6.4 Performance Evaluation of BICM via Gaussian Approximations

Two major handicaps of the numerical integration we discussed at the end of Section 6.3.1 were removed in Sections 6.3.2 and 6.3.3. However, the problem of a direct connection between the PEP and the parameters of the constellation and/or labeling still remains. In order to solve this problem, in this section we use the simplified Gaussian forms for the PDF we derived in Section 5.5.

6.4.1 PEP Calculation using Gaussian Approximations

For PAM and PSK constellations labeled by the BRGC, we showed in Section 5.5 that we may use Gaussian functions to approximate the PDF of the L-values conditioned on the transmitted symbols. More specifically, for a given transmitted symbol , we have

6.368

where , and depend on the transmitted symbol , bit position , constellation , labeling , and adopted approximation model (consistent model (CoM) or zero-crossing model (ZcM)). The mean values and variances and in (6.368) are given in Tables 5.1 and 5.2 for PAM and PSK, respectively.

The PDF of the L-values conditioned on the transmitted bit can then be obtained via marginalization, i.e.,

6.369

6.370

where (6.370) follows from (6.368) and (2.77). In (6.370) we use a factor to take into account that the same PDF approximation will be obtained in each of “groups” of symbols in the constellation labeled by the BRGC for a given (see more details in Section 5.5.4).

By inspecting Tables 5.1–5.2, it is possible to see that for a given , the possible values of and obtained for include those obtained for , which in turn include those obtained for , and so on. In otherwords, the set of mean and variances obtained for covers the mean and variances for . Thus, for simplicity, and assuming there are at most different Gaussian PDFs, we define and as the mean and variance of the th Gaussian PDF. Furthermore, we assume , and .¹⁴ We can then express the PDFs (6.370) as the following Gaussian mixture

6.371

where the proportion of th Gaussian PDFs in the mixture, denoted by , can be interpreted as the probability that the L-value is distributed according to the th Gaussian PDF.

We can gather the weighting factors in a matrix

6.374

The parameters of the th Gaussian PDF uniquely depend on the ED between the symbol and the closest symbol in , and therefore, the elements of may be seen as a generalization of the EDS defined in Section 2.5.1. We will thus refer to as a (normalized) constellation bit-wise Euclidean distance spectrum (CBEDS).

Under the quasirandom interleaving assumption, the PDF in (6.227) can be expressed using (6.371) as

6.376

6.377

where

6.378

has a meaning of the probability that an L-value passed to the decoder is distributed according to the th Gaussian PDF. Later in this section we compute for PAM and PSK constellations.

With a closed-form approximation for the PDF of in (6.377), we are ready to compute the PDF of in (6.283). This is done as follows:

6.379

6.380

6.381

where (6.381) follows from reorganizing the terms in (6.380), and where in denotes the number of L-values distributed according to the th Gaussian PDF.

Using (6.381) in (6.282) we find

6.382

6.383

The PEP approximation in (6.383) is in closed-form; however, for large values of and/or , the enumeration of all the terms in (6.383) becomes tedious. This can be simplified by taking only the Q-function with the smallest argument, which is an approximation that will be tight for . This dominant Q-function is obtained for , i.e., when all the L-values are distributed according to the Gaussian PDF with the smallest mean value (). The PEP in (6.383) is then approximated as

6.384

Finally, from (6.225) we obtain a closed-form approximation for the BEP in BICM

6.385

where can be evaluated using either (6.383) or (6.384). To evaluate (6.385), we need the IDS of the binary encoder, the weighting coefficients and the parameters of the Gaussian approximations and . In the following, we particularize the results in this section to PAM and PSK constellations.

6.4.2 MPAM Constellations

For PAM labeled by the BRGC, we have and by generalizing Example 6.33, we obtain

6.386

for . The mean values and variances are obtained from Table 5.1 as

6.387

6.388

and in (6.378) is

6.389

6.390

6.391

where .

Using (6.391) we express (6.383) as

6.392

for CoM and

6.393

for ZcM.

The simplified PEP approximation in (6.384) (the same result is obtained by applying CoM and ZcM) for PAM is then given by

6.394

6.395

where we used (2.47). We can thus conclude that an increase on the size of the constellation reduces the multiplicative factor before the Q-function. More importantly, an increase of by one is equivalent to decreasing the SNR by . This SNR shift will dominate the behavior of the BEP at high SNR.

