrix H is low. Further, decoding performance of the LDPC coding is superior to other coding schemes, which is adjacent to the theoretical limit of

Shannon. However, the LDPC coding scheme could not be implemented with the hardware technique at the time of proposal by Gallegar, so that the LDPC coding scheme has not been paid attention to by the people for 30 years. A repetitive decoding scheme using graphs was developed in early 1980's and a couple of decoding algorithms for the LDPC code have been developed using the repetitive decoding scheme. One of them is a sum-product algorithm.

Hereinafter, features of the LDPC coding will be described. The LDPC coding has a superior error correction capability, thereby improving communication speed and capacity. When combined with a multi-input and multi-output (MIMO) scheme, the LDPC coding can be applied to a high speed wireless LAN having a data transmission speed of several hundred Mbit/s, a high speed mobile communication system having a data transmission speed of one Mbit/s or greater for a user moving at a speed of 250 km/h, and an optical communication system having a data transmission speed of 40 Gbit/s or greater. In addition, the LDPC coding can enable quantum encryption communication diminishing the number of retransmission times on a communication path having low quality since transmission quality is improved due to its high error correction capability. Further, data packets having errors can be easily recovered due to the low complexity and superior loss compensation capability of the LDPC coding, so that contents having quality equal to that of TV can be transmitted through the Internet and the mobile communication system. 1OG BASE-T transmission within a range of 100m which has been considered impossible earlier can be realized owing to wide application range and large capacity which are advantages of the LDPC coding. In addition, transmission capacity of a single satellite transmitter having 36 MHz bandwidth can be increased up to 80 Mbit/s which is 1.3 times of usual transmission capacity. Hereinafter, a structured LDPC scheme will be described.

A parity check matrix H is used to use the LDPC code, and includes most of elements 0 and some elements 1. Since the matrix H has a great size of 10⁵ bits or greater, a large sized memory is required to express the matrix H. The structured LDPC scheme expresses the elements of the matrix H used for LDPC encoding and decoding as a sub-block of a certain size as shown in FIG. 2.

In IEEE 802.16e, the sub-block is identified by one integer index to decrease the size of the memory required to store the matrix H. Various kinds of matrixes could be the sub- block. For example, a permutation matrix of a certain size could be the sub-block. If the structured LDPC scheme is used, one integer (i.e., index) is only needed to be stored in a specific memory instead of a certain sized matrix consisting of elements of 1 or 0. Accordingly, the size of the memory required to express the matrix H can be decreased.

For example, if a size of a codeword reflected in the IEEE802.16e standard is 2304 and a coding rate is 2/3, a model matrix used for LDPC encoding/decoding is as shown in FIG. 3. The model matrix means a parity check matrix consisting of at least one sub-block expressed by shift numbers which will be described later. The model matrix can be generated by being extended to the parity check matrix by a method which will be described later.

Accordingly, performing encoding and decoding with a specific model matrix means that encoding and decoding performs with the parity check matrix generated by being extended from the corresponding model matrix.

As shown in FIG 3, the structured LDPC matrix can be expressed by -1, 0 and positive integers, wherein index of -1 represents a zero matrix of a specific size, and index of

0 represents an identity matrix of a specific size. Index of the positive integers excluding -1 and 0 represent shift numbers. If a sub-block is expressed by ' 1,' the corresponding sub-block is one time shifted to a specific direction in the identity matrix.

FIG. 4 illustrates a method of expressing a matrix according to the aforementioned positive integers, i.e., shift numbers. It is assumed that a specific parity check matrix is identified by a structured matrix of 4x4 (i.e., sub-block). In this case, if the specific sub block is identified by 3, the sub-block becomes the matrix of FIG. 4. Hereinafter, LDPC encoding method will be described.

According to a general LDPC encoding method, information bits are encoded by using a generation matrix G derived from an LDPC parity check matrix H. In order to derive the generation matrix G, the parity check matrix H is configured in the form of [ P^τ : I ] by using a Gaussian reduction method. Assuming the number of the information bits is 'k' and a size of an encoded codeword is 'n', the 'P' is a matrix having 'k' number of rows and '(n- k)' number of columns and the T is an identity matrix having 'k' number of rows and

columns.

When the parity check matrix H is represented in the form of [ P^τ : I ], the generation matrix G has the form of [ I : P ]. The information bits of k bits to be encoded can be represented as a matrix 'x' having one row and 'k' number of columns. In this case, the codeword 'c' is represented in the form of the following Equation.

[Equation 1]

<?=xG= [x:xP] In the above Equation, x represents an information part (or a systematic part), and xP

represents a parity part.

Meanwhile, if encoding is performed by the Gaussian Reduction method as above, since a calculation amount is increased, the form of the matrix H is designed in a specific structure so that a method of directly encoding data in the matrix H without deriving the matrix G is used. In other words, if the Equation 1 is multiplied by H^τ by using a feature (i.e., GH^T=0) that a product between the matrix G and a transpose H^τ for the matrix H is equal to 0, the following Equation 2 can be obtained. The codeword 'c' can be obtained by adding a parity bit suitable for the Equation 2 next to the information bit 'x.' [Equation 2]

The aforementioned method will be described with reference to the accompanying drawing.

