IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 59, NO. 11, NOVEMBER 2011

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Transactions Letters Design of Irregular Weighted Nonbinary Repeat-Accumulate Codes over GF(𝑞 ) with 𝑞 -ary Orthogonal Modulation Using a Gaussian Approximation Yongsang Kim, Kyungwhoon Cheun, Member, IEEE, and Hyuntack Lim

Abstract—Using a Gaussian approximation to the distribution of the message vectors under density evolution, we design irregular weighted nonbinary repeat-accumulate codes over GF(𝑞) with 𝑞-ary orthogonal modulation. The resulting codes achieve a frame error rate of 10−1 within 0.56 to 0.91 dB of channel capacity under the AWGN channel. Index Terms—Channel capacity, density evolution, orthogonal modulation, repeat-accumulate codes.

I. I NTRODUCTION

I

N [1], weighted nonbinary (regular) repeat-accumulate (WNRA) codes over GF(𝑞) were proposed as a nonbinary extension of regular binary RA codes allowing very simple encoder/decoder structures. Subsequently in [2], an approximate maximum likelihood (ML) decoding threshold of WNRA codes over GF(𝑞) with 𝑞-ary orthogonal modulation and coherent detection under the AWGN channel was derived. Numerical results in [2] suggest that the value of the derived approximate upper bound to the ML decoding threshold falls very close to the channel capacity with decreasing code rate and increasing 𝑞 [2]. However, the performance of WNRA codes over GF(𝑞) with 𝑞-ary orthogonal modulation rapidly degrades with increasing code rate [2]. For example, for 4-ary frequency-shift keying (FSK) modulation, simulation results indicate that a rate-1/2 WNRA code using a random weighter [2] with an information frame length of 2520 bits requires a signal-to-noise ratio (SNR) in excess of 2.5 dB above the channel capacity for both coherent (CFSK) and noncoherent detection (NFSK). In order to design efficient high-rate WNRA codes, irregular WNRA (IWNRA) codes

Paper approved by F. Fekri, the Editor for LDPC Codes and Applications of the IEEE Communications Society. Manuscript received January 15, 2010; revised February 7, 2011. This research was supported by the MKE (The Ministry of Knowledge Economy), Korea, under the ITRC (Information Technology Research Center) support program supervised by the NIPA (National IT Industry Promotion Agency), (NIPA-2011-(C1090-1111-0011)), and the World Class University program funded by the Ministry of Education, Science, and Technology through the National Research Foundation of Korea (R31-10100). Y. Kim was with the Division of Electrical and Computer Engineering, POSTECH. He is now with Samsung Electronics Co., Ltd., Republic of Korea (e-mail: [email protected]). K. Cheun is with WCU (Division of ITCE, POSTECH), Pohang, Republic of Korea (e-mail: [email protected]). H. Lim is with the Division of Electrical and Computer Engineering, POSTECH, Republic of Korea (e-mail: [email protected]). Digital Object Identifier 10.1109/TCOMM.2011.063011.100012

may be considered similar to irregular binary RA codes considered in [5], [6]. Traditionally, density evolution [3] and its Gaussian approximated versions [4]–[6] are employed in the design of capacity approaching irregular binary low-density parity-check (LDPC)/RA codes. However, unlike binary codes, the application of the exact density evolution technique for codes over GF(𝑞) is practically feasible only for very small values of 𝑞 [8]1 . In order to simplify the design of LDPC codes over GF(𝑞), [7] and [8] proposed two Gaussian approximations to the distribution of the message vectors. In [7], as an extension of the density evolution under the binaryinput output-symmetric channel [4], the distribution of the message vectors was approximated to follow the Gaussian distribution characterized only by its mean vector under 𝑞-aryinput output-symmetric channels. In [8], in order to simplify the code design using the extrinsic information transfer (EXIT) chart [9], the distribution of the message vectors was further approximated by a Gaussian distribution characterized by a single scalar parameter. This technique was applied in the design of coset LDPC codes over GF(𝑞), 𝑞 = 4, 32, 64 with appropriately designed signal constellations achieving channel capacity within 1.0 dB at bit error rates below 10−5 [8]. Based on the single scalar parameter Gaussian approximation of the message vectors developed in [8], we develop a Gaussian approximated density evolution algorithm for the 𝑞ary belief-propagation (BP) decoding algorithm of IWNRA codes over GF(𝑞) with 𝑞-ary orthogonal modulation. We then use this to compute the decoding thresholds and design efficient IWNRA codes over GF(𝑞) for 𝑞 between 4 and 256. The designed IWNRA codes of a rate approximately equal to 1/2 with an information frame length of 2520 bits achieve a frame error rate (FER) of 10−1 within 0.71 to 0.91 dB and 0.56 to 0.64 dB of channel capacity under the AWGN channel with 𝑞-ary CFSK and NFSK modulation, respectively, for 𝑞 between 4 and 128. II. T HE S TRUCTURE OF IWNRA C ODES OVER GF(𝑞) Though we may consider non-systematic IWNRA codes, we focus on systematic IWNRA codes since non-systematic IWNRA codes run into decoder convergence problems similar 1 This

is not in the case for the binary erasure channel [24].

c 2011 IEEE 0090-6778/11$25.00 ⃝

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IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 59, NO. 11, NOVEMBER 2011

variable repeater

weighter

interleaver

( f1 , f2 ,", fJ ) Fig. 1. 𝑎.

serial to parallel convertor

f1

f2

fJ

a … …

accumulator

D

1

1

1

…1

IWNRA encoder over GF(𝑞) with parameters (𝑓1 , 𝑓2 , ⋅ ⋅ ⋅, 𝑓𝐽 ) and

to those of irregular binary RA codes [5]. Figs. 1 and 2 show the encoder structure of systematic IWNRA codes and its Tanner graph representation [10]. A fraction, 0 ≤ 𝑓𝑖 ≤ 1, ∑ 𝑖 = 1, 2, ⋅ ⋅ ⋅, 𝐽 with 𝐽𝑖=1 𝑓𝑖 = 1 of the information symbols are repeated 𝑖 times where 𝐽 denotes the maximum degree of an information node, and the vector, (𝑓1 , 𝑓2 , ⋅ ⋅ ⋅, 𝑓𝐽 ) is referred to as the repetition profile [5]. The output of the repeater is first weighted, i.e., multiplied by weighting values taking on values of nonzero elements in GF(𝑞) with equal probability, after which it is interleaved and then fed into an accumulator. The accumulator then accumulates consecutive 𝑎 symbols of the interleaver output where 𝑎 is referred to as the grouping factor [5]. Let us denote the fraction of the edges connected to the information nodes of degree 𝑖 as 𝜆𝑖 . Then, the rate of the systematic IWNRA codes, 𝑟 is given by [6] ∑∞ 𝑎 𝑖=1 𝜆𝑖 /𝑖 ∑∞ (1) 𝑟= 1 + 𝑎 𝑖=1 𝜆𝑖 /𝑖 ∑𝐽 where the vector, (𝜆1 , 𝜆2 , ⋅ ⋅ ⋅, 𝜆𝐽 ) such that 𝑖=1 𝜆𝑖 = 1 is referred to as the degree distribution from the edge perspective. III. D ENSITY E VOLUTION U SING A G AUSSIAN A PPROXIMATION Following the standard definition used in [8], let x ≜ (𝑥0 , . . . , 𝑥𝑞−1 ) denote the 𝑞-dimensional probability vector passed in a message passing decoder. Then the probability vector x may also be equivalently represented by a (𝑞 − 1)dimensional log-likelihood vector, w ≜ (𝑤1 , . . . , 𝑤𝑞−1 ) with ( ) 𝑥0 𝑤𝑖 ≜ log , 𝑖 = 1, . . . , 𝑞 − 1. 𝑥𝑖 As in [8], we denote the mapping from a 𝑞-dimensional probability vector to a (𝑞 − 1)-dimensional LLR vector and vice versa as LLR(⋅) and LLR−1 (⋅), respectively. Also, let DFTq (⋅) and IDFTq (⋅) denote the 𝑞-dimensional discrete Fourier transform (DFT) and the inverse DFT operations, respectively. Then, the update rule for the outgoing message vectors2 from the check nodes, denoted as u, for the BP decoder can be written as [11] ( )) (𝑑 −1 𝑐 ∏ ) ( −1 u = LLR IDFT𝑞 DFT𝑞 LLR (v𝑖 ) (2) 𝑖=1

where v𝑖 denotes the 𝑖th incoming message vector and 𝑑𝑐 is the degree of the check node. The update rule for the outgoing 2 In this letter, we consider the message vectors as the (𝑞 − 1)-dimensional LLR vectors as defined in [8].