The following example show the approximations for particular values of .

6.4.3 MPSK Constellations

In the case of PSK, we know that and from Table 5.2 we read

6.402

6.403

and knowing that the PDF of is the same as the PDF of (see also Example 5.7), we have

6.404

and

6.405

for .

In analogy to (6.391) we can find as

6.406

6.407

6.408

where .

The results above show that in the case of PSK constellations, the L-values can again be approximated as a Gaussian mixture, where the parameters of the Gaussian PDFs as well as the weights are known in closed form. Using these closed-form expressions, it is possible to repeat the analysis we presented before, which we do not include here as it is mostly a repetition of the developments for PAM in Section 6.4.2.

6.4.4 Case Study: BEP for Constellation Rearrangement

We consider here transmission with the so-called constellation rearrangement (CoRe), which is used in hybrid automatic repeat request (HARQ). When errors are detected in the received codeword, the same codeword is retransmitted, but the binary labeling of the constellation is changed. The constellation is therefore “rearranged”, hence the name CoRe. In what follows we briefly outline the principles of CoRe in the case of 4PAM; this is equivalent to using 16QAM labeled by the BRGC.

We will use the Gaussian model of the L-values we showed in Example 5.19. More specifically, we reorganize the term from (5.197) and (5.198) and explicitly condition on the bits and :

6.409

6.410

Using (6.409) and (6.410), we make two key observations:

• The L-value has a “high-protection” distribution if and a “low-protection” distribution if .
• The L-value has always a low-protection distribution, irrespective of the value of the transmitted bits.

CoRe can be then seen as a process that equalizes the “protection” experienced by the bits passing through the different bit positions in different transmissions. This is possible because in each transmission the same bits and are transmitted. More specifically, CoRe is based on two operations: negation of the bit at position (i.e., negation of the second row of the matrix ) and swapping the position of the labels of and (i.e., swapping the first and second row of ).

The negation operation is connected with the first observation we made above: as depending on the value of the L-value changes its distribution, using in the first transmission and in the next one, we can guarantee that “high” protection is offered to the bit in one out of the two transmissions.

The swapping responds to the second observation we made above, and is meant to transmit the bits at position so that can take advantage of the “high” protection offered in that bit position.

At the receiver, the L-values for different transmissions are calculated, negated and/or swapped (if necessary), and then added to form aggregated CoRe L-values .¹⁵ These L-values are then passed to the decoder. After transmissions, we obtain the following distributions for

6.411

and for

6.412

The PDF of the L-values passed to the decoder is then given by

6.413

6.414

where

Since the PDF in (6.414) is again a Gaussian mixture, an approximation similar to the one in (6.398) may be used

6.415

In Fig. 6.23 the results obtained via numerical simulations are contrasted against the approximations of the PEP in (6.392) and (6.393). Unlike in Fig. 6.22, the differences between ZcM and CoM are now very clear. The ZcM provides a tight approximation on the coded BEP, especially when the number of transmissions increases. In particular, for , and have the same PDF and and have to be used in the model. Thus, HARQ accentuates the importance of the adequate modeling of the “high-protection” effect. Note that without HARQ and CoRe, the effect of “high-protection” is less pronounced and can be even neglected, e.g., using the one-term “low-protection” approximation (6.399). In the presence of CoRe we cannot do this because for we have , i.e., the “low-protection” is entirely removed.

6.5 Bibliographical Notes

Performance evaluation in uncoded transmission has been the focus of research for many decades. The initial approximations of the SEP [1] were later replaced by calculations for regularly spaced constellations QAM and/or PSK [2 3]. In this chapter we showed expression for . The calculation of the BEP for 3D constellations was considered in [4].

The BEP for uncoded transmission in (6.28) has been studied in detail in [5 6], where the asymptotic optimality of the BRGC for regular constellations is proved. Significant efforts have been made to evaluate the BEP in fading channels, i.e., to average the expressions for the AWGN channel over the fading distribution. This was considered, e.g., in [7–10]. Some of the formulas we presented in this chapter were shown, for integer in [11, eq. (6)], for half-integer in [12, eq. (15)], and for arbitrary (using hypergeometric functions) in [13]. The literature is abundant in this area, so it is in fact quite difficult to recognize all the contributions. For a hopefully more complete list, we refer the reader to [14].