FIG. 5 is a block diagram illustrating an LDPC encoding method. As shown, if a single parity check matrix exists, information bits [ao, ai, a₂, a₃] generate parity bits [p₀, pi, p₂, p₃] in accordance with Equations suggested in a lower part of

FIG. 5. In FIG. 5, ' ^^¹ represents exclusive 'OR' operation.

As a result of FIG. 5, codewords [ao, a_l5 a₂, a₃, p₀, pi, p₂, p₃] are generated. Encoding may be performed using the parity check matrix as illustrated in FIG 5, or encoding may be performed using the generation matrix G.

Hereinafter, an LDPC decoding method will be described.

Data encoded in a communication system includes noise when passing through a wireless channel of FIG. 1. A receiver represents a decoding procedure of data through a procedure as shown in FIG 6. A decoding block of the receiver obtains an information bit 'x' from a receiving signal c' having the encoded codeword 'c' added with noise by using the feature of cH^T=0. In other words, assuming that the received codeword is c', a value of c'H^T is calculated. As a result, if the value of c'H^T is 0, first k number of bits from c' are determined as the information bit x. If the value of c'H is not 0, c' which satisfies c'H of 0 is searched by using a decoding method such as a sum-product algorithm through a graph, thereby restoring the information bit x.

Hereinafter, a coding rate of LDPC coding will be described.

Generally, when the size of the information bit is k and the size of the codeword which is actually transmitted is n, a coding rate (R) is as follows. [Equation 3] R = k/n

When the matrix H necessary for LDPC encoding and decoding has a row size of m and a column size of n, a coding rate is as follows. [Equation 4]

R = I - m/n

As described above, since the related art LDPC code is encoded and decoded by the matrix H, the structure of the matrix H is very important. In other words, since encoding and decoding performance is greatly affected by the structure of the matrix H, design of the matrix H is more important than anything else.

Hereinafter, cycle 4 which is a problem of the related art will be described. FIG. 7 illustrates a bipartite graph that expresses a parity check matrix. FIG 7 illustrates a bipartite graph that expresses a parity check matrix of the following Equation 5. [Equation 5]

1 0 1 01

H= 0 0 0 0

1 0 1 0_.

The first row of the Equation 5 corresponds to a first check node Cl, its second row corresponds to a second check node C2, and its third row corresponds to a third check node

C3. Also, the first column of the Equation 5 corresponds to a first variable node Vl, its second column corresponds to a second variable node V2, its third column corresponds to a third variable node C3, and its fourth column corresponds to a fourth variable node V4.

If connection of the bipartite graph is as shown, it is regarded that cycle 4 exists. In other words, if a connection line is drawn from one node to return to the node through four lines, it is regarded that cycle 4 exists.

If cycle 4 exists as shown, a problem may occur in performing LDPC decoding. In other words, when LDPC decoding is performed, an incorrect receiving signal should be propagated by a belief propagation (BP) scheme. In this case, if cycle 4 exists, propagation of the incorrect signal is not performed, whereby performance of LDPC decoding is degraded.

DISCLOSURE OF THE INVENTION

Accordingly, the present invention is suggested to substantially obviate one or more problems due to limitations and disadvantages of the related art. An object of the present invention is to provide a method of encoding data using a low density parity check (LDPC) code, of which performance can be improved.

Another object of the present invention is to provide a method of encoding data using a low density parity check (LDPC) code, which can easily control cycle features of a parity check matrix.

To achieve these objects and other advantages and in accordance with the purpose of the invention, as embodied and broadly described herein, a method of encoding data using a low density parity check (LDPC) code includes providing an information bit stream for encoding; and encoding the information bit stream using a parity check matrix which includes a plurality of sub-blocks, wherein at least one of the sub-blocks of the parity check matrix is generated by overlapping at least two sub-blocks.

In another aspect of the present invention, a method of decoding data using a low density parity check (LDPC) code includes receiving a codeword encoded by a parity check matrix which includes a plurality of sub-blocks; and decoding the codeword using the parity check matrix, wherein at least one of the sub-blocks of the parity check matrix is generated by overlapping at least two sub-blocks.

The embodiment according to the present invention suggests a method of configuring a new model matrix by overlapping a first model matrix with a second model matrix. Preferably, these model matrixes are extended to the parity check matrix and used for LDPC encoding and decoding. There is no limitation in the number of model matrixes which undergo overlap. More preferably, the first model matrix is a model matrix which is previously given. For example, it is preferable that the first model matrix is a model matrix suggested in the IEEE 802.16e standard. The model matrixes which include the first model matrix and the second model matrix are extended by an identity matrix of a specific size. In other words, the model matrixes are extended by the identity matrix of a specific size to configure a parity check matrix. Preferably, the specific size is z*z.

