Performance Tradeoff Analysis on Inner and Outer Code Rates for Concatenated Codes in Gaussian...

ISSN: 2395-0560 International Research Journal of Innovative Engineering

www.irjie.com Volume1, Issue 2 of February 2015

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Performance Tradeoff Analysis on Inner and Outer Code Rates for Concatenated Codes in Gaussian Binary

Symmetric Channels Ye Hoon Lee

Department of Electronic and IT Media Engineering, Seoul National University of Science and Technology, Seoul, 139-743, Korea

Abstract In this paper, we address a problem of finding the optimum inner and outer code rates for a concatenated code in Gaussian binary symmetric channels. Clearly, as the inner code rate decreases, the error detection capability of the inner code increases. However, decreasing the inner code rate implies a decrease in error-correction capability of the outer code when overall code rate is fixed. With this notion in mind, we examine the optimum distribution of redundancy on the outer and inner codes to achieve a maximum performance gain in the concatenated coding scheme. Our analysis shows that the maximum coding gain can be obtained when the inner code rate is maximized and the outer code rate is minimized under the constraint of total code rate is fixed.

Keywords Concatenated code, optimal code rate, Gaussian channel, binary symmetric channel.

1. Introduction Concatenated codes form a class of error-correcting codes that are derived by combining an inner code and an outer code

[1]. They were conceived as a solution to the problem of finding a code that has both exponentially decreasing error prob-ability with increasing block length and polynomial-time decoding complexity. Concatenated codes became widely used in space communications in the 1970s and recently adopted in Digital Television Terrestrial Broadcasting (DTTB) [2].

The most natural choice for outer codes are Reed-Solomon (RS) codes in concatenated codes. Because the RS codes, being maximum-distance-separable codes, make highly efficient use of redundancy, and well suited to burst error correc-tion [3]. We will use RS codes as outer codes throughout this work. The inner code we consider in this paper is error de-tecting or correcting binary block code. The inner code corrects ec errors and detects ed errors provided 2ec + ed < dmin, where dmin is the minimum distance of the inner code. When an error is detected, every symbol of the inner code is erased. There are, however, errors that are not detected nor corrected by the inner code, which results in errors at the output of the inner decoder. The purpose of the outer code is to correct the errors and erasures of the inner code.

When the overall code rate is fixed, decreasing the inner code rate results in an increase of the error detection capability of the inner code. However, decreasing the inner code rate also implies a decrease in error-correction capability of the outer code. Thus, we expect there to be optimum inner and outer code rates for the overall block error probability to be mini-mized when the overall code rate is fixed. We will now examine the optimum rates in the Gaussian binary symmetric channels.

This paper is organized as follows. The Gaussian binary symmetric channel is presented in Section 2. The concatenated coding scheme is illustrated in Section 3. The optimum code rate is analysed in Section 4. The numerical results and dis-cussions are given in Section 5. Finally, a conclusion is made in Section 6.

2. Gaussian Binary Symmetric Channel The channel has input alphabet AX and channel output alphabet AY, both equal to the set of all real numbers. If Xi , i = 1,

2, are the inputs to the channel at time i, then the corresponding outputs Yi , i = 1, 2, are given by Yi =Xi+Zi , where Zi , i = 1, 2, are independent, identically distributed Gaussian random variables with mean zero and variance 2 . This channel is shown in Figure 1.




