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 c06-math-1293PAM and c06-math-1294PSK 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 c06-math-1295, we have

where c06-math-1297, c06-math-1298 and c06-math-1299 depend on the transmitted symbol c06-math-1300, bit position c06-math-1301, constellation c06-math-1302, labeling c06-math-1303, and adopted approximation model (consistent model (CoM) or zero-crossing model (ZcM)). The mean values and variances c06-math-1304 and c06-math-1305 in (6.368) are given in Tables 5.1 and 5.2 for c06-math-1306PAM and c06-math-1307PSK, respectively.

The PDF of the L-values conditioned on the transmitted bit c06-math-1308 can then be obtained via marginalization, i.e.,

6.369 equation

where (6.370) follows from (6.368) and (2.77). In (6.370) we use a factor c06-math-1311 to take into account that the same PDF approximation will be obtained in each of c06-math-1312 “groups” of symbols in the constellation labeled by the BRGC for a given c06-math-1313 (see more details in Section 5.5.4).

By inspecting Tables 5.15.2, it is possible to see that for a given c06-math-1314, the possible values of c06-math-1315 and c06-math-1316 obtained for c06-math-1317 include those obtained for c06-math-1318, which in turn include those obtained for c06-math-1319, and so on. In otherwords, the set of mean and variances obtained for c06-math-1320 covers the mean and variances for c06-math-1321. Thus, for simplicity, and assuming there are at most c06-math-1322 different Gaussian PDFs, we define c06-math-1323 and c06-math-1324 as the mean and variance of the c06-math-1325th Gaussian PDF. Furthermore, we assume c06-math-1326, and c06-math-1327.14 We can then express the PDFs (6.370) as the following Gaussian mixture

where the proportion of c06-math-1332th Gaussian PDFs in the mixture, denoted by c06-math-1333, can be interpreted as the probability that the L-value c06-math-1334 is distributed according to the c06-math-1335th Gaussian PDF.

We can gather the weighting factors c06-math-1353 in a matrix

6.374 equation

The parameters of the c06-math-1355th Gaussian PDF uniquely depend on the ED between the symbol c06-math-1356 and the closest symbol in c06-math-1357, and therefore, the elements of c06-math-1358 may be seen as a generalization of the EDS c06-math-1359 defined in Section 2.5.1. We will thus refer to c06-math-1360 as a (normalized) constellation bit-wise Euclidean distance spectrum (CBEDS).

c06f021

Figure 6.21 8PAM constellation labeled by the BRGC. The EDs that are relevant from the point of view of obtaining the CBEDS c06-math-1367 are shown for the subconstellation c06-math-1368 for (a) c06-math-1369, (b) c06-math-1370, and (c) c06-math-1371

Under the quasirandom interleaving assumption, the PDF c06-math-1372 in (6.227) can be expressed using (6.371) as

6.376 equation

where

has a meaning of the probability that an L-value passed to the decoder is distributed according to the c06-math-1376th Gaussian PDF. Later in this section we compute c06-math-1377 for c06-math-1378PAM and c06-math-1379PSK constellations.

With a closed-form approximation for the PDF of c06-math-1380 in (6.377), we are ready to compute the PDF of c06-math-1381 in (6.283). This is done as follows:

6.379 equation

where (6.381) follows from reorganizing the terms in (6.380), and where c06-math-1385 in c06-math-1386 denotes the number of L-values distributed according to the c06-math-1387th Gaussian PDF.

Using (6.381) in (6.282) we find

6.382 equation

The PEP approximation in (6.383) is in closed-form; however, for large values of c06-math-1390 and/or c06-math-1391, 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 c06-math-1392. This dominant Q-function is obtained for c06-math-1393, i.e., when all the c06-math-1394 L-values are distributed according to the Gaussian PDF with the smallest mean value (c06-math-1395). The PEP in (6.383) is then approximated as

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

where c06-math-1398 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 c06-math-1399 and the parameters of the Gaussian approximations c06-math-1400 and c06-math-1401. In the following, we particularize the results in this section to c06-math-1402PAM and c06-math-1403PSK constellations.

