[ieee 2006 international multi-symposiums on computer and computational sciences (imsccs) -...

Constructions of Euclidean Geometry LDPC Codes and Performance Analysis in MIMO-OFDM System

Fangni Chen, Lei Shen, and Shiju LiDept. of ISEE

Zhejiang University [email protected]; [email protected]; [email protected]

Abstract

In this paper, we introduce the history and basic characteristic of Low density parity check codes, then emphasize on the construction of Euclidean geometry LDPC code and take it as the channel code into IEEE802.16d MIMO-OFDM system simulation and compare it with RS-CC codes. Finally we makes a conclusion that Euclidean geometry LDPC codes are strong competitors to FEC codes for error control in communication systems.

1. Introduction

Since LDPC (Low Density Parity Check) codes were first discovered by Gallager [l] in 1962, they haven’t received enough recognition for thirty years. But MacKay and Neal [2] rediscovered the excellent LDPC codes in 1996. Since then the researches on LDPC codes are increasing and hot.

LDPC codes have the following properties: In binary input AWGN channel, the error performance of the best irregular LDPC code (with length 107, rate R=1/2) is 0.04dB away from the Shannon limit, it is outperform all the Turbo codes ever known [3]; LDPC codes are flexible and have lower error floors; they can be described simply and validated for strict theoretic analysis; The decoding complexity is lower than Turbo codes.

This paper starts with the structure of LDPC codes. Then emphasize on the constructions of Euclidean geometry LDPC code and take it as the channel code into IEEE802.16d MIMO-OFDM system simulation and compared it with RS-CC codes.

2. The structure of LDPC codes

LDPC codes are linear block codes, they can be described using the results of the parity equations

HCT=0, the matrix H is the parity check matrix of the code C. An LDPC code of length n is defined by a J *n parity check matrix H with the following structural properties: (1) each row consists of “ones”; (2) eachcolumn consists of “ones”; (3) the number of “ones” in common between any two columns is no greater than one; and (4) both and are small compared to nand J (the number of rows in H).

An LDPC codes can also be expressed by the Bi-Partite graph or the Tanner graph [4], which is corresponding with a parity check matrix. Figure 1 shows the parity check matrix H and the Tanner graph of an (6, 2, 3) LDPC code. The Tanner graph consists of two node sets v and c: v=v1, v2, … v6 called variable nodes or code bit nodes; c=c1,c2,c3,,c4 called parity check nodes.

1 1 0 1 0 00 0 1 1 1 01 0 0 0 1 10 1 1 0 0 1

H

1v 2v 3v 4v 5v 6v

1c 2c 3c 4c

Figure 1. LDPC code check matrix and Bi-Partite graph

A line in the Tanner graph which connects a variable node vn and a parity check node cm if and only if Hmn=1, is denoted (vn, cm). From Figure 1 we can see a cycle of length 6 with bold black lines. For these short cycles of length 4 or 6 are adverse for the widely

Proceedings of the First International Multi-Symposiums on Computer and Computational Sciences (IMSCCS'06) 0-7695-2581-4/06 $20.00 © 2006 IEEE

used and the best LDPC decoding algorithm BP(Belief Propagation) algorithm, we should reduce the appearance of these short cycles when we do encoding.

Euclidean geometry LDPC codes introduced in the next sections are so-called good codes which don’t have cycles of length 4.

3. Euclidean geometry LDPC codes

Although LDPC codes have outstanding error performance and many methods have been found for constructing these codes, such as Gallager construction, MacKay construction, Xiaoyu Hu’s PEG construction and so on. But these methods are all random constructions, and good LDPC codes that have been constructed are largely computer generated, especially long codes. In 2000, Yu Kou and Shu Lin provided a new algebraic construction of LDPC codes base on finite geometry. Codes of this method are either cyclic or quasi-cyclic, therefore, their encoding is very simple because it can be implemented with linear feedback shift registers based on their generator polynomials. Euclidean geometry (EG) and projective geometry (PG) are two family of finite geometry, this paper emphasized on the former one.