At least one index, which configures the second model matrix, i.e., a shift number is preferably determined by a size 'z' of a matrix used when the model matrix is extended to the parity check matrix.

Furthermore, at least one index, which configures the second model matrix, i.e., a shift number is preferably determined by the relation between the index constituting the first model matrix and the index constituting the second model matrix. For example, the index of the second model matrix can be determined by the difference between the index constituting the first model matrix and the index constituting the second model matrix.

Furthermore, if a third model matrix additionally undergoes overlap, index of the third model matrix is preferably determined by the relation among the indexes of the first, second and third model matrixes. For example, the index of the third model matrix can be determined by the difference between the index of the first model matrix and the index of the second model matrix.

BRIEF DESCRIPTION OF THE DRAWINGS FIG. 1 illustrates a structure of a mobile communication channel to which embodiment according to the present invention and the related art are applied;

FIG. 2 illustrates a concept of a sub-block having a predetermined size on a parity check matrix;

FIG. 3 illustrates an example of a model matrix used for LDPC encoding/decoding; FIG. 4 illustrates a method of expressing a matrix depending on indexes, i.e., shift numbers;

FIG. 5 is a block diagram illustrating an LDPC encoding method; FIG. 6 is a block diagram illustrating an LDPC decoding method; FIG. 7 illustrates a bipartite graph that expresses a parity check matrix; FIG. 8 illustrates an example of a model matrix which undergoes overlap;

FIG. 9 illustrates another example of a model matrix which undergoes overlap; FIG. 1OA illustrates a parity check matrix that expresses a sub-block when a shift number of the sub-block is greater than 0;

FIG. 1OB illustrates location of an element having a weight in the sub-block of FIG 1 OA, which is expressed by Equation;

FIG 11 is a block diagram illustrating a case where girth 4 is generated within a sub- block;

FIG. 12 is a block diagram illustrating another case where girth 4 is generated within a sub-block; 495

FIG. 13 is a block diagram illustrating another case where girth 4 is generated within different sub-blocks;

FIG. 14 is a block diagram illustrating another case where girth 4 is generated within two sub-blocks which exist in a vertical direction; FIG. 15 to FIG. 18 illustrate examples of matrixes generated by overlapping two matrixes in accordance with embodiments of the present invention; and

FIG. 19A to FIG. 21C are block diagrams illustrating examples of model matrixes according to embodiments of the present invention.

BEST MODE FOR CARRYING OUT THE INVENTION

Hereinafter, structures, operations, and advantages of the present invention will be understood readily by the preferred embodiments of the present invention, examples of which are illustrated in the accompanying drawings.

The embodiment of the present invention suggests that at least two model matrixes are overlapped with each other.

FIG. 8 illustrates an example of a model matrix which undergoes overlap. Also, FIG. 9 illustrates another example of a model matrix which undergoes overlap.

The embodiment of the present invention suggests combination of the model matrixes illustrated in FIG 8 and FIG. 9. When the model matrixes are overlapped with each other, the following advantages can be obtained.

When two model matrixes are overlapped with each other, a first model matrix will be referred to as a basic matrix and a second model matrix will be referred to as an overlap matrix. In this case, the basic matrix could be a model matrix previously suggested as illustrated in FIG. 8. When the basic matrix is referred to as H_b and the overlap matrix is referred to as H_s, a model matrix H_r is newly configured by overlap. [Equation 6]

H_r=H_b+H_s The basic matrix, the overlap matrix, and the newly configured model matrix satisfy the relation of the above Equation 6.

In this case, a codeword encoded by the matrix H_r satisfies the following Equation. [Equation 7]

When a model matrix is configured by using the basic matrix and at least one overlap matrix, a new model matrix having advantages of the basic matrix can be configured. The overlap matrix can have weights in given regions only as illustrated in FIG 9. In this case, in addition to the basic matrix having regular features, a model matrix having irregular features can be configured. In other words, a regular LDPC parity check matrix can be converted into an irregular LDPC parity check matrix by using the overlap matrix.

When performance of irregular LDPC code is evaluated in accordance with degree distribution, it is known that performance becomes better as maximum degree of a variable node becomes great. Also, it is known that performance of the irregular LDPC code is good if a rate occupied by maximum degree is greater than a certain rate.

Since the overlap matrix is used in the embodiment of the present invention, the embodiment of the present invention is advantageous in that maximum degree can freely be adjusted by adjusting location of index values which are overlapped. Although it is advantageous as aforementioned when the model matrixes are overlapped with each other, a problem may occur in that the newly configured model matrix has a structure of cycle 4.

For example, a weight of row or column for the sub-block constituting the conventional model matrix is ' 1' as illustrated in FIG. 4. In this case, the structure of cycle 4 cannot exist within one sub-block. However, if the model matrixes are overlapped with each other, the structure of cycle 4 may exist even within one sub-block. Also, even though the structure of cycle 4 does not exist in the model matrix which is previously suggested, the structure of cycle 4 may newly be generated by overlap.