…

a

…

2

2

a

J

weighter 2 interleaver

…

2

u L(l )

…

u R(l )

…

J …

…

v L(l )

…

J …

a

J

…

v R(l )

information node check node parity node Fig. 2. Tanner graph representation of IWNRA code over GF(𝑞) with parameters (𝑓1 , 𝑓2 , ⋅ ⋅ ⋅, 𝑓𝐽 ) and 𝑎.

message vectors of the variable nodes, denoted v, is given as [11] v = u0 +

𝑑∑ 𝑣 −1

u𝑖

(3)

𝑖=1

where u0 is the initial message vector generated by the channel observation, u𝑖 denotes the 𝑖th incoming message vector from the check nodes and 𝑑𝑣 is the degree of the variable node. Since we consider codes over GF(𝑞) with 𝑞-ary orthogonal modulation under the AWGN channel, the channel is outputsymmetric and thus, the single scalar parameter Gaussian approximation to the distribution of the message vectors developed in [8] may be applied. For IWNRA codes, let us denote the message vectors emanating from the check nodes to (𝑙) the information nodes in the 𝑙th decoder iteration as u𝐿 , those (𝑙) from the information nodes to the check nodes as v𝐿 , those (𝑙) from the check nodes to the parity nodes as u𝑅 and those (𝑙) from the parity nodes to the check nodes as v𝑅 (see Fig. 2). Also, the initial message vectors generated by the channel observation are denoted as u0 . Following [8], we approximate the distribution of the (𝑞 − 1)-dimensional message vectors (𝑙) (𝑙) (𝑙) (𝑙) u𝐿 , v𝐿 , u𝑅 , v𝑅 , u0 as following the single scalar parameter (𝑙) (𝑙) (𝑙) (𝑙) Gaussian distribution. Let Ω(𝑙) ≜ {u𝐿 , v𝐿 , u𝑅 , v𝑅 , u0 } and let mw(𝑙) and Σw(𝑙) denote the mean vector and the covariance matrix of a vector w(𝑙) ∈ Ω(𝑙) with elements 2 𝑚w(𝑙) ,𝑖 and Σw(𝑙) ,𝑖,𝑗 , respectively. Then, 𝑚w(𝑙) ,𝑖 = 𝜎w (𝑙) /2, 2 2 𝑖 = 1, . . ., 𝑞 − 1 and Σw(𝑙) ,𝑖,𝑗 = 𝜎w(𝑙) if 𝑖 = 𝑗 and 𝜎w (𝑙) /2,

KIM et al.: DESIGN OF IRREGULAR WEIGHTED NONBINARY REPEAT-ACCUMULATE CODES OVER GF(𝑄) WITH 𝑄-ARY ORTHOGONAL . . .

otherwise for a given positive real number 𝜎w(𝑙) [8]. Note that the joint probability distribution function of the message vectors is now characterized by a single scalar parameter 𝜎w(𝑙) which we will refer to as the Gaussian parameter of the message vector w(𝑙) . The readers are referred to [8] for a rigorous mathematical justification of the above single scalar parameter Gaussian approximation. We now derive the recursion equations for the Gaussian parameters, 𝜎u(𝑙) and 𝜎u(𝑙) which will lead to a sufficient 𝐿 𝑅 condition on the SNR for asymptotically achieving arbitrary small error probability as 𝑙 approaches infinity, i.e., the decoding threshold. Using the check node update rule of (2) and (𝑙) (𝑙) (𝑙) (𝑙) the definition of the message vectors, u𝐿 , v𝐿 , u𝑅 , v𝑅 , we may write,

(𝑙)

where 𝑤𝑖 , 𝑖 = 1, 2, ⋅ ⋅ ⋅, 𝑞 − 1 denotes the 𝑖th element of w(𝑙) . Then, the check node update equations (6) and (7) may be rewritten as ( ) [ ( )]𝑎−1 [ ( )]2 𝜙𝑖 𝜎v(𝑙) 𝜙𝑖 𝜎u(𝑙+1) = 𝜙𝑖 𝜎v(𝑙) , 𝑖 = 2, 3, . . ., 𝑞, 𝐿 𝐿 𝑅 (12) ( ) [ ( )]𝑎 [ ( )] 𝜙𝑖 𝜎u(𝑙+1) = 𝜙𝑖 𝜎v(𝑙) 𝜙𝑖 𝜎v(𝑙) , 𝑖 = 2, 3, . . ., 𝑞. 𝑅 𝐿 𝑅 (13)

Similarly, the variable node update rule of (3) may be written as follows: 𝐽 ( ) ∑ ( ) 𝜙𝑖 𝜎v(𝑙+1) = 𝜆𝑗 𝜙𝑖 (𝑗 − 1)𝜎u(𝑙) + 𝜎u0 , 𝑖 = 2, 3, . . ., 𝑞, 𝐿

(

(𝑙+1)

DFT𝑞 LLR−1 (u𝐿

(

)

) =

(𝑙+1)

DFT𝑞 LLR−1 (u𝑅

( [ )]𝑎−1 (𝑙) DFT𝑞 LLR−1 (v𝐿 ) ( [ )]2 (𝑙) × DFT𝑞 LLR−1 (v𝑅 ) , (4) ( [ )]𝑎 (𝑙) DFT𝑞 LLR−1 (v𝐿 ) ( ) (𝑙) ×DFT𝑞 LLR−1 (v𝑅 ) .

)

) =

Furthermore, invoking in [8], [12], we have,

the

assumption3

independence

[ ( )] (𝑙+1) 𝐸u(l+1) DFT𝑞 LLR−1 (u𝐿 ) L [ ( )]𝑎−1 (𝑙) = 𝐸v(l) DFT𝑞 LLR−1 (v𝐿 ) L [ ( )]2 (𝑙) × 𝐸v(l) DFT𝑞 LLR−1 (v𝑅 ) , R

[ ( )] (𝑙+1) 𝐸u(l+1) DFT𝑞 LLR−1 (u𝑅 ) R [ ( )]𝑎 (𝑙) = 𝐸v(l) DFT𝑞 LLR−1 (v𝐿 ) L [ ( )] (𝑙) × 𝐸v(l) DFT𝑞 LLR−1 (v𝑅 ) . R

≜

(

𝜙𝑖 𝜎v(𝑙+1)

(6)

𝑅

𝜙2 (𝜎w(𝑙) ) = 𝜙3 (𝜎w(𝑙) ) =

(7)

