6.2.4 BICM Decoder

The decoding metric in (6.129) for the BICM decoder is defined in (3.22), and thus,

6.183 equation

where

6.184 equation
6.185 equation

The PEP in (6.139) is then calculated as

6.186 equation
6.188 equation

where c06-math-0659 are the random variables modeling the L-values c06-math-0660, c06-math-0661 was expressed as c06-math-0662, and where, because of the linearity of the code, the error codeword c06-math-0663 satisfies c06-math-0664.

For any c06-math-0665 such that c06-math-0666 (i.e., when c06-math-0667), the L-values c06-math-0668 do not affect the PEP calculation. Therefore, there are only c06-math-0669 pairs c06-math-0670 that are relevant for the PEP calculation in (6.187), and the PEP depends solely on c06-math-0671 and c06-math-0672. We can thus write

6.189 equation

where

and where c06-math-0676 and c06-math-0677, c06-math-0678 are reindexed versions of c06-math-0679, and c06-math-0680, respectively. The reindexing enumerates only the elements with indices c06-math-0681 for which c06-math-0682, i.e., when c06-math-0683.

Moreover, knowing that the bits c06-math-0684 are an interleaved version of the bits c06-math-0685 and the L-values c06-math-0686 are obtained via deinterleaving of c06-math-0687, we can write

where c06-math-0689 and c06-math-0690 are the interleaved/deinterleaved versions of c06-math-0691 and c06-math-0692, respectively.

The main reason to use (6.192) instead of (6.191) is that we already know how to model the L-values c06-math-0698 (see Chapter 5). We now have to address the more delicate issue of the joint probabilistic model of the L-values c06-math-0699. The situation which is the simplest to analyze occurs when the L-values c06-math-0700 are independent, which we illustrate in Fig. 6.13 (a). In this case, the L-values c06-math-0701 are obtained from c06-math-0702 different channel outcomes c06-math-0703. A more complicated situation to deal with arises when we can find at least two indices c06-math-0704 such that c06-math-0705 and c06-math-0706 correspond to c06-math-0707 and c06-math-0708, i.e., when c06-math-0709 and c06-math-0710 are L-values calculated at the same time c06-math-0711 (and thus, they are dependent). The case when the L-values are independent as well as the requirements on the interleaver for this assumption to hold are discussed below. The case where the L-values are not independent is discussed in Chapter 9.

c06f013

Figure 6.13 Codewords c06-math-0693 corresponding to different interleaved error codewords c06-math-0694. The L-values c06-math-0695 in (6.192) (indicated by rectangles) are (a) independent or (b) dependent random variables. The L-values, which are calculated using the same c06-math-0696, i.e., at the same time c06-math-0697, are indicated with dark-gray rectangles

In what follows we provide a lemma that will simplify the discussions thereafter.

Lemma 6.17 allows us to simplify the WEP expression in (6.141) as follows:

6.205 equation
6.206 equation
6.207 equation

where the approximation in (6.208) follows from the truncation we make in the number of Hamming weights (HWs) to consider (c06-math-0749). To transform (6.208) into a more suitable form we use a simple observation given by the following lemma.

To illustrate the result of Lemma 6.18 we show in Fig. 6.14 two error codewords c06-math-0762 with the same HW c06-math-0763 interleaved into codewords with different GHWs c06-math-0764.

c06f014

Figure 6.14 The codewords c06-math-0765 with the same HW c06-math-0766 are interleaved into codewords with different GHWs c06-math-0767: (a) GHW c06-math-0768 and (b) GHW c06-math-0769

We now develop (6.208) as

6.212 equation
6.213 equation

where (6.214) follows from (6.209) and the third condition of the quasirandom interleaver in Definition 2.46, and where

The condition of having the spectrum c06-math-0778 bounded for given c06-math-0779 and growing c06-math-0780 is satisfied for turbo codes (TCs) (in fact, for TCs c06-math-0781 actually decreases with c06-math-0782). On the other hand, for convolutional codes (CCs), the spectrum c06-math-0783 may increase with c06-math-0784, but as discussed in Section 6.2.5, we may still neglect the effect of c06-math-0785 for high SNR and large c06-math-0786. Thus, to streamline further discussions, from now on we do not consider c06-math-0787 in (6.214). Furthermore, if we assume the first condition of quasirandomness in Definition 2.46 to be satisfied, we can express (6.214) as

Using steps similar to those that lead to (6.208) and then to (6.214), we can develop (6.144) as follows:

6.219 equation
6.221 equation

where c06-math-0794 in (6.220) is the input sequence corresponding to the codeword c06-math-0795, c06-math-0796 is the generalized input–output distance spectrum (GIODS) (see Definition 2.37), and to pass from (6.222) to (6.223) we used (2.114).