The embodiment of the present invention suggests a method of preventing cycle 4 from being generated when model matrixes are overlapped with each other to configure a new model matrix. In other words, the model matrix according to the embodiment of the present invention or the parity check matrix generated from the model matrix according to the embodiment of the present invention does not have the structure of cycle 4. Namely, girth of the model matrix or the parity check matrix according to the embodiment of the present invention exceeds 4. Namely, the model matrix or the parity check matrix according to the embodiment of the present invention has girth greater than 6.

"Girth" means a cycle having the shortest length on the bipartite graph. In other words, the smallest cycle existing in one model matrix or parity check matrix is referred to as

"girth." If girth exceeds 4, a structure of cycle 4 does not exist in a corresponding model matrix or parity check matrix. Accordingly, it is possible to obtain improved performance during LDPC decoding.

Hereinafter, a method of preventing a structure of cycle 4 within one sub-block, a method of preventing a structure of cycle 4 within a plurality of sub-blocks, and a method of preventing girth 4 from being generated within one sub-block will be described. If the overlap scheme is not used, degree is determined as ' 1 ' within one sub-block having a shift number greater than '0.' Accordingly, the relation among different sub-blocks has been considered when girth is calculated. However, if overlap is used, sub-blocks having a shift number greater than '0' may be additionally provided. In other words, degree may be greater than '2' within one sub-block.

If two sub-blocks are additionally provided, it is checked whether girth 4 can be generated. Also, a method of mathematically expressing location of an element having a weight within one sub-block will be described.

FIG. 1OA illustrates a result of a sub-block expressed by a parity check matrix when a shift number of the sub-block is greater than 0. Also, FIG. 1OB illustrates location of an element having a weight in the sub-block of FIG. 1OA, which is expressed by an Equation.

As shown in a left side of FIG. 1OA, a shift number could be '0' and a size of one sub-block (otherwise referred to as 'z factor') could be '6.' In other words, one sub-block could be extended to a matrix of 6*6. In this case, the corresponding sub-block is extended to a parity check matrix as expressed in a left side of FIG 1OA. The corresponding sub-block is expressed as illustrated in a left side of FIG. 1OB. In other words, when the shift number is '0' and the size of the sub-block is '6,' location of an element having a weight in the corresponding sub-block is expressed by [x, x]. As shown in a right side of FIG. 1OA, when a shift number could be '3' and a size of one sub-block (otherwise referred to as 'z factor') could be '6,' the corresponding sub-block is expressed as illustrated in a right side of FIG. 1OB.

In other words, when the shift number is '3' and the size of the sub-block is '6,' location of an element having a weight in the corresponding sub-block is expressed by [x, (x- 3) mod 6], wherein 'mod' represents the modulo operation. In short, when the shift number 'ai' and the size of the sub-block is 'z,' an x coordinate value of the element having a weight in the corresponding sub-block is 'x' and its y coordinate value is '(x-aj) mod z.'

Hereinafter, the case where girth 4 is generated within one sub-block will be described.

When a first model matrix illustrated in the following Equation 8a and a second model matrix illustrated in the following Equation 8b are overlapped with each other, a new model matrix illustrated in the following Equation 8c can be obtained. In other words, the Equation 8a could be referred to as the first model matrix, and the Equation 8b could be referred to as the second model matrix. In this case, the first shift number (a{) of the first model matrix is overlapped with the second shift number (a₂) of the second model matrix. In other words, the sub-block corresponding to the first shift number is overlapped with the sub- block corresponding to the second shift number.

However, since the element having a weight in the second model matrix is a₂ only, the shift number x of the first model matrix is not overlapped with the shift number of the second model matrix.

[Equation 8a]

[Equation 8b] - 1 - 1 - 1

- 1 α₂ - 1

- 1 - 1 - 1

Equation 8c]

The Equation 8c can be expressed as illustrated in FIG 11. FIG. 11 is a block diagram illustrating the case where girth 4 is generated within one sub-block.

As shown, an element 1101 represents an element generated in accordance with the first shift number aj. Also, an element 1102 represents an element generated in accordance with the second shift number a₂. To generate girth 4, an element having a weight should be located in the form of quadrangle. Accordingly, an element 1103 is required. It is assumed that the element 1103 is shifted from the element 1102 by -a₂ + a^ Since an element 1104 should have the same x coordinate value as that of the element 1103, its x coordinate value is determined as '(x-a₂+a_!) mod z.' In this case, since the element 1104 is generated in accordance with the shift number a₂, its y coordinate value is determined as '(x-a₂+a_!-a₂) mod z.'