−𝑤1

1+𝑒

3 Asymptotically,

−𝑤2

+𝑒

−𝑤3

+𝑒

(

𝐿

𝑅

)]𝑎−1

+ 𝜎u0 𝑗=1 𝜆𝑗 𝜙𝑖 (𝑗 − 1)𝜎u(𝑙) 𝐿 )]2 [ ( × 𝜙𝑖 𝜎u(𝑙) + 𝜎u0 , 𝑖 = 2, 3, . . ., 𝑞,(16) 𝑅 ( ) [∑ ( )]𝑎 𝐽 (𝑗 − 1)𝜎 𝜆 𝜙 𝜙𝑖 𝜎u(𝑙+1) = (𝑙) + 𝜎u0 𝑗 𝑖 𝑗=1 u 𝑅 [ ( )] 𝐿 × 𝜙𝑖 𝜎u(𝑙) + 𝜎u0 , 𝑖 = 2, 3, . . ., 𝑞.(17)

In order to asymptotically achieve arbitrary small error probability as 𝑙 approaches infinity, 𝜎u(𝑙) and 𝜎u(𝑙) in (16) and (17) 𝐿 𝑅 must approach infinity as 𝑙 approaches infinity [12][17]. (𝑙) Let us define 𝑥𝑖 as 𝐽 ∑ 𝑗=1

( ) 𝜆𝑗 𝜙𝑖 (𝑗 − 1)𝜎u(𝑙) + 𝜎u0 , 𝑖 = 2, 3, ⋅ ⋅ ⋅, 𝑞. (18) 𝐿

By straightforward extension of the binary case [4], [5], we have that for 𝑧 > 0, 𝜙𝑖 (𝑧), 𝑖 = 2, 3, ⋅ ⋅ ⋅, 𝑞 is continuous and monotonically increasing with lim 𝜙𝑖 (𝑧) = 0 and 𝑧→0 lim 𝜙𝑖 (𝑧) = 1. Hence, from (16) and (17), 𝜎u(𝑙) and 𝜎u(𝑙) 𝑧→∞ 𝐿 𝑅 approach infinity as 𝑙 approaches infinity if we have, (𝑙)

𝑥𝑖 =

𝐽 ∑ 𝑗=1

(𝑙+1)

< 𝑥𝑖

( ) 𝜆𝑗 𝜙𝑖 (𝑗 − 1)𝜎u(𝑙) + 𝜎u0 𝐿

=

𝐽 ∑ 𝑗=1

( ) 𝜆𝑗 𝜙𝑖 (𝑗 − 1)𝜎u(𝑙+1) + 𝜎u0 , 𝑖 = 2, 3, ⋅ ⋅ ⋅, 𝑞 𝐿

(19)

(𝑙) 𝑥𝑖

if 𝑖 = 1

for any and 𝑙. Dividing (16) by the square of (17) and then substituting (18) into the result, we have,

if 𝑖 = 2 if 𝑖 = 3

[∑ 𝐽

𝐿

(10)

and that 0 < 𝜙𝑖 (𝜎w(𝑙) ) < 1, 𝑖 = 2, 3, ⋅ ⋅ ⋅, 𝑞. For example, for 𝑞 = 4, ⎧ 1, [ ] (𝑙) (𝑙) (𝑙) −𝑤1 −𝑤2 −𝑤3 1−𝑒 +𝑒 −𝑒 𝐸w(𝑙) 1+𝑒−𝑤1(𝑙) +𝑒−𝑤2(𝑙) +𝑒−𝑤3(𝑙) , ⎨ [ ] (𝑙) (𝑙) (𝑙) −𝑤1 −𝑤2 −𝑤3 𝜙𝑖 (𝜎w(𝑙) ) = 1+𝑒 −𝑒 −𝑒 , 𝐸 (𝑙) (𝑙) (𝑙) (𝑙) w −𝑤1 −𝑤2 −𝑤3 1+𝑒 +𝑒 +𝑒 [ ] (𝑙) (𝑙) (𝑙) −𝑤 −𝑤 −𝑤3 ⎩ 𝐸w(𝑙) 1−𝑒 1(𝑙) −𝑒 2(𝑙) +𝑒 (𝑙) ,

(15)

𝑅

𝜙𝑖 𝜎u(𝑙+1) =

(𝑙)

(9)

⋅ ⋅ ⋅ = 𝜙𝑞 (𝜎w(𝑙) )

(14)

)

= 𝜙𝑖 𝜎u(𝑙) + 𝜎u0 , 𝑖 = 2, 3, . . ., 𝑞.

)

Then, it is relatively simple to verify that 1,

(

Substituting (14) and (15) into (12) and (13), we have the following final recursion equations for 𝜎u(𝑙) and 𝜎u(𝑙) :

𝑥𝑖 ≜

[𝜙1 (𝜎w(𝑙) ), 𝜙2 (𝜎w(𝑙) ), ⋅ ⋅ ⋅, 𝜙𝑞 (𝜎w(𝑙) )]𝑇 )] [ ( 𝐸w(𝑙) DFT𝑞 LLR−1 (w(𝑙) ) . (8) 𝜙1 (𝜎w(𝑙) ) =

)

𝑅

In order to simplify the presentation, let us define Φ(𝜎w(𝑙) ) as Φ(𝜎w(𝑙) ) =

𝐿

𝑗=1

(

(5)

2931

if 𝑖 = 4.

at large frame length, all incoming and outgoing message vectors at any particular node may be assumed to be independent with high probability [8], [12].

( )⎞ (𝑙) 𝜙2𝑖 𝑓 (𝑥𝑖 ) ⎝ ⎠ , 𝑖 = 2, 3, ⋅ ⋅ ⋅, 𝑞 = 𝜙−1 𝑖 (𝑙) (𝑥𝑖 )𝑎+1 ⎛

(11)

𝜎u(𝑙+1) 𝐿

(20)

(𝑙)

where 𝜎u(𝑙+1) is written as 𝑓 (𝑥𝑖 ) in order to highlight the fact 𝑅

(𝑙)

that 𝜎u(𝑙+1) is a function of 𝑥𝑖 . Substituting (20) into (19), 𝑅

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IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 59, NO. 11, NOVEMBER 2011

TABLE I E XAMPLES OF DEGREE DISTRIBUTIONS FROM THE EDGE PERSPECTIVE OF IWNRA CODES OVER GF(𝑞), 𝑞 = 4, 8, 16, 32, 64, 128, 256 WITH 𝑎 = 4 AND RATE APPROXIMATELY EQUAL TO 1/2 ALONG WITH THE CORRESPONDING DECODING THRESHOLDS AND CHANNEL CAPACITIES FOR CFSK UNDER THE AWGN CHANNEL . 𝑞 𝜆3 𝜆6 𝜆10 Code rate Decoding threshold [dB] Channel capacity [dB]

4 0.544653 0.445517 0.00982993 0.506696 1.59 1.08

8 0.583891 0.35786 0.0582495 0.509899 0.42 0.15

16 0.535024 0.456354 0.00862201 0.505208 -0.27 -0.44

0

q=2 q=4 q=8 q = 16 q = 32 q = 64 q = 128

256 0.579292 0.342104 0.0786049 0.50785 -1.29 -1.26

Channel capacity Decoding threshold

Nonbinary Turbo codes

q=128 q=64 q=32

q=128

10

q=8

-3

0

q=64 q=32

q=4

1

Nonbinary Turbo codes

q=16

-2

q=2

q=16

-1

-1

10

FER

FER

10

10

128 0.582495 0.344523 0.0729828 0.508729 -1.20 -1.16

10

-1

-2

64 0.615574 0.115267 0.269159 0.501315 -1.05 -1.01

0

10

10

32 0.543999 0.376714 0.0792867 0.502039 -0.72 -0.75

q=8

q=2 q=4 q=8 q = 16 q = 32 q = 64 q = 128 Channel capacity

q=4

q=2

-3

2 Eb/N0 [dB]

3

4

5

10

1

2

3

4 5 Eb/N0 [dB]

6

7

8

Fig. 3. FER performances of IWNRA codes over GF(𝑞), 𝑞= 4, 8, 16, 32, 64, 128, of a rate approximately equal to 1/2 in Table I. Information frame length is 2520 bits with 𝑞-ary CFSK modulation under the AWGN channel.