We note that (6.218) and (6.223) hold for any interleaver that satisfies the first and the third conditions in Definition 2.46. The second condition of quasirandomness is not exploited yet and, as we show in the following theorem, it simplifies the analysis further.

The key difference between the bounds for the ML decoder in (6.169) and (6.171) with those for the BICM decoder in (6.224) and (6.225), is that the performance of BICM depends on the DS of the binary code c06-math-0819, and not on the DS of the code c06-math-0820. Further, because of the numerous approximations we made, the expressions (6.224) and (6.225) are not true bounds, but only approximations. Nevertheless, we will later show that these approximations indeed give accurate predictions of the WEP and BEP for BICM.

The somewhat lengthy derivations above may be simplified using the model of independent and random assignment of the bits c06-math-0821 to the positions of the mapper in Fig. 2.24, and assuming that all codewords achieve their maximum diversity. Then, c06-math-0822 is immediately obtained as the average of the PDF of the L-values over the different bit positions. Our objective here was to show that, in order to use such a random-assignment model, the interleaver (which is a deterministic element of the transceiver) must comply with all conditions of the quasirandomness in Definition 2.46.

The quasirandomness condition is not automatically satisfied by all interleavers. For example, we showed in Example 2.47 that for a convolutional code (CC), a rectangular interleaver does not satisfy the second condition of quasirandomness, which allowed us to replace c06-math-0823 by c06-math-0824 in (6.228). Therefore, while the first and the third condition of the quasirandomness hold for the rectangular interleaver and we may apply(6.218),10 we cannot use Theorem 6.20 in this case. The importance of the second condition of the quasirandomness is illustrated in the following example.

c06f015

Figure 6.15 The analytical bound obtained using (6.225) and (6.236) (solid line) and the simulation results (markers) for pseudorandom and rectangular interleavers

For the remainder of this chapter, we use only pseudorandom interleavers, and thus, assume the conditions of quasirandomness in Definition 2.46 are fulfilled. In analogy to Example 6.15, the following example shows the performance of BICM with CENCs.

The IWD of the CENC c06-math-0875 in (6.245) is the total HW of all input sequences that generate error events with HW c06-math-0876, where an error event is defined as a path in the trellis representation of the code that leaves the zero state and remerges with it after an arbitrary number of trellis stages. In the following example, we show how to calculate this IWD.

c06f016

Figure 6.16 CENC c06-math-0874: (a) encoder and (b) state machine

We note that the enumeration of the diverging and merging paths in the trellis we analyzed in Example 6.23 does not consider all the codewords with HW c06-math-0898 because we do not take into account paths that diverged/merged multiple times. Assigning a codeword c06-math-0899 to the c06-math-0900th diverging/merging path, we see that all error codewords are mutually orthogonal and their sum c06-math-0901 is also a validcodeword. Thus, they comply with the conditions we defined in Section 6.2.2, which lets us expurgate the codeword c06-math-0902 from the bound. In other words, the enumeration of the uniquely-diverging codewords provides us with an expurgated bound. Therefore, for CENCs we have

6.254 equation
6.255 equation

where the approximation is due to the expurgation.

Since the performance of the receiver is directly affected by the characteristics of the CENC (via c06-math-0905 and c06-math-0906), efforts should be made to optimize the CENC. In fact, the CENCs we show in Table 2.1 are the encoders optimal in terms of the IWD c06-math-0907. To find them, an exhaustive search over the CENC universe c06-math-0908 is performed: first, the free Hamming distance (FHD) c06-math-0909 is maximized and next, the corresponding term c06-math-0910 is minimized. If multiple encoders with the same c06-math-0911 and the same c06-math-0912 are found, those that minimize c06-math-0913 are chosen, then c06-math-0914 is analyzed, and so on. We refer to these optimized CENCs shown in Table 2.1 as “ODS CENCs” even if the literature uses the name “ODS codes.” We use the name ODS CENCs because c06-math-0915 depends on the encoder and not only on the code.

c06f017

Figure 6.17 Simulation results (circles) are compared to the bounds (6.263) and (6.266) (solid lines) and the approximations (6.244) and (6.242) for the ODS CENC with c06-math-0940; the WEP results are shown for two values of c06-math-0941. The bounds and approximations are only shown in the range of SNR where (6.259) is guaranteed to converge, i.e., for c06-math-0942

6.2.5 BICM Decoder Revisited

To obtain the results in Theorem 6.20, we removed from (6.214) the contributions of the codewords that do not achieve their maximum diversity. Here, we want to have another look at this issue, but instead of proofs, we outline a few assumptions that hold indeed in practice and are sufficient to explain why we can neglect the factor c06-math-0943 in (6.214).

