Abstract—In this paper we study the use of repeat-

puncture superorthogonal convolutional turbo codes in an additive white Gaussian noise (AWGN) channel. The performance of turbo codes (TC) have been shown to be near the theoretical limit in the AWGN channel. By using orthogonal signaling, which allows for bandwidth expansion, the performance of the turbo coding scheme can be improved even further. Since this is a low-rate code, the code is mainly suitable for spread-spectrum modulation applications. In classical turbo codes the frame length is set equal to the interleaver size, however, the codeword distance spectrum of turbo codes improves with an increasing interleaver size. It has been reported that the performance of turbo codes can be improved by using repetition and puncturing. Repeat-puncture turbo codes (RPTC) have shown a significant increase in performance at moderate to high signal-to-noise ratios. In this paper, we study the use of orthogonal signaling and parallel concatenation together with repetition and puncturing to improve the performance of superorthogonal convolutional turbo codes for reliable and effective communications. Simulation results in AWGN channel are presented together with analytical upper bounds, which have been derived using transfer function bounding techniques.

Index Terms—orthogonal, repetition, puncturing, turbo codes, parallel concatenation.

I. INTRODUCTION ultipath fading, interference and noise are all deleterious inputs to signal fidelity at the receiver. In

a digital communication system this signal degradation translates to abundance in bit errors. If these bit errors are not controlled then the errors would render the system useless. Fortunately, the information bits transmitted over a digital wireless link can be protected by channel coding. A turbo code can be visualized as a “two-dimensional” or a “multi-dimensional” forward error correcting convolutional code.

The performance of turbo codes have been reported to be near the Shannon limit in [1]. Typically for large frame lengths (N=16384) and with iterative decoding, ob NE /

values of -0.15 dB at a bit-error rate level of 310− have been reported. At the expense of bandwidth expansion, in the link budget, the performance can be improved even further. For a frame length of N=4000, ob NE / values of -0.5 dB has

been reported at a bit-error rate (BER) of 310− with superorthogonal convolutional turbo codes (SCTC), which, is a low-rate code that achieves good distance properties. Due to the resultant low-rate code, these codes are mainly suited for spread-spectrum applications. In conventional turbo codes the frame length is set equal to the interleaver size, however, if the information frame length is increased then the resultant performance increases due to a larger interleaver size. This makes it difficult to use turbo codes for, e.g. voice transmission, since the inherent delay would be too great. Repeated-punctured turbo codes (RPTCs) proposed in [7] make use of interleavers of sizes larger than the information frame length and show a significant increase in performance at moderate to high SNRs. For N=1024, simulation results have shown that RPTC has approximately 1.5 dB coding gain over conventional turbo codes at a bit-error rate of 310− . Motivated by [7] we propose repeat-puncture superorthogonal convolutional turbo codes.

In this paper, we study the use of orthogonal signaling and parallel concatenation together with repetition and puncturing to improve the performance of superorthogonal convolutional turbo codes for reliable and effective communications. Simulation results in AWGN channel are presented together with analytical upper bounds, which have been derived using transfer function bounding techniques. The paper is structured such that, in Section II, we discuss the principals of repeat-puncture superorthogonal convolutional turbo codes. Section III describes the advantages and disadvantages of the Max-log-MAP algorithm, Section IV presents the performance analysis and the simulation results over the AWGN channel. Finally, conclusions are drawn in Section V.

II. REPEAT-PUNCTURE SUPERORTHOGONAL CONVOLUTIONAL TURBO CODES

A. Encoder The structure of the repeat-puncture superorthogonal

convolutional turbo code (RPSCTC) encoder is shown in Fig. 2 and the superorthogonal convolutional turbo code (SCTC) encoder structure shown in Fig. 1. The form is of a parallel concatenated code (PCC) where, two recursive systematic constituent code (RSCC) encoders, RSCC1 and RSCC2, are linked together by an interleaver. Let the number of information bits/frame be N.

The encoder for the first constituent code is the same as in superorthogonal convolutional turbo codes studied in [2], [5]. The structure of any one constituent code for memory, m=4, is presented in Fig.3, employing feedback

Repeat-Puncture Superorthogonal Convolutional Turbo Codes in AWGN Channel

Narushan Pillay, Hongjun Xu, Member, IEEE, Fambirai Takawira, Member, IEEE

M

1-4244-0987-X/07/$25.00 ©2007 IEEE.

polynomial 431)( DDDg ++= . If the memory of the code is m then the length of the output sequence from RSCC1 is 12 −m N. The code rate of such a constituent code is 1/ 12 −m . RSCC2 differs a little in RPSCTC from the normal SCTC scheme.

RSCC 1

RSCC 2

kd

π

)(2,11,10,1 1..., Nmyyy −

)(2,21,20,2 1..., Nmyyy −

Constituent Code 1

Constituent Code 2

)(2 1 Nm− )(2 1 Nm−

)(2 1 Nm−)(2 1 Nm−

N

)(2 1 Nm−

Fig.1. SCTC encoder structure.