6.4.2 MPAM Constellations

For c06-math-1404PAM labeled by the BRGC, we have c06-math-1405 and by generalizing Example 6.33, we obtain

6.386 equation

for c06-math-1407. The mean values and variances are obtained from Table 5.1 as

and c06-math-1410 in (6.378) is

6.389 equation
6.390 equation

where c06-math-1414.

Using (6.391) we express (6.383) as

for CoM and

for ZcM.

The simplified PEP approximation in (6.384) (the same result is obtained by applying CoM and ZcM) for c06-math-1417PAM is then given by

6.395 equation

where we used (2.47). We can thus conclude that an increase on the size of the constellation c06-math-1420 reduces the multiplicative factor before the Q-function. More importantly, an increase of c06-math-1421 by one is equivalent to decreasing the SNR by c06-math-1422. This SNR shift will dominate the behavior of the BEP at high SNR.

The following example show the approximations for particular values of c06-math-1423.

c06f022

Figure 6.22 BEP approximations (lines) and simulations (markers) for a CENC with c06-math-1453 over the AWGN channel: 4PAM (circles), 8PAM (triangles) and 16PAM (squares). The dashed and dashed-dotted lines are the approximations obtained using (6.392) and (6.393) and the “1-term” approximation is obtained using the single-term expressions for the PEP shown in (6.399)–(6.401)

6.4.3 MPSK Constellations

In the case of c06-math-1454PSK, we know that c06-math-1455 and from Table 5.2 we read

6.402 equation
6.403 equation

and knowing that the PDF of c06-math-1458 is the same as the PDF of c06-math-1459 (see also Example 5.7), we have

6.404 equation

and

6.405 equation

for c06-math-1462.

In analogy to (6.391) we can find c06-math-1463 as

6.406 equation
6.407 equation
6.408 equation

where c06-math-1467.

The results above show that in the case of c06-math-1468PSK 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 c06-math-1469PAM 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 c06-math-1470 and c06-math-1471:

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

  • The L-value c06-math-1474 has a “high-protection” distribution c06-math-1475 if c06-math-1476 and a “low-protection” distribution c06-math-1477 if c06-math-1478.
  • The L-value c06-math-1479 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 c06-math-1480 and c06-math-1481 are transmitted. More specifically, CoRe is based on two operations: negation of the bit at position c06-math-1482 (i.e., negation of the second row of the matrix c06-math-1483) and swapping the position of the labels of c06-math-1484 and c06-math-1485 (i.e., swapping the first and second row of c06-math-1486).

The negation operation is connected with the first observation we made above: as depending on the value of c06-math-1487 the L-value c06-math-1488 changes its distribution, using c06-math-1489 in the first transmission and c06-math-1490 in the next one, we can guarantee that “high” protection is offered to the bit c06-math-1491 in one out of the two transmissions.

The swapping responds to the second observation we made above, and is meant to transmit the bits c06-math-1492 at position c06-math-1493 so that c06-math-1494 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 c06-math-1495.15 These L-values are then passed to the decoder. After c06-math-1496 transmissions, we obtain the following distributions for c06-math-1497

6.411 equation

and for c06-math-1499

6.412 equation

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

6.413 equation

where

equation

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 equation
c06f023

Figure 6.23 BEP approximations using the ZcM (solid lines) and the CoM (dashed lines) for a CENC with c06-math-1505 over an AWGN channel with CoRe and c06-math-1506 retransmissions. The simulation results are shown with markers: c06-math-1507 (triangles), c06-math-1508 (circles), c06-math-1509 (diamonds), and c06-math-1510 (squares)

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 c06-math-1511, c06-math-1512 and c06-math-1513 have the same PDF and c06-math-1514 and c06-math-1515 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 c06-math-1516 we have c06-math-1517, 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 c06-math-1518QAM and/or c06-math-1519PSK [2 3]. In this chapter we showed expression for c06-math-1520. 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 c06-math-1521 in [11, eq. (6)], for half-integer c06-math-1522 in [12, eq. (15)], and for arbitrary c06-math-1523 (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 c06-math-1524 of such codes decreases with c06-math-1525, 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 c06-math-1526. For large values of c06-math-1527 the Bayesian evolutionary analysis by sampling trees (BEAST) algorithm recently introduced in [54] (see also [55]) is more appropriate.

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