Fig. 4. FER performances of IWNRA codes over GF(𝑞), 𝑞= 4, 8, 16, 32, 64, 128, of a rate approximately equal to 1/2 in Table I. Information frame length is 2520 bits with 𝑞-ary NFSK modulation under the AWGN channel.

we have the following final sufficient condition for achieving arbitrary small error probability as 𝑙 approaches infinity:

IV. N UMERICAL R ESULTS

(𝑙) 𝑥𝑖

𝑥, 𝑥𝑎+1 𝑗=1 𝐽 ∑

∀𝑥 ∈ [𝑥0 , 1),

𝑖 = 2, 3, ⋅ ⋅ ⋅, 𝑞

here 𝑥0 > 0 is the value of 𝑥 at the first iteration.

In Table I, we show examples of optimum degree distributions from the edge perspective, (𝜆1 , 𝜆2 , ⋅ ⋅ ⋅, 𝜆𝐽 ) found via the single scalar parameter Gaussian approximation for IWNRA codes derived in Section III over GF(𝑞) for 𝑞 = 4, 8, 16, 32, 64, 128, 256 with grouping factor, 𝑎 = 4 and rate approximately equal to 1/2. We also show the corresponding decoding thresholds for CFSK modulation under the AWGN channel.4 The search for the degree distributions from the edge perspective was accomplished using a variant of the downhill simplex method [13], [14] in order to reduce the search space for the values of (𝜆1 , 𝜆2 , ⋅ ⋅ ⋅, 𝜆𝐽 ) for which 𝐹 (⋅) needs to be computed. We evaluated the expectation operation in the computation of Φ(⋅) via the Monte-Carlo method. For comparison, we also show the AWGN channel capacities for CFSK modulation computed using the method derived in [15] in Table I. We observe that the decoding thresholds of the resulting IWNRA code designs with CFSK modulation under the AWGN channel approach the channel capacity within 0.03 to 0.51 dB for 𝑞 = 4, 8, 16, 32. We note that the decoding thresholds for 𝑞 = 64, 128 and 256 are smaller than the corresponding channel capacities, which is due to the fact 4 It is easy to show that for CFSK modulation under the AWGN channel, the distribution of the initial message vectors generated by the channel observation is Gaussian with the mean vector, mw(0) and the covariance matrix, Σw(0) .

KIM et al.: DESIGN OF IRREGULAR WEIGHTED NONBINARY REPEAT-ACCUMULATE CODES OVER GF(𝑄) WITH 𝑄-ARY ORTHOGONAL . . .

that the decoding thresholds are computed under the Gaussian approximation and not the exact distributions. Figs. 3 and 4 show the simulated FER curves of the IWNRA code designs in Table I for 𝑞 =4, 8, 16, 32, 64, 128, using a random weighter [2] with 𝑞-ary CFSK and NFSK modulation under the AWGN channel along with the corresponding channel capacities and decoding thresholds. For comparison, we also show the simulated FER curves of irregular binary RA code with rate approximately equal to 1/2 designed in [5] and the nonbinary Turbo codes of [23] with rate 1/2 in Figs. 3 and 4. Simulations were carried out using the BP decoding algorithm [16] with 40 iterations per frame with a frame length of 2520 information bits. We observe that the IWNRA code designs achieve an FER of 10−1 within 0.71 to 0.91 dB and 0.56 to 0.64 dB of channel capacity under the AWGN channel for 𝑞-ary CFSK and NFSK modulation, respectively. Also, we observe that the designed IWNRA codes consistently exhibit performance superior to that of the nonbinary Turbo codes of [23]. Additionally, other simulation results (not presented here) indicate that IWNRA code designs with additional information node degrees and/or grouping factors do not offer significant performance improvement compared to those given in Table I. V. C ONCLUSIONS In this letter, we have developed a single scalar parameter Gaussian approximated density evolution algorithm for the 𝑞ary BP decoding algorithm of IWNRA codes over GF(𝑞) with 𝑞-ary orthogonal modulation. This was then used to compute the decoding threshold and design efficient IWNRA codes for 𝑞 between 4 and 256. Simulation results indicate that the resulting IWNRA code designs with a frame length of 2520 information bits achieved an FER of 10−1 within 0.56 to 0.91 dB of channel capacity under the AWGN channel with 𝑞-ary CFSK and NFSK . R EFERENCES [1] K. Yang, “Weighted nonbinary repeat-accumulate codes,” IEEE Trans. Inf. Theory, vol. 50, no. 3, pp. 527–531, Mar. 2004. [2] Y. Kim, K. Cheun, and H. Lim, “Performance of weighted nonbinary repeat-accumulate codes over GF(q) with q-ary orthogonal modulation,” to be published. [3] T. J. Richardson, M. A. Shokrollahi, and R. L. Urbanke, “Design of capacity-approaching irregular low-density parity-check codes,” IEEE Trans. Inf. Theory, vol. 47, no. 2, pp. 619–637, Feb. 2001. [4] S. Chung, T. J. Richardson, and R. L. Urbanke, “Analysis of sumproduct decoding of low-density parity-check codes using a Gaussian approximation,” IEEE Trans. Inf. Theory, vol. 47, no. 2, pp. 657–670, Feb. 2001.