If girth 4 is generated within one sub-block, the case where y coordinate value of the element 1101 and y coordinate value of the element 1104 are the same as each other should exist. Whether such a case exists will be explained by the following Equation 9. [Equation 9] 495

(a₁-a₂ ⁺a_l-a₂) mod z=0 (2α_r2α₂) mod z=0

In short, when (2a_r2_a2) mod z=0 which is the result of the Equation 9 is satisfied, girth 4 is generated. Accordingly, when the first shift number a_\ corresponding to the first model matrix and the second shift number a₂ corresponding to the second model matrix are overlapped with each other, girth 4 is not generated if 2(ai-a₂) mod z is not satisfied.

If z is an odd number, the Equation 9 cannot be achieved regardless of aj and a₂. In other words, if z is an odd number, the structure of FIG. 12 cannot exist regardless of values of a_t and a₂.

However, if z is an even number, it is preferable that a multiple of z/2 is not obtained as the result of ai-a₂. For example, in case of z=10, it is preferable to determine aj.a₂ so as not to obtain 5, 10, 15, and 20. If a_\ has been already determined, it is preferable that the value of a₂ is adjusted so as not to satisfy the Equation 9.

If overlap is made in accordance with the embodiment of the present invention, there is no limitation in the number of model matrixes which undergo overlap. Hereinafter, a method of preventing girth 4 when three model matrixes are overlapped with one another will be described. FIG. 12 is a block diagram illustrating the case where girth 4 is generated within one sub-block. As shown in FIG. 12, an element 1201 represents an element generated in accordance with the second shift number a₂. Also, an element 1202 represents an element generated in accordance with the third shift number a₃.

To generate girth 4, an element having a weight should be located in the form of quadrangle. Accordingly, an element 1203 is required. It is assumed that the element 1203 is shifted from the element 1202 by a₂ - a₃. Since an element 1204 should have the same x coordinate value as that of the element 1203, its x coordinate value is determined as '(x+a₂-a₃) mod z.' In this case, if it is assumed that the element 1204 is generated in accordance with the first shift number ai, y coordinate value of the element 1204 is determined as '(x-a_!+a₂-a₃) mod z.'

To generate girth 4 by overlapping three shift numbers, the case where y coordinate value of the element 1201 and y coordinate value of the element 1204 are the same as each other should exist. Whether such a case exists will be explained by the following Equation 10.

[Equation 10]

(a_Λ+aΛ mod z=2a₀ mod z

(a, +a₃) mod z=2a₂ mod z

In short, if which is the result of the

Equation 10 is satisfied, girth 4 is generated. Accordingly, when the three model matrixes are overlapped with one another, girth 4 is generated if a sum of the first shift number and the third shift number is twice of the second shift number. If three model matrixes are overlapped with one another, the first to third shift numbers are preferably determined so that modulo operation for the sum result of the first shift number and the third shift number is different from that for twice of the second shift number. At least any one of the first to third shift numbers could be a value which has been already determined. In this case, it is preferable to determine a shift number which does not satisfy the Equation 10, by adjusting the other values.

Hereinafter, the case where girth 4 is generated among different sub-blocks will be described.

When a first model matrix illustrated in the following Equation 11a and a second model matrix illustrated in the following Equation lib are overlapped with each other, a new model matrix illustrated in the following Equation lie can be obtained. In other words, the Equation 11a could be referred to as the first model matrix, and the Equation lib could be referred to as the second model matrix. In this case, the first shift number (ai) of the first model matrix is overlapped with the second shift number (a₂) of the second model matrix. In other words, the sub-block corresponding to the first shift number is overlapped with the sub-block corresponding to the second shift number. Also, the fourth shift number fø) of the first model matrix is overlapped with the third shift number (a₃) of the second model matrix. Each of the Equations 11a and lib does not have the structure of girth 4. However, if overlap is made, each of the Equations 11a and lib can have the structure of girth 4.

[Equation Ha]

- 1 - 1 - 1 - 1 a_x a_$ - 1 - 1 - 1

[Equation lib] - 1 - 1 - 1 - 1 a₂ a₃ - 1 - 1 - 1

[Equation lie]

The Equation lie can be expressed as illustrated in FIG. 13. FIG. 13 is a block diagram illustrating the case where girth 4 is generated within different sub-blocks.

As shown, an element 1301 represents an element generated in accordance with the first shift number ai. Also, an element 1302 represents an element generated in accordance with the second shift number a₂. To generate girth 4, an element having a weight should be located in the form of quadrangle. Accordingly, an element 1303 is required. It is assumed that the element 1303 is shifted from the element 1302 by -a₂ + a₃. Since an element 1304 should have the same x coordinate value as that of the element 1303, its x coordinate value is determined as '(x-a₂+a₃) mod z.' In this case, if it is assumed that the element 1304 is generated in accordance with the shift number a₄, y coordinate value of the element 1304 is determined as '(x-a₂+ai-a₄) mod z.¹

If girth 4 is generated, the case where y coordinate value of the element 1301 and y coordinate value of the element 1304 are the same as each other should exist. Whether such a case exists will be explained by the following Equation 12.