From the view point of digital communication, modulator, demodulator and waveform channel are characterized by a set of input symbols, output symbols, and crossover probabilities [4]. In the simplest case, the channel crossover probabilities are time invariant and independent from symbol to symbol. This is so-called discrete memoryless channel (DMC). The most commonly encountered case of the DMC is the so-called binary symmetric channel (BSC) as shown in Figure 2. In this case the channel crossover probability, p, for binary phase-shift keying (BPSK) signalling is given by

0

.2 bEp Q N

(1)

where

2 212

t

xQ x e dt

/( ) . (2)

Figure 1. The Gaussian channel

Figure 2. The binary symmetric channel

3. Concatenated Coding System The basic concept of concatenated codes is illustrated in Figure 3, which provides the general outline of a powerful class

of code. Information to be transmitted is first encoded with an (n,k) outer code. The symbols from the outer encoder are further encoded with an (N,K) inner code. At the receiver side, the demodulated data is first decoded with the inner decoder, and then the symbols from the inner decoder are decoded with the outer decoder. The combination of inner encoder, chan-nel, and inner decoder is referred to as the super channel. The alphabet sizes of the inner code and outer code are denoted as Mi and Mo, respectively.

Figure 3. The block diagram of a concatenated coding system




Several characteristics of concatenated codes are evident from Figure 3. First, the resulting concatenated code has overall length of nN channel symbols (Mi-ary) with kK information symbols per overall codeword and with overall code rate rR = kK/nN, where R = K/N and r = k/n. Although the overall length of the code is nN, the structure imposed by the concatena-tion allow decoding operation to be performed by two decoders for codes of length N and n, respectively. This allow a sig-nificant reduction is complexity over that which would be required to provide the same overall error rate with a single level of coding. The most natural choice for outer codes are Reed-Solomon (RS) codes. Because the RS codes, being maxi-mum-distance-separable codes, make highly efficient use of redundancy, and well suited to burst error correction. We will use RS codes as outer codes throughout this work.

The inner code we consider in this paper is error detecting or correcting binary (Mi = 2) block code. The inner code cor-rects ec errors and detects ed errors provided 2ec + ed < dmin, where dmin is the minimum distance of the inner code. When an error is detected, every symbol of the inner code is erased. There are, however, errors that are not detected nor corrected by the inner code, which results in errors at the output of the inner decoder. The purpose of the outer code is to correct the er-rors and erasures of the inner code.

Figure 4. M-ary erasures and errors channel model

When the inner code is used for detecting errors only, the receiver will decode symbols incorrectly if and only if a chan-nel-error pattern is the same as a nonzero codeword [5]. For a binary symmetric channel, the probability that a channel er-ror pattern will match any particular codeword of weight i in a code of length N is equal to pi(1-p)N-i. Thus the probability of undetected error is given by

1

1N

j N jj

jud A p pP

( ) (3)

where Aj is the weight distribution of the inner code. And the probability of corrected error is given by

1 Nc pP ( ) (4)

Thus the probability of detected error can be obtained as

1d ud cP P P . (5)

When an inner code is used for detecting errors, the super channel created by the inner code can be modelled by M-ary erasures and errors channel as in Figure 4. The purpose of the outer code is to correct the errors and erasures of the inner code. From the minimum distance property, the (n, k) Reed-Solomon code with bounded distance decoding can correct up to e = n k erasures or up to t = (n k)/2 errors. More generally, it can correct any combination of l erasures and m errors provided that 2m + l does not exceed n k. Thus the probability of overall block (an outer codeword) error, PE , for the memoryless channel of Figure 4 is given by




21e t n t eE d ud d ud

t e n k

nP P P P Pt e

( ), . (6)

where

n n

t e t e n t e

!

, ! ! !. (7)

4. Optimum Inner and Outer Code Rate In order to analyse the optimum rates, we approximate the block error probability PE in (6) as

2 2

1 2

4 4 4

d ud

Ed d ud ud d ud

knn

P PP Q

P P P P P P

. (8)

We will assume the probability of undetected error is much lower than that of detected error (i.e., PdPud ). This assumption is valid in most cases of practical interests. We can now ignore the Pud terms in (8) and obtain

1 dE

d

knn

PP Q

P

. (9)

Therefore, to minimize the block error probability, 1 d dk n P P ( / ) / should be maximized, which means Pd and k/n should both be minimized. This can be achieved by minimizing the outer code rate and maximizing the inner code rate with the overall code rate fixed. 5. Numerical Results and Discussions