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[5] H. Jin, A. Khandekar, and R. J. McEliece, “Irregular repeat-accumulate codes,” in Proc. 2nd Int. Symp. Turbo Codes Related Topics, Sep. 2000, pp. 1–8. [6] A. Roumy, S. Guemghar, G. Caire, and S. Verdú, “Design methods for irregular repeat-accumulate codes,” IEEE Trans. Inf. Theory, vol. 50, no. 8, pp. 1711–1727, Aug. 2004. [7] G. Li, I. J. Fair, and W. A. Krzymie`n, “Analysis of nonbinary LDPC codes using Gaussian approximation,” in Proc. IEEE ISIT, June–July 2003, p. 234. [8] A. Bennatan and D. Burshtein, “Design and analysis of nonbinary LDPC codes for arbitrary discrete-memoryless channels,” IEEE Trans. Inf. Theory, vol. 52, no. 2, pp. 549–583, Feb. 2006. [9] S. ten Brink, “Convergence of iterative decoding,” Electron. Lett., vol. 35, no. 13, pp. 806–808, May 1999. [10] F. R. Kschischang, B. J. Frey, and H. A. Loeliger, “Factor graphs and the sum-product algorithm,” IEEE Trans. Inf. Theory, vol. 47, no. 2, pp. 498–519, Feb. 2001. [11] M. C. Davey, “Error-correction using low-density parity-check codes,” Ph.D. dissertation, Univ. Cambridge, Cambridge, U.K., 1999. [12] T. Richardson and R. Urbanke, “The capacity of low-density paritycheck codes under message-passing decoding,” IEEE Trans. Inf. Theory, vol. 47, no. 2, pp. 599–618, Feb. 2001. [13] W. H. Press, B. P. Flannery, S. A. Teukolsk, and W. T. Vetterling, Numerical Recipes in C. Cambridge, 1988. [14] Y. Kim and K. Cheun, “A reduced-complexity tree search detection algorithm for MIMO systems,” IEEE Trans. Signal Process., vol. 57, no. 6, pp. 2420–2424, June 2009. [15] W. E. Stark, “Capacity and cutoff rate of noncoherent FSK with nonselective Rician fading,” IEEE Trans. Commun., vol. 33, no. 11, pp. 1153–1159, Nov. 1985. [16] D. J. C. MacKay, “Good error-correcting codes based on very sparse matrices,” IEEE Trans. Inf. Theory, vol. 45, no. 2, pp. 399–431, Mar. 1999. [17] H. Jin, “Analysis and design of turbo-like codes,” Ph.D. dissertation, Calififornia Institute of Technology, Pasadena, 2001. [18] M. Ardakani and F. R. Kschischang, “A more accurate one-dimensional analysis and design of irregular LDPC codes,” IEEE Trans. Commun., vol. 52, no. 12, Dec. 2004. [19] H. Sankar and N. Sindhushayana, “Design of low-density parity-check (LDPC) codes for high order constellations,” in Proc. IEEE GLOBECOM, Dec. 2004, vol. 5, no. 12, pp. 3113–3117. [20] G. Liva, W. E. Ryan, and M. Chiani, “Quasi-cyclic generalized LDPC codes with low error floors,” IEEE Trans. Commun., vol. 56, no. 1, pp. 49–57, Jan. 2008. [21] S. ten Brink and G. Kramer, “Design of repeat-accumulate codes for iterative detection and decoding,” IEEE Trans. Signal Process., vol. 51, no. 11, pp. 2764–2772, Nov. 2003. [22] S. ten Brink, G. Kramer, and A. Ashikhmin, “Design of low-density parity-check codes for modulation and detection,” IEEE Trans. Commun., vol. 52, no. 4, pp. 670–678, Apr. 2004. [23] Y. Kim, K. Cheun, K. Yang, and M. Sagong, “Design of turbo codes over GF(q) with q-ary orthogonal modulation,” to be published. [24] V. Rathi and R. Urbanke, “Density evolution, threshold and the stability condition for non-binary LDPC codes,” IEE Proc. - Commun., vol. 152, no. 6, pp. 1069–1074, Dec. 2005. [25] G. Caire, G. Taricco, and E. Biglieri, “Bit-interleaved coded modulation,” IEEE Trans. Inf. Theory, vol. 44, no. 3, pp. 927–946, May 1998. [26] M. C. Valenti and S. Cheng, “Iterative demodulation and decoding of turbo-coded 𝑀 -ary noncoherent orthogonal modulation,” IEEE J. Sel. Areas Commun., vol. 23, no. 9, pp. 1739–1747, Sep. 2005.

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Transactions Letters Design of Irregular Weighted Nonbinary Repeat-Accumulate Codes over GF(𝑞 ) with 𝑞 -ary Orthogonal Modulation Using a Gaussian Approximation Yongsang Kim, Kyungwhoon Cheun, Member, IEEE, and Hyuntack Lim

Abstract—Using a Gaussian approximation to the distribution of the message vectors under density evolution, we design irregular weighted nonbinary repeat-accumulate codes over GF(𝑞) with 𝑞-ary orthogonal modulation. The resulting codes achieve a frame error rate of 10−1 within 0.56 to 0.91 dB of channel capacity under the AWGN channel. Index Terms—Channel capacity, density evolution, orthogonal modulation, repeat-accumulate codes.

I. I NTRODUCTION

I

N [1], weighted nonbinary (regular) repeat-accumulate (WNRA) codes over GF(𝑞) were proposed as a nonbinary extension of regular binary RA codes allowing very simple encoder/decoder structures. Subsequently in [2], an approximate maximum likelihood (ML) decoding threshold of WNRA codes over GF(𝑞) with 𝑞-ary orthogonal modulation and coherent detection under the AWGN channel was derived. Numerical results in [2] suggest that the value of the derived approximate upper bound to the ML decoding threshold falls very close to the channel capacity with decreasing code rate and increasing 𝑞 [2]. However, the performance of WNRA codes over GF(𝑞) with 𝑞-ary orthogonal modulation rapidly degrades with increasing code rate [2]. For example, for 4-ary frequency-shift keying (FSK) modulation, simulation results indicate that a rate-1/2 WNRA code using a random weighter [2] with an information frame length of 2520 bits requires a signal-to-noise ratio (SNR) in excess of 2.5 dB above the channel capacity for both coherent (CFSK) and noncoherent detection (NFSK). In order to design efficient high-rate WNRA codes, irregular WNRA (IWNRA) codes

Paper approved by F. Fekri, the Editor for LDPC Codes and Applications of the IEEE Communications Society. Manuscript received January 15, 2010; revised February 7, 2011. This research was supported by the MKE (The Ministry of Knowledge Economy), Korea, under the ITRC (Information Technology Research Center) support program supervised by the NIPA (National IT Industry Promotion Agency), (NIPA-2011-(C1090-1111-0011)), and the World Class University program funded by the Ministry of Education, Science, and Technology through the National Research Foundation of Korea (R31-10100). Y. Kim was with the Division of Electrical and Computer Engineering, POSTECH. He is now with Samsung Electronics Co., Ltd., Republic of Korea (e-mail: [email protected]). K. Cheun is with WCU (Division of ITCE, POSTECH), Pohang, Republic of Korea (e-mail: [email protected]). H. Lim is with the Division of Electrical and Computer Engineering, POSTECH, Republic of Korea (e-mail: [email protected]). Digital Object Identifier 10.1109/TCOMM.2011.063011.100012

may be considered similar to irregular binary RA codes considered in [5], [6]. Traditionally, density evolution [3] and its Gaussian approximated versions [4]–[6] are employed in the design of capacity approaching irregular binary low-density parity-check (LDPC)/RA codes. However, unlike binary codes, the application of the exact density evolution technique for codes over GF(𝑞) is practically feasible only for very small values of 𝑞 [8]1 . In order to simplify the design of LDPC codes over GF(𝑞), [7] and [8] proposed two Gaussian approximations to the distribution of the message vectors. In [7], as an extension of the density evolution under the binaryinput output-symmetric channel [4], the distribution of the message vectors was approximated to follow the Gaussian distribution characterized only by its mean vector under 𝑞-aryinput output-symmetric channels. In [8], in order to simplify the code design using the extrinsic information transfer (EXIT) chart [9], the distribution of the message vectors was further approximated by a Gaussian distribution characterized by a single scalar parameter. This technique was applied in the design of coset LDPC codes over GF(𝑞), 𝑞 = 4, 32, 64 with appropriately designed signal constellations achieving channel capacity within 1.0 dB at bit error rates below 10−5 [8]. Based on the single scalar parameter Gaussian approximation of the message vectors developed in [8], we develop a Gaussian approximated density evolution algorithm for the 𝑞ary belief-propagation (BP) decoding algorithm of IWNRA codes over GF(𝑞) with 𝑞-ary orthogonal modulation. We then use this to compute the decoding thresholds and design efficient IWNRA codes over GF(𝑞) for 𝑞 between 4 and 256. The designed IWNRA codes of a rate approximately equal to 1/2 with an information frame length of 2520 bits achieve a frame error rate (FER) of 10−1 within 0.71 to 0.91 dB and 0.56 to 0.64 dB of channel capacity under the AWGN channel with 𝑞-ary CFSK and NFSK modulation, respectively, for 𝑞 between 4 and 128. II. T HE S TRUCTURE OF IWNRA C ODES OVER GF(𝑞) Though we may consider non-systematic IWNRA codes, we focus on systematic IWNRA codes since non-systematic IWNRA codes run into decoder convergence problems similar 1 This

is not in the case for the binary erasure channel [24].

c 2011 IEEE 0090-6778/11$25.00 ⃝

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IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 59, NO. 11, NOVEMBER 2011

variable repeater

weighter

interleaver

( f1 , f2 ,", fJ ) Fig. 1. 𝑎.