[Equation 12] (x-aι)vcLθά z

z

(a_l-a₂+a₃-a₄) mod, z⁼0

(^r₁ -α₂) mod z=(α₄-α₃) mod z

In short, if (a₁-a₂) mod z=(<2₄-α₃) mod z which is the result of the Equation 12 is satisfied, girth 4 is generated.

Accordingly, when the model matrixes are overlapped with one another, the shift numbers are preferably determined so as not to satisfy the Equation 12. At least any one of the first to fourth shift numbers could be a value which has been already determined. In this case, it is preferable to determine a shift number which does not satisfy the Equation 12, by adjusting the other values.

In the aforementioned example, two shift numbers are overlapped with each other within one sub-block. There is no limitation in the number of shift numbers which undergo overlap. For example, to identify whether girth 4 is generated among different sub-blocks as three shift numbers are overlapped with one another within one sub-block, it is preferable to check whether girth 4 exists for two of the three shift numbers, as described above.

FIG. 13 relates to the case where girth 4 is generated within two sub-blocks which exist in a horizontal direction. Hereinafter, the case where girth 4 is generated within two sub-blocks which exist in a vertical direction will be described. When a first model matrix illustrated in the following Equation 13a and a second model matrix illustrated in the following Equation 13b are overlapped with each other, a new model matrix illustrated in the following Equation 13c can be obtained. In other words, the Equation 13a could be referred to as the first model matrix, and the Equation 13b could be referred to as the second model matrix. In this case, the first shift number (a^ of the first model matrix is overlapped with the second shift number (a₂) of the second model matrix. In other words, the sub-block corresponding to the first shift number is overlapped with the sub-block corresponding to the second shift number. Also, the fourth shift number (a*) of the first model matrix is overlapped with the third shift number (a₃) of the second model matrix.

Each of the Equations 13a and 13b does not have the structure of girth 4. However, if overlap is made, each of the Equations 13a and 13b can have the structure of girth 4.

[Equation 13 a]

- 1 a_λ - 1 - 1 a₃ - 1 - 1 - 1 - 1

[Equation 13b]

- 1 a₂ - 1 - 1 a₄ - 1 -1 - 1 - 1

[Equation 13 c]

The Equation 13c can be expressed as illustrated in FIG. 14.

FIG 14 is a block diagram illustrating the case where girth 4 is generated within two sub-blocks which exist in a vertical direction.

As shown, an element 1401 represents an element generated in accordance with the first shift number a_\. Also, an element 1402 represents an element generated in accordance with the third shift number a₃. To generate girth 4, an element having a weight should be located in the form of quadrangle. Accordingly, an element 1403 is required. It is assumed that the element 1403 is shifted from the element 1402 by -a₃ + a₄. Since an element 1404 should have the same x coordinate value as that of the element 1403, its x coordinate value is determined as '(x-a₃+a₄) mod z.' In this case, if it is assumed that the element 1404 is generated in accordance with the shift number a₂, y coordinate value of the element 1404 is determined as '(x-a₃+a₄-a₂) mod z.'

If girth 4 is generated, the case where y coordinate value of the element 1401 and y coordinate value of the element 1404 are the same as each other should exist. Whether such a case exists will be explained by the following Equation 14.

[Equation 14]

(α₁-α₃+α₂-«₄) mod z=0

Ca₁-^₃) mod Z=(U₄-Ci₂) mod z

The result of the Equation 14 is the same as that obtained by exchanging parameters of the Equation 12. Accordingly, girth 4 can be prevented from being generated in the same manner as that checked in the example of FIG. 13.

Hereinafter, the example that two matrixes are overlapped with each other in accordance with the method suggested in the embodiment of the present invention will be described.

FIG. 15 illustrates an example of two matrixes overlapped in accordance with the method suggested in the embodiment of the present invention. In FIG. 15, two shift numbers are overlapped with each other within one sub-block. As shown, the model matrix of FIG. 15 includes has a column weight of maximum 6.

FIG. 16 illustrates another example of two matrixes overlapped in accordance with the method suggested in the embodiment of the present invention. In FIG. 16, two shift numbers are overlapped with each other within two sub-blocks. As shown, the model matrix of FIG 16 includes has a column weight of maximum 8. FIG. 17 illustrates another example of two matrixes overlapped in accordance with the method suggested in the embodiment of the present invention. In FIG. 17, two shift numbers are overlapped with each other within two sub-blocks. As shown, the model matrix of FIG. 17 includes has a column weight of maximum 8.

FIG. 18 illustrates another example of two matrixes overlapped in accordance with the method suggested in the embodiment of the present invention. In FIG. 18, two shift numbers are overlapped with each other within four sub-blocks. In the aforementioned examples, girth 4 is not generated.

Hereinafter, the result of the embodiment according to the present invention will be identified in accordance with one of various methods of checking performance of an LDPC code.

