Bit by bit row/column interleaving of block accumulate codes Sang Ik Han, J.P. Fonseka and E.M. Dowling A novel bit by bit row/column interleaver (BBRCI) design technique is proposed to significantly increase the minimum Hamming distance (MHD) of block accumulate codes (BACs), making them suitable for optical applications. The proposed BBRCI design method employs a separate row/column interleaver (RCI) for the transmission of each bit position of the outer block code. The numbers of rows of these individual RCIs are properly selected to increase the MHD of the resulting BAC and to reduce the frame size. It is demonstrated that BACs constructed with a BBRCI can be designed to have significantly higher MHD than similar BACs constructed with uniform interleavers (UI) and S type interleavers. Simulations show example BACs constructed with BBRCIs that perform significantly better than those BACs constructed with a UI.
Introduction: It is known that serially concatenated codes (SCCs) can perform better than parallel concatenated codes (PCCs) [1, 2]. SCCs are constructed by passing the message bits through an outer code, an interleaver, and an inner code in a serial manner to generate the coded sequence. It is known that the outer code is preferably a block code or a non-recursive convolutional code, while the inner code is preferably a recursive convolutional code [1]. Block accumulate codes (BACs), which fall under SCCs and employ an outer block code and an inner rate-1 recursive convolutional code (R1RCC) that has G(D) = [1/(1 + D)], which is also known as an accumulator [2], can generate powerful high rate codes [1, 2]. As with any SCC, the interleaver in a BAC can be designed to improve the performance. In particular, with an inner R1RCC, the interleaver can be designed to generate a high minimum Hamming distance (MHD), dBAC , for the resulting BAC. Many interleaver designs have been considered for concatenated codes in the literature, however, mostly with PCCs, and some with SCCs [1, 3]. If a uniform interleaver (UI) is used with a BAC, dBAC is limited to ⌈do /2⌉, as any codeword of the outer code with weight 2⌈do /2⌉ can generate a weight-⌈do /2⌉ coded sequence after the inner R1RCC, where do is the MHD of the outer block code. Similarly, S type interleavers [2, 3] perform well with PCCs, but only generate a MHD of do if used with a BAC. This is because they place the nonzero bits of two nonzero codewords each with weight do in pairs at the input of the R1RCC generating a weight-do sequence in order not to violate the spread constraint imposed by the S type interleaver. Hence, it is important to find good interleaver design methods for BACs that can increase their dBAC value thereby lowering their error floor and making them applicable for low error rate applications such as in optical communications. In this Letter, we propose a bit by bit row/column interleaver (BBRCI) design technique that focuses on increasing dBAC .
maintain a high weight of the output sequence corresponding to any combination of two nonzero codewords of the outer code, it is additionally necessary to vary the order of the transmission of codewords among different bit positions by varying the number of rows of the RCIs used for different bit positions. Hence, during the transmission of any pth coded bit position, the BBRCI employs a RCI with L p rows and Nc /L p columns used to determine the order of transmission of the Nc codewords. Specifically, the RCI of the pth bit is filled by integers 1 to Nc in an orderly manner along rows and reads the interleaved sequence out along columns from bottom to top starting from the first column. For example, Fig. 1 illustrates the order of transmission of the four coded bits of 12 codewords (which are denoted by numbers 1 to 12) using a BBRCI with L = [1, 3, 4, 6]. Without loss of generality, we consider all L j values in the ascending order L j ≤ L j+1 for j = 1, 2, . . . , (n − 1). Since the number of columns, Nc /L p , of every pth individual RCI has to be an integer, the value of Nc has to be an integer multiple of every L p value. To minimise the interleaver size, we choose Nc = LCM (L1 , L2 , . . . , Ln )
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The selection of the L p values, which can be expressed in the form of a vector L = (L1 , L2 , . . . , Ln ), and the corresponding value of Nc in (1) uniquely define the BBRCI. Considering the minimum weight generated by two adjacent nonzero codewords of the outer code, the MHD of a BAC with a BBRCI, dBAC−BBRCI , can be bounded by dBAC−BBRCI ≤