serial to parallel convertor

f1

f2

fJ

a … …

accumulator

D

1

1

1

…1

IWNRA encoder over GF(𝑞) with parameters (𝑓1 , 𝑓2 , ⋅ ⋅ ⋅, 𝑓𝐽 ) and

to those of irregular binary RA codes [5]. Figs. 1 and 2 show the encoder structure of systematic IWNRA codes and its Tanner graph representation [10]. A fraction, 0 ≤ 𝑓𝑖 ≤ 1, ∑ 𝑖 = 1, 2, ⋅ ⋅ ⋅, 𝐽 with 𝐽𝑖=1 𝑓𝑖 = 1 of the information symbols are repeated 𝑖 times where 𝐽 denotes the maximum degree of an information node, and the vector, (𝑓1 , 𝑓2 , ⋅ ⋅ ⋅, 𝑓𝐽 ) is referred to as the repetition profile [5]. The output of the repeater is first weighted, i.e., multiplied by weighting values taking on values of nonzero elements in GF(𝑞) with equal probability, after which it is interleaved and then fed into an accumulator. The accumulator then accumulates consecutive 𝑎 symbols of the interleaver output where 𝑎 is referred to as the grouping factor [5]. Let us denote the fraction of the edges connected to the information nodes of degree 𝑖 as 𝜆𝑖 . Then, the rate of the systematic IWNRA codes, 𝑟 is given by [6] ∑∞ 𝑎 𝑖=1 𝜆𝑖 /𝑖 ∑∞ (1) 𝑟= 1 + 𝑎 𝑖=1 𝜆𝑖 /𝑖 ∑𝐽 where the vector, (𝜆1 , 𝜆2 , ⋅ ⋅ ⋅, 𝜆𝐽 ) such that 𝑖=1 𝜆𝑖 = 1 is referred to as the degree distribution from the edge perspective. III. D ENSITY E VOLUTION U SING A G AUSSIAN A PPROXIMATION Following the standard definition used in [8], let x ≜ (𝑥0 , . . . , 𝑥𝑞−1 ) denote the 𝑞-dimensional probability vector passed in a message passing decoder. Then the probability vector x may also be equivalently represented by a (𝑞 − 1)dimensional log-likelihood vector, w ≜ (𝑤1 , . . . , 𝑤𝑞−1 ) with ( ) 𝑥0 𝑤𝑖 ≜ log , 𝑖 = 1, . . . , 𝑞 − 1. 𝑥𝑖 As in [8], we denote the mapping from a 𝑞-dimensional probability vector to a (𝑞 − 1)-dimensional LLR vector and vice versa as LLR(⋅) and LLR−1 (⋅), respectively. Also, let DFTq (⋅) and IDFTq (⋅) denote the 𝑞-dimensional discrete Fourier transform (DFT) and the inverse DFT operations, respectively. Then, the update rule for the outgoing message vectors2 from the check nodes, denoted as u, for the BP decoder can be written as [11] ( )) (𝑑 −1 𝑐 ∏ ) ( −1 u = LLR IDFT𝑞 DFT𝑞 LLR (v𝑖 ) (2) 𝑖=1

where v𝑖 denotes the 𝑖th incoming message vector and 𝑑𝑐 is the degree of the check node. The update rule for the outgoing 2 In this letter, we consider the message vectors as the (𝑞 − 1)-dimensional LLR vectors as defined in [8].

…

a

…

2

2

a

J

weighter 2 interleaver

…

2

u L(l )

…

u R(l )

…

J …

…

v L(l )

…

J …

a

J

…

v R(l )

information node check node parity node Fig. 2. Tanner graph representation of IWNRA code over GF(𝑞) with parameters (𝑓1 , 𝑓2 , ⋅ ⋅ ⋅, 𝑓𝐽 ) and 𝑎.

message vectors of the variable nodes, denoted v, is given as [11] v = u0 +

𝑑∑ 𝑣 −1

u𝑖

(3)

𝑖=1

where u0 is the initial message vector generated by the channel observation, u𝑖 denotes the 𝑖th incoming message vector from the check nodes and 𝑑𝑣 is the degree of the variable node. Since we consider codes over GF(𝑞) with 𝑞-ary orthogonal modulation under the AWGN channel, the channel is outputsymmetric and thus, the single scalar parameter Gaussian approximation to the distribution of the message vectors developed in [8] may be applied. For IWNRA codes, let us denote the message vectors emanating from the check nodes to (𝑙) the information nodes in the 𝑙th decoder iteration as u𝐿 , those (𝑙) from the information nodes to the check nodes as v𝐿 , those (𝑙) from the check nodes to the parity nodes as u𝑅 and those (𝑙) from the parity nodes to the check nodes as v𝑅 (see Fig. 2). Also, the initial message vectors generated by the channel observation are denoted as u0 . Following [8], we approximate the distribution of the (𝑞 − 1)-dimensional message vectors (𝑙) (𝑙) (𝑙) (𝑙) u𝐿 , v𝐿 , u𝑅 , v𝑅 , u0 as following the single scalar parameter (𝑙) (𝑙) (𝑙) (𝑙) Gaussian distribution. Let Ω(𝑙) ≜ {u𝐿 , v𝐿 , u𝑅 , v𝑅 , u0 } and let mw(𝑙) and Σw(𝑙) denote the mean vector and the covariance matrix of a vector w(𝑙) ∈ Ω(𝑙) with elements 2 𝑚w(𝑙) ,𝑖 and Σw(𝑙) ,𝑖,𝑗 , respectively. Then, 𝑚w(𝑙) ,𝑖 = 𝜎w (𝑙) /2, 2 2 𝑖 = 1, . . ., 𝑞 − 1 and Σw(𝑙) ,𝑖,𝑗 = 𝜎w(𝑙) if 𝑖 = 𝑗 and 𝜎w (𝑙) /2,

KIM et al.: DESIGN OF IRREGULAR WEIGHTED NONBINARY REPEAT-ACCUMULATE CODES OVER GF(𝑄) WITH 𝑄-ARY ORTHOGONAL . . .

otherwise for a given positive real number 𝜎w(𝑙) [8]. Note that the joint probability distribution function of the message vectors is now characterized by a single scalar parameter 𝜎w(𝑙) which we will refer to as the Gaussian parameter of the message vector w(𝑙) . The readers are referred to [8] for a rigorous mathematical justification of the above single scalar parameter Gaussian approximation. We now derive the recursion equations for the Gaussian parameters, 𝜎u(𝑙) and 𝜎u(𝑙) which will lead to a sufficient 𝐿 𝑅 condition on the SNR for asymptotically achieving arbitrary small error probability as 𝑙 approaches infinity, i.e., the decoding threshold. Using the check node update rule of (2) and (𝑙) (𝑙) (𝑙) (𝑙) the definition of the message vectors, u𝐿 , v𝐿 , u𝑅 , v𝑅 , we may write,

(𝑙)

where 𝑤𝑖 , 𝑖 = 1, 2, ⋅ ⋅ ⋅, 𝑞 − 1 denotes the 𝑖th element of w(𝑙) . Then, the check node update equations (6) and (7) may be rewritten as ( ) [ ( )]𝑎−1 [ ( )]2 𝜙𝑖 𝜎v(𝑙) 𝜙𝑖 𝜎u(𝑙+1) = 𝜙𝑖 𝜎v(𝑙) , 𝑖 = 2, 3, . . ., 𝑞, 𝐿 𝐿 𝑅 (12) ( ) [ ( )]𝑎 [ ( )] 𝜙𝑖 𝜎u(𝑙+1) = 𝜙𝑖 𝜎v(𝑙) 𝜙𝑖 𝜎v(𝑙) , 𝑖 = 2, 3, . . ., 𝑞. 𝑅 𝐿 𝑅 (13)

Similarly, the variable node update rule of (3) may be written as follows: 𝐽 ( ) ∑ ( ) 𝜙𝑖 𝜎v(𝑙+1) = 𝜆𝑗 𝜙𝑖 (𝑗 − 1)𝜎u(𝑙) + 𝜎u0 , 𝑖 = 2, 3, . . ., 𝑞, 𝐿

(

(𝑙+1)

DFT𝑞 LLR−1 (u𝐿

(

)

) =

(𝑙+1)

DFT𝑞 LLR−1 (u𝑅

( [ )]𝑎−1 (𝑙) DFT𝑞 LLR−1 (v𝐿 ) ( [ )]2 (𝑙) × DFT𝑞 LLR−1 (v𝑅 ) , (4) ( [ )]𝑎 (𝑙) DFT𝑞 LLR−1 (v𝐿 ) ( ) (𝑙) ×DFT𝑞 LLR−1 (v𝑅 ) .