Degree distribution suggested by Richardson et al. can be used to check performance of an LDPC code. Hereinafter, degree distribution of a parity check matrix adopted by the IEEE 802.16e standard will be calculated, and degree distribution of matrixes overlapped in accordance with the embodiment of the present invention will be compared. The parity check matrix adopted by the IEEE 802.16e standard has degree distribution as follows. [Equation 15]

2XIlXz ₌ 22

² 2XllXz+3X8Xz+6X5Xz 76

3X8Xz ₌ 24 3 2XllXz+3X8Xz+6X5Xz 76

6X5Xz ₌ 30

⁶ 2XllXz+3X8Xz+6X5Xz 76

represents a rate of a column having a weight of 2, represents a rate of

column having a weight of 3 , and represents a rate of a column having a weight of 6.

The matrix of FIG 15 according to the embodiment of the present invention has degree distribution as follows. [Equation 16]

λ - 2XIlXz _ 22

² 2X11XZ+3X8XZ+6X4XZ+7X1XZ 77

λ^δ~ 2XllXz+3X8Xz+6X4Xz+7XlXz ^~ 77 ^~3l2%

The matrix of FIG. 18 according to the embodiment of the present invention has degree distribution as follows. [Equation 17]

8X2Xz ₌ 22_

L= . ...:.. :..... =≡r =20.0%

2X ll Xz+3 X8Xz+6X3 Xz.+8 X2Xz 80

It is noted from the Equation 16 and Equation 17 that degree distribution can be adjusted when overlap is made in accordance with the embodiment of the present invention.

For example, if degree distribution is adjusted using overlap, a model matrix approximate to optimized values, which have been suggested by Richardson et al., can be configured.

Hereinafter, the following Equation 18 illustrates an example that a model matrix with maximum degree distribution of 8 is adjusted to approximate to optimized values suggested by Richardson et al. [Equation 18] 2X12Xz ₌ 24 ₌

² 2X12Xz+3X8Xz+8X4Xz 80 ^v '

^{3X8Xz 24} =30.0% (28.4%)

³ 2X12Xz+3X8Xz+8X4Xz 80

8X4Xz = 32 =40.0% (41.6%)

⁸ 2X12Xz+3X8Xz+8X4Xz. 80 ^{v ;}

In the Equation 18, numerical values in parenthesis represent optimized values suggested by Richardson et al.

It is noted that degree distribution of the model matrix is approximate to the optimized values suggested by Richardson et al. when the model matrix has degree distribution suggested in the Equation 18.

Hereinafter, the following Equation 19 illustrates an example that a model matrix with maximum degree distribution of 12 is adjusted to approximate to the optimized values suggested by Richardson et al. [Equation 19]

% =0.0% (l.i%) 4

^λ5~2XllXz+3X9Xz+5XlXz+12X3Xz^~90^{~5"6% (5"5%)} λ₈=o.α% (i.5%)

λ₁₀==O.O% (1.3%)

In the Equation 19, numerical values in parenthesis represent optimized values suggested by Richardson et al.

It is noted that degree distribution of the model matrix is approximate to the optimized values suggested by Richardson et al. when the model matrix has degree distribution suggested in the Equation 19.

Hereinafter, the following Equation 20 illustrates an example that a model matrix with maximum degree distribution of 15 is adjusted to approximate to the optimized values suggested by Richardson et al. [Equation 20]

λ^r 2X12Xz+3X8Xz+4XlXz+5X2Xz+l5X2Xz ^~ 90 ^{~U" %} ^ ²^ λ_τ=0.0% (1.6%)

λ_M=0.0% (0.5%)

In the Equation 20, numerical values in parenthesis represent optimized values suggested by Richardson et al.

It is noted that degree distribution of the model matrix is approximate to the optimized values suggested by Richardson et al. when the model matrix has degree distribution suggested in the Equation 20.

Examples of the model matrixes according to degree distribution of the aforementioned Equations 18 to 20 will be described with reference to FIG. 19Ato FIG. 21C. FIG 19A to FIG. 21 C are block diagrams illustrating examples of the model matrixes according to the embodiment of the present invention.

When the model matrixes of FIG 19A and FIG 19B are overlapped with each other, the model matrix which satisfies the conditions of the Equation 18 to 20 can be obtained. FIG 19A illustrates the first model matrix, and FIG. 19B illustrates the second model matrix.

When the model matrixes of FIG 2OA and FIG 2OB are overlapped with each other, the model matrix which satisfies the conditions of the Equation 18 to 20 can be obtained. FIG 2OA illustrates the first model matrix, and FIG 2OB illustrates the second model matrix.