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Bit by bit row/column interleaver design: The proposed BBRCI is developed based on the observation that an R1RCC output starts a sequence of 1s at a 1 in the input sequence and ends at the next 1 in the input sequence. With that observation, a BBRCI is designed to ensure a high weight of the coded sequence of a BAC corresponding to all input sequences with one and two nonzero codewords of the outer block code. By doing so, a BBRCI creates the possibility of generating a high MHD of the BAC that is the minimum weight of all its coded sequences corresponding to all combinations of any number of nonzero codewords of the outer code. Even though it is not guaranteed in all cases, it is demonstrated here that the proposed BBRCI design, which focuses on cases with one and two nonzero codewords of the outer code, can indeed generate BACs with high MHD values. Let us consider a BBRCI design of nNc bits generated by Nc codewords of the outer (n, k) block code with MHD do. To ensure that any single nonzero codeword of the outer code generates a high weight coded sequence, the BBRCI spreads out coded bits of every codeword by (a) transmitting each coded bit position of all codewords together starting from the set of first bits of all codewords followed by the set of second bits of all codewords and so on up to the set of the last nth bit of all codewords, and (b) employing a row/column interleaver (RCI) at each bit position. At high values of Nc found in practice, the above design features (a) and (b) can generate a very high weight for all single nonzero codewords of the outer code. To simultaneously
In the numerical results, we present examples of three separate BBRCI designs with three separate outer codes along with their corresponding L sequences that can reach the bound in (2). It is noted that, even though the bound in (2) cannot be reached for all outer code and L sequence combinations, the BBRCI can generate a high dBAC−BBRCI value in many cases, specifically for the outer code with a large MHD. To reach the MHD bound in (2), it may sometimes be necessary to add a padding of zeros with Nzero number of zeros after the transmission of every pth bit position before moving to the ( p + 1)th bit position. The value of Nzero depends on dBAC−BBRCI and do. The challenge in the BBRCI design for a given (n, k) outer code is to find a set of values L j , j = 1, 2, . . . , n, without drastically increasing the size of the interleaver n(Nc + Nzero ). In this Letter, we present examples of attractive BBRCI designs with reasonable interleaver sizes and demonstrate that such designs can perform significantly better than uniform interleaving (UI). Further, in order to limit the size of the interleaver, the BBRCI is more suitable for small to medium size outer codes. Numerical results: Table 1 lists the properties of three BACs constructed using a BBRCI that reaches the MHD bound in (2). It is seen that a BBRCI can generate BACs with much higher MHD values than those created using a UI (which is ⌈do /2⌉) or an S type interleaver (which is do) while maintaining reasonable frame sizes. Among the
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three schemes listed in Table 1 with a BBRCI, the first scheme with the (7,6) SPC outer code requires zero padding with Nzero = 1 to reach the bound in (2) (without which MHD drops to 3), while the other two schemes with the (7,4) and the (8,4) outer codes do not require any zero padding to reach (2). To examine their performance, Fig. 2 shows the bit error rate (BER) variation of these BACs with Eb /N0 over an additive white Gaussian noise channel with two-sided power spectral density N0 /2, where Eb is the bit energy. For comparison, the performance variations of the same codes with a UI at the same interleaver sizes are also plotted in Fig. 2. It is seen in Fig. 2 that BACs with a BBRCI do not suffer from error floor due to their higher MHD while those with a UI suffer from the error floor effect due to their much lower MHD. As a result, in all cases presented in Fig. 2, a BBRCI can achieve significantly higher coding gain than a UI and that gain increases as the operating BER lowers.
© The Institution of Engineering and Technology 2013 15 March 2013 doi: 10.1049/el.2013.0876 Sang Ik Han and J.P. Fonseka (The University of Texas at Dallas, Richardson, TX 75080, USA) E-mail:
[email protected]
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E.M. Dowling (Trellis Phase Communications, Marshall, TX 756704281, USA) References
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Conclusions: We have proposed a novel bit by bit row/column interleaver design technique that can significantly increase the MHD of BACs over BACs constructed with UI and S type interleavers. We have constructed BACs with the BBRCI interleaver design technique and demonstrated via simulation that they can perform significantly better than those constructed with a UI, particularly at low error rates for applications such as in optical transmissions. Even though the proposed interleaver design technique has been tailored for BACs, it can be used with other serial concatenations with suitable modifications.
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1 Benedetto, S., Divsalar, D., Montorsi, G., and Pollara, F.: ‘Serial concatenation of interleaved codes: performance analysis, design, and iterative decoding’, IEEE Trans. Inf. Theory, 1998, 44, (3), pp. 909–926 2 Ryan, W.E., and Lin, S.: ‘Channel codes: classical and modern’ (Cambridge University Press, 2009) 3 Daneshgaran, F., Laddomada, M., and Mondin, M.: ‘Interleaver design for serially concatenated convolutional codes: theory and application’, IEEE Trans. Inf. Theory, 2004, 50, (6), pp. 1177–1188
Fig. 2 Numerical results of BACs
Table 1: Properties of BACs constructed with BBRCI Outer code do L dBAC−BBRCI (7,6) SPC 2 [1, 3, 4, 5, 9, 15, 36] 4 (7,4) BCH 3 [2, 3, 7, 10, 12, 60, 105] 12 (8,4) eBCH 4 [2, 3, 7, 10, 12, 60, 105, 140] 22
Nc Nzero 180 1 420 0 420 0
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