To explicitly show the dependence of the PEP on the SNR, we use c06-math-0944 to denote the PEP in (6.226), c06-math-0945 to denote the PEP in (6.193), and c06-math-0946 to denote (6.210).

We then assume that c06-math-0947 is monotonically decreasing in both arguments, i.e.,

In other words, we assert that increasing the HD between the codewords or increasing the SNR decreases the PEP.

Now we are able to relate (6.268) to the PEP expression c06-math-0950. In general, c06-math-0951, but because c06-math-0952 is finite, for any c06-math-0953 and c06-math-0954, and thanks to (6.268), we can always find c06-math-0955 such that

where c06-math-0957.

Further, we assume that if c06-math-0958 is a “subcodeword” of c06-math-0959,12 i.e., c06-math-0961, and c06-math-0962 is such that it achieves its maximum diversity, i.e., c06-math-0963, then

where c06-math-0965 and where (6.270) holds for any c06-math-0966 and any c06-math-0967.

Since c06-math-0968, where c06-math-0969 is the sum of L-values corresponding to the c06-math-0970 nonzero bits in c06-math-0971, to obtain c06-math-0972 we may need to remove some elements from c06-math-0973. Thus, the expression in (6.270) shows that removing one (or more) L-values from the sum c06-math-0974 yields a larger PEP. The intuition behind such an assertion is the following: each L-value contributes to improve the reliability of the detection so removing any element should produce a larger PEP. When the sum c06-math-0975 is composed of i.i.d. L-vales c06-math-0976, this can be better understood from Sections 6.3.3 and 6.3.4, where we showed that an upper bound on the PEP is determined by the exponential of the sum of the cumulant-generating functions (CGFs) corresponding to each of the L-values. As the CGFs are negative, removing one L-value corresponds to increasing the bound. On the other hand, if some of the L-values in c06-math-0977 are not i.i.d., our assertion is slightly less obvious. We treat such a case in more detail in Chapter 9, where we show that adding dependent L-values decreases the PEP, and thus, removing L-values from the sum must increase the PEP.

Combining (6.270) and (6.269) we have that for any c06-math-0978 and c06-math-0979, we can find c06-math-0980 such that

6.272 equation
6.273 equation
6.274 equation

where c06-math-0985 and where we used the fact that we can always find a codeword c06-math-0986 with interleaving diversity c06-math-0987 satisfying c06-math-0988 (see (2.116)).

We further make a simple extension to the conditions of quasirandomness we introduced in Section 2.7. Namely, we assume that if the interleaver is quasirandom, then the diversity efficiency c06-math-0989 decreases monotonically with c06-math-0990, i.e.,

This property holds for random interleavers as we showed in (2.127). Here we extend it to the case of quasirandom (but fixed) interleavers, as we already did with other properties of the interleavers in Definition 2.46.

Let us recall now the formula for the WEP given by (6.224), where we reconsider the term c06-math-0992 in (6.215). Then, the WEP can be expressed as

where

6.277 equation

denotes the WEP approximation in (6.224), which ignores the codewords that do not achieve maximum diversity.

The WEP in (6.276) can be expressed as

where c06-math-0999. In order to obtain (6.278) we used (6.271) and to obtain (6.279) we used (2.117). To obtain (6.280) we used (6.267) and (6.268), where c06-math-1000 should be sufficiently small so as to compensate for the increase of the first argument of the PEP (from c06-math-1001 to c06-math-1002). Here we also used (6.275), which implies that c06-math-1003.

From (6.281) we conclude that for sufficiently high SNR, we can neglect the term c06-math-1004 provided that we increase c06-math-1005 so as to make the term c06-math-1006 sufficiently small. This condition is fulfilled when using quasirandom interleavers. An alternative interpretation of this result is that the true WEP for a finite c06-math-1007 is formed by two terms, where the second term c06-math-1008 corresponds to a WEP obtained for an equivalent SNR c06-math-1009. This factor might be relevant for the evaluation of the WEP for small values of c06-math-1010.

It is worth noting that all the considerations in this section may be repeated in the case of a BEP analysis, as well as for fading channels. For the former, the PEP should be weighted by the number of bits in error, and for the latter, c06-math-1011 should be replaced by c06-math-1012.

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