When the model matrixes of FIG 21 A and FIG. 2 IB are overlapped with each other, the model matrix which satisfies the conditions of the Equation 18 to 20 can be obtained. FIG. 21 A illustrates the first model matrix, and FIG. 21 B illustrates the second model matrix. If encoding is performed using the parity check matrixes generated by the embodiments of the present invention, input source data can be encoded using a generation matrix. In other words, input source data S₁ X_k of k bits is encoded in the generation matrix and becomes a codeword X₁X_k of n bits. The codeword x has a configuration of x=[s p]-[so, S₁, ..., s_k-1, po, P₁, ... , p_m-i], wherein (p₀, pi, ... , p_m-0 represents parity check bits and (s₀, S₁, ... , S_k-O represents systematic bits.

However, the encoding method using the generation matrix is very complicated.

Accordingly, in order to reduce such complexity, it is preferable to directly encode input source data by using the parity check matrix H without using the generation matrix. In other words, since x=[s p], H ^• x = H ^• [s p] = 0 can be obtained using a feature of H ^• x = 0.

Since the parity check bit p can be obtained from H ^• x = H - [S pJ = O, the codeword x=[s p] can be obtained.

A method of performing decoding using the parity check matrix generated in accordance with the embodiments of the present invention will be described. A decoder obtains an information bit 'x' from the codeword, which is the result of encoding, by using a feature of cH^τ=0. In other words, assuming that the codeword received from the receiver is c', a value of c'H^T is calculated. As a result, if the value of c'H^T is 0, first k number of bits from c' are determined as the decoded information bit. If the value of c'H^T is not 0, c' which satisfies c'H^T of 0 is searched by using a sum-product algorithm through a graph and a belief propagation algorithm, thereby restoring the information bit x.

If the parity check matrix is configured in accordance with the present invention, it is advantageous in that, although overlap is used, the model matrix having excellent cycle features can newly be configured. It will be apparent to those skilled in the art that the present invention can be embodied in other specific forms without departing from the spirit and essential characteristics of the invention. Thus, the above embodiments are to be considered in all respects as illustrative and not restrictive. The scope of the invention should be determined by reasonable interpretation of the appended claims and all change which comes within the equivalent scope of the invention are included in the scope of the invention.

INDUSTRIAL APPLICABILITY

The present invention can be applied to every field where encoding and decoding are used, as well as a wireless communication system such as a mobile communication system or a wireless Internet system.

Claims

[CLAIMS]

1. A method of encoding data using a low density parity check (LDPC) code, the method comprising: providing an information bit stream for encoding; and encoding the information bit stream using a parity check matrix which includes a plurality of sub-blocks, wherein at least one of the sub-blocks of the parity check matrix is generated by overlapping at least two sub-blocks.

2. The method of claim 1, wherein a first sub-block among the at least two sub- blocks which undergo overlap is included in a first model matrix, and a second sub-block among the at least two sub-blocks is included in a second model matrix.

3. The method of claim 2, wherein a shift number of the second sub-block is determined by a shift number of the first sub-block and sizes of the first and second sub- blocks.

4. The method of claim 2, wherein a shift number of the second sub-block is determined by a result of a mathematical operation using a shift number of the first sub-block and a shift number of the second sub-block and sizes of the first and second sub-blocks.

5. The method of claim 2, wherein a shift number of the second sub-block is determined by modulo operation for sizes of the first and second sub-blocks and a result value obtained through mathematical operation using the shift number of the first sub-block and the shift number of the second sub-block.

6. The method of claim 2, wherein when a shift number of the first sub-block is ai, a shift number of the second sub-block is a₂, and sizes of the first and second sub-blocks are z* z, ai and a₂ are determined to satisfy the following Equation:

(2α₁-2α₂)mod z≠O

7. The method of claim 2, wherein when a shift number of the first sub-block is ai, a shift number of the second sub-block is a₂, a shift number of the third sub-block is a₃, a shift number of the fourth sub-block is a₄, and the first, second, third, and fourth sub-blocks are z* z matrixes, a_l5 a₂, a₃ and a₄ are determined to satisfy the following Equation:

Ca₁-^₂) mod z≠(a₄-a₃) mod z

8. The method of claim 2, wherein when a shift number of the first sub-block is ai, a shift number of the second sub-block is a₂, a shift number of the third sub-block is a₃, a shift number of the fourth sub-block is a₄, and the first, second, third, and fourth sub-blocks are z* z matrixes, a_ls a₂, a₃ and a_t are determined to satisfy the following Equation:

Ca₁-^₂) mod z≠(a₃-a₄) mod z

9. The method of claim 1 , wherein the parity check matrix has girth greater than 6.

10. A method of decoding data using a low density parity check (LDPC) code, the method comprising: receiving a codeword encoded by a parity check matrix which includes a plurality of sub-blocks; and decoding the codeword using the parity check matrix, wherein at least one of the sub-blocks of the parity check matrix is generated by overlapping at least two sub-blocks.

11. The method of claim 10, wherein a first sub-block among the at least two sub- blocks which undergo overlap is included in a first model matrix, and a second sub-block among the at least two sub-blocks is included in a second model matrix.

12. The method of claim 11, wherein a shift number of the second sub-block is determined by a shift number of the first sub-block and sizes of the first and second sub- blocks.