)

) =

Furthermore, invoking in [8], [12], we have,

the

assumption3

independence

[ ( )] (𝑙+1) 𝐸u(l+1) DFT𝑞 LLR−1 (u𝐿 ) L [ ( )]𝑎−1 (𝑙) = 𝐸v(l) DFT𝑞 LLR−1 (v𝐿 ) L [ ( )]2 (𝑙) × 𝐸v(l) DFT𝑞 LLR−1 (v𝑅 ) , R

[ ( )] (𝑙+1) 𝐸u(l+1) DFT𝑞 LLR−1 (u𝑅 ) R [ ( )]𝑎 (𝑙) = 𝐸v(l) DFT𝑞 LLR−1 (v𝐿 ) L [ ( )] (𝑙) × 𝐸v(l) DFT𝑞 LLR−1 (v𝑅 ) . R

≜

(

𝜙𝑖 𝜎v(𝑙+1)

(6)

𝑅

𝜙2 (𝜎w(𝑙) ) = 𝜙3 (𝜎w(𝑙) ) =

(7)

−𝑤1

1+𝑒

3 Asymptotically,

−𝑤2

+𝑒

−𝑤3

+𝑒

(

𝐿

𝑅

)]𝑎−1

+ 𝜎u0 𝑗=1 𝜆𝑗 𝜙𝑖 (𝑗 − 1)𝜎u(𝑙) 𝐿 )]2 [ ( × 𝜙𝑖 𝜎u(𝑙) + 𝜎u0 , 𝑖 = 2, 3, . . ., 𝑞,(16) 𝑅 ( ) [∑ ( )]𝑎 𝐽 (𝑗 − 1)𝜎 𝜆 𝜙 𝜙𝑖 𝜎u(𝑙+1) = (𝑙) + 𝜎u0 𝑗 𝑖 𝑗=1 u 𝑅 [ ( )] 𝐿 × 𝜙𝑖 𝜎u(𝑙) + 𝜎u0 , 𝑖 = 2, 3, . . ., 𝑞.(17)

In order to asymptotically achieve arbitrary small error probability as 𝑙 approaches infinity, 𝜎u(𝑙) and 𝜎u(𝑙) in (16) and (17) 𝐿 𝑅 must approach infinity as 𝑙 approaches infinity [12][17]. (𝑙) Let us define 𝑥𝑖 as 𝐽 ∑ 𝑗=1

( ) 𝜆𝑗 𝜙𝑖 (𝑗 − 1)𝜎u(𝑙) + 𝜎u0 , 𝑖 = 2, 3, ⋅ ⋅ ⋅, 𝑞. (18) 𝐿

By straightforward extension of the binary case [4], [5], we have that for 𝑧 > 0, 𝜙𝑖 (𝑧), 𝑖 = 2, 3, ⋅ ⋅ ⋅, 𝑞 is continuous and monotonically increasing with lim 𝜙𝑖 (𝑧) = 0 and 𝑧→0 lim 𝜙𝑖 (𝑧) = 1. Hence, from (16) and (17), 𝜎u(𝑙) and 𝜎u(𝑙) 𝑧→∞ 𝐿 𝑅 approach infinity as 𝑙 approaches infinity if we have, (𝑙)

𝑥𝑖 =

𝐽 ∑ 𝑗=1

(𝑙+1)

< 𝑥𝑖

( ) 𝜆𝑗 𝜙𝑖 (𝑗 − 1)𝜎u(𝑙) + 𝜎u0 𝐿

=

𝐽 ∑ 𝑗=1

( ) 𝜆𝑗 𝜙𝑖 (𝑗 − 1)𝜎u(𝑙+1) + 𝜎u0 , 𝑖 = 2, 3, ⋅ ⋅ ⋅, 𝑞 𝐿

(19)

(𝑙) 𝑥𝑖

if 𝑖 = 1

for any and 𝑙. Dividing (16) by the square of (17) and then substituting (18) into the result, we have,

if 𝑖 = 2 if 𝑖 = 3

[∑ 𝐽

𝐿

(10)

and that 0 < 𝜙𝑖 (𝜎w(𝑙) ) < 1, 𝑖 = 2, 3, ⋅ ⋅ ⋅, 𝑞. For example, for 𝑞 = 4, ⎧ 1, [ ] (𝑙) (𝑙) (𝑙) −𝑤1 −𝑤2 −𝑤3 1−𝑒 +𝑒 −𝑒 𝐸w(𝑙) 1+𝑒−𝑤1(𝑙) +𝑒−𝑤2(𝑙) +𝑒−𝑤3(𝑙) , ⎨ [ ] (𝑙) (𝑙) (𝑙) −𝑤1 −𝑤2 −𝑤3 𝜙𝑖 (𝜎w(𝑙) ) = 1+𝑒 −𝑒 −𝑒 , 𝐸 (𝑙) (𝑙) (𝑙) (𝑙) w −𝑤1 −𝑤2 −𝑤3 1+𝑒 +𝑒 +𝑒 [ ] (𝑙) (𝑙) (𝑙) −𝑤 −𝑤 −𝑤3 ⎩ 𝐸w(𝑙) 1−𝑒 1(𝑙) −𝑒 2(𝑙) +𝑒 (𝑙) ,

(15)

𝑅

𝜙𝑖 𝜎u(𝑙+1) =

(𝑙)

(9)

⋅ ⋅ ⋅ = 𝜙𝑞 (𝜎w(𝑙) )

(14)

)

= 𝜙𝑖 𝜎u(𝑙) + 𝜎u0 , 𝑖 = 2, 3, . . ., 𝑞.

)

Then, it is relatively simple to verify that 1,

(

Substituting (14) and (15) into (12) and (13), we have the following final recursion equations for 𝜎u(𝑙) and 𝜎u(𝑙) :

𝑥𝑖 ≜

[𝜙1 (𝜎w(𝑙) ), 𝜙2 (𝜎w(𝑙) ), ⋅ ⋅ ⋅, 𝜙𝑞 (𝜎w(𝑙) )]𝑇 )] [ ( 𝐸w(𝑙) DFT𝑞 LLR−1 (w(𝑙) ) . (8) 𝜙1 (𝜎w(𝑙) ) =

)

𝑅

In order to simplify the presentation, let us define Φ(𝜎w(𝑙) ) as Φ(𝜎w(𝑙) ) =

𝐿

𝑗=1

(

(5)

2931

if 𝑖 = 4.

at large frame length, all incoming and outgoing message vectors at any particular node may be assumed to be independent with high probability [8], [12].