In this paper, we examine the numerical results obtained by using parity check codes as inner codes. The weight distribu-tion of the parity check code is well known as [6]

= even

0 otherwisei

Ni

A i

,

,. (10)

Figure 5. The structure of the parity-check codes for varying the inner code length N




The structure of the parity-check codes is illustrated in Figure 5, where the inner code length N varies from 6 to 10. Based on the use of parity-check codes in Figure 5, the probability of undetected error Pud is given by

6

2,4,6

2 4

2,4

2

2 2

32 2

66, 1

3 47, 1 1 (1 ) 1 (1 )

2

38, 1 1 1 (1 )

2

39, 1 1 1 (1 )

2

9,

iiud

i

i iud

i

ud

ud

N P p pi

N P p p p pi

N P p p p

N P p p p

N

521 1udP p

. (11)

In Table 3.1 and 3.2, we have listed Pud , Pd and PE of (N, 5) parity check inner codes when overall cade rate rR = 0.3 and 0.5, respectively. We can see that the condition PdPud is satisfied, and the optimum point exists at N=6, i.e. when the redundancy of the inner code is minimum as has been predicted based on the theoretical analysis in Section 4. Therefore the distribution of redundancies between outer and inner code is optimum when the outer code has maximum redundancy and the inner code has minimum redundancy respectively.

Table 1. Pud , Pd and PE when inner codes are (N,5) parity check codes with over all code rate = 0.3. (Outer codes are (32, k) Reed-Solomon codes in Gaussian channel with p = 0.01)

N k Pud Pd PE

6 11 1.441e-03 5.708e-02 3.976e-17

7 13 8.849e-04 6.705e-02 1.228e-14

8 15 6.939e-04 7.656e-02 8.056e-12

9 17 5.969e-04 8.589e-02 3.465e-09

10 19 4.999e-04 9.512e-02 6.936e-07

Table 2. Pud , Pd and PE when inner codes are (N,5) parity check codes with over all code rate = 0.5. (Outer codes are (32, k) Reed-Solomon codes in Gaussian channel with p = 0.01)

N k Pud Pd PE

6 19 1.441e-03 5.708e-02 1.028e-08

7 22 8.849e-04 6.705e-02 9.413e-06

8 25 6.939e-04 7.656e-02 2.952e-03

9 28 5.969e-04 8.589e-02 1.429e-01

10 32 4.999e-04 9.512e-02 9.599e-01

6. Conclusion In this paper, we analysed the optimum inner and outer code rates for a concatenated code over Gaussian binary sym-

metric channels. We found that based on theoretical and numerical analysis, the optimum redundancy distribution is to al-locate the maximum redundancy to the outer code and correspondingly minimum redundancy to the inner code. This indi-cates that when the overall code rate of the concatenated codes is fixed, it is desirable to assign the maximum possible re-dundancy to the outer code to achieve the maximum overall coding gain in concatenated codes.




ACKNOWLEDGEMENT This study was supported by the Research Program funded by the Seoul National University of Science and Technology (2014-1699).

REFERENCES [1] G. D. Forney, Concatenated Codes, MIT Press, Cambridge, MA, 1966. [2] Z. Yang, M. Han, C. Pan, A. Men, and L. Yang, A novel scheme of coding and modulation for digital television terrestrial broad-

casting, Proc. of IEEE PIMRC, pp. 376-379, 2003. [3] T. Blahut, Theory and Practice of Error Control Codes, Addison Wesley Pub. Co, 1983. [4] G. C. Clark and J. B. Cain, Error Control Coding for Digital Communications, Plenum Press, New York, 1982. [5] R. Berlekamp, Algebraic Coding Theory, Aegean Park Press, 1984. [6] J. H. Wolf, et. All, On the probability of undetected error for linear block codes, IEEE trans. on Communions, vol. 30, pp. 317-324,

Feb. 1982.

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