RSCC 1

RSCC 2

kd

π

)(2,11,10,1 1..., Nmyyy −

)(2,21,20,2 1..., LNmyyy −

Constituent Code 1

Constituent Code 2

N )(2 1 Nm−

)(2 1 LNm−LN

N

LN

Repeat 'L'

LN

Fig.2. RPSCTC encoder structure.

0a 1a 2a 3a

Walsh-Hadarmard Generator

First paritysequence

Datainput

Fig.3. RPSCTC constituent encoder structure.

This is because prior to interleaving, to allow for a larger

interleaver size, each input information bit is repeated L times thus rendering a larger frame, i.e., LN, to the interleaver and the constituent encoder 2. This results in a sequence of length 12 −m LN at the output of RSCC2. This sequence needs to be punctured to control the overall code rate of the encoder. A simple puncturing pattern was considered, where, the first n bits out of every L bits in the second parity sequence, were transmitted. Corresponding to an input information stream, Nddddd ,...,,, 321= the output from RSCC1 is ),...,,( 12,12,11,11 Nmyyyy −= and the

output from the second constituent encoder prior to puncturing is ),...,,( 12,22,21,22 LNmyyyy −= . The trellis for

memory, m=2 is depicted in Fig. 4, in which, the recursive nature of the code is clearly evident.

00

01

10

11

0011

1101

0010

1001

Fig.4. Trellis for RPSCTC, m=2.

In the case of RPSCTC and SCTC there is no common channel output stream as in conventional turbo codes because the systematic information bit is not explicitly sent via the channel, but with appropriate mapping of the Walsh functions the outputs can be made systematic. Walsh functions obtained from Hadarmard matrices were used to generate the set of orthogonal sequences for each constituent code.

B. Decoder Unlike the Viterbi algorithm (VA) which finds the most

likely sequence to have been transmitted, the maximum a-posteriori (MAP) algorithm determines the most likely information bit to have been transmitted at each bit time k. Using a-posteriori probabilities (APP), for small probability of bit-error ( bP ) the performance difference between MAP and soft-output Viterbi algorithm (SOVA) is very small. However, at low ob NE / and high bP values MAP outperforms SOVA [6]. The MAP algorithm proceeds somewhat like the VA in two directions, forward and backward, over a block of code bits. Assuming an AWGN channel we start with the log-likelihood ratio (LLR) )ˆ( kdL .

⎥⎥⎦

⎤

⎢⎢⎣

⎡−

⎥⎥⎦

⎤

⎢⎢⎣

⎡= ∑∑

m

mk

m

mkkdL ,0,1 loglog)ˆ( λλ (1)

where,

∑∑ +=⎥⎥⎦

⎤

⎢⎢⎣

⎡

m

mfk

mk

mk

m

mk

),1(1

,1,1 βδαλ (2)

and

∑∑ +=⎥⎥⎦

⎤

⎢⎢⎣

⎡

m

mfk

mk

mk

m

mk

),0(1

,0,0 βδαλ . (3)

(2) represents the conditional probability )|1( xdP = and (3) represents the conditional probability )|0( xdP = , where

x is the bit received at time k. mkα , m

k,1δ and ),1(

1mf

k +β are the forward-state metric, reverse-state metric and branch metric respectively for )|1( xdP = , where m is the encoder state at bit time k and ),1( mf is the next state also for

1=d . The decoding structure as shown in Fig. 5 makes use of an iterative technique. An outer decoder and an inner decoder both in co-operation produce soft decisions which are mutually exchanged, thus increasing the reliability of the decisions. At the output of decoder 2 LN soft decisions are computed but only N decisions are needed. To make this of good use, each pair of decisions is averaged to produce N soft decisions. In the first step of the decoding the corrupted first parity sequence from the channel is sent to decoder 1 to produce N LLR (soft) estimates of the sequence using no a-priori information. Using equation (4) the extrinsic values are computed then repeated L times and permuted by the same mapping at the encoder and then sent to decoder 2 as a-priori information. This sharing of information to improve the decision is somewhat like co-operation. Decoder 2 then produces LN extrinsic decisions as described by equation (5).

)()()()( 11 kakckdecoderke dLxLdLdL −−= (4)

)()()()( 122 kekckdecoderke dLxLdLdL −−= (5) These extrinsic decisions are de-interleaved and sent to

decoder 1 as a-priori information. This process is iterated several times after which LN LLRs are taken from decoder 2, de-interleaved, then averaged (equation (6)) to produce N LLRs and compared to yield hard decisions. The sign of the LLR represents the hard decision and the amplitude of the LLR represents the degree of reliability. Hard decisions are never used as input to a decoder since this will degrade performance drastically.

212,2,

,++

= ieieie

LLL

average , i=0, 1…N-1 (6)

decoder 1 decoder 2repeat 'L'

Corruptedparitysequence 1 1eL

π

1−π

1−π

average

averageDecisions

Corrupted interleavedparitysequence 2

1eL′ 2eL′

2eL

)( kdL )( kdL′

Fig. 5. RPSCTC decoder structure.