( )⎞ (𝑙) 𝜙2𝑖 𝑓 (𝑥𝑖 ) ⎝ ⎠ , 𝑖 = 2, 3, ⋅ ⋅ ⋅, 𝑞 = 𝜙−1 𝑖 (𝑙) (𝑥𝑖 )𝑎+1 ⎛

(11)

𝜎u(𝑙+1) 𝐿

(20)

(𝑙)

where 𝜎u(𝑙+1) is written as 𝑓 (𝑥𝑖 ) in order to highlight the fact 𝑅

(𝑙)

that 𝜎u(𝑙+1) is a function of 𝑥𝑖 . Substituting (20) into (19), 𝑅

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IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 59, NO. 11, NOVEMBER 2011

TABLE I E XAMPLES OF DEGREE DISTRIBUTIONS FROM THE EDGE PERSPECTIVE OF IWNRA CODES OVER GF(𝑞), 𝑞 = 4, 8, 16, 32, 64, 128, 256 WITH 𝑎 = 4 AND RATE APPROXIMATELY EQUAL TO 1/2 ALONG WITH THE CORRESPONDING DECODING THRESHOLDS AND CHANNEL CAPACITIES FOR CFSK UNDER THE AWGN CHANNEL . 𝑞 𝜆3 𝜆6 𝜆10 Code rate Decoding threshold [dB] Channel capacity [dB]

4 0.544653 0.445517 0.00982993 0.506696 1.59 1.08

8 0.583891 0.35786 0.0582495 0.509899 0.42 0.15

16 0.535024 0.456354 0.00862201 0.505208 -0.27 -0.44

0

q=2 q=4 q=8 q = 16 q = 32 q = 64 q = 128

256 0.579292 0.342104 0.0786049 0.50785 -1.29 -1.26

Channel capacity Decoding threshold

Nonbinary Turbo codes

q=128 q=64 q=32

q=128

10

q=8

-3

0

q=64 q=32

q=4

1

Nonbinary Turbo codes

q=16

-2

q=2

q=16

-1

-1

10

FER

FER

10

10

128 0.582495 0.344523 0.0729828 0.508729 -1.20 -1.16

10

-1

-2

64 0.615574 0.115267 0.269159 0.501315 -1.05 -1.01

0

10

10

32 0.543999 0.376714 0.0792867 0.502039 -0.72 -0.75

q=8

q=2 q=4 q=8 q = 16 q = 32 q = 64 q = 128 Channel capacity

q=4

q=2

-3

2 Eb/N0 [dB]

3

4

5

10

1

2

3

4 5 Eb/N0 [dB]

6

7

8

Fig. 3. FER performances of IWNRA codes over GF(𝑞), 𝑞= 4, 8, 16, 32, 64, 128, of a rate approximately equal to 1/2 in Table I. Information frame length is 2520 bits with 𝑞-ary CFSK modulation under the AWGN channel.

Fig. 4. FER performances of IWNRA codes over GF(𝑞), 𝑞= 4, 8, 16, 32, 64, 128, of a rate approximately equal to 1/2 in Table I. Information frame length is 2520 bits with 𝑞-ary NFSK modulation under the AWGN channel.

we have the following final sufficient condition for achieving arbitrary small error probability as 𝑙 approaches infinity:

IV. N UMERICAL R ESULTS

(𝑙) 𝑥𝑖

𝑥, 𝑥𝑎+1 𝑗=1 𝐽 ∑

∀𝑥 ∈ [𝑥0 , 1),

𝑖 = 2, 3, ⋅ ⋅ ⋅, 𝑞

here 𝑥0 > 0 is the value of 𝑥 at the first iteration.

In Table I, we show examples of optimum degree distributions from the edge perspective, (𝜆1 , 𝜆2 , ⋅ ⋅ ⋅, 𝜆𝐽 ) found via the single scalar parameter Gaussian approximation for IWNRA codes derived in Section III over GF(𝑞) for 𝑞 = 4, 8, 16, 32, 64, 128, 256 with grouping factor, 𝑎 = 4 and rate approximately equal to 1/2. We also show the corresponding decoding thresholds for CFSK modulation under the AWGN channel.4 The search for the degree distributions from the edge perspective was accomplished using a variant of the downhill simplex method [13], [14] in order to reduce the search space for the values of (𝜆1 , 𝜆2 , ⋅ ⋅ ⋅, 𝜆𝐽 ) for which 𝐹 (⋅) needs to be computed. We evaluated the expectation operation in the computation of Φ(⋅) via the Monte-Carlo method. For comparison, we also show the AWGN channel capacities for CFSK modulation computed using the method derived in [15] in Table I. We observe that the decoding thresholds of the resulting IWNRA code designs with CFSK modulation under the AWGN channel approach the channel capacity within 0.03 to 0.51 dB for 𝑞 = 4, 8, 16, 32. We note that the decoding thresholds for 𝑞 = 64, 128 and 256 are smaller than the corresponding channel capacities, which is due to the fact 4 It is easy to show that for CFSK modulation under the AWGN channel, the distribution of the initial message vectors generated by the channel observation is Gaussian with the mean vector, mw(0) and the covariance matrix, Σw(0) .

KIM et al.: DESIGN OF IRREGULAR WEIGHTED NONBINARY REPEAT-ACCUMULATE CODES OVER GF(𝑄) WITH 𝑄-ARY ORTHOGONAL . . .

that the decoding thresholds are computed under the Gaussian approximation and not the exact distributions. Figs. 3 and 4 show the simulated FER curves of the IWNRA code designs in Table I for 𝑞 =4, 8, 16, 32, 64, 128, using a random weighter [2] with 𝑞-ary CFSK and NFSK modulation under the AWGN channel along with the corresponding channel capacities and decoding thresholds. For comparison, we also show the simulated FER curves of irregular binary RA code with rate approximately equal to 1/2 designed in [5] and the nonbinary Turbo codes of [23] with rate 1/2 in Figs. 3 and 4. Simulations were carried out using the BP decoding algorithm [16] with 40 iterations per frame with a frame length of 2520 information bits. We observe that the IWNRA code designs achieve an FER of 10−1 within 0.71 to 0.91 dB and 0.56 to 0.64 dB of channel capacity under the AWGN channel for 𝑞-ary CFSK and NFSK modulation, respectively. Also, we observe that the designed IWNRA codes consistently exhibit performance superior to that of the nonbinary Turbo codes of [23]. Additionally, other simulation results (not presented here) indicate that IWNRA code designs with additional information node degrees and/or grouping factors do not offer significant performance improvement compared to those given in Table I. V. C ONCLUSIONS In this letter, we have developed a single scalar parameter Gaussian approximated density evolution algorithm for the 𝑞ary BP decoding algorithm of IWNRA codes over GF(𝑞) with 𝑞-ary orthogonal modulation. This was then used to compute the decoding threshold and design efficient IWNRA codes for 𝑞 between 4 and 256. Simulation results indicate that the resulting IWNRA code designs with a frame length of 2520 information bits achieved an FER of 10−1 within 0.56 to 0.91 dB of channel capacity under the AWGN channel with 𝑞-ary CFSK and NFSK . R EFERENCES [1] K. Yang, “Weighted nonbinary repeat-accumulate codes,” IEEE Trans. Inf. Theory, vol. 50, no. 3, pp. 527–531, Mar. 2004. [2] Y. Kim, K. Cheun, and H. Lim, “Performance of weighted nonbinary repeat-accumulate codes over GF(q) with q-ary orthogonal modulation,” to be published. [3] T. J. Richardson, M. A. Shokrollahi, and R. L. Urbanke, “Design of capacity-approaching irregular low-density parity-check codes,” IEEE Trans. Inf. Theory, vol. 47, no. 2, pp. 619–637, Feb. 2001. [4] S. Chung, T. J. Richardson, and R. L. Urbanke, “Analysis of sumproduct decoding of low-density parity-check codes using a Gaussian approximation,” IEEE Trans. Inf. Theory, vol. 47, no. 2, pp. 657–670, Feb. 2001.

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