3.1. EG-LDPC codes

Let ( , 2 )sEG m be an m-dimensional Euclidean

geometry over the Galois field (2 )sGF . This geometry

consists of 2 points, each point is simply an m-tuple over

ms

(2 )sGF . The all-zero m-tuple point is called the

origin. There are lines

in

( 1)2 (2 1) /(2m s ms s

Let i and j be two linearly independent points in ( , 2 )sEG m . Then the collection of the following

points: form a

line in

{ } { : (2i j i j sGF )}

( , 2 )sEG m that passes through the point i .Lines do not have any point in common and hence they are parallel. ( , 2 )sEG m consists of J lines do not pass

through the origin, ( 1)(2 1)(2 1) /(2 1)m s ms sJ .

Let be a matrix over GF(2) whose rows are the incidence vectors of all the lines in

(1) ( , 0, )EGH m s

( , 2 )sEG m that do not pass through the origin and whose columns correspond to the nonorigin points in ( , 2 )sEG m . The matrix (1) ( , 0, )EGH m s has these

parameters: row weight 2s , column weight

(2 1) /(2 1) 1ms s , density which is small as m 2 and s 2. Therefore,

is a low-density matrix and the null space

of it gives a regular LDPC code of length

2 /(2 1)s msr

(1) ( , 0, )EGH m s

2 1msn ,called the type-I m-dimensional EG-LDPC code, denoted . This code is cyclic and its

minimum distance is at least . The number of parity bits is equal to the degree of its generator polynomial, especially for type-I 2-dimensional EG-LDPC code, its number of parity bits is

(1) ( , 0, )EGC m s

(2 1) /(2 1)ms s

3 1sn k .Example 1: Construct an EG-LDPC 255 175

code.1)

( , 2 )sEG m , and 2s points in every line. There are

lines intersecting at one point.

Let be the extension field of

(2 1) /(2 1) 1ms s

1) Firstly, we know EG-LDPC 255 175 code

is based on Euclidean geometry , i.e. m=2,

s=4. The points in are correspond to the

elements in , so we can first

construct the Galois field , which can be generated by its primitive polynomial

4(2, 2 )EG4(2, 2 )EG

8(2 ) (2 )msGF GF8(2 )GF

(2 )msGF (2 )sGF and can be regarded as a realization of Euclidean geometry

( , 2 )sEG m .Therefore, the elements in may be

regarded as the points in

(2 )msGF

2ms ( , 2 )sEG m .The element

zero in is the origin of(2 )msGF ( , 2 )sEG m . Let be a

primitive element of , then the nonzero

elements in

(2 )msGF

(2 )msGF 0 1 2 21, , , ...,ms

2 3 4( ) 1 8p x x x x x .2) Secondly, we should find a line which pass

through one point but do not pass through the origin. Known that the points of a line that pass through the point i have the following form: 2 form

the (2 nonorigin points in1)ms ( , 2 )sEG m . { } { : (i j i j sGF 2 )} .


Let i=2, so we should find out the points which satisfy 2{ }ja , let

(2 1) /(2 1) 17ms s,

so2 14{0,1, , , ..., } .

Then we could get a line consists of 16 points. 3) Thirdly, we should find the incidence vector

from the line. The incidence vector consists of 255 points, if the point is on the line, the value of the corresponding place in the vector is 1, and otherwise, the value is 0.

4) Finally, we take the incidence vector as the first row of the parity check matrix. This vector and its 254 cyclic shifts form the parity check matrix of the 2-dimensional EG-LDPC code with length 255 and 175 information bits.

From this example, we find that the construction of EG-LDPC code is very simple as long as we get one line or one incidence vector. The next step is just cyclic shift.

The companion code of the m-dimensional type-I EG-LDPC code is the null space of the parity check matrix as follows:

(1) ( , 0, )EGC m s

(2) ( , 0, )EGH m s = (1) ( , 0, )[ ]TEGH m s .

This LDPC code is

long and its minimum distance is at least

( 1)(2 1)(2 1) /(2 1)m s ms s

2 1sd .We call this code the m-dimensional type-II EG-LDPC code, denoted . This code is not cyclic but can be put in quasi-cyclic form. It has the same number of parity check symbols as the type-I code .