III. MAX-LOG MAP ALGORITHM Although the MAP algorithm yields the best overall performance in the decoding of turbo codes the implementation of RPSCTC in hardware would lead to a lot of problems because of the increase in complexity. When we use the maximization approximation instead, for the forward and backward recursions, i.e., kα and kβ , this leads to an approximation error in the computation of these variables. For high SNRs this error is comparable to noise, thus degrading performance, and for low SNRs it is much less than the noise power. Thus by using the Max-Log-MAP algorithm we are settling for a degraded performance but a significantly lower complexity for a hardware implementation.

IV. PERFORMANCE ANALYSIS

A. Analytical BER bounds Transfer function bounding techniques studied in [8] was

used to derive the BER bound for RPSCTC. Starting with the state transition matrix shown in equation (7)

⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜

⎝

⎛

=jz

dilz

dil

jdildil

DILDIL

DILDILDILA

,0,

,00,0

),,(L

MOM

L

(7)

where, in a monomial dil DIL , l is always equal to 1, and i and d are either zero or one depending on the input and output weights respectively for the thz to the thj state of

the state diagram. For an encoder with m2 states with the last m edges as termination edges, the generating function is

given by (8).

∑∑∑≥ ≥ ≥

=0 0 0

),,(),,(l i d

dil diltDILDILT (8)

Since 132 )(... −−=++++ AIAAAI (9) Then the transfer function for each constituent encoder can be expressed in the form of (10)

mmDILAIDILT0,0

1 ])),,([(),,( −−= (10)

Denote the probability of producing a codeword fragment of weight d given a randomly selected input sequence of weight i by (11) for constituent encoder 1.

.),,(

),,(),,(

)|( 1

1

11

1 ⎟⎟⎠

⎞⎜⎜⎝

⎛≈=

∑iN

diNtdiNt

diNtidp

d

(11)

and (12) for constituent encoder 2.

.),,(

)|( 22

⎟⎟⎠

⎞⎜⎜⎝

⎛=

LiLN

dLiLNtidp (12)

The bound can be obtained from (13)

∑ ∑∑∑=

⎟⎟⎠

⎞⎜⎜⎝

⎛≤

n

dd i d dbit dpidpidp

iN

Nip

min 1 2221 )()|()|( (13)

where )/2()(2 ob NdREQdp ≤ (14)

B. Simulation in AWGN channel In order to evaluate the performance of repeat-puncture

superorthogonal convolutional turbo codes simulations were run and the results were plotted. Figure 6 shows the plot of BER versus ob NE / for memory, m=4, for SCTC

and RPSCTC which corresponds to the code rate R =151 .

The code rate of 151 is due to the first bit of the Walsh

sequences being punctured in the second parity sequence from the encoder. The union bound is also shown in Fig.6. A uniform interleaver was chosen and a straightforward puncturing pattern was used although other extravagant patterns could be investigated to yield a better performance. A frame length of N=200 was chosen and the stopping criterion was set at 30 frame errors. 18 iterations were chosen for the decoder although this could be decreased at higher SNRs. The RPSCTC scheme exhibited a higher computational complexity than SCTC in the simulation; this is mostly due to the second decoder handling an input sequence twice the size than that of the decoders in superorthogonal convolutional turbo codes. It

is shown in Fig.6 that approximately 0.5dB gain is obtained for RPSCTC compared to SCTC at BER of 310− and approximately 1.5dB gain at BER level of 610− . It is also shown that simulation results match the theoretic union bounds.

-1 0 1 2 3 4 510

-12

10-10

10-8

10-6

10-4

10-2

100

SNR

BE

R

SCTC and RPSCTC simulation for N=200

N=200 SCTC m=4

N=200 RPSCTC m=4

bound SCTC m=4bound RPSCTC m=4

Fig.6. SCTC and RPSCTC simulation with bounds.

V. CONCLUSION In this paper we have shown how repetition and

puncturing can be used to improve the performance of superorthogonal convolutional turbo codes in an AWGN channel. The QoS was presented in terms of the BER and the increase in performance between the scheme presented and SCTC was evident. The concreteness of the transfer function bounding technique was also proved in the derivation of the bound for the RPSCTC scheme.

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[3] L. R. Bahl, J. Cocke, F. Jelinek, and J. Raviv, “Optimal decoding of linear codes for minimizing symbol error rate,” IEEE Trans. Inform. Theory, Mar. 1974, vol. IT-20, pp. 284-287.

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[6] B.Sklar, Digital Communications. Fundamentals and Applications. Beijing 2001

[7] Y. Kim, J. Cho, W. Oh and K. Cheun, “Improving the performance of turbo codes by repetition and puncturing,” Division of Electrical and Computer Engineering, Pohang University of Science and Technology.

[8] D. Divsalar, S. Dolinar, and F. Pollara, “Transfer Function Bounds on the Performance of Turbo Codes,” TDA Progress Report 42-122, August 15, 1995, Communications Systems and Research Section, R. J. McEliece California Institute of Technology, pp. 44-55.

[9] D. Divsalar and F. Pollara, “Multiple Turbo Codes for Deep-Space Communications,” The Telecommunications and Data Acquisition Progress Report 42-121, January-March 1995, Jet Propulsion Laboratory, Pasadena, California, May 15, 1995, pp. 66-77.

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