An alternative representation of the bivariate Q-function (6.99) was shown in [15] and the identity (6.102), simplified with respect to [16, eq. (11)], via the use of Q-functions. The form (6.99) is called Craig's form after the author of [17], who derived first a simple alternative form of Q-function we showed in (6.108).

The error event appearing in (6.173) is sometimes called “first-error event” [18, Section 6.2], [19, Section I], [20, Section 12.2] or error probability per node [21, Section 4.3]. The upper bound on the WEP in the case of the TCM transmissions can be found in [18, eq. ((6.6))], [20, eq. (12.20)], [22, eq. ((4.1))]. The expressions for the WEP in (6.178) and (6.241) are straightforward generalizations of the bound presented in [23] for CCs. TCM encoders with optimal distance spectrum (similar to the ones we used in Example 6.16) were presented in [24].

The expression in (6.243) is the most common expression for the upper bound for BICM, cf. [25, eq. (26)], [26, eq. (4.12)]. The upper bound in (6.243) can be found in almost any existing book on digital communications or coding (see, e.g., [20, eq. (12.28)], [27, eq. (7.9)], [28, eq. ((8.2)–19)], and it was originally defined for channels in which the metrics for the code bits passed to the decoder are i.i.d., e.g., 2PAM over the AWGN channel, in which the conditional L-values follow a Gaussian distribution as shown in(3.63).

The performance analysis of BICM transmission under random infinite-length interleaving proposed in [25] has been widely adopted in the literature. As we have seen, this analysis can be used in the case of a fixed interleaver if the assumptions of quasirandomness are fulfilled. On the other hand, finite-length interleaving has received much less attention. PEP calculations for infinite-length (but random) interleaving have been presented in [26, Chapter 4.3].

A formal analysis of the relationship between the spectrum of the code, the finite-length interleaving, and the performance in terms of WEP/BEP still seems to be missing in the literature. However, while this issue may be interesting from a theoretical point of view, its practical importance is often negligible as we argued in Section 6.2.5. This is particularly true for capacity-approaching codes such as TCs, for which we can eliminate the finite-length related terms from the WEP expression in (6.214) (see Lemma 6.19). This follows from the fact that the spectrum of such codes decreases with , which has been shown, e.g., in [29, Fig. 10].

Insights into the gains of BICM over TCM were first shown in [30] via bounding techniques. In [31] the PEP was evaluated via direct/inverse Laplace transforms. This idea was refined in [32 33]. The formal derivation of the SPA we presented in Section 6.3.2 may be found in [34, Chapter 2], and the intuitive approach we showed was presented in [33, Appendix I]. The SPA was used for PEP evaluation in [33], where Monte Carlo integration was suggested to calculate the MGF and its derivatives. The SPA was then used in [35] for 2PAM and fading channels and later reused in [36 37], where closed-form formulas were obtained thanks to the analytical description of the PDFs of the L-values. The use of (zero-crossing or consistent) Gaussian approximations to simplify the integration was made popular in [38]. The zero-crossing approximation is due to [39], where it was first applied to analyze uncoded HARQ transmission based on the CoRe.

The CBEDS we used in this chapter was first introduced in [40 41] where all the binary labelings for 8PSK having a different CBEDS were classified. The same concept was also presented in [42, Chapter 4]. The CBEDS in fact corresponds to a generalization of the ED spectrum of [43] in the sense that it considers the bit positions separately.

Mapping diversity has been studied, e.g., in [44–47]. CoRe was recommended by the third-generation partnership project (3GPP) working group because of its simplicity [48] and is only slightly suboptimal when compared to metrics calculation based on the outcomes of all transmissions (as required in other mapping diversity schemes). Moredetails about CoRe may be found in [49]. Various mapping diversity schemes are analyzed from an information-theoretic point of view in [50].

The WD and the IWD we used for CENCs in Example 6.22 can be extended to turbo encoders (TENCs) using the concept of uniform and random interleaver introduced in [51 52]. For the numerical results in this chapter, we used a breadth-first search algorithm [53]. Alternatively, a transfer function approach could be used, which works well for small values of memories . For large values of the Bayesian evolutionary analysis by sampling trees (BEAST) algorithm recently introduced in [54] (see also [55]) is more appropriate.

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