(2) ( , 0, )EGC m s

(1) ( , 0, )EGC m s

3.2. Error performance analysis

Fig. 2 gives the bit error performance of EG-LDPC (255,175) code and MacKay-LDPC (255,175). These two LDPC codes have the same BPSK modulation, transmit to the AWGN channel and decoded with BP algorithm. For EG-LDPC codes, the maximum number of decoding iterations is set to 10. MacKay LDPC codes use 1A construction algorithm [2] and the maximum number of decoding iterations is set to 20. From Figure 2, we see the EG-LDPC code gives the better error performance than MacKay-LDPC although it has fewer iteration numbers. And from the simulation time, we get to know EG-LDPC has almost half of the time that MacKay LDPC code needs. As the decoding algorithms are the same, it shows that

Euclidean geometry encoding is simple than MacKay encoding because the former one takes advantage of the cyclic code.

Reference [5] proposed that an extended or shortened Euclidean geometry LDPC code can get excellent error performance: only a few tenths of a decibel away from the Shannon limit. That’s so amazing.

Figure 2. EG and MacKay LDPC codes performance analysis

4. Simulation and performance analysis

4.1. Simulation system and parameters

We adopt the base band system of IEEE 802.16d standard released in October, 2004 [6]. For the sake of simpleness, we omit the steps of randomization

interleave synchronization channel estimation and equalization. Fig. 3 illustrates the steps involved in the simulation system. Source data shall first be channel encoded and modulated (constellation mapped). The modulated symbols shall be space time encoded, and OFDM modulated: include insert pilot and add cyclic prefix IFFT. Then the base band symbols shall be transmitted to the channel, and do the reverse operations at the receiver, which includes OFDM demodulation space time decoding data demodulation and channel decoding.

The simulation parameters are listed in Table 1.

Table 1. Simulation parametersSubcarrier number 256FFT number 256

Guard interval 1/4


Channel encode EG-LDPC(255,175) RS-CC(64,48,2/3)

modulation BPSK 16QAMChannel model SUI: Standford University Interim

channel 2 3

Antenna number 2×2STC STBC

Dataout FFT

FFT

remove pilot and

cyclic prefix

remove pilot and

cyclic prefix

channel decode STC

decode

demod

STC

Datain insert pilot and

add cyclic prefix

insert pilot and

add cyclic prefix

Channel

encode

IFFT

IFFT

mod

Figure 3. Simulation system

4.2. System simulation result

Fig. 4 and Fig. 5 give the system simulation results. From the figure, we can see EG-LDPC (255,175) code perform a little worse than the RS-CC code when the modulation is BPSK. But when the modulation is 16QAM, the two curves are very close. It is believed that if we use a longer good EG-LDPC code, it will perform better. So we are confident with the EG-LDPC code take a very important position in the next wireless communication technology.

Figure 4. System performance graph (BPSK)


Figure 5. System performance graph (16QAM)

5. Conclusion

In this paper, we started with the development and the structure of LDPC codes. Then we constructed a Euclidean geometry LDPC code, made a comparison with a MacKay-LDPC code and found EG-LDPC code outperforms the latter one. As the EG-LDPC codes are cyclic or quasi-cyclic, they can be simply encoded and implemented. Finally, we took the EG-LDPC code as channel code into IEEE802.16d MIMO-OFDM system simulation and compared it with RS-CC code. And we find that Euclidean geometry LDPC codes are strong competitors to FEC codes for error control in communication systems and they may be adopted as one of the key technologies in the next wireless communication systems.

6. References

[1] R.G. Gallager, “Low-density parity-check codes,” IRETrans. Inform. Theory, IT-8, Jan. 1962, pp. 21-28.[2] D.J.C. MacKay and R.M. Neal, “Near shannon limit performance of low density parity check codes,” Electronic Letters, vol. 32, Aug. 1996, pp. 1645-1646. [3] D.J.C. MacKay, “Good error-correcting codes based on very sparse matrices,” IEEE Trans. Inform. Theory, vol. IT-45, 1999, pp. 399-431. [4] R. M. Tanner, “A recursive approach to low complexity codes,” IEEE Trans. Inform. Theory, vol. IT-27, Sept. 1981, pp. 533-547. [5] Y. Kou, S. Lin, and M. Fossorier, “Low-density parity-check codes based on finite geometries: a rediscovery and new results,” IEEE Trans. Inform. Theory, vol. 47, Nov. 2001, pp. 2711-2736. [6] IEEE Std 802